LYAPUNOV-BASED CONTROL METHODS FOR NEUROMUSCULAR ELECTRICAL
STIMULATION
By
NITIN SHARMA
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2010
S2010 Nitin Sharma
To my loving wife, Deepti, my dear parents, Neena and Balwinder Sharma, and my
affectionate sister, Nitika for their unwavering support
ACKNOWLEDGMENTS
I would like to express sincere gratitude to my advisor, Dr. Warren E. Dixon, for
giving me the opportunity to work with him. I thank him for exposing me to vast and
exciting research area of nonlinear control and motivating me to work on Neuromuscular
Electrical Stimulation (NMES) control problem. I have learnt tremendously from his
experience and appreciate his significant role in developing my professional skills and
contributing to my academic success.
I would also like to thank my co-advisor Dr. ('!n -' Gregory for answering my queries
related to muscle physiology and for guiding me in building correct protocols during
NMES experiments. I also appreciate my committee members Dr. Scott Banks, Dr. Carl
D. Crane III and Dr. Jacob Hammer for the time and help they provided.
I would like to thank my colleagues for their support and appreciate their steadfast
volunteering in NMES experiments.
I would like to thank my wife for her love and patience. Also, I would like to attribute
my overall success to my mother who took her time and effort to teach me during my
childhood. Finally, I would like to thank my father for his belief in me.
TABLE OF CONTENTS
page
ACKNOW LEDGMENTS .................................
LIST O F TABLES . . .
LIST OF FIGURES . . .
A B ST R A C T . . .
CHAPTER
1 INTRODUCTION ..................................
1.1 Motivation and Problem Statement ......................
1.2 C contributions . . .
2 MUSCLE ACTIVATION AND LIMB MODEL .................
3 NONLINEAR NEUROMUSCULAR ELECTRIC
TRACKING CONTROL OF A HUMAN LIME
Introduction .. ............
Control Development .........
Nonlinear NMES Control of a Human
of Error (RISE) method .......
Limr
3.3.1 Stability Analysis .........
3.3.2 Experimental Results .......
3.3.2.1 Testbed and protocol
3.3.2.2 Results and discussion
3.3.3 Conclusion .............
3.4 Modified Neural Network-based Electrical
Tracking . .
3.4.1 Open-Loop Error System .....
3.4.2 Closed-Loop Error System .
3.4.3 Stability Analysis .. .......
3.4.4 Experimental Results .......
3.4.4.1 Testbed and protocol
3.4.4.2 Results and discussion
3.4.5 Limitations .. .........
3.4.6 Conclusion .. ...........
CAL STIMULATION (NMES)
ib via Robust Integral of Signum
Stimulation for Human Limb
4 NONLINEAR CONTROL OF NMES: INCORPORATING FATIGUE AND
CALCIUM DYNAMICS .. .........................
4.1 Introduction . . .
4.2 Muscle Activation and Limb Model .. ..................
4.3 Control Development .................. ........... .. 76
4.3.1 Open-Loop Error System .................. ... .. 77
4.3.2 Closed-Loop Error System .................. .. 79
4.3.3 Backstepping Error System ...... .......... .. 81
4.4 Stability Analysis ............... ............ .. 81
4.5 Simulations ............... ............ .. 84
4.6 Conclusion ................ .............. .. 85
5 PREDICTOR-BASED CONTROL FOR AN UNCERTAIN EULER-LAGRANGE
SYSTEM WITH INPUT DELAY .................. ........ .. 89
5.1 Introduction .................. ................ .. 89
5.2 Dynamic Model and Properties .................. .... .. 90
5.3 Control Development .................. ........... .. 91
5.3.1 Objective. .................. ........... .. 91
5.3.2 Control development given a Known Inertia Matrix ... 91
5.3.3 Control development with an Unknown Inertia Matrix ... 97
5.4 Experimental Results and Discussion ..... ... 103
5.5 Delay compensation in NMES through Predictor-based Control ...... 106
5.5.1 Experiments: Input Delay Characterization 107
5.5.2 Experiments: PD Controller with Delay Compensation ...... .113
5.6 Conclusion .. .... .. 117
6 RISE-BASED ADAPTIVE CONTROL OF AN UNCERTAIN NONLINEAR
SYSTEM WITH UNKNOWN STATE DELAYS ..... 120
6.1 Introduction .................. ................ .. 120
6.2 Problem Formulation .................. ........... 120
6.3 Error System Development .................. ........ .. 121
6.4 Stability Analysis .................. ........... .. 125
6.5 Simulations .................. ................ .. 130
6.6 Conclusion .. .... .. 131
7 CONCLUSION AND FUTURE WORK ................... 135
7.1 Conclusion .. .... .. 135
7.2 Future W ork .................. ................ .. 136
APPENDIX
A PREDICTOR-BASED CONTROL FOR AN UNCERTAIN EULER-LAGRANGE
SYSTEM WITH INPUT DELAY ................... ...... .. 139
B RISE-BASED ADAPTIVE CONTROL OF AN UNCERTAIN NONLINEAR
SYSTEM WITH UNKNOWN STATE DELAYS ..... ..... 141
REFERENCES .................. ................ .. .. 143
BIOGRAPHICAL SKETCH ................... .......... 153
LIST OF TABLES
Table page
3-1 Tabulated results indicate that the test subject was not learning the desired
trajectory since the RMS errors are relatively equal for each trial. ... 40
3-2 Experimental results for two period desired trajectory .............. .41
3-3 Summarized experimental results for multiple, higher frequencies and higher
range of m otion. .. .. .. ... .. .. .. ... ... ... .. ... .. 43
3-4 Summarized experimental results and P values of one tailed paired T-test for a
1.5 second period desired trajectory. .................. ..... 63
3-5 Summarized experimental results and P values of one tailed paired T-test for
dual periodic (4-6 second) desired trajectory. .................. 66
3-6 Experimental results for step response and changing loads .... 66
3-7 The table shows the RMS errors during extension and flexion phase of the leg
movement across different subjects, .................. ..... .. 71
5-1 Summarized experimental results of traditional PID/PD controllers and the PID/PD
controllers with delay compensation. .................. ..... 108
5-2 Results compare performance of the PD controller with delay compensation,
when the B gain matrix is varied from the known inverse inertia matrix. 108
5-3 Experimental results when the input delay has uncertainty. The input delay
value was selected as 100 ms. .................. .. .. .. 109
5-4 Summarized input delay values of a 1 1, i.,l!: individual across different stimulation
parameters .................. ................... .. 116
5-5 Table compares the experimental results obtained from the traditional PD controller
and the PD controller with /. /.,;/ compensation. ................ 119
LIST OF FIGURES
Figure page
2-1 Muscle activation and limb model. .................. ..... 25
2-2 The left image illustrates a person's left leg in a relaxed state. .... 27
3-1 Top plots: Actual left limb trajectory of a subject (solid line) versus the desired
two periodic trajectory (dashed line) input. .................. 42
3-2 Top plot: Actual limb trajectory (solid line) versus the desired triple periodic
trajectory (dashed line). ............... ........... 43
3-3 Top plot: Actual limb trajectory (solid line) versus the desired constant period
(2 sec) trajectory (dashed line). .................. ........ .. 44
3-4 Top plot: Actual limb trajectory (solid line) versus the triple periodic desired
trajectory with higher range of motion (dashed line). 45
3-5 Top plot: Actual limb trajectory (solid line) versus the desired constant period
(6 sec) trajectory (dashed line). .................. ........ .. 46
3-6 Top plot : Actual limb trajectory (solid line) versus desired step trajectory (dashed
line). .... .. .. .. 47
3-7 The top plot shows the actual limb trajectory (solid line) obtained from the RISE
controller versus the desired 1.5 second period desired trajectory (dashed line). 61
3-8 The top plot shows the actual limb trajectory (solid line) obtained from the NN+RISE
controller versus the desired 1.5 second period desired trajectory (dashed line). 62
3-9 The top plot shows the actual limb trajectory (solid line) obtained from the RISE
controller versus the dual periodic desired trajectory (dashed line). ...... ..64
3-10 The top plot shows the actual limb trajectory (solid line) obtained from the NN+RISE
controller versus the dual periodic desired trajectory (dashed line). ...... ..65
3-11 Experimental plots for step change and load addition obtained from NN+RISE
controller .................. ............. .. .. 67
3-12 Initial sitting position during sit-to-stand experiments. The knee-angle was measured
using a goniometer attached around the knee-axis of the subject's leg. 68
3-13 The top plot shows the actual leg angle trajectory (solid line) versus desired
trajectory (dotted line) obtained during the standing experiment. ... 69
4-1 An uncertain fatigue model is incorporated in the control design to address muscle
fatigue. Best guess estimates are used for unknown model parameters. 76
4-2 Top plot shows the knee angle error for a 6 second period trajectory using the
proposed controller. .................. ... .......... 84
4-3 Top plot shows the knee angle error for a 2 second period trajectory using the
proposed controller. .................. ... .......... 85
4-4 Top plot shows the knee angle error for a 6 second period trajectory using the
RISE controller .................. ................. .. 86
4-5 Top plot shows the knee angle error for a 2 second period trajectory using the
RISE controller .................. ................. .. 87
4-6 RISE controller with fatigue in the dynamics ................ 87
4-7 Performance of the proposed controller .................. ...... 88
4-8 Fatigue variable .................. ............... .. .. 88
5-1 Experimental testbed consiting of a 2-link robot. The input delay in the system
was artificially inserted in the control software. ................ 103
5-2 The plot shows three torque terms .................. ..... 107
5-3 The top-left and bottom-left plots show the errors of Link 1 and Link 2 ..... ..110
5-4 The top-left and bottom-left plots show the torques of Link 1 and Link 2 111
5-5 Typical input delay during NMES in a healthy individual. ..... 112
5-6 Average input d,1 iv values across different frequencies. ............. .113
5-7 Average input d,1 iv values across different voltages. ............... ..114
5-8 Average input d,1 iv values across different pulsewidths. ............. .115
5-9 Top plot: Actual limb trajectory of a subject (solid line) versus the desired trajectory
(dashed line) input obtained with the PD controller with /. /.,;/ compensation... 117
5-10 Top plot: Actual limb trajectory of a subject (solid line) versus the desired trajectory
(dashed line) input ............... ........... .. .. 118
6-1 Tracking error for the case r = 3 s. ............... .. ..... 132
6-2 Control input for the case r = 3 s ................. ..... 132
6-3 Parameter estimates for the case T = 3 s. ................ ...... 133
6-4 Tracking error for the case T = 10 s. .............. .... 133
6-5 Control input for the case T = 10 s. ................ .. ... 134
6-6 Parameter estimates for the case T = 10 s. ............. 134
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
LYAPUNOV-BASED CONTROL METHODS FOR NEUROMUSCULAR ELECTRICAL
STIMULATION
By
Nitin Sharma
August 2010
('C! i': Warren E. Dixon
Major: Mechanical Engineering
Neuromuscular electrical stimulation (NMES) is the application of a potential field
across a muscle in order to produce a desired muscle contraction. NMES is a promising
treatment that has the potential to restore functional tasks in persons with movement
disorders. Towards this goal, the research objective in the dissertation is to develop NMES
controllers that will enable a person's lower shank to track a continuous desired trajectory
(or constant setpoint).
A nonlinear musculoskeletal model is developed in C!i lpter 2 which describes muscle
activation and contraction dynamics and body segmental dynamics during NMES. The
definitions of various components in the musculoskeletal dynamics are provided but are
not required for control implementation. Instead, the structure of the relationships is used
to define properties and make assumptions for control development.
A nonlinear control method is developed in C(i lpter 3 to control the human
quadriceps femoris muscle undergoing non-isometric contractions. The developed
controller does not require a muscle model and can be proven to yield .,-i~,iiil'I ic stability
for a nonlinear muscle model in the presence of bounded nonlinear disturbances. The
performance of the controller is demonstrated through a series of closed-loop experiments
on healthy normal volunteers. The experiments illustrate the ability of the controller to
enable the shank to follow trajectories with different periods and ranges of motion, and
also track desired step changes with changing loads.
The most promising and popular control methods for NMES are neural network
(NN)-based methods since these methods can be used to learn nonlinear muscle force
to length and velocity relationship, and the inherent unstructured and time-varying
uncertainties in available models. Further efforts in C'! lpter 3 focus on the use of a NN
feedforward controller that is augmented with a continuous robust feedback term to yield
an .i-i-,ii !ll ic result (in lieu of typical uniformly ultimately bounded (UUB) stability).
Specifically, a NN-based controller and Lyapunov-based stability a'i ,1 i-i; are provided
to enable semi-global .i-vmptotic tracking of a desired time-varying limb trajectory (i.e.,
non-isometric contractions). The added value of incorporating a NN feedforward term is
illustrated through experiments on healthy normal volunteers that compare the developed
controller with the pure RISE-based feedback controller.
A pervasive problem with current NMES technology is the rapid onset of the
unavoidable muscle fatigue during NMES. In closed-loop NMES control, disturbances such
as muscle fatigue are often tackled through high-gain feedback which can overstimulate
the muscle which further intensifies the fatigue onset. In C'! lpter 4, a NMES controller
is developed that incorporates the effects of muscle fatigue through an uncertain function
of the calcium dynamics. A NN-based estimate of the fatigue model mismatch is
incorporated in a nonlinear controller through a backstepping method to control the
human quadriceps femoris muscle undergoing non-isometric contractions. The developed
controller is proven to yield UUB stability for an uncertain nonlinear muscle model
in the presence of bounded nonlinear disturbances (e.g., spasticity, d-.1 'v, changing
load dynamics). Simulations are provided to illustrate the performance of the proposed
controller. Continued efforts will focus on achieving .I-vmptotic tracking versus the UUB
result, and on validating the controller through experiments.
Another impediment in NMES control is the presence of input or actuator delay.
Control of nonlinear systems with actuator delay is a challenging problem because of
the need to develop some form of prediction of the nonlinear dynamics. The problem
becomes more difficult for systems with uncertain dynamics. Motivated to address the
input delay problem in NMES control and the absence of non-model based controllers for
a nonlinear system with input delay in the literature, tracking controllers are developed
in C'!i lpter 5 for an Euler-Lagrange system with time-d, 1 i-, .1 actuation, parametric
uncertainty, and additive bounded disturbances. One controller is developed under
the assumption that the inertia is known, and a second controller is developed when
the inertia is unknown. For each case a predictor-like method is developed to address
the time delay in the control input. Lyapunov-Krasovskii functionals are used within
a Lyapunov-based stability analysis to prove semi-global UUB tracking. Extensive
experiments show better performance compared to traditional PD/PID controller as well
as robustness to uncertainty in the inertia matrix and time delay value. Experiments
are performed on l. ,lr i!:r normal individuals to show the feasibility, performance, and
robustness of the developed controller.
In addition to efforts focused on input d. 1 i, .1 nonlinear systems, a parallel
motivation exists to address another class of time d. 1 -i, i1 systems which consist of
nonlinear systems with unknown state d.-1 iv. A continuous robust adaptive control
method is designed in ('!i lpter 6 for a class of uncertain nonlinear systems with unknown
constant time-d--. 1- in the states. Specifically, the robust adaptive control method, a
gradient-based desired compensation adaptation law (DCAL), and a Lyapunov-Kravoskii
(LK) functional-based delay control term are utilized to compensate for unknown
time-d-. 1 -, linearly parameterizable uncertainties, and additive bounded disturbances
for a general nonlinear system. Despite these disturbances, a Lyapunov-based analysis
is used to conclude that the system output .,-i-!ii11i .1 ically tracks a desired time varying
bounded trajectory.
('!i lpter 7 concludes the dissertation with a discussion of the developed contributions
and future efforts.
CHAPTER 1
INTRODUCTION
1.1 Motivation and Problem Statement
Neuromuscular electrical stimulation (NMES) is the application of a potential field
across a muscle to produce a desired muscle contraction (for functional tasks, NMES
is described as functional electrical stimulation (FES)). Efforts in NMES facilitate
improved limb control and functionality for patients with stroke, spinal cord injuries,
and other neurological impairments [1, 2]. Although most NMES procedures in physical
therapy clinics consist of tabulated open-loop application of electrical stimulation, a
significant market exists for the development of noninvasive closed-loop methods.
However, the application and development of NMES control have been stymied by
several technical challenges. Specifically, due to a variety of uncertainties in muscle
physiology (e.g., temperature, pH, and architecture), predicting the exact contraction
force exerted by the muscle is difficult. One cause of this difficulty is that there is an
unknown mapping between the generated muscle force and stimulation parameters.
There are additional problems with delivering consistent stimulation energy to the
muscle due to a variety of factors including: muscle fatigue, input delay, electrode
placement, hyperactive somatosensory reflexes, inter- and intra-subject variability in
muscle properties, changing muscle geometry under the electrodes in non-isometric
conditions, percentage of subcutaneous body fat, overall body hydration, etc.
Given the uncertainties in the structure of the muscle model and the parametric
uncertainty for specific muscles, some investigators have explored various linear PID-based
methods (cf. [3-8] and the references therein). Typically, these approaches have only
been empirically investigated and no analytical stability analysis has been developed that
provides an indication of the performance, robustness or stability of these control methods.
The development of a stability analysis for previous PID-based NMES controllers has been
evasive because of the fact that the governing equations for a muscle contraction/limb
motion are nonlinear with unstructured uncertainties. Some efforts have focused on
analytical control development for linear controllers (e.g., [6, 9, 10]); however, the
governing equations are typically linearized to accommodate a gain scheduling or linear
optimal controller approach.
Motivated to develop effective NMES control in light of these challenges, the first
result in ('! Ilpter 3 develops an open-loop error system for a general uncertain nonlinear
muscle model based on available analytical and empirical data [11, 12]) that facilitates
the development of a new continuous feedback method (coined RISE for Robust Integral
of the Sign of the Error). Through this error-system development, the continuous RISE
controller is proven (through a Lyapunov-based stability analysis) to yield an .,-vmptotic
stability result despite the uncertain nonlinear muscle model and the presence of additive
bounded disturbances (e.g., muscle spasticity, fatigue, changing loads in functional tasks,
and unmodeled muscle behavior).
Seminal work in [13-18] continue to inspire new investigations (cf. [19-26] and
the references therein) in neural network (NN)-based NMES control development.
One motivation for NN-based controllers is the desire to augment feedback methods
with an adaptive element that can adjust to the uncertain muscle model, rather than
only relying on feedback to dominate the uncertainty based on worse case scenarios.
NN-based control methods have attracted more attention in NMES than other adaptive
feedforward methods because of the nature of the unstructured uncertainty and the
universal approximation property of NNs. However, since NNs can only approximate a
function within some residual approximation error, all previous NN-based controllers yield
uniformly ultimately bounded stability (i.e., the errors converge to a region of bounded
steady-state error).
The result in the third section of ('! Ilpter 3 focuses on the development of a
RISE-based NMES controller and the associated analytical stability il i ,i--- that
yields .,-i-! iiil'itic tracking in the presence of a nonlinear uncertain muscle model with
nonvanishing additive disturbances. This result uses feedback and an implicit learning
mechanism to dominate uncertainty and disturbances. However, the RISE method as well
as the previous linear feedback methods inherently rely on high gains or high frequency
to dominate the model uncertainty, potentially resulting in overstimulation. Recent
results from general control systems literature [27] indicate that the RISE-based feedback
structure can be augmented with a NN feedforward term to yield .,-i- !,.l .l ic tracking
for some classes of systems. Based on these general results, an extension is provided in
the fourth section of C'! lpter 3 where the RISE-based method is modified with a NN to
develop a new NMES controller for the uncertain muscle model.
While efforts in C'! lpter 3, provide an inroad to the development of analytical
NMES controllers for the nonlinear muscle model, these results do not account for muscle
fatigue, which is a primary factor to consider to yield some functional results in many
rehabilitation applications. Heuristically, muscle fatigue is a decrease in the muscle
force output for a given input and is a complex, multifactorial phenomenon [28-30]. In
general, some of the factors associated with the onset of fatigue are failure of excitation
of motor neurons, impairment of action potential propagation in the muscle membrane
and conductivity of sarcoplasmic reticulum to Ca2+ ion concentration, and the change in
concentration of catabolites and metabolites [31]. Factors such as the stimulation method,
muscle fibre composition, state of training of the muscle, and the duration and task to
be performed have been noticed to affect fatigue during NMES. Given the impact of
fatigue effects during NMES, researchers have proposed different stimulation strategies
[30, 32, 33] to delay the onset of fatigue such as choosing different stimulation patterns
and parameters, improving fatigue resistance through muscle retraining, sequential
stimulation, and size order recruitment.
Controllers can be designed with some feedforward knowledge to approximate the
fatigue onset or employ some assumed mathematical model of the fatigue in the control
design. Researchers in [34-38] developed various mathematical models for fatigue. In
[34], a musculotendon model for a quadriceps muscle undergoing isometric contractions
during functional electrical stimulation (FES) was proposed. The model incorporated
fatigue based on the intracellular pH level where the fatigue parameters for a typical
subject were found through metabolic information, experimentation and curve fitting. A
more general mathematical model for dynamic fatigue defined as a function of normalized
muscle activation variable (Ca2+ dynamics) was proposed in [35, 36]. The fatigue was
introduced as a fitness function that varies according to the increase or decrease in muscle
activation during electrical stimulation. The fatigue time parameters were estimated
from stimulation experiments. Models in [37] and [38] predict force due to the effect of
stimulation patterns and resting times with changing physiological conditions, where
model parameterization required investigating experimental forces generated from a
standardized stimulation protocol. Although these mathematical models for fatigue
prediction are present in literature, few researchers have utilized these assumed fatigue
models in closed-loop NMES control. Results in [36] and [39] use the fatigue model
proposed in [35] and [36] for a FES controller, where patient specific parameters (e.g.
fatigue time constants) are assumed to be known along with exact model knowledge of the
calcium dynamics. The difficulty involved in the control design using calcium dynamics or
intracellular pH level is that these states cannot be measured easily for real-time control.
Therefore, these states (calcium dynamics or pH level) are modeled as a first or second
order ordinary differential equation (cf., [34, 36, 39]) and the parameters in the equations
are estimated from experimentation or are based on data from past studies.
The focus of Chapter 4 is to address muscle fatigue by incorporating an uncertain
fatigue model (i.e., the model developed in [35]) in the NMES controller. The uncertain
fatigue model is defined as a function of a normalized muscle activation variable. The
normalized muscle activation variable denotes the calcium (Ca2+ ion) dynamics which
act as an intermediate variable between contractile machinery and external stimulus.
The calcium dynamics are modeled as a first order differential equation based on [6] and
[39]. A backstepping approach is utilized to design virtual control input that consists
of NN-based feedforward signal and feedback signal. The developed controller yields a
uniformly ultimately bounded stability result given an unknown nonlinear muscle model
with uncertain fatigue and calcium dynamics.
Another technical challenge that hampers the satisfactory NMES control performance
is electromechanical delay in muscle force generation which is defined as the difference in
time from the arrival of action potential at the neuromuscular junction to the development
of tension in the muscle [8]. In NMES control, the electromechanical delay is modeled as
an input delay in the musculoskeletal dynamics [6] and occurs due to finite conduction
velocities of the chemical ions in the muscle in response to the external electrical input
[36]. Input delay can cause performance degradation as was observed during NMES
experimental trials on volunteer subjects with RISE and NN+RISE controllers and has
also been reported to potentially cause instability during human stance experiments with
NMES [40]. Time delay in the control input (also known as dead time, or input delay)
is a pervasive problem in control applications other than NMES control. C('. 111- I1 and
combustion processes, telerobotic systems, vehicle platoons, and communication networks
[41-44] often encounter d-1i-, in the control input. Such d--1 are often attributed to
sensor measurement delay, transport lags, communication d4-1 i- or task prioritization,
and can lead to poor performance and potential instability.
Motivated by performance and stability problems, various methods have been
developed for linear systems with input d (1civ{ (cf. [45-57] and the references therein). As
discussed in [45, 46], an outcome of these results is the development and use of prediction
techniques such as Artstein model reduction [48], finite spectrum assignment [51], and
continuous pole placement [58]. The concept of predictive control originated from classic
Smith predictor methods [59]. The Smith predictor requires a plant model for output
prediction and has been widely studied and modified for control purposes (cf. [60-67]
and references therein). However, the Smith predictor does not provide good closed-loop
performance in the presence of model mismatch and can only be applied for stable plants
[42, 46]. Contrary to the Smith predictor, finite spectrum assignment or Artstein model
reduction techniques and their extensions [47-53, 68-71] can be applied to unstable or
multivariable linear plants. These predictor-based methods utilize finite integrals over past
control values to reduce the d. 1 i,, .1 system to a delay free system.
Another approach to develop predictive controllers is based on the fact that input
delay systems can be represented by hyperbolic partial differential equations (cf. [45, 46]
and references therein). This fact is exploited in [54-57] to design controllers for actuator
d. 1 i-, .1 linear systems. These novel methods model the time d. 1 li .1 system as an
ordinary differential equation (ODE)-partial differential equation (PDE) cascade where
the non-d, 1 .i, d1 input acts at the PDE boundary. The controller is then designed by
employing a backstepping type approach for PDE control [72].
Predictor techniques have also been extended to adaptive control of unknown linear
plants in [41, 56, 73]. In [41, 73] the controller utilizes a modified Smith predictor type
structure to achieve a semi-global result. In [56] (and the companion paper [55]), a global
adaptive controller is developed that compensates for uncertain plant parameters and a
possibly large unknown delay.
In comparison to input d. 1 l' .1 linear systems, fewer results are available for nonlinear
systems. Approaches for input d, 1 -i, .1 nonlinear systems such as [74, 75] utilize a Smith
predictor-based globally linearizing control method and require a known nonlinear
plant model for time delay compensation. In [42], a specific technique is developed for a
telerobotic system with constant input and feedback d,41-i where a Smith predictor for
a locally linearized subsystem is used in combination with a neural network controller
for a remotely located uncertain nonlinear plant. In [76], an approach to construct
Lyapunov-Krasovskii (LK) functionals for input d. 1 i- .'1 nonlinear system in feedback
form is provided, and the control method in [77] utilizes a composite Lyapunov function
containing an integral cross term and LK functional for stabilizing nonlinear cascade
systems, where time d 1 iv can be either in the input or the states. The robustness of input
to state stabilizability is proven in [78] for nonlinear finite-dimensional control systems
in presence of small input d.- 1i- by utilizing a Razumikhin-type theorem. In [79], the
backstepping approach that utilizes ODE-PDE cascade transformation for input d. 1 .i1 ,
systems is extended to a scalar nonlinear system with actuator delay of unrestricted
length. However, to the best of our knowledge, no attempt has been made towards
stabilizing an input d, 1 '1 nonlinear system with parametric uncertainty and/or additive
bounded disturbances.
Motivated by the lack of NMES controllers that compensate for input delay and
the desire to develop non-model based controllers for nonlinear systems with input delay
C'! lpter 5 focuses on the development of a tracking controller for an uncertain nonlinear
Euler-Lagrange system with input delay. The input time delay is assumed to be a known
constant and can be arbitrary large. The dynamics are assumed to contain parametric
uncertainty and additive bounded disturbances. The first developed controller is based on
the assumption that the mass inertia is known, whereas the second controller is based on
the assumption that the mass inertia is unknown. The key contributions of this effort is
the design of a d. 1 li compensating auxiliary signal to obtain a time delay free open-loop
error system and the construction of LK functionals to cancel time d,1 li. 1 terms. The
auxiliary signal leads to the development of a predictor-based controller that contains
a finite integral of past control values. This d, 1 li,. 1 state to d, 1 li free transformation
is analogous to the Artstein model reduction approach, where a similar predictor-based
control is obtained. LK functionals containing finite integrals of control input values
are used in a Lyapunov-based analysis that proves the tracking errors are semi-global
uniformly ultimately bounded.
Another class of time-d, 1 .i, systems which are also endemic to engineering systems
and can cause degraded control performance and make closed-loop stabilization difficult
are systems with state d.-1 'i-. In time-d, 1 .i, d systems, the dynamics not only depends
on the current system states but also depends on the past state values. These systems
occur in many industrial and manufacturing systems (e.g., metal cutting process, rolling
mill, and chemical processes [46, 80].) A desire parallel to NMES research existed to
address this class of time delay systems. Various controllers have been developed to
address time-delay induced performance and stability issues as described in the survey
papers [45, 46] and in recent results that target control of uncertain systems with state
d-.1i- (cf. [80-86] and references therein). Control synthesis and stability analysis
methods for nonlinear time-d, 1 i,- d systems are often based on Lyapunov techniques in
conjunction with a Lyapunov-Kravoskii (LK) functional (cf. [82, 83, 85, 87]). For example,
in [82], an iterative procedure utilizing LK functionals for robust stabilization of a class
of nonlinear systems with triangular structure is developed. However, as stated in [88],
the controller cannot be constructed from the given iterative procedure. Semi-global
uniformly ultimately bounded (SUUB) results have been developed for time-d, 1 li,
nonlinear systems [83, 85] by utilizing neural network-based control, where appropriate LK
functionals are utilized to remove time d. 1 -i, i states. A discontinuous adaptive controller
was recently developed in [87] for a nonlinear system with an unknown time delay to
achieve a UUB result with the aid of LK functionals. However, controllers designed in
[83, 87] can become singular when the controlled state reaches zero and an ad hoc control
strategy is proposed to overcome the problem. Moreover, as stated in [89] and [90], the
control design procedure described in [85] cannot be generalized for nth order nonlinear
systems.
Sliding mode control (SMC) has also been utilized for time d, 1 1 systems in [80,
91-94]. However, utilizing SMC still poses a challenging design and computation problem
when d,-1 -- are present in states [45, 46]. Moreover, the discontinuous sign function
present in SMC controller often gives rise to the undesirable chattering phenomenon
during practical applications. To overcome the limitations of discontinuity in SMC, a
continuous adaptive sliding mode strategy is designed in [95] for nonlinear plants with
unknown state d.-1 i- where an LK functional along with a discontinuous Lyapunov
function is proposed for the stability analysis.
The development in ('! Ilpter 6 is motivated by the lack of continuous robust
controllers that can achieve .,-,il:-l .1.ic stability for a class of uncertain time-d, 1 li- 1
nonlinear systems with additive bounded disturbances. The approach described in the
current effort uses a continuous implicit learning [96] based Robust Integral of the Sign of
the Error (RISE) structure [11, 27]. Due to the added benefit of reduced control effort and
improved control performance, an adaptive controller in conjunction with RISE feedback
structure is designed. However, since the time dl 1iv value is not .liv .,i- known, it becomes
challenging to design a delay free adaptive control law. Through the use of a desired
compensation adaptive law (DCAL) based technique and segregating the appropriate
terms in the open loop error system, the dependence of parameter estimate laws on the
time d. 1 .i-, 1 unknown regression matrix is removed. Contrary to previous results, there is
no singularity in the developed controller. A Lyapunov-based stability analysis is provided
that uses an LK functional along with Young's inequality to remove time d. 1 I. terms
and achieves .i-vmptotic tracking.
1.2 Contributions
This dissertation focuses on developing nonlinear controllers for a musculoskeletal
system excited by NMES. The controllers are developed to account for various technical
challenges hampering an effective NMES control performance such as unknown nonlinear
muscle model, muscle fatigue, input and measurement delay. The contributions of
C'!i lpters 3-6 are as follows.
1. C'!I iter 3, Nonlinear Neuromuscular Electrical Stimulation Tracking Control of
a Human Limb: The contribution of this chapter is to illustrate how a recently
developed continuous feedback method called robust integral of signum of the error
(coined as RISE) can be applied for NMES systems. The muscle model developed
in C'!i lpter 2 is rewritten in a form that adheres to RISE-based Lyapunov stability
analysis. Through this error-system development, the continuous RISE controller
is proven (through a Lyapunov-based stability analysis) to yield an .,i- i !,1 l)tic
stability result despite the uncertain nonlinear muscle model and the presence
of additive bounded disturbances (e.g., muscle spasticity, fatigue, changing loads
in functional tasks, and d-.1 i-). The performance of the nonlinear controller is
experimentally verified for a human leg tracking on a leg extension machine by
applying the controller as a voltage potential across external electrodes attached
to the distal-medial and proximal-lateral portion of the quadriceps femoris muscle
group. The RISE controller is implemented by a voltage modulation scheme with a
fixed frequency and a fixed pulse width. Other modulation strategies (e.g., frequency
or pulse-width modulation) could have also been implemented (and applied to other
skeletal muscle groups) without loss of generality. The experiments illustrate the
ability of the controller to enable the shank to track single and multiple period
trajectories with different ranges of motion, and also track desired step changes with
changing loads.
The second result in the chapter focuses on blending NN-based feedforward
technique with RISE based feedback method which was shown to yield .,- ii!! 1 .i i,
tracking in the presence of a nonlinear uncertain muscle model with nonvanishing
additive disturbances. The first result uses feedback and an implicit learning
mechanism to dominate uncertainty and disturbances. Recent results from general
control systems literature [27] indicate that the RISE-based feedback structure
can be augmented with a NN feedforward term to yield .,-,il i l'1 ic tracking for
some classes of systems. Based on these general results, the RISE-based method
is modified with a multi 1 ,-, i t1 NN to develop a new NMES controller for the
uncertain muscle model. The experimental results indicate that the addition of the
NN reduces the root mean squared (RMS) tracking error for similar stimulation
effort when compared to the first method developed in the chapter(RISE method
without the NN feedforward component). Additional experiments were conducted to
depict that the NN-based feedforward technique holds promise in clinical-type tasks.
Specifically, a preliminary sit-to-stand experiment was performed to show controller's
feasibility for any functional task.
2. C'!i lpter 4, Nonlinear Control of NMES: Incorporating Fatigue and Calcium Dy-
namics An open-loop error system for an uncertain nonlinear muscle model is
developed that includes the fatigue and calcium dynamics. A virtual control
input is designed using nonlinear backstepping technique which is composed of a
NN based feedforward signal and an error based feedback signal. The NN based
control structure is exploited not only to feedforward muscle dynamics but also to
approximate the error generated due to parametric uncertainties in the assumed
fatigue model. The actual external control input (applied voltage) is designed based
on the backstepping error. Through this error-system development, the continuous
NN based controller is proven (through a Lyapunov-based stability analysis) to yield
an uniformly ultimately bounded stability result despite the uncertain nonlinear
muscle model and the presence of additive bounded disturbances (e.g., muscle
spasticity, changing loads in functional tasks, and d,-1 i-).
3. C'! i lter 5, Predictor-Based Control for an Uncertain Euler-Lagr gi, System with
Input D. 1.ai; This chapter focuses on the development of a tracking controller for
an uncertain nonlinear Euler-Lagrange system with input delay. The input time
delay is assumed to be a known constant and can be arbitrary large. The dynamics
are assumed to contain parametric uncertainty and additive bounded disturbances.
The first developed controller is based on the assumption that the mass inertia is
known, whereas the second controller is based on the assumption that the mass
inertia is unknown. The key contributions of this effort is the design of a delay
compensating auxiliary signal to obtain a time d. 1 iv free open-loop error system
and the construction of LK functionals to cancel time d, 1 li. d1 terms. The auxiliary
signal leads to the development of a predictor-based controller that contains a finite
integral of past control values. This d, 1 li .1 state to delay free transformation is
analogous to the Artstein model reduction approach, where a similar predictor-based
control is obtained. LK functionals containing finite integrals of control input values
are used in a Lyapunov-based analysis that proves the tracking errors are semi-global
uniformly ultimately bounded. Extensive experiments were performed to show the
controller's better performance in comparison to traditional PID/PD controllers and
robustness to uncertainty in time delay and inertia matrix. Additional experiments
show that the developed controller can be applied to compensate input delay in
NMES.
4. C'! lpter 6, RISE-Based Adaptive Control of an Uncertain Nonlinear System with
Unknown State D. /l; The development in this chapter is motivated by the lack
of continuous robust controllers that can achieve .,-vmptotic stability for a class
of uncertain time-d, 1 ,li .1 nonlinear systems with additive bounded disturbances.
The approach described in the current effort uses a continuous implicit learning
[96] based Robust Integral of the Sign of the Error (RISE) structure [11, 27]. Due
to the added benefit of reduced control effort and improved control performance,
an adaptive controller in conjunction with RISE feedback structure is designed.
However, since the time d. 1 iv value is not ah--bi-i known, it becomes challenging to
design a delay free adaptive control law. Through the use of a desired compensation
adaptive law (DCAL) based technique and segregating the appropriate terms in
the open loop error system, the dependence of parameter estimate laws on the time
d. 1 .i-, 1 unknown regression matrix is removed. Contrary to previous results, there
is no singularity in the developed controller. A Lyapunov-based stability analysis is
provided that uses an LK functional along with Young's inequality to remove time
d, 1 li, '1 terms and achieves .i-i-i!1 il Iic tracking.
CHAPTER 2
MUSCLE ACTIVATION AND LIMB MODEL
The following model development represents the musculoskeletal dynamics during
neuromuscular electrical stimulation performed on human quadriceps muscle. The model
simulates limb dynamics when external voltage is applied on the muscle. The total muscle
knee joint model can be categorized into body segmental dynamics and muscle activation
and contraction dynamics. The muscle activation and contraction dynamics explains the
force generation in the muscle while the body segmental dynamics considers the active
moment and passive joint moments.
The total knee-joint dynamics can be modeled as [6]
M1 + + 11[, + r T.. + =. (2-1)
In (2-1), MI(q) E R denotes the inertial effects of the shank-foot complex about the
Contraction and Activation Dynamics Body Segmental Dynamics
Dy Recruitment Calcium ertial
Curve Dynamics Passive Force Force
Gravitational
Fatie Force
Model
1oltage Input
Controller
Figure 2-1. Muscle activation and limb model. The force generating contraction and
activation dynamics in the muscle is denoted by an unknown nonlinear function
Tr(q, q) E R in the dynamics. The detailed contraction and activation dynamics including
fatigue and calcium dynamics are introduced in C'!h plter 4.
knee-joint, i (q) E R denotes the elastic effects due to joint stiffness, .1.,(q) E R denotes
the gravitational component, i .(q) E R denotes the viscous effects due to damping in the
musculotendon complex [97], rd(t) E R is considered as an unknown bounded disturbance
which represents an unmodeled reflex activation of the muscle (e.g., muscle spasticity)
and other unknown unmodeled phenomena (e.g., dynamic fatigue, electromechanical
d-i-' ), and r(t) E R denotes the torque produced at the knee joint. In the subsequent
development, the unknown disturbance Td(t) is assumed to be bounded and its first and
second time derivatives are assumed to exist and be bounded. These are reasonable
assumptions for typical disturbances such as muscle spasticity, fatigue, and load changes
during functional tasks. For simplicity, the passive damping and elastic force of muscle
and joints are considered together. The inertial and gravitational effects in (2-1) can be
modeled as
M1((t)) = J(t), -,(q(t)) = -mglsin(q(t)), (2-2)
where q(t), q(t), q(t) E R denote the angular position, velocity, and acceleration of the
lower shank about the knee-joint, respectively (see Fig. 2-2), JE R denotes the unknown
inertia of the combined shank and foot, mE R denotes the unknown combined mass of
the shank and foot, E R is the unknown distance between the knee-joint and the lumped
center of mass of the shank and foot, and g E R denotes the gravitational acceleration.
The elastic effects are modeled on the empirical findings by Ferrarin and Pedotti in [97] as
ii. (q) = -k(exp(-k2q(t)))(q(t) k3), (2-3)
where kl, k2, k3 E R are unknown positive coefficients. As shown in [6], the viscous
moment 1.(q) can be modelled as
Vif..((t)) B1 tanh(-B2(t)) B3(t), (2-4)
where B1, B2, and B3 E R are unknown positive constants.
Figure 2-2. The left image illustrates a person's left leg in a relaxed state. The right image
shows the left leg during stimulation. The angle q(t) is measured with respect to the
vertical line as shown.
The torque produced about the knee is controlled through muscle forces that are
elicited by NMES. For simplicity (and without loss of generality), the development in
this chapter focuses on producing knee torque through muscle tendon forces, denoted by
Fr(t) E I, generated by electrical stimulation of the quadriceps (i.e., antagonistic muscle
forces are not considered). The knee torque is related to the muscle tendon force as
r(t) = ((q(t))FT(t), (2-5)
where ((q(t)) IR denotes a positive moment arm that changes with the extension and
flexion of the leg as shown in studies by [98] and [99]. The tendon force FT(t) in (2-5) is
defined as
F F cos a(q) (2-6)
where a(q(t)) is defined as the pennation angle between the tendon and the muscle. The
pennation angle of human quadriceps muscle changes monotonically during quadriceps
contraction and is a continuously differentiable, positive, monotonic, and bounded function
with a bounded first time derivative [100]. The relationship between muscle force and
applied voltage is denoted by the unknown function q((q, q) E R as
F(t) = (q,j)V(t), (2-7)
where V(t) E R is the voltage applied to the quadriceps muscle by electrical stimulation.
While exact force versus voltage models are debatable and contain parametric uncertainty,
the generally accepted empirical relationship between the applied voltage (or similarly,
current, frequency [101, 102], or pulse width) is well established. The empirical data in
[101] and [102] indicates the function rq(q, q) is a continuously differentiable, non-zero,
positive, monotonic, and bounded function, and its first time derivative is bounded.
The total force generated at the tendon could be considered as the sum of forces
generated by an active element (often denoted by FCE), the tension generated by a passive
elastic element (often denoted by FpE), and forces generated by viscous fluids (often
denoted by FVE). These forces have dynamic characteristics. For example, the passive
element increases with increasing muscle length, and the muscle stiffness has been reported
to change by greater than two orders of magnitude [34] under dynamic contractions. The
muscle model in the chapter considers the total muscle force composed of the sum of
these elements as the function of an unknown nonlinear function q](q, q) and an applied
voltage V(t). The introduction of the unknown nonlinear function rq(q, q) enables the
muscle contraction to be considered under general dynamic conditions in the subsequent
control development. Expressing the muscle contraction forces in this manner enables the
development of a control method that is robust to changes in the forces, because these
effects are included in the uncertain nonlinear muscle model that is incorporated in the
stability analysis. The model developed in (2-1)-(2-7) is used to examine the stability
of the subsequently developed controller, but the controller does not explicitly depend
on these models. The following assumptions are used to facilitate the subsequent control
development and stability analysis.
Assumption 1: The moment arm ((q) is assumed to be a non-zero, positive,
bounded function [98, 99] whose first two time derivatives exist, and based on the
empirical data [101, 102], the function rq(q, q) is assumed to be a non-zero, positive,
and bounded function with a bounded first and second time derivatives.
Assumption 2: The auxiliary non-zero unknown scalar function Q(q, q) E R is
defined as
Q = ( cosa, (2-8)
where the first and second time derivatives of Q(q, q) are assumed to exist and be bounded
(see Assumption 1).
Assumption 3: The unknown disturbance -d(t) is bounded and its first and second
derivatives with respect to time exist and are bounded. Based on Assumptions 1 and 2,
the ratio Td(t)/l((q, q) is also assumed to be bounded and its first and second derivatives
with respect to time exist and are bounded.
CHAPTER 3
NONLINEAR NEUROMUSCULAR ELECTRICAL STIMULATION (NMES)
TRACKING CONTROL OF A HUMAN LIMB
3.1 Introduction
An open-loop error system for a general uncertain nonlinear muscle model is
developed in the chapter by grouping terms in a manner that facilitates the development
of a new continuous feedback method (coined RISE for Robust Integral of the Sign
of the Error in [11, 12])and its extension through combining NN-based feedforward
method. Through this error-system development, the continuous RISE controller and
its modification is proven (through a Lyapunov-based stability analysis) to yield an
.ii-i:!1ll .I ic stability result despite the uncertain nonlinear muscle model and the presence
of additive bounded disturbances (e.g., muscle spasticity, fatigue, changing loads in
functional tasks). The performance of the two nonlinear controllers is experimentally
verified for human leg tracking by applying the controller as a voltage potential across
external electrodes attached to the distal-medial and proximal-lateral portion of the
quadriceps femoris muscle group. The RISE and NN + RISE controllers are implemented
by a voltage modulation scheme with a fixed frequency and a fixed pulse width. Other
modulation strategies (e.g., frequency or pulse-width modulation) could have also been
implemented (and applied to other skeletal muscle groups) without loss of generality.
Third section of the chapter discusses the development of RISE controller for
uncertain nonlinear muscle model. The experiments illustrate the ability of the controller
to enable the leg shank to track single and multiple period trajectories with different
periods and ranges of motion, and also track desired step changes with changing loads.
In fourth section the RISE-based method is modified with a NN to develop a new NMES
controller for the uncertain muscle model. The experimental results indicate that the
addition of the NN reduces the root mean squared (RMS) tracking error for similar
stimulation effort when compared to the first result (RISE method without the NN
feedforward component). A preliminary test was also conducted on a healthy volunteer to
test the capability of the controller to enable the person to perform a sit-to-stand task.
3.2 Control Development
A high-level objective of NMES is to enable a person to achieve some functional
task (i.e., functional electrical stimulation (FES)). Towards this goal, the objective of
the current effort is to develop a NMES controller to produce a knee position trajectory
that will enable a human shank to track a desired trajectory, denoted by qd(t) e R. The
desired trajectory can be any continuous signal (or a simple constant setpoint). In the
subsequent experimental results, the desired trajectories were selected as periodic signals
(for simplicity and without loss of generality) of different frequencies and step functions
with changes in the dynamic load. Although such trajectories may not be truly functional,
trajectory-based movements are necessary for the performance of many FES augmented
tasks (e.g., repetitive stepping during walking). Whether the desired trajectories are based
on limb position, as in the current result, or other information (e.g., desired joint kinetics
or kinematics), the ability to precisely track a desired pattern is fundamental to eliciting
reproducible movement patterns during functional tasks.
To quantify the objective, a position tracking error, denoted by ei(t) E R, is defined
as
ei(t) = qd(t)- q(t), (3-1)
where qd(t) is an a priori trajectory which is designed such that qd(t), qj(t) E ,,where
qd(t) denotes the ithderivative for i = 1, 2, 3, 4. To facilitate the subsequent analysis,
filtered tracking errors, denoted by e2(t) and r(t) E R, are defined as
e2(t) = et) + ae,(t), (3-2)
r(t) = e2(t) + a2e2(t), (3-3)
where ac, a2 E R denote positive constants. The filtered tracking error r(t) is introduced
to facilitate the closed-loop error system development and stability analysis but is not used
in the controller because of a dependence on acceleration measurements.
3.3 Nonlinear NMES Control of a Human Limb via Robust Integral of
Signum of Error (RISE) method
After multiplying (3-3) by J and utilizing the expressions in (2-1) and (2-5)-(3-2),
the following expression can be obtained:
Jr = W QV + rd, (3-4)
where W(eI, e2, t) E R is an auxiliary signal defined as
W = J(qd + a1~i + a262) + i + 1-, + 11. (3-5)
and the continuous, positive, monotonic, and bounded auxiliary function Q(q, t) E R is
defined in (2-8). After multiplying (3-4) by -l (q, t) e R, the following expression is
obtained:
JQr = WQ V + dn, (3-6)
where J(q, t) E R, Trd(q, t) E R, and W(eli,e2,t) E R are defined as
J rd
JQ = TdQ (3-7)
W
WQ = W Jn(qd + ae + a2e2) + + ... + (38)
To facilitate the subsequent stability analysis, the open-loop error system for (3-6) can be
determined as
Jr = r +N- V e2, (3-9)
where N(el, e2, r, t) E R denotes the unmeasurable auxiliary term
1.
N =W + e2 Qr + -dQ(q, t). (3 10)
2
To further facilitate the analysis, another unmeasurable auxiliary term, Nd(qd, qd, qd, qd, t) E
R, is defined as
Nd = (qd)qd + J (qd) d + .(q) + i.. .(qd) + d. .(q) + d (qd, t). (3-11)
After adding and subtracting (3-11) to (3-9), the open-loop error system can be expressed
as
JA -V e2 + d + Nd tJr, (3-12)
2
where the unmeasurable auxiliary term ]N(el, e2, r, t) E R is defined as
N(t) = N Nd. (3-13)
Motivation for expressing the open-loop error system as in (3-12) is given by the
desire to segregate the uncertain nonlinearities and disturbances from the model into
terms that are bounded by state-dependent bounds and terms that are upper bounded
by constants. Specifically, the Mean Value Theorem can be applied to upper bound
N(ei,e2, r, t) by state-dependent terms as
N < p (|| ) | | (3-14)
where z(t) E R3 is defined as
z(t) ^ [eT e r]T, (3-15)
and the bounding function p (|| ||) is a positive, globally invertible, nondecreasing
function. The fact that qd(t), qd(t) E V i = 1,2, 3, 4 can be used to upper bound
Nd(qd, qd, gd, qd, t) as
w||N < ( Nd < (. (3-16)
where (Nd and (Nd c R are known positive constants.
Based on the dynamics given in (2-1)-(2-7), the RISE-based voltage control input
V(t) is designed as
V(t) A (k, + l)e2(t) (k + 1)e2(0) + v(t), (3-17)
where ks E R denotes positive constant adjustable control gain, and v(t) E R is the
generalized solution to
v(t) = (k, + l)a22(t) + 3sgn(e2(t)), v(0) 0, (3-18)
where E R denotes positive constant adjustable control gain, and sgn(-) denotes the
signum function. Although the control input is present in the open-loop error system
in (3-4), an extra derivative is used to develop the open-loop error system in (3-12)
to facilitate the design of the RISE-based controller. Specifically, the time-derivative
of the RISE input in (3-17) looks like a discontinuous sliding mode controller. Sliding
mode control is desirable because it is a method that can be used to reject the additive
bounded disturbances present in the muscle dynamics (e.g., muscle spasticity, load
changes, electromechanical d.1 ,i-,) while still obtaining an .,-i- ii!ll ic stability result. The
disadvantage of a sliding mode controller is that it is discontinuous. By structuring the
open-loop error system as in (3-12), the RISE controller in (3-17) can be implemented as
a continuous controller (i.e., the unique integral of the sign of the error) and still yield an
.,-i-~, iill ic stability result. Without loss of generality, the developed voltage control input
can be implemented through various modulation methods (i.e., voltage, frequency, or pulse
width modulation).
3.3.1 Stability Analysis
Theorem 1. The controller given in (3 17) ensures that all system -.:l,,l- are bounded
under closed-loop operation. The position tracking error is n, gi.il.l I in the sense that
I|le(t)| 0 as t oc, (319)
and the controller ;./,. 1.1 semi-l.1- lrl .',/ ',,;/,I1. .: tracking provided the control gain ks,
introduced in (3 17) is selected suffi.. .: l, '/7./, and/3 is selected according to the
following sufficient condition:
S> (N, (3-20)
where (Nd and (g are introduced in (3-16).
Proof for Theorem 1: Let D C R4 be a domain containing y(t) = 0, where
y(t) e R3+1 is defined as
(t) A [zT (t) I (3-21)
and the auxiliary function P (t) E R is the generalized solution to the differential equation
P (t) = -L (t) P (0) /3 p e2 (0)| e2 (0) Nd (0). (3-22)
The auxiliary function L (t) E R in (3-22) is defined as
L (t) r (Nd (t)- psgn (e)) (3-23)
Provided the sufficient conditions stated in Theorem 1 are satisfied, then P (t) > 0.
Let VL (y, t) : D x [0, oo) -+ R denote a Lipschitz continuous regular positive definite
functional defined as
11
VL (, t) A eTe, + e2e2 + rJr + P, (3-24)
2 2
which satisfy the inequalities
U, (y)< VL (y, t)< U2 (y), (3-25)
provided the sufficient condition introduced Theorem 1 is satisfied, where Ui (y) U2 (y)
R are continuous, positive definite functions. After taking the time derivative of (3-24),
VL (y, t) can be expressed as
1 1
21 (y, t) 1 2eej + Ce262 + Jor+ Jr2 + P. (3-26)
2 2
From (3-2), (3-3), (3-12), (3-22), and (3-23), some of the differential equations
describing the closed-loop system for which the stability analysis is being performed
have discontinuous right-hand sides as
ei = e2 ale1, (3-27a)
r2 = r- a22, (3-27b)
Jr = -Jlr + N + Nd- e2 (k, + 1)r psgn(e2), (3-27c)
P (t) -r (Nd (t)- sgn (e2)) (3-27d)
Let f(y, t) E R4 denote the right hand side of (3-27). Since the subsequent analysis
requires that a solution exists for y = f(y, t), it is important to show the existence of
the solution to (3-27). As described in [103-106], the existence of Filippov's generalized
solution can be established for (3-27). First, note that f(y, t) is continuous except in the
set {(/y,t)|e2 = 0}. From [103-106], an absolute continuous Filippov solution y(t) exists
almost everywhere (a.e.) so that
y K[f](y, t) a.e. (3-28)
Except the points on the discontinuous surface {(y, t)e2 = 0}, the Filippov set-valued
map includes unique solution. Under Filippov's framework, a generalized Lyapunov
stability theory can be used (see [106-109] for further details) to establish strong stability
of the closed-loop system. The generalized time derivative of (3-24) exists a.e., and
VL(y, t) Eae VL (y,t) where
T
VL = vEL(y,t) K 2 1p-P
VL Kr 2 2 iP T
V LTK 1 2 7 P-2'P 1
C 2e 2 r1J 2 jP r2 K \ e, 22
2 2 -'P
where OV is the generalized gradient of V [107], and K[.] is defined as [108, 109]
K[f](y) cof (B(x, ) N),
6>0 N=0
where n denotes the intersection of all sets N of Lebesgue measure zero, co denotes
pN=0
convex closure, and B(x, 6) represents a ball of radius 6 around x. After utilizing (3-2),
(3-3), (3-12), (3-17), (3-18), (3-22), and (3-23)
VL (y,t) C 2ele2-2ace2 +e2r-( '.'+ Jr +rN + d r re2- C (ks+1)r2
2
-prK[sgn(e2) Jr2 rNd(t) + 3rK[sgn(e2)], (3-29)
2
where [108]
K[sgn(e2)] = SGN(e2), (3-30)
such that
1 e2 >0
SGN(e2) [-1,1] 2 e 0 (3-31)
-1 e2 <0
Cancelling common terms and based on the fact that
2ele2< <6l2 + 6111 (3-32)
(3-29) can be written as
VL (y, t) C -(2ai 1)e2 (a2 1)e r2 + r kr2. (3 33)
As shown in (3-29)-(3-33), the unique integral signum term in the RISE controller
is used to compensate for the disturbance terms included in Nd(qd, qd, ld, t', t), provided
the control gain 3 is selected according to (3-20). Using (3-14), the term r(t)NV(ei, 2, r, t),
can be upper bounded by following inequality:
rN < p(| : ) |||| ||r (3 34)
to obtain
VL (y,t) C -min{2ai 1, 1,1} + [p (|| ||) || | ||r|| k, ||r||2]. (3-35)
Completing the squares for the bracketed terms in (3-35) yields
VL (Y,t) C min{2ao -,- t, 2 ,1} 1 +, (3-36)
4k8
The following expression can be obtained from (3-36):
S(y, t)c -U(y), (3-37)
where U (y) is a continuous positive definite function, provided k, is selected sufficiently
large based on the initial conditions of the system. That is, the region of attraction can
be made arbitrarily large to include any initial conditions by increasing the control gain k,
(i.e., a semi-global type of stability result), and hence
c 1||(t)| 2 0 as t oo Vy(0) e S. (3-38)
Based on the definition of z(t) in (3 15), (3-38) can be used to show that
I||e(t) | 0 as t -oo Vy(0) S. (3-39)
3.3.2 Experimental Results
Experiments were performed using the RISE controller given in (3 17). The voltage
controller was implemented through an amplitude modulation scheme composed of a
variable amplitude positive square wave with a fixed pulse width of 100 p sec and fixed
frequency of 30 Hz. The 100 psec pulse width and the 30 Hz stimulation frequency were
chosen a-priori and represent parametric settings that are within the ranges typically
reported during NMES studies. During stimulation at 100 psec pulse widths, human
skeletal muscle response to changes in stimulation amplitude (force-amplitude relationship)
and frequency (force-frequency relationship) are highly predictable and thus deemed
appropriate for use in the present study. The 30 Hz stimulation was selected based on
force-frequency curves [110] which show that as stimulation frequency is increased muscle
force increases to a saturation limit. Higher frequencies can be chosen to generate more
force up to a saturation limit, but muscles tend to fatigue faster at higher frequencies.
The 30 Hz pulse wave yields reduced fatigue in comparison to higher frequencies but
lower frequencies tend to produce rippled knee motion [35, 110]. Therefore stimulation
frequencies in the range of 30-40 Hz is an optimal choice for conducting external electrical
stimulation. The following results indicate that the RISE algorithm was able to minimize
the knee angle error while dynamically tracking a desired trajectory.
3.3.2.1 Testbed and protocol
The tested consists of a custom computer controlled stimulation circuit and a
modified leg extension machine (LEM). The LEM was modified to include optical
encoders. The LEM allows seating adjustments to ensure the rotation of the knee is
about the encoder axis. A 4.5 kg (10 lb.) load was attached to the weight bar of the LEM,
and a mechanical stop was used to prevent hyperextension.
In the experiment, bipolar self-adhesive neuromuscular stimulation electrodes were
placed over the distal-medial and proximal-lateral portion of the quadriceps femoris muscle
group and connected to the custom stimulation circuitry. Prior to participating in the
study, written informed consent was obtained from all the subjects, as approved by the
Institutional Review Board at the University of Florida. Tracking experiments for a two
period desired trajectory were conducted on both legs of five subjects. The subjects
included two healthy females and three healthy males in the age group of 22 to 26 years.
The electrical stimulation responses of healthy subjects have been reported as similar to
paraplegic subjects' responses [16, 22, 39, 111]. Therefore healthy subjects were used in
NMES experiments as substitute for paraplegic patients which were not available. As
described in Section 3.3.2.2, the results were approximately equal across the subjects
(i.e., a standard deviation of 0.53 degrees of Root Mean Squared (RMS) tracking error).
Therefore, additional experiments were conducted on a single subject's leg to illustrate the
applicability of the controller for different conditions.
During the experiments each subject was instructed to relax and to allow the
stimulation to control the limb motion (i.e., the subjects were not supposed to influence
the leg motion voluntarily and were not allowed to observe the desired trajectory).
Varying the time period and range of motion may also help to reduce any possible
trajectory learning and anticipation by a healthy subject. To experimentally examine if
any trajectory learning occurred, four successive tests were conducted on a healthy subject
with a two minute interval between trials. The experiments were conducted for 15 seconds
on a dual period trajectory of 4 and 6 seconds. The resulting RMS errors are given in
Table 3-1. The results in Table 3-1 illustrate that trajectory learning by the subject is not
apparent since the standard deviation between the successive trials is 0.039 degrees.
Trial RMS error ( in deg.)
1 4.35
2 4.28
3 4.26
4 4.29
Table 3-1. Tabulated results indicate that the test subject was not learning the desired
trajectory since the RMS errors are relatively equal for each trial.
3.3.2.2 Results and discussion
The experimental results of five subjects tested for the two period desired trajectory
depicted in Fig. 3-1, are summarized in Table 3-2. In Table 3-2, the maximum steady-state
error is defined as the maximum absolute value of error that occurs after 4 seconds of the
trial. The maximum steady-state errors range from 4.25 to 7.55 degrees with a mean of
6.32 degrees and a standard deviation of 1.18 degrees. The RMS tracking errors range
from 20 to 3.47 with a mean RMS error of 2.75 degrees and a standard deviation of 0.53
degrees. The tracking error results for Subject B and the corresponding output voltages
computed by the RISE method (prior to voltage modulation) are shown in Fig. 3-1. The
results successfully illustrate the ability of the RISE controller to track the desired two
period trajectory.
Subject Leg RMS Max.
Error Steady
State Error
A Left 2.890 7.550
A Right 2.360 7.140
B Left 2.000 5.400
B Right 2.350 6.990
C Left 2.070 4.250
C Right 2.940 4.51
D Left 3.470 7.300
D Right 2.890 6.940
E Left 3.110 6.800
E Right 3.450 6.300
Mean 2.750 6.320
Std. Dev. 0.530 1.180
Table 3-2. Experimental results for two period desired trajectory
To further illustrate the performance of the controller, experiments were also
conducted for trajectories with faster and slower periods and larger ranges of motion.
Specifically, the controller's performance was tested for a desired trajectory with a
constant 2 second period, a constant 6 second period, a triple periodic trajectory
with cycles of 2, 4, and 6 seconds and for a higher range of motion of 65 degrees. As
indicated in Table 3-1, the results for the two period trajectory yielded similar results
for all the subjects. Hence, these additional tests were performed on a single individual
simply to illustrate the capabilities of the controller, with the understanding that some
variations would be apparent when implemented on different individuals. The RMS
tracking errors and maximum steady-state errors are provided in Table 3-3. The RMS
error and the maximum steady state errors are lowest for a constant 6 second period
desired trajectory and higher for faster trajectories and higher range of motion. These
results are an expected outcome since tracking more .-- ressive trajectories generally yields
more error. The triple periodic trajectory consists of a mix of slower and faster period
-60 -60
0 0
40 40
A --5ii-, 5
0 10 20 30 0 10 20 30
Time (se) Time (sec)
35 35
201 20 '
0 10 20 30 0 10 20 30
Time (sec) Time (sec)
Figure 3-1. Top plots: Actual left limb trajectory of a subject (solid line) versus the
desired two periodic trajectory (dashed line) input. (left leg top left plot and right leg
- top right plot). Middle plots: The tracking error (desired angle minus actual angle) of
a subject's leg tracking a two periodic desired trajectory. (left leg middle left plot and
right leg middle right plot). Bottom plots: The computed RISE voltage during knee joint
tracking for the case of two period trajectory (left leg bottom left plot and right leg -
bottom right plot).
trajectories, therefore the RMS and the maximum steady state errors are in between the
respective errors obtained for more .,.-.-ressive 2 second period and higher range of motion
desired trajectories. Figs. 3-2 3-5 depict the errors for the experiments summarized in
Table 3-3.
Additional experiments were also conducted to examine the performance of the
controller in response to step changes and changing loads. Specifically, a desired trajectory
of a step input was commanded with a 10 pound load attached to the LEM. An additional
Trajectory A B
Constant 6 sec. 2.88 6.13
Constant 2 sec. 4.11 10.67
Triple periodic (6, 4, 2) sec. 3.27 7.82
Triple periodic (6, 4, 2) sec with 5.46 12.48
higher range of motion
Table 3-3. Summarized experimental results for multiple, higher frequencies and higher
range of motion. Column (A) indicates RMS error in degrees, and column (B) indicates
maximum steady state error in degrees.
V60
40
20
i-40
Time (s)
0 10 20 30
Time (sec)
Figure 3-2. Top plot: Actual limb trajectory (solid line) versus the desired triple periodic
trajectory (dashed line). Bottom plot: The limb tracking error (desired angle minus actual
angle) of a subject tracking a triple periodic desired trajectory.
10 pound load was added once the limb stabilized after a step down of 15 degrees. The
limb was again commanded to perform a step response to raise the limb back up an
additional 15 degrees with the total load of 20 pounds. The results are shown in Fig. 3-6.
The steady state error was within 1 degree. A maximum error of 3 degrees was observed
when the external load was added. The results give some indication of the controller's
40
0I P
0 10 20 30
Time (s)
0 10 20 30
Time (sec)
Figure 3-3. Top plot: Actual limb trajectory (solid line) versus the desired constant period
(2 sec) trajectory (dashed line). Bottom plot: The limb tracking error (desired angle
minus actual angle) of a subject tracking a constant period (2 sec) desired trajectory.
ability to adapt to changes in load and step inputs and motivate possible future case
studies with neurologically impaired individuals that express muscle spasticity.
For each experiment, the computed voltage input was modulated by a fixed pulse
width of 100 p sec and fixed frequency of 30 Hz. The stimulation frequency was selected
based on subject comfort and to minimize fatigue. During preliminary experiments
with stimulation frequencies of 100 Hz, the subjects fatigued approximately two times
faster than in the current results. The results also indicate that a 100 f sec pulse width
was acceptable, though future studies will investigate higher pulse widths in the range
of 300 350/1 sec which recruit more slow fatiguing motor units [110]. Our previous
preliminary experiments indicated that longer pulse widths (e.g., 1 msec) produced similar
effects as a direct current voltage.
280,.
0
0 10 20 30
Time (s)
S-10 .....
0 10 20 30
Time (sec)
Figure 3-4. Top plot: Actual limb trajectory (solid line) versus the triple periodic desired
trajectory with higher range of motion (dashed line). Bottom plot: The limb tracking
error (desired angle minus actual angle) of a subject tracking a triple periodic desired
trajectory with higher range of motion
The use of the RISE control structure is motivated by its implicit learning characteristics
[96] and its ability to compensate for additive system disturbances and parametric
uncertainties in the system. The advantage of the RISE controller is that it does not
require muscle model knowledge and guarantees .,-i-!,iill1 ic stability of the nonlinear
system. The experimental results indicate that this feedback method may have promise in
some clinical applications.
Although the RISE controller was successfully implemented, the performance of the
controller may be improved by including a feedforward control structure such as neural
networks (a black box function approximation technique) or physiological/phenomenological
muscle models. Since the RISE controller is a high gain feedback controller that yields
.,-i-ii! ,ll. ic performance, adding a feedforward control element may improve transient and
0
0 10 20 30
Time (s)
10
S 0
0 10 20 30
Time (sec)
Figure 3-5. Top plot: Actual limb trajectory (solid line) versus the desired constant period
(6 sec) trajectory (dashed line). Bottom plot: The limb tracking error (desired angle
minus actual angle) of a subject tracking a constant period (6 sec) desired trajectory.
steady state performance and reduce the overall control effort, thereby reducing muscle
fatigue. Another possible improvement to the controller is to account for fatigue. Fatigue
can be reduced for short durations by selecting optimal stimulation parameters, but
functional electrical stimulation (FES) may require a controller that adapts with fatigue
to yield performance gains for longer time durations. Therefore our future goal will be to
include a fatigue model in the system to enhance the controller performance.
3.3.3 Conclusion
A Lyapunov-based stability analysis indicates that the closed-loop nonlinear control
method yields .i, i!!Ill i tracking for a nonlinear muscle activation and limb dynamics,
even in the presence of additive disturbances. Experiments using external electrodes
on human subjects demonstrated the ability of the RISE controller to enable a limb to
track a desired trajectory composed of varying amplitude and frequency sinusoids, step
0 ----. ......
2o 4 00 .. ....................... ......... ........
0
0 10 20 30
Time (s)
S 1 0 .. . . . .
0 10 20 30
Time (sec)
Figure 3-6. Top plot : Actual limb trajectory (solid line) versus desired step trajectory
(dashed line). The limb is tested for two step inputs. The load is added once the limb
stabilizes ( between 23 and 24 second interval). Bottom plot: The limb tracking error for
step inputs.
changes, and changes in the load. Specifically, the experimental results indicated that
with no muscle model (and only voltage amplitude modulation), the RISE algorithm
could determine the appropriate stimulation voltage for the tracking objective. For the
fastest tested trajectory the maximum steady-state tracking errors were approximately
10 degrees, whereas the maximum steady-state error in slower trajectories were as little
as approximately 4 degrees. An advantage of this controller is that it can be applied
without knowledge of patient specific parameters like limb mass or inertia, limb center
of gravity location, parameters that model passive and elastic force elements. Thus, its
application would not require specific expertise or extensive testing prior to use. The
control development also accounts for unmodeled disturbance (e.g. muscle spasticity) that
are commonly observed in clinical populations. The proposed strategy holds promise for
clinical implementation of the controller as a therapeutic tool to enhance muscle function
during isolated joint movements. However, results have yet to demonstrate functional
movements (e.g. walking) in populations without the ability to voluntarily activate their
muscles. As such, future directions will focus on studies to demonstrate the effectiveness
of the controller under such conditions. Although the trajectories used in the experiments
may not be truly functional, the controller can be applied to any continuous trajectory.
This is clinically relevant because trajectory-based movements are necessary for the
performance of many FES augmented tasks (e.g., repetitive stepping during walking).
Whether the desired trajectories are based on limb position, as in the current result,
or other information (e.g., desired joint kinetics or kinematics), the ability to precisely
track a desired pattern is fundamental to eliciting reproducible movement patterns during
functional tasks. An advantage of the control development is that it allows for inter- as
well as intra-individual variations in trajectory tracking (i.e. task performance) to be
accounted for both within and between sessions (e.g. during rehabilitation training),
thus potentially providing a tool to aid in the future advancement of rehabilitation. A
possible disadvantage of the controller is that high gains are used to achieve the robustness
to disturbances and unmodeled effects. The next section will investigate augmenting
the RISE structure with feedforward control architectures that can accommodate for
disturbances without requiring high gain feedback.
3.4 Modified Neural Network-based Electrical Stimulation for Human Limb
Tracking
NN-based estimation methods are well suited for NMES because the muscle model
contains unstructured nonlinear disturbances as given in (2 1). Due to the universal
approximation property, NN-based estimation methods can be used to represent the
unknown nonlinear muscle model by a three-ii -r NN as [112]
f(x) =WTa(UTx)+ c(x), (3-40)
for some input x(t) E RIN1+. In (3-40), U e R(Ni+1)xN2 and W e R(WN2+)xn are
bounded constant ideal weight matrices for the first-to-second and second-to-third l v. rs
respectively, where N1 is the number of neurons in the input l ..V-r, N2 is the number
of neurons in the hidden lI.-r, and n is the number of neurons in the output l -.,r.
The sigmoid activation function in (3-40) is denoted by a(-) : RN*+1 -- RN2+, and
e(x) : R N1+1 -+ IR is the functional reconstruction error. The additional term "1" in the
input vector x(t) and activation term (-.) allows for thresholds to be included as the first
columns of the weight matrices [112]. Thus, any tuning of W and U then includes tuning
of thresholds. Based on (3-40), the typical three l-v-r NN approximation for f(x) is given
as [112]
f(x)= WTa(UTx), (3-41)
where U e R(N,+1)xN2 and We R( N2+)xn are subsequently designed estimates of the
ideal weight matrices. The estimate mismatch for the ideal weight matrices, denoted by
U(t) e R(N+I1)xN2 and W(t) e R(N2+1)xn, are defined as
S= U U, W W (3-42)
and the mismatch for the hidden-li-v-r output error for a given x(t), denoted by a(x) E
RN2+1, is defined as
a = = a(UTx) a(UTx). (3-43)
The NN estimate has ceratin properties and assumptions that facilitate the subsequent
development.
Property 1: (Taylor Series Approximation) The Taylor series expansion for a(UTy)
for a given y(t) may be written as [112]
a(UTY) = a(^Ty) + a'(UTy)UTy + O(UTy)2, (344)
where o'((UTy) = da(UTy)/d(UTy)UlTy= Ty and O(UTy)2 denotes the higher order terms.
After substituting (3-44) into (3-43) the following expression can be obtained:
a = &Ty + O(UTy)2 (3 45)
where a' = a'(UTy).
Assumption 1: (Boundedness of the Ideal Weights) The ideal weights are assumed
to exist and are bounded by known positive values so that
IlU12 = tr(UTU)= ec(U)TVec(U) < Us, (3-46)
I11II l= tr(WTW)= ec(W)Tvec(W) < WB, (347)
where I||-|F is the Frobenius norm of a matrix, tr(-) is the trace of a matrix. The
ideal weights in a NN are bounded, but knowledge of this bound is a non-standard
assumption in typical NN literature (although this assumption is also used in textbooks
such as [112, 113]). If the ideal weights are constrained to stay within some predefined
threshold, then the function reconstruction error will be larger. Typically, this would
yield a larger ultimate steady-state bound. Yet, in the current result, the mismatch
resulting from limiting the magnitude of the weights is compensated through the RISE
feedback structure (i.e., the RISE structure eliminates the disturbance due to the function
reconstruction error).
3.4.1 Open-Loop Error System
The open-loop tracking error system can be developed by multiplying (3-3) by J and
by utilizing the expressions in (2-1) and (2-5)-(3-2) as
Jr = J(a2eC + ane + d) + + i + i .. QV + Td, (3-48)
where Q(q, q) is defined in (2-8). The dynamics in (3-48) can be rewritten as
JQr =fd + S V + -d (3-49)
where the auxiliary functions fd(qdqd,d qd) E R and S(q, qd, q, qd, qd) E R are defined as
fd = LQ(qd, qd) + J (qd)qd, (3-50)
S = JQ(q)(a2e2+ alel) + JQ(qqd J(qd)qd + LQ(q,q) LQ(qdA, d)
and JQ(q, <) E R, LQ(q, ) E R, and Tdn(q, t) E R are defined as
T J 7Td i + 1[,+i (+ ).
JQ dan La = (3-51)
The expression in (3-50) can be represented by a three-li -r NN as
fdA WTa(UTXd) + C(xd), (3-52)
where Xd(t) E R4 is defined as Xd(t) = [1 qd(t) qd(t) qd(t)]. Based on the assumption that
the desired trajectory is bounded, the following inequalities hold
||(X)|| < ,i b I (Xd)\ < 42 &(X) < eb3, (3-53)
where Cb,, i I'l 3 E R are known positive constants.
3.4.2 Closed-Loop Error System
The control development in this section is motivated by several technical challenges
related to blend the NN feedforward term with the RISE feedback method. One of the
challenges is that the NN structure must be developed in terms if the desired trajectories
to avoid the use of acceleration measurements. Also, while the NN estimates are upper
bounded by constants, the time derivatives of these terms are state dependent, and hence
violate the traditional RISE assumptions. To address this issue, the closed-loop error
system development requires a strategic separation and regrouping of terms. In this
section, the control is designed and the closed-loop error system is presented. Based on the
open-loop error system in (3-49) and the subsequent stability an 1 ,-i- the control torque
input is designed as [27]
V =d+l (3-54)
where fd(t) E R is the three-i V-r NN feedforward estimate designed as
f WT(U^Txd) (3-55)
and p(t) E R is the RISE feedback term designed as [11, 96, 114, 115]
p(t) A (k, + l)e2(t) (k, + l)e2(0) + v(t). (3-56)
The estimates for the NN weights in (3-55) are generated on-line using a projection
algorithm as
W =proj (oa17'T ; U = Proj (2d ( 2 TWe (3-57)
) T)
where F1 e R(N2+1)x(N2+1) and F2 E 4x4 are constant, positive definite, symmetric gain
matrices. In (3-56), k, E IR denotes positive constant adjustable control gain, and v(t) E R
is the generalized solution to
v(t) = (k, + 1)a2e2(t) + j 2sgn(e2(t)), v(0) = 0, (3-58)
where fl E R denotes positive constant adjustable control gain, and sgn(-) denotes the
signum function. The closed-loop tracking error system can be developed by substituting
(3-54) into (3-49) as
Jr = fd + S- p + TdQ, (3-59)
where
fd(Xd) = fd fd. (3-60)
To facilitate subsequent closed loop stability analysis, the time derivative of (3-59) can be
determined as
JQ = -Jr + fd + S i + d. (3-61)
Although the voltage control input V(t) is present in the open loop error system in (3-49),
an additional derivative is taken to facilitate the design of the RISE-based feedback
controller. After substituting the time derivative of (3-60) into (3-61) by using (3-52) and
(3-55), the closed loop system can be expressed as
.T .T
J. =- -Jr + WT' (UTXd)UTX d (- W Tx) WTa'(uTxdu) uTd wU wWT,'(uT.Xd) J Xd
+ (Xd) + S- + dn, (3-62)
where a'(UTx) = d(UTx)/d(UTx) IUTX (T1 After adding and subtracting the terms
WT&'/VTXd + WT&'/VTXd to (3-62), the following expression can be obtained:
JQ =-JQr + WT' VTaXd + WTa'fVT'Xd WT, 'VTXd -WTa'VTXd (3-63)
.T T
+ WTa'UTXd + (Xd) Wa' Xd-- + S i + Tn,
where the notation & (.) is introduced in (3-43). Using the NN weight tuning laws
described in (3-57), the expression in (3-63) can be rewritten as
1.
JQ. = 2-J r + N + N e2 (k, + 1) r psgn(e2), (3-64)
where the unmeasurable auxiliary terms N(e,e2, r, t) and N(W, U, d, t) E R given in
(3-64) are defined as
N(t) = J2r + +2 -- poj (r ,, Pj W 'pTOj (F2 d (rwTC2 Xd
(3-65)
N = NB + Nd. (3-66)
In (3-66), Nd(q, q,xd,x_, t) E R is defined as
Nd WTaUTXd + t(xd) + rd, (3-67)
while NB(W, U, xd,Jd,t) C R is defined as
NB= NB1 + NB,, (3-68)
where NB,(W, U, xd, xd,t) and NB2(W, U, xd, xd,t) E R are defined as
B, = -WTa'T 'X WTUUiTXd, (3-69)
and
NB, = W 'TU + WTl'UTXd. (3-70)
Motivation for the definitions in (3-65)-(3-67) are based on the need to segregate terms
that are bounded by state-dependent bounds and terms that are upper bounded by
constants for the development of the NN weight update laws and the subsequent stability
analysis. The auxiliary term in (3-68) is further segregated to develop gain conditions
in the stability analysis. Based on the segregation of terms in (3-65), the Mean Value
Theorem can be applied to upper bound N(el,e2, r, t) as
N < p (| |) | (3-71)
where z(t) E R3 is defined as
z(t) ^ [e e rT]T, (3-72)
and the bounding function p (||l |) E R is a positive globally invertible nondecreasing
function. Based on Assumption 3 in ('!i plter 2, (3-46), (3-47), (3-53), and (3-68)-(3-70),
the following inequalities can be developed [27]:
INd <_ 1 NBl < (2 Nd <(3 (373)
NB
where ( E R ,(i = 1, 2, ...5) are known positive constants.
3.4.3 Stability Analysis
Theorem 2. The composite NN and RISE controller given in (3-54)-(3-58) ensures that
all system -:,,it..,l are bounded under closed-loop operation and that the position tracking
error is ,, i.,l., .l in the sense that
||ei(t)| 0 as t oo, (3-74)
within some set S containing the initial conditions of the system, provided the control
gains in (3 56) and (3 58) are selected suff .'. i.l, '/1/; 7
Proof for Theorem 2: Let D C R5 be a domain containing y(t) = 0, where
y(t) E R5 is defined as
y(t) (3-75)
where the auxiliary function Q (t) E R is defined as
Q (t) A tr (IW W + 2tr (UT (376)
F22 J) 2(3-76
and P (t) E R is the generalized solution to the differential equation
P (t) = -L (t) P (0) 1 pl2 (0) e2 (0) N (0). (3-77)
Since Fi and F2 in (3-76) are constant, symmetric, and positive definite matrices, and
c2 > 0, it is straightforward that Q (t) > 0. The auxiliary function L (t) E R in (3-77) is
defined as
L (t) A r (NB (t) + Nd(t) Assgn (e2)) + C2NB (t) 2e2(t)2, (3 78)
where si, /32 E R introduced in (3-58) and (3-78) respectively, are positive constants
chosen according to the following sufficient conditions
1 > (1 + (2 + C3 + 4, /2 > C5, (3-79)
Q2 Q2
where (i E R ,(i = 1, 2,..., 5) are known positive constants introduced in (3-73). Provided
the sufficient conditions in (3-79) are satisfied, then P (t) > 0.
Let VL (y, t) : D x [0, oo) -- R denote a Lipschitz continuous regular positive definite
functional defined as
VL (y, t) 2 + t 2 + P+JQ, (3-80)
which satisfies the inequalities
S(y) < VL (Y, t) < U2 (), (381)
provided the sufficient conditions in (3-79) are satisfied, where Ui (y) U2 (y) c R are
continuous, positive definite functions defined as
U, (y) = A y112, U2 (y) A2 IX12, (3-82)
where A1, A2 E R are known positive functions or constants. From (3-2), (3-3), (3-64),
(3-77), (3-78), and after taking the time derivative of (3-76), some of the differential
equations describing the closed-loop system for which the stability analysis is being
performed have discontinuous right-hand sides as
e = e2 ale1, (3-83a)
e2 = r a2e2, (3-83b)
J,, = Jr + 1N + N e2 (k + 1) r- sgn(e2), (3-83c)
2
P (t) -r (NB, (t) + Nd(t) isgn (e2))- 2NB, (t) + 22(t)2, (3-83d)
Q (t) = (at 2 WT 1-/) + (trau2 T2 ) (3-83e)
Let f(y, t) e R5 denote the right hand side of (3-83). f(y, t) is continuous except in the
set {(y,t)le2 = 0}. From [103-106], an absolute continuous Filippov solution y(t) exists
almost everywhere (a.e.) so that
E K[f](y,t) a.e.
The generalized time derivative of (3-80) exists a.e., and VL(y, t) e VL (y, t) where
a 1s
VL (y, t) = 10n K elr +2 (7k +)- IQ-Q g(3-84)
T
= VLT 2 7 1P-P 1Q- ,
r 1 K t
C 2e, e2 rJQ 2P 2 2Q 1K 2 _P 1
For more details of the notations used in 3-83 to 3-84 and discussion, see Section 3.3.1.
After utilizing (3-2), (3-3), (3-64), (3-77), (3-78), the expression in 3-84 can be rewritten
as
VL (y, t) C 2eie2 2ale + e2r ae2 + .]^ + rN + rN e2 (ks + 1) r2 prK[sgn(eC2
2
jr2 rNB, rNd(t) + prK[sgn(eC2) BeMN(t + 0 + tr a2 W 1W
+tr (a2 fTF-u (3-85)
Using (3-57), (3-66), (3-68), (3-70), cancelling common terms, and based on the fact
that
2ele2< IJ 11 2+ 211 2
(3-85) can be written as
VL (y,t) C -(2ai l)e (a2 32 l)e| r2 + rN k,2. (386)
As shown in (3-85)-(3-86), the unique integral signum term in the RISE controller
is used to compensate for the disturbance terms included in Nd(qd, qd, qd, 9d, t) and
NB, (W, U, xd, Xd, t), provided the control gain Pl and f2 are selected according to
(3-79). Further the term NB2(W, U, xd, xd, t) is partially rejected by the unique integral
signum term and partially cancelled by adaptive update law. Using (3-71), the term
rT(t)N(ei, e2, r, t), can be upper bounded by following inequality:
rN < p (11 11) 11 1 r ,
to obtain
VL (y,t) c -min {2ai a 32 ,2 } '+ [p (11 : ) | : I||I r kr2].
Completing the squares for the bracketed terms in (3-87) yields
p2 ( : ) :
VL (y, t) C min {2ai 1, a2 -- 2 ,1} II 1 +
4kg
(3-87)
(3-88)
The following expression can be obtained from (3-88):
VL (y, t) C -U(y),
(3-89)
where U (y) = c ||:|, for some positive constant c E R, is a continuous positive
semi-definite function that is defined on the following domain:
D y Rhe I I, < d a s d a )
where A3 A min{2i 1, a2 /2 1, 1}. Let S C D denote a set defined as follows:
S y(t)c 2 (y(t)) < A 1 (2VA3k)2)
(3-90)
where S C D is introduced in Theorem 2. The region of attraction in (3-90) can be made
arbitrarily large to include any initial conditions by increasing the control gain k, (i.e., a
semi-global type of stability result), and hence
cll :(t)112
as t oo
Vy(O) e S.
(3-91)
Based on the definition of z(t) in (3-72), (3-91) can be used to show that
Vy(O) E S.
(3-92)
as t oo
Ilel(t II -
3.4.4 Experimental Results
Results are provided in this section that examine the performance of the controller
given in (3-54)-(3-58) in experiments with volunteer subjects. These results were
compared with the previous results in [116] that used the RISE feedback structure without
the NN feedforward term. The NMES controller was implemented as an amplitude
modulated voltage composed of a positive rectangular pulse with a fixed width of 400
p sec and fixed frequency of 30 Hz. The a priori chosen stimulation parameters are within
the ranges typically reported during NMES studies [110, 116]. Without loss of generality,
the controller is applicable to different stimulation protocols (i.e., voltage, frequency, or
pulse width modulation). The following results indicate that the developed controller
(henceforth denoted as NN+RISE) was able to minimize the knee angle error while
dynamically tracking a desired trajectory.
3.4.4.1 Testbed and protocol
The tested consists of a custom computer controlled stimulation circuit and a
modified leg extension machine (LEM). The LEM was modified to include optical
encoders. The LEM allows seating adjustments to ensure the rotation of the knee is
about the encoder axis. A 4.5 kg (10 lb.) load was attached to the weight bar of the LEM
and a mechanical stop was used to prevent hyperextension.
The objective in one set of experiments was to enable the knee and lower leg to
follow an angular trajectory, whereas, the objective of a second set of experiments was to
regulate the knee and lower leg to a constant desired setpoint. An additional preliminary
test was also performed to test the capability of the controller for a sit-to-stand task. For
each set of experiments, bipolar self-adhesive neuromuscular stimulation electrodes were
placed over the distal-medial and proximal-lateral portion of the quadriceps femoris muscle
group of volunteers and connected to custom stimulation circuitry. The experiments
were conducted on non-impaired male and female subjects (as in our previous study in
[116]) with age ranges of 20 to 35 years, with written informed consent as approved by
the Institutional Review Board at the University of Florida. The electrical stimulation
responses of non-impaired subjects have been reported as similar to paraplegic subjects'
responses [16, 22, 39, 111]. The volunteers were instructed to relax as much as possible
and to allow the stimulation to control the limb motion (i.e., the subject was not supposed
to influence the leg motion voluntarily and was not allowed to see the desired trajectory).
The NN+RISE controller was implemented with a three input 1i,-vr neurons,
twenty-five hidden 1i,-vr neurons, and one output 1-v r neuron. The neural network
weights were estimated on-line according to the adaptive algorithm in (3-57). For each
experiment, the computed voltage input was modulated by a fixed pulse width of 400
p sec and fixed frequency of 30 Hz. The stimulation frequency was selected based on
subject comfort and to minimize fatigue. Nine subjects (8 males, 1 female) were included
in the study. The study was conducted for different types of desired trajectories including:
a 1.5 second periodic trajectory, a dual periodic trajectory (4-6 second), and a step
trajectory. For the 1.5 second periodic trajectory, controllers were implemented on both
legs of four subjects, while the rest of the tests were performed on only one leg of the
other three subjects since they were not available for further testing. Three subjects (1
male, 1 female (both legs); 1 male (one leg)) were asked to volunteer for the dual periodic
desired trajectory tests while regulation tests were performed on one of the legs of two
subjects. Each subject participated in one trial per criteria (e.g., one result was obtained
in a session for a given desired trajectory). For each session, a pre-trial test was performed
on each volunteer to find the appropriate initial voltage for the controller to reduce the
initial transient error. After the pre-trial test, the RISE controller was implemented on
each subject for a thirty second duration and its performance was recorded. A rest period
of five minutes was provided before the NN+RISE controller was implemented for an
additional thirty second duration.
Time [sec]
15
0 5
0 .................
0 5 10 15 20 25 30
Time [sec]
30
I25
middle plot shows the tracking error (desired angle minus actual angle). The maximum
steady state error obtained is 5.95 (at 20.7 sec.). The bottom plot shows the computed
15
0 5 10 15 20 25 30
Time [sec]
Figure 3-7. The top plot shows the actual limb trajectory (solid line) obtained from the
RISE controller versus the desired 1.5 second period desired trajectory (dashed line). The
middle plot shows the tracking error (desired angle minus actual angle). The maximum
steady state error obtained is 5.95' (at 20.7 sec.). The bottom plot shows the computed
RISE voltage. The maximum steady state voltage obtained is 28.1 V (at 21.47 sec.).
3.4.4.2 Results and discussion
The knee/lower limb tracking results for a representative subject with stimulation
from the RISE and the NN+RISE controllers are shown in Figs. 3-7-3-8 and are
summarized in Table 3-4. In Table 3-4, the maximum steady state voltage (SSV) and
maximum steady state error (SSE) are defined as the computed voltage and absolute value
of error respectively, that occur after 1.5 seconds of the trial. Paired one tailed t-tests
(across the subject group) were performed with a level of significance set at a = 0.05. The
results indicate that the developed controller demonstrates the ability of the knee angle
to track a desired trajectory with a mean (for eleven tests) RMS error of 2.92 degrees
with a mean maximum steady state error of 7.01 degrees. Combining the NN with the
RISE feedback structure in [116] yields (statistically significant) reduced mean RMS error
0 5 10 15 20 25 30
Time [sec]
15
7 1 0 ... ..:. .. .. .. .. . . . ..
0 .
0 5 10 15 20 25 30
Time [sec]
30
r 25
0 5 10 15 20 25 30
Time [sec]
Figure 3-8. The top plot shows the actual limb trajectory (solid line) obtained from the
NN+RISE controller versus the desired 1.5 second period desired trajectory (dashed
line). The middle plot shows the tracking error (desired angle minus actual angle). The
maximum steady state error obtained is 4.240 (at 28.6 sec.). The bottom plot shows the
computed NN+RISE voltage. The maximum steady state voltage obtained is 26.95 V (at
29.1 sec.).
for approximately the same input stimulus. The maximum steady state voltages for the
RISE and NN+RISE controllers revealed no statistical differences. To illustrate that the
performance of NN+RISE controller (in comparison to the RISE controller alone) can
be more significant for different desired trajectories, both controllers were implemented
on three subjects (2 male, 1 female) with the control objective to track a dual periodic
(4 6 second) desired trajectory with a higher range of motion. The stimulation results
from the RISE and the NN+RISE controllers are shown in Figs. 3-9 and 3-10 and are
summarized in Table 3-5. In Table 3-5, the maximum SSV and SSE were observed after
4 seconds of the trial. The results illustrate NN+RISE controller yields reduced mean
RMS error (across the group) and reduced mean maximum SSE (across the group) for
Subject Leg RMS Error Max SSE RMS Voltage [V] Max SSV [V]
RISE NNR RISE NNR RISE NNR RISE NNR
A Left 3.590 2.920 12.420 7.590 22.91 23.98 29.5 31
A Right 2.600 2.630 5.740 6.510 27.70 25.40 32.95 31.5
B Left 2.470 2.230 5.950 4.240 22.41 22.81 28.1 26.95
B Right 2.830 2.740 6.280 6.760 25.10 23.03 29.8 30.5
C Left 3.180 2.460 8.1 6.170 41.35 40.14 48.9 44.8
C Right 2.970 3.010 6.90 9.630 36.32 35.15 46.4 42.3
D Left 3.230 3.71 6.040 5.860 25.25 28.24 30 34.1
D Right 3.530 2.960 8.80 7.580 13.62 14.95 24.2 23.4
E Left 3.920 3.260 11.150 7.920 30.89 31.46 45 40.5
F Left 3.380 2.830 7.990 6.41 26.15 28.13 31.8 34.1
G Left 3.520 3.320 8.20 8.450 41.59 43.44 49.8 50
Mean 3.200 *2.920 7.960 7.010 28.48 28.79 36.04 35.38
Std. Dev. 0.450 0.410 2.180 1.440 8.49 8.29 9.44 8.08
p-value 0.02 0.08 0.28 0.22
Table 3-4. Summarized experimental results and P values of one tailed paired T-test for
a 1.5 second period desired trajectory. indicates statistical difference. NNR stands for
NN+RISE controller.
approximately the same input stimulus. Paired one tailed t-tests (across the subject
group) were performed with a level of significance set at a = 0.05. The results show that
the difference in mean RMS error and mean maximum SSE were statistically significant.
The P value for the mean RMS error (0.00043) and mean maximum SSE (0.0033) t-test
obtained in the case of dual periodic trajectory is smaller when compared to the P values
(0.02 and 0.08, respectively) obtained for the 1.5 second trajectory. This difference
indicates the increased role of the NN for slower trajectories (where the adaptation gains
can be increased).
As in [117], additional experiments were also conducted to examine the performance
of the NN+RISE controller in response to step changes and changing loads. Specifically,
a desired trajectory of a step input was commanded with a 10 pound load attached to
the LEM. An additional 10 pound load was added once the limb stabilized at 15 degrees.
The limb was again commanded to perform a step response to raise the limb back up an
Time [sec]
0 5 10 15 20 25 30
Time [sec]
30
0
0 5 10 15 20 25 30
Time [sec]
Figure 3-9. The top plot shows the actual limb trajectory (solid line) obtained from the
RISE controller versus the dual periodic desired trajectory (dashed line). The middle plot
shows the tracking error (desired angle minus actual angle). The maximum steady state
error obtained is 6.560 (at 21 sec). The bottom plot shows the computed RISE volatge.
The maximum steady state voltage obtained is 29.67 V (at 26.7 sec.).
additional 15 degrees with the total load of 20 pounds. The results from a representative
subject using NN+RISE controller are shown in Fig. 3-11. The experimental results
for the step response and load addition are given in Table 3-6. The results give some
indication of the controller's ability to adapt to changes in load and step inputs and
motivate possible future case studies.
Experiments were also performed to test the NN+RISE controller for a sit-to-stand
task. These tests were conducted on a l. ,ril!:v individual initially seated on a chair (see
Fig. 3-12). The knee angle was measured using a goniometer (manufactured by Biometrics
0 5 10 15 20 25 30
Time [sec]
10 5 1 1 2 2 30
-5
Ti0 ....
0 5 10 15 20 25 30
Time [sec]
3 0 '"
0 5 10 15 20 25 30
Time [sec]
Figure 3-10. The top plot shows the actual limb trajectory (solid line) obtained from
the NN+RISE controller versus the dual periodic desired trajectory (dashed line). The
middle plot shows the tracking error (desired angle minus actual angle). The maximum
steady state error obtained is 4.57 (at 10.5 sec.). The bottom plot shows the computed
NN+RISE volatge. The maximum steady state voltage obtained is 29.68 V (at 26.9 sec.
Ltd.) attached to both sides of the subject's knee, where the initial knee angle is set
to zero (sitting position). The goniometer was interfaced with the custom computer
controlled stimulation circuit via an angle display unit (ADU301). The objective was to
control the angular knee trajectory that resulted in the volunteer rising from a seated
position, with a final desired angle of 900 (standing position). The error, voltage, and
desired versus actual knee angle plots are shown in Fig. 3-13. The RMS error and voltage
during this experiment were obtained as 2.92 and 26.88 V, respectively. The final steady
state error reached within -0.50, the maximum transient error was observed as 8.23,
Subject Leg RMS Error Max SSE RMS Voltage [V] Max SSV [V]
RISE NNR RISE NNR RISE NNR RISE NNR
A Left 2.350 1.850 6.120 4.300 29.08 29.19 34.10 34.09
A Right 1.730 1.260 4.490 3.90 30.00 29.67 35.75 34.62
B Left 3.520 2.620 6.450 5.640 37.09 36.34 44.04 43.47
B Right 3.390 2.890 6.530 6.000 37.88 38.57 45.30 46.19
C Right 3.840 2.820 6.560 4.570 23.99 24.09 29.67 29.68
Mean 2.970 *2.290 6.030 *4.880 31.61 31.57 37.77 37.61
Std. Dev. 0.890 0.710 0.880 0.900 5.84 5.85 6.69 6.93
p-value 0.00043 0.0033 0.43 0.29
Table 3-5. Summarized experimental results and P values of one tailed paired T-test for
dual periodic (4-6 second) desired trajectory. indicates statistical difference. NNR stands
for NN+RISE controller.
Subject Leg Max. SSE Max. Tran- Max. Error Max. SSV (af-
(after step sient Error (during dis- ter step input)
input) turbance) [Volts]
A Left 0.70 9.50 2.80 42.2
B Right 0.60 9.520 2.00 19.2
Table 3-6. Experimental results for step response and changing loads
and the maximum voltage was obtained as 35.1 V. The significance of these tests is to
depict the applicability of the controller on clinical tasks such as sit to stand maneuvers.
Although the experiments were conducted on a healthy individual, these preliminary
results show that the controller holds promise to provide satisfactory performance on
patients in a clinical-type scenario.
The NN+RISE structure is motivated by the desire to blend a NN-based feedforward
method with a continuous feedback RISE structure to obtain .i-, i!ill, i ic limb tracking
despite an uncertain nonlinear muscle response. The ability of the neural networks to
learn uncertain and unknown muscle dynamics is complemented by the ability of RISE to
compensate for additive system disturbances (hyperactive somatosensory reflexes that may
be present in impaired individuals) and NN approximation error. Although the NN+RISE
controller was successfully implemented and compared to RISE controller in the present
work, the performance of the controller may be further improved in efforts to reduce the
20 5
10 U ....
0 5 10 15 20 25 30
3Time [sec]
0 5 10 15 20 25 30
Time [sec]
Figure 3-11. Experimental plots for step change and load addition obtained from
NN+RISE controller. Top plot shows actual limb trajectory (solid line) versus desired step
trajectory (dashed line). The load is added once the limb stabilizes (between 13-15 second
interval). After load addition the limb is tested for the step input. Middle plot shows the
limb tracking error obtained during the experiment. Bottom plot shows computed voltage
for the experiment.
effects of muscle fatigue in future studies. Fatigue can be reduced for short durations by
selecting optimal stimulation parameters, but functional electrical stimulation (FES) may
require a controller that adapts with fatigue to yield performance gains for longer time
durations. Therefore our future goal will be to include a fatigue model and incorporating
calcium dynamics in the muscle dynamics to enhance the controller performance.
Figure 3-12. Initial sitting position during sit-to-stand experiments. The knee-angle was
measured using a goniometer attached around the knee-axis of the subject's leg.
3.4.5 Limitations
The results illustrate the added value of including a NN feedforward component
in comparison to only using the RISE feedback structure in [116]. However, several
limitations exist in the experimental study. The contribution from the NN component
was observed to increase but the RISE contribution did not decline proportionally. A
possible reason for this observation is that the 1.5 second period desired trajectory has
a large desired acceleration qd(t), which is an input to the NN that can lead to large
voltage swings during the transient stage. To reduce large voltage variants during the
transient due to qd(t), the update law gains are reduced in comparison to gains that could
be employ, ,1 during less .. .-essive trajectories. The experimental results with slower
trajectories (dual periodic 4-6 second period) illustrate that the NN component can pl i,
a larger role depending on the trajectory. Specifically, the dual periodic trajectory results
indicate that the RMS error obtained with the NN+RISE controller is lower than the
RMS error obtained with the RISE controller with a lower P value (0.00043) compared to
the P value (0.02) obtained with the 1.5 second period trajectory.
I I |
... .. .. .. .
0 -'
0 1 2 3 4
Time [sec]
S10,,
to
0 1 2 3 4
3 5 ... . ... ... .
S25
0 1 2 3 4
Time [sec]
Figure 3-13. The top plot shows the actual leg angle trajectory (solid line) versus desired
trajectory (dotted line) obtained during the standing experiment. The middle plot shows
the error obtained during the experiment. The bottom plot shows the voltage produced
during the experiment.
Since a trajectory for a specific functional task was not provided, the desired
trajectory used in the first set of experiments was simply selected as a continuous sinusoid
with a constant 1.5 second period. The desired trajectory was arbitrarily selected, but
the period of the sinusoid is inspired by a typical walking gait trajectory. As the work
transitions to applications where a specific functional trajectory is generated, the control
results should directly translate. Furthermore, some clinical goals may be better expressed
as a desired force profile rather than a desired limb trajectory. The results from this work
could be directly applied to these cases by altering the control objective and open-loop
error system, but the form of the control method (i.e., NN+RISE) would remain intact.
An analysis of RMS errors during extension and flexion phase of the leg movements
across different subjects, trajectories (1.5 second and dual periodic), and both controllers
showed that the mean RMS error is more when leg is moving upwards (extension phase)
compared to periods when leg is moving downwards (flexion phase). A t-test analysis
showed that the results are statistically significant with p values of 0.00013 and 0.0014
obtained from RISE and NN+RISE controllers, respectively. The mean RMS errors during
extension phase for RISE and NN+RISE controllers were 3.49 and 2.680, respectively
while mean RMS errors during flexion phase for RISE and NN+RISE controllers were
2.960 and 2.420, respectively. Summarized RMS errors for both phases are shown in
Table 3-7. An increased error during extension phase can be attributed to higher control
effort required during extension. The performance during the extension phase can also
be ...: i ivated by increased time delay and muscle fatigue due to the requirement for
higher muscle force compared to the flexion phase. This analysis indicates a possible need
for separate control strategies during extension and flexion phase of the leg movement.
Particularly, future efforts will investigate a hybrid control approach for each phase of
motion.
Currently the experiments were performed on non-impaired persons. In future studies
with impaired individuals, our untested hypothesis is that the added value of the NN
feedforward component will be even more pronounced (and that the controller will remain
stable) as disturbances due to more rapid fatigue and more sensitive somatosensory
reflexes may be present in impaired individuals. To delay the onset of fatigue, different
researchers have proposed different stimulation strategies [32, 33, 118] such as choosing
different stimulation patterns and parameters. The NMES controller in this study was
Subject Leg Trajectory RMS Error (RISE) RMS Error (NN+RISE)
Extension Flexion Extension Flexion
A Left Dual period 4.350 2.41 3.300 1.680
A Right Dual period 3.980 2.680 3.390 2.280
B Left Dual period 2.740 1.860 1.770 1.920
B Right Dual period 1.780 1.690 1.350 1.170
C Right Dual period 4.220 3.430 3.270 2.280
D Left 1.5 second 2.870 2.000 2.540 1.880
D Right 1.5 second 3.21 2.380 3.070 2.380
E Left 1.5 second 3.870 3.300 3.300 2.490
E Right 1.5 second 2.560 2.650 2.340 2.880
F Left 1.5 second 3.81 2.51 4.000 3.400
F Right 1.5 second 3.590 3.47 2.960 2.960
G Left 1.5 second 3.930 2.180 2.860 1.970
G Right 1.5 second 2.980 2.950 2.820 3.190
H Left 1.5 second 4.180 2.700 3.920 2.580
I Left 1.5 second 3.970 2.660 3.110 2.51
J Right 1.5 second 3.790 4.050 3.380 3.130
Mean 3.490 2.680 2.960 2.420
p-value 0.00013 0.0014
Table 3-7. The table shows the RMS errors during extension and flexion phase of the
leg movement across different subjects, trajectories (1.5 second and dual periodic), and
controllers (RISE/NN+RISE). The results show that the mean RMS error is more during
the extension phase than during the flexion phase.
implemented using constant pulse width amplitude modulation of the voltage. However,
the controller can be implemented using other modulation schemes such as pulse width
and frequency modulation without any implications on the stability analysis, but the
effects of using frequency modulation or varying pulse trains (e.g. a pulse train containing
doublets) remain to be investigated clinically.
3.4.6 Conclusion
A Lyapunov-based stability analysis indicates that the developed closed-loop
nonlinear NMES control method yields .,-vmptotic tracking for a unknown nonlinear
muscle activation and limb dynamics, even in the presence of uncertain additive
disturbances. Experiments using external electrodes on non-impaired volunteers
demonstrated the ability of the NN+RISE controller to enable the knee and lower leg
to track a desired trajectory composed of sinusoids, step changes, and changes in the load.
Statistical analysis of the experimental results indicates that the NN+RISE algorithm
yields reduced RMS tracking error when compared to the RISE controller for statistically
insignificant differences in voltage input. A preliminary experiment (a sit-to-stand task)
to test the controller for a clinical-type functional task showed a promising control
performance. These experiments -i-i-. -1 that future efforts can be made to test the
performance on patients with movement disorders. Specifically, experiments should be
conducted for functional tasks such as walking and sit-to-stand maneuvers.
CHAPTER 4
NONLINEAR CONTROL OF NMES: INCORPORATING FATIGUE AND CALCIUM
DYNAMICS
4.1 Introduction
The focus of this chapter is to address muscle fatigue by incorporating an uncertain
fatigue model (i.e., the model developed in [35]) in the NMES controller. The contribution
of the method is that only best guess estimates of patient specific fatigue time constants
and natural frequency of calcium dynamics are required and the mismatch between the
estimated parameters and actual parameters is included in a stability analysis. The
fatigue model is defined as a function of a normalized muscle activation variable. The
normalized muscle activation variable denotes the calcium (Ca2+ ion) dynamics which
act as an intermediate variable between contractile machinery and external stimulus.
The calcium dynamics are modeled as a first order differential equation based on [6] and
[39]. An open-loop error system for an uncertain nonlinear muscle model is developed
that includes the fatigue and calcium dynamics. A virtual control input is designed using
nonlinear backstepping technique which is composed of a NN based feedforward signal
and an error based feedback signal. The NN based control structure is exploited not
only to feedforward muscle dynamics but also to approximate the error generated due to
parametric uncertainties in the assumed fatigue model. The actual external control input
(applied voltage) is designed based on the backstepping error. Through this error-system
development, the continuous NN based controller is proven (through a Lyapunov-based
stability analysis) to yield an uniformly ultimately bounded stability result despite the
uncertain nonlinear muscle model and the presence of additive bounded disturbances (e.g.,
muscle spasticity, changing loads in functional tasks, and d-.1-).
4.2 Muscle Activation and Limb Model
The musculoskeletal model given in C(i Ipter 2 is modified to consider calcium and
fatigue dynamics during neuromuscular electrical stimulation. The additional dynamics
of calcium ions and muscle fatigue are incorporated in the contraction and activation
dynamics while the body segmental dynamics remains the same as provided in C'! lpter 2.
The torque produced about the knee is generated through muscle forces that are
elicited by NMES. The active moment generating force at the knee joint is the tendon
force Fr(t) c R defined as [119]
FT = F cos a, (4-1)
where a(q(t)) E R is defined as the pennation angle between the tendon and the muscle,
where q(t), q(t) E R denote the angular position and velocity of the lower shank about
the knee-joint, respectively (see Fig. 2-2). The pennation angle of the human quadriceps
muscle changes monotonically during quadriceps contraction and is a continuously
differentiable, positive, monotonic, and bounded function with a bounded first time
derivative [100]. The muscle force F(t) E R in (4-1) is defined as [36]
F = Fm, T2P(x)x, (4-2)
where F, E R is the maximum isometric force generated by the muscle. The uncertain
nonlinear functions 1i(q), ry2(q, q) E R in (4-2) are force-length and force-velocity
relationships, respectively, defined as [36, 120, 121]
(q) exp ( b 1)2) (4 3)
h2(q, q) = arctan(c (q, q) + C3) + C4 (4 4)
where b, l(q) e R in (4-3) denote the unknown shape factor and the normalized length
with respect to the optimal muscle length, respectively, and v(q, q) E R is an unknown
non-negative normalized velocity with respect to the maximal contraction velocity of the
muscle, and ci, c2, C3, C4 are unknown, bounded, positive constants.
Assumption: The force-velocity relationship 92 is lower bounded by a known
constant E,. The lower bound on the force-velocity relationship is practical in the sense
that rT2(q, ) = 0 (i.e., no force output) only occurs when the muscle shortening velocity (a
concentric contraction) is at the maximum rate.
The definitions in (4-3) and (4-4) are not directly used in the control development.
Instead, the structure of the relationships in (4-3) and (4-4) is used to conclude that r1q(q)
and q2(q, q) are continuously differentiable, non-zero, positive, monotonic, and bounded
functions, with bounded first time derivatives. The muscle force in (4-2) is coupled to the
actual external voltage control input V(t) E R through an intermediate normalized muscle
activation variable x(t) E R. The muscle activation variable is governed by following
differential equation [34, 119]
2x = -wx + wsat[V(t)l, (4-5)
where w E R is the constant natural frequency of the calcium dynamics. The function
sat[V(t)]E R (i.e., recruitment curve) is denoted by a piecewise linear function as
0 V < Vmi,
sat[V(t)] = V-Vmin Vmin < V < Vmax (46)
Vmax Vmin
1 V > Vmax,
where Vmin E R is the minimum voltage required to generate noticeable movement or force
production in a muscle, and Vmax E R is the voltage of the muscle at which no considerable
increase in force or movement is observed. Based on (4-5) and (4-6), a linear differential
inequality can be developed to show that x(t) E [0, 1]. Muscle fatigue is included in (4-2)
through the invertible, positive, bounded fatigue function p(x) E R that is generated from
the first order differential equation [35, 36]
1 1
=7 ( -+min P)x + (4-7)
If T,
where pmin is the unknown minimum fatigue constant of the muscle, and Tf, T, are
unknown time constants for fatigue and recovery in the muscle, respectively.
Active muscle Netactive
force force
Uncertain
Fatigue Model
Figure 4-1. An uncertain fatigue model is incorporated in the control design to address
muscle fatigue. Best guess estimates are used for unknown model parameters.
4.3 Control Development
The objective is to develop a NMES controller to produce a knee torque trajectory
that will enable a human shank to track a desired trajectory, denoted by qd(t) E R, despite
the uncertain fatigue effects and coupled muscle force and calcium dynamics. Without
loss of generality, the developed controller is applicable to different stimulation protocols
(i.e., voltage, frequency, or pulse width modulation). To quantify the objective, a position
tracking error, denoted by e(t) E R, is defined as
e(t) qd(t) q(t), (4-8)
where qd(t) is an a priori trajectory which is designed such that qd(t), qj(t) E L,, where
q) (t) denotes the ithderivative for i = 1, 2, 3,4. To facilitate the subsequent analysis, a
filtered tracking error, denoted by r(t), is defined as
r(t) e(t) + ae(t), (4-9)
where a E R denotes a positive constant.
4.3.1 Open-Loop Error System
The open-loop tracking error system can be developed by taking the time derivative
of (4-9), multiplying the resulting expression by J, and then utilizing the expressions in
(4-1), (4-2), (2-1), (2-5) and (4-8) as
Ji = J(ae + gd) + Vi + 1., + iV .. + rd pcx, (4-10)
where the auxiliary function p(q, q) E R is defined as
p = cos(a)FmlT7i72 (4 11)
After multiplying (4-10) by p-l(q, q) E R, the following expression is obtained:
Jpi = Jp(ae + id) + Lp + Tdp ox, (4-12)
where J(q, t), Tdp(q, t), Lp(q, q) E R are defined as
Jp = -lJ, Tdp = p-Td,
LP p- 1(. + 3, + f ..).
Property 3: Based on the assumptions and properties (in Section 4.2), p(q, q) is
continuously differentiable, positive, monotonic, and bounded. Also the function p- (q, q)
is bounded. The first time derivatives of p(q, q) and p- (q, q) exist and are bounded. The
inertia function Jp is positive definite and can be upper and lower bounded as
ai I12 < 7TJ7 < a2 2 V7 (4 13)
where a,, a2 E R are some known positive constants. Also using the boundedness of
p(q,q), P(q, ), P-(q,)
Jp < \Tdp < (4-14)
where ij, R E R are some known positive constants.
Based on (4-7) a positive estimate p(x) is generated as
1 1
(7min ) ( + -(1- (- (4-15)
Tf T,
1 > (0) > 0,
where Tf, T E R denote constant best guess estimates of the time constants Tf and T,
respectively, (mjin E R is a non zero positive constant, and x(t) E R is the estimated
normalized muscle activation variable which is generated based on (4-5) as
2x = -iw + wsat[V(t)], (4-16)
where w c R denotes the constant best guess estimate of natural frequency of calcium
dynamics w. The estimated function ('(x) is upper bounded by a positive constant p E R.
Specifically, p can be determined as
T
S= (0) + + 1 + mn. (4-17)
Tf
The algorithm used in (4-15) ensures that (p(x) remains strictly positive. Based on (4-6)
and (4-16), a linear differential inequality can be developed to show that x(t) c [0, 1].
To facilitate the control development, the terms (p(x)x +ip(x)x + p(x)x are added and
subtracted to (4-12) to yield
Jp = S + Tdp ejTr -- p pe 'px, (4-18)
where the auxiliary function S(q, q, qd, e, r, x) E R is defined as
S Jp(qd + oae) + L,(q, q) + 2r + e yx (4-19)
and the error functions pr(x,.), 9((x), x(t) E R are denoted as
G(x) = (x') (x'), (4-20)
e(x, X) = (x) ( ), (4-21)
Sx x. (4-22)
Since ip(x) and p(x) are bounded functions, the error function y(t) can be upper bounded
as
I e < (4-23)
where R E R is some known positive constant. The auxiliary function S(q, q, qd, e, r, ') can
be represented by a three-i i'r NN as
S WTo(UTy) + c(y), (4-24)
where y(t) E R7 is defined as
y( ) [1 q(t) d() t) (t) r(t) x(t) (4-25)
and c(y) is a functional reconstruction error that is bounded by a constant as
I y) < 6. (4-26)
4.3.2 Closed-Loop Error System
Since a direct control input does not appear in the open-loop system in (4-18), a
backstepping-based approach is used to inject a virtual control input Xd(t) E R (i.e.,
desired calcium dynamics) as
1
Jpt = S + Tdp -- X )Pe C jr e CX+ + CXd CXd. (4-27)
Based on (4-27), the virtual control input is designed as a three 1-v-r NN feedforward
term plus a feedback term as
Xd + -1(S + kr), (4-28)
where k, E R denotes a positive constant adjustable control gain. The feedforward NN
component in (4-28), denoted by S(t) E R is generated as
S = iTao(UT y). (4-29)
The estimates for the NN weights in (4-29) are generated on-line using projection
algorithm as [27]
W = proj(Fl6rT'), U = proj(F2y(oTWr)T), (4-30)
where F1 e R(N2+1)x (N2+1) and F2 cE R(Ni+1)x(Ni+1) are constant, positive definite,
symmetric gain matrices. The closed-loop tracking error system can be developed by
substituting (4-28) into (4-27) as
1
J1i = 2-jr e + S + dp x x kr ., (431)
where S(y) E R is defined as
S(y) =S -S, (4-32)
and e,(t) E R is the backstepping error defined as
eC, = Xd. (4 33)
The closed loop system can be expressed as
jPr = 2- r e + WTa (UTy) IWT oa(Ty) + C(y) + rdp x cx kr ;-. ... (4-34)
After adding and subtracting the terms WT1- + WTT to (4-34), the following expression
can be obtained:
Jp, = 2-jr e + WTo + + + eWT) + WT + (Y) p r x kc r k per (4-35)
where the notations o (.) and a (.) are introduced in (3-43). The Taylor series approximation
described in (3-44) and (3-45) can now be used to rewrite (4-35) as
Jpr = 2- r e + N + TIa + WT'-Ty kr ,., (4-36)
2 ,,.
where cr'(UTy) = do(UTy)/d(UTy)ulTy- UTy. The unmeasurable auxiliary term N(W, U, y, p-l,t)
E R is defined as
N V= 'U + Wy + WO(y) + (y) + dp (4 37)
Based on (4-14), (4-23), (4-26), (4-30), the fact that x(t), x(t) e [0,1], and the
assumption that desired trajectories are bounded, the following inequality can be
developed [122]:
INI I ( + (2 1:11, (438)
where ( e R, (i = 1,2) are known positive constants and z e R2 is defined as
z [ r]T. (4-39)
4.3.3 Backstepping Error System
To facilitate the subsequent stability analysis, the time derivative of the backstepping
error (4-33) can be determined by using (4-16) as
e --x + -sat[V(t)]- Xd. (4-40)
2 2
Based on (4-6) and (4-40), and assumption that control input remains below the
saturation voltage Vmax, the control input (Voltage input) V(t) E R is designed as
V(t) = (Vmax Vmin) (- d + c-x + r kex) + Vmin, (4-41)
where k E R denotes a positive constant adjustable control gain. Substituting (4-41) into
(4-40), yields
e = cr ke. (4-42)
4.4 Stability Analysis
Theorem 3. The controller given in (4 -.') and (4-41) ensures that all system .:-,il. are
bounded under closed-loop operation and that the position tracking error is ,, ,l.rla. in the
sense that
|e(t)| < co exp(-cit) +C2, (4-43)
provided the control gains a, k, introduced in (4-9), (449), (4-50) are selected according
to the following sufficient condition:
min(a, ks,) > (2, (4-44)
where co, 1, C2 E R denote positive constants, and (2 is a known positive constant
introduced in (4 38).
Proof: Let VL (t) E R denote a continuously differentiable, non negative, radially
unbounded function defined as
1T 1T 1 1T
VL(t) A e2e + r Jr + e e, + -trW F ) + ttrUF J). (4-45)
2 2 2 2 2
By using (4-13) and typical NN properties [112], VL (t) can be upper and lower bounded
as
A1 X2 VL (t) < A2 IX2 + 0, (446)
where A1,A2, 0 E R are known positive constants, and X(t) e R3 is defined as
T
X(t) A Z(t) eX(t) (447)
Taking the time derivative of (4-45), utilizing (4-9), (4-36), (4-42), and canceling similar
terms yields
VL = -eae + rTN rTkr + rTWT + rTWT'y eTke, r T(j Jp)r
tr(WTF1 W) tr(T F2 ). (4-48)
Using (4-14) and (4-38), the expression in (4-48) can be upper bounded as
VL < -ae2 k1r2 + (2 III Irl + [Irl (i ksr2] ke2 + rTWTcr + rTWTr'OTy tr(WWTFi )
tr(UT F2 1), (4-49)
where k,,k k,2 E R are positive constant gains that satisfy
k, = k,l + k,,. (4-50)
Completing the squares for the bracketed term in (4-49) and using the update laws in
(4-30) yields
VL < -[min(al, k,,) 2] III ke + 2 (4-51)
4k,"
The inequality in (4-46) can be used to rewrite (4-51) as
VL< VL + (4-52)
A2
where E IR is a positive constant defined as
E= + 0, (4-53)
4k,2 A2
and p E R is defined as
/ = min[(min(ao, k,,) (2), k]. (4-54)
The linear differential inequality in (4-52) can be solved as
VL(t)< V()e^ +E C (4-55)
Provided the sufficient condition in (4-44) is satisfied, the expressions in (4-45) and (4-55)
indicate that e(t), r(t), ex(t), W(t), U(t) e ,. Given that e(t), r(t), qd(t), qd(t) E oo,
(4-8) and (4-9) indicate that q(t), q(t) e L,. Since W(t), U(t) e L, (3-42) and
Assumption 1 (3.4) can be used to conclude that W(t), U(t) e /,. Based on (4-5), it
can be shown that x(t) e [0, 1]. Given that qd(t), e(t), r(t), q(t), q(t), x(t) e o, the NN
input vector y(t) E L, from (4-25). Since ex(t), x(t) e /,, (4-33) can be used to show
that Xd(t) e oo. Given that r(t), W(t), U(t), Xd(t) e L, (4-28) and (4-29) indicate that
S(t), 1(t) e C. Since e(t), r(t), W(t), W(t), U(t),ex(t) (t) e L, (4-36) and (4-38)
indicate that r(t) e L,. As r(t), y(t), W(t) e L,, the update laws W(t), U(t) e L,.
Since ((t), L(t) E o, it can be shown that p(t) e o,. Given that the (t), ~-l(t), i(t),
r(t), W(t), U(t), W(t), U(t) e L,, it can be shown that Xd(t) e ,. Because o(t), Xd(t),
r(t), x(t), e,(t) c L, it can be concluded that the voltage control input V(t) is bounded.
4.5 Simulations
Simulations are performed to illustrate the performance of the controller. The model
parameters were chosen from [6, 36, 123]. The RISE and the proposed controller are tested
for two different desired trajectories: 1) slow trajectory with 6 second period, 2) fast
trajectory with 2 second period.
1
0 10 20 30 40
.. 5 0 0 -.. .. ... ...
0
0 10 20 30 40
00
0 10 20 30 40
5 0 | ----------------
Time [sec.]
Figure 4-2. Top plot shows the knee angle error for a 6 second period trajectory using
the proposed controller. Middle plot shows the pulsewidth computed by the proposed
controller. Bottom plot shows the actual leg angle (dashed line) vs desired trajectory
(solid line).
From the results shown in Figs. 4-2-4-8, it is clear that the proposed controller tracks
both time varying desired trajectories better than the RISE controller. Figs. 4-4 and 4-5
illustrate the performance of the RISE controller when implemented on muscle dynamics
without including the fatigue dynamics. The steady state error from the RISE controller
is between 80 for desired trajectory with period 6 seconds. The steady state error in the
case of RISE controller increases to 140 when faster trajectory with period 2 seconds
is used. Fig. 4-6 depicts that the control performance degrades later in time when RISE
-2
0 10 20 30 40
1500
1 0 0 0................. ................
S 500- .-
0 10 20 30 40
-_50
o0
0 10 20 30 40
Time [sec.]
Figure 4-3. Top plot shows the knee angle error for a 2 second period trajectory using
the proposed controller. Middle plot shows the pulsewidth computed by the proposed
controller. Bottom plot shows the actual leg angle (dashed line) vs desired trajectory
(solid line).
controller is implemented on muscle dynamics with fatigue model included. The proposed
controller was implemented on the complete muscle dynamics that included the fatigue
dynamics. Figs. 4-2, 4-3 and 4-7 show that the steady state error in the case of proposed
controller remains within 0.50 for both slow and fast trajectories. Fig. 4-8 shows how the
fatigue variable evolves with time as a deceasing input gain. The proposed controller is
able to compensate for the decreasing control gain, and the performance does not degrade
over time as shown in Fig. 4-7.
4.6 Conclusion
A NN based nonlinear control algorithm is developed to elicit non-isometric
contractions of the human quadriceps muscle via NMES. The primary objective of the
developed method is to incorporate an uncertain muscle fatigue model and unknown
calcium dynamics in the nonlinear muscle dynamics. The unknown muscle model and
the parametric uncertainties in the fatigue model are approximated by the NN structure
through an estimate of the calcium dynamics. A Lyapunov based stability analysis is
-20
0 10 20 30 40
1000
500 -.. ........ .-. ..... -..... ... .... ... -........ ....
0
0 10 20 30 40
-750
/ oy/
0 10 20 30 40
Time [sec.]
Figure 4-4. Top plot shows the knee angle error for a 6 second period trajectory using
the RISE controller. Middle plot shows the pulsewidth computed by the RISE controller.
Bottom plot shows the actual leg angle (dashed line) vs desired trajectory (solid line).
performed to prove uniformly ultimately bounded result in the presence of bounded
disturbances (e.g muscle spasticity), parametric uncertainties. Simulation results clearly
illustrate that the proposed controller performs better in terms of reduced error in
comparison to the RISE controller. However, the performance of the controller on
volunteers or patients remains to be seen. The controller's dependence on acceleration
and mathematical fatigue and calcium models hinder its implementation on volunteers.
The mathematical calcium and fatigue models were incorporated due to the fact that the
measurement of actual fatigue state and calcium variable is difficult. Future efforts can
be made to incorporate an observer-based design in the controller in order to estimate the
fatigue and calcium states.
4U
0 2 0.... : .... ... ..... .... ....
0 10 20 30 40
2000
1000
-10 10 20 30 40
0 10 20 30 40
50 ,
50
0 10 20 30 40
Time [sec.]
Figure 4-5. Top plot shows the knee angle error for a 2 second period trajectory using
the RISE controller. Middle plot shows the pulsewidth computed by the RISE controller.
Bottom plot shows the actual leg angle (dashed line) vs desired trajectory (solid line).
201, ,
20
-20 --------
60 70 80 90 100
S2000
1001
0 10 20 30 40
50
60 70 80 90 100
Time [sec.]
Figure 4-6. RISE controller with fatigue in the dynamics: Top plot shows the knee angle
error for a 6 second period trajectory using the RISE controller. Middle plot shows the
pulsewidth computed by the RISE controller. Bottom plot shows the actual leg angle
(dashed line) vs desired trajectory (solid line).
.J !
0
60 70 80 90 100
S 5 0 0 .. .. ... .. .. .. .. .. .
500
0 10 20 30 40
-7 50.
0
60 70 80 90 100
Time [sec.]
Figure 4-7. Performance of the proposed controller: Top plot shows the knee angle error
for a 6 second period trajectory using the proposed controller. Middle plot shows the
pulsewidth computed by the proposed controller. Bottom plot shows the actual leg angle
(dashed line) vs desired trajectory (solid line).
0.98
0.96
S0.94
u 0 .9 2 . .. . . .
0 .9 .. .. .
0 .8 8 .. ... .... ..
0.86
0 20 40 60 80 100
Time [sec.]
Figure 4-8. Fatigue variable
CHAPTER 5
PREDICTOR-BASED CONTROL FOR AN UNCERTAIN EULER-LAGRANGE
SYSTEM WITH INPUT DELAY
5.1 Introduction
This chapter focuses on the development of tracking controllers for an uncertain
nonlinear Euler-Lagrange system with input delay. The input time delay is assumed to
be a known constant and can be arbitrarily large. The dynamics are assumed to contain
parametric uncertainty and additive bounded disturbances. The first developed controller
is based on the assumption that the inertia matrix is known. The known inertia case is
provided to illustrate how a proportional integral (PID) controller can be augmented to
compensate for input delay. The second controller removes the assumption that inertia
matrix is known, and different design/analysis efforts are used to yield a PD controller
with an augmented predictor component. The key contributions of this effort is the design
of a delay compensating auxiliary signal to obtain a time delay free open-loop error system
and the construction of LK functionals to cancel the time d. 1 i-, .1 terms. The auxiliary
signal leads to the development of a predictor-based controller that contains a finite
integral of past control values. This d, 1 i, .1 state to delay free transformation is analogous
to the Artstein model reduction approach, where a similar predictor-based control is
obtained. LK functionals containing finite integrals of control input values are used in
a Lyapunov-based analysis that proves the tracking errors are semi-global uniformly
ultimately bounded. Experimental results are obtained for a two-link direct drive robot.
The results illustrate the robustness and added value of the developed predictor-based
controllers.
The primary motive of this research is to develop and implement a controller that
compensates for electromechanical delay (EMD) in NMES. The last section of the chapter
focuses on characterizing EMD during NMES. Experiments results obtained from
I. i,11!:r volunteers are provided which describe the effect of stimulation parameters on the
EMD during NMES. Finally, a PD controller with an augmented predictor component
is implemented on the healthy volunteers. Experiments show that the controller can be
applied to compensate EMD in NMES. A comparison with the traditional PD controller
shows that the PD controller with delay compensation provides a better performance.
5.2 Dynamic Model and Properties
Consider the following input d, 1 i. I1 Euler-Lagrange dynamics
M(q)q + Vm(q, q)q + G(q) + F(q) + d(t) = u(t r). (5-1)
In (5-1), M(q) E denotes a generalized inertia matrix, Vm(q, q) E T. denotes a
generalized centripetal-Coriolis matrix, G(q) E R" denotes a generalized gravity vector,
F(q) E PR denotes generalized friction, d(t) E R" denotes an exogenous disturbance (e.g.,
unmodeled effects), u(t r) E R" represents the generalized d. 1 ,i .1 input control vector,
where r E R is a constant time delay, and q(t), q(t), q(t) E R" denote the generalized
states. The subsequent development is based on the assumptions that q(t) and q(t) are
measurable, Vm(q, q), G(q), F(q), d(t) are unknown, the time d,1 iv constant Tr E R is
known1 and the control input vector u(t) and its past values (i.e., u(t 0) V 0 E [0 r])
are measurable. For the controller developed in Section 5.3.2, M(q) is assumed to be
known to illustrate the development of a PID-like controller. In Section 5.3.3, this
assumption is removed and a PD-like controller is developed. Throughout the paper, a
time dependent d. 1 li-,- function is denoted as x(t r) (or as x,) and a time dependent
function (without time delay) is denoted as x(t) (or as x). The following assumptions are
used in the subsequent development.
1 Experimental results (where the time d 1 iv is artificially injected in a desired manner)
illustrate the performance of the developed controllers when the time delay has as much as
10lOi' error between the assumed and actual delay.
Assumption 1: The inertia matrix M(q) is symmetric, positive definite, and satisfies
the following inequality V (t) E R" :
mi |1112 < TM < m2 ,1112, (5-2)
where mi, m2 e R+ are known constants and I||-| denotes the standard Euclidean norm.
Assumption 2: The desired trajectory qd(t) is designed such that qd(t), q) (t) E oo,
where qd)(t) denotes the 1th time derivative for i = 1,2,3.
Assumption 3: If q(t), q(t) E L,, then M(q), V,(q, q), G(q), and F(q) are bounded.
Moreover, if q(t), q(t), q(t) e L, then the first time derivatives of M(q), Vm,(q, ), G(q),
F(q) exist and are bounded. The infinity norm of M(q) and its inverse can be upper
bounded as
||-1(q)1|l < (1 (--(q) < 2, (5-3)
where (1, (2 e R+ are known constants.
Assumption 4: The nonlinear disturbance term and its first time derivative are
bounded, i.e., d(t), d(t) c L,.
5.3 Control Development
5.3.1 Objective
The objective is to develop a controller that will enable the input d, 1 i 1 system in
(5-1) to track a desired trajectory, denoted by qd(t) E R". To quantify the objective, a
position tracking error, denoted by ei(t) E R", is defined as
el = qd(t) q(t). (5-4)
5.3.2 Control development given a Known Inertia Matrix
To facilitate the subsequent analysis, a filtered tracking error, denoted by e2(t) E R",
is defined as
e2 e1 + iei, (5-5)
where acl Rc + denotes a constant. To reduce the input d. 1 i .1 system in (5-1) to an
input delay free system, an auxiliary signal denoted by r(t) E R", is also defined as
r 2 + a262 + M-1(q)(u(t r) u(t)), (5-6)
where a2 E R+ denotes a constant. The auxiliary signal r(t) is only introduced to
facilitate the subsequent analysis, and is not used in the control design since the
expression in (5-6) depends on the unmeasurable generalized state q(t).
After multiplying (5-6) by M(q) and utilizing the expressions in (5-1), (5-4), and
(5-5), the transformed open-loop tracking error system can be expressed in an input delay
free form as
M(q)r = M(q)qd + Vm(q, q)q + G(q) + F(q) + aM(q)e, + a2M(q)e2 + d u(t). (5-7)
Based on (5-7) and the subsequent stability analysis, the control input u(t) E R" is
designed as
u- ka (2 + 0 a22(0) + M-(0O)(u(O T) u(O))dO) ka2(0), (5-8)
where k, E R+ is a known constant that can be expanded as
k = k, + k,2 + 1, (5-9)
to facilitate the subsequent stability analysis, where k,,, ka, E R+ are known constants.
The controller u(t) in (5-8) is a proportional integral derivative (PID) controller modified
by a predictor like feedback term for time delay compensation. Although the control input
u(t) is present in the open loop error system in (5-7), an additional derivative is taken to
facilitate the subsequent stability analysis. The time derivative of (5-7) can be expressed
as
M(q)r --M(q)r + N + d- kr, (5-10)
2
where N(el, e2, r, t) CE R is an auxiliary term defined as
1
N = ()r + M(q)q ++ (q)qqd+V,(q,q)q+V(q,q)q + (q)+F(q) (5-11)
2
+ (aci + a2) M(q)r al2M(q)e2 a M(q)ei a M(q)e2 + O iMV(q)
+a2M(q)e2 (ac + a2) (U, ) M(q)ei,
and (5-6) is used to write the time derivative of (5-8) as
u = kar.
After adding and subtracting the auxiliary function Nd(qd, qd, ld, "d, t) E R" defined as
Nd M(qd)qd) + qd)9d + ) + (qd, qd)qd + K((qd, gd) + dG (qd) + F(qd),
to (5-10), the following expression is obtained:
1 .
M(q)i = --M(q)r + N + S e2 kar,
2
(5-12)
where the auxiliary functions N(ex,e2, r, t) CE R and S(qd, qd, d, q d, t) E R" are defined as
N N- Nd+e2,
S = Nd + d.
(5-13)
Some terms in the closed-loop dynamics in (5-12) are segregated into auxiliary terms
in (5-13) because of differences in how the terms can be upper bounded. For example,
Assumptions 2, 3 and 4, can be used to upper bound S(qd, qd, qd, 4d, t) as
(5-14)
where E1 E R+ is a known constant and the Mean Value Theorem can be used to upper
bound N(e, e2, r, t) as
N < pi(II | ) 1 5 1 ,
(5-15)
where z e -. is defined as
z = e eT rT eT (5-16)
and the bounding function pl (||l |) E R is a known positive globally invertible
nondecreasing function. In (5-16), e, e R" is defined as
ez = u UT (0)d,
t-T
based on the Leibnitz-Newton formula.
Theorem 4. The controller given in (5-8) ensures semi-ill..1' l/// ;,,'.. il,,,;, ;,ll.:i,,i,;/. 1
bounded (SUUB) tracking in the sense that
|ei(t)|| < coexp(-cit) + C2, (5-17)
where co, C1, C2 C R+ denote constants, provided the control gains ac, a2, and ka introduced
in (5-5), (5-6), and (5-8), ,' ./.. /.:,. /;, are selected according to the following sufficient
conditions:
1 (722 1
c > -, a2 > 1 + k< w2 > 27, (5-18)
Proof: Let y(t) c D C L be defined as
T
y(t) e ej rT (5-19)
where Q(t) E R is defined as [45, 76]
Q j It(0) 12 dO ds, (5-20)
9-7
where w E R+ is a known constant. A positive definite Lyapunov functional candidate
V (y, t) : D x [0 oo) R is defined as
V (y, t) A ee + i e22 + 2 rT (q)r + Q, (5-21)
2 2
and satisfies the following inequalities
A 112 < V < A2 ll 2, (5-22)
where Ai, A2 E R+ are known constants defined as
1 1
A1 = min[ml,1], A2 = max[ 2, 1], (5-23)
2 2
where m, and m2 are defined in (5-2).
After utilizing (5-5), (5-6), and (5-12) and cancelling the similar terms, the time
derivative of (5-21) is
V 2e]e2-2aeT e1-a 2e 2- karTr+eCTM-l(q)e +rTS+rTN+wT 112I2-_ IWJ (0)2 d0,
t-T
(5-24)
where the Leibniz integral rule was applied to determine the time derivative of Q(t) in
(5-20) (see the Appendix 7.2). The expression in (5-24) can be upper bounded by using
(5-3), (5-14) and (5-15) as
V < -(2a, 1) ||el12 (a2 1) 162 2 k I|r||2 + 6211 ||e1 (5-25)
t-
wT .112 +1 rI+Il+(1) 1r_ W t(O) 2d0.
The following term in (5-25) can be upper bounded by using Young's inequality:
2 1 2 I 2 2112 + 11 2, (5-26)
where 7 e R+ is a known constant. Further, by using the Cauchy Schwarz inequality, the
following term in (5-26) can be upper bounded as
(527)
Adding and subtracting Jft_ Iit0)l 2 dO in (5 25) yields
V < -(2ai 1) Ile 112 (2 1) li2112 k rIIr22 + 2I11 1ell + IIl12 (5-28)
T T
|1 rI pi(||ll / (lll) 11H II|- r f ) ) it(O) l2 dO 1 t(0) 2d0.
E- 7 1 J-T 7 2T
Utilizing (5-9) and the bounds given in (5-26) and (5-27), the inequality in (5-28) can be
upper bounded as
222 2
V < -(2ai 1) ||el2 a 2 1 2) 2 ( ... 1.) 22 27) 2
4 ) 11
+Pl(ll l) ll| ||r|| ka ||r 2 k., |r||2 IIt(O) l2d0. (5-29)
After completing the squares, the inequality in (5-29) can be upper bounded as
'T I lt (0) 1 2 1 2 1
V < -1 | j (0) ll2 d0l- (5-30)
7 4k ai 4k,,2
where p E R+ is defined as
1 = min (a2- 1 2 ), (2ai 1), (1 ..1 2 .
4 ,2- .
Since
/ t(
t-" Y
the expression in (5
V_<-
V < -
Using the definition
expressed as
it|(0)112d0 ds < S sup [ (0 \ 112 ( d0] j (0)11 2d0,
s [t,t-r] Js t-
30) can be rewritten as
p )1 2 t t F 2
0,1 4 ) 11 2 I T j(0) 2 dO + (5-31)
4k, 4 /1 )Jt-T UJs 44ka2
of z(t) in (5-16) and y(t) in (5-19), the expression in (5-31) can be
V < -< |_ ||Il 2 3 (ll l) lllC 2 k+ 1 (5-32)
/4 k('l 4kai2 (
where Pl(|| I|) R+ is defined as
/i mmin (i
4kll II, y21
By further utilizing (5-22), the inequality in (5-32) can be upper bounded as
S< V +-. (5-33)
A2 4k,,
Consider a set S defined as
S,4, ,< < -
^{z(t)R4 c RI
In S, Pl(|| :|) can be lower bounded by a constant 61 E IR+ as
61 < 0(ll: Il). (5-35)
Based on (5-35), the linear differential equation in (5-33) can be solved as
LiE2 1 t
V(y, t) < V(0)e -^ + e- ^ (5-36)
4k,,26, Ik II
provided I|||| < p1 (2 /3ika/ ) From (5-36), if z(0) E S then ka can be chosen according
to the sufficient conditions in (5-18) (i.e. a semi-global result) to yield the result in (5-17).
Based on definition of y(t), it can be concluded that el(t), e2(t), r(t) E L in S. Given
that el(t), e(t) qd(t), d(t) E in S, (5-4) and (5-5) indicate that q(t), q(t) E L, in
S. Since r(t), e2(t), q(t), (t), qd(t), qd(t) c L in S, and u(t) u(t 7) ft, it(0)dO =
k, t, r(O)dO (by Leibnitz-Newton formula)e LC, in S, then (5-6) and Assumption 3
indicate that q(t) E L, in S. Given that r(t), e2(t), q(t), t(t), qd(t) qd(t) E L" in S, (5-7)
and Assumptions 3 and 4 indicate that u(t) E L in S.
5.3.3 Control development with an Unknown Inertia Matrix
To facilitate the subsequent control design and stability analysis for the uncertain
inertia problem, the auxiliary signal, e2(t) E IR" is redefined as
e2(t) -e + Cae B u(O)dO, (5-37)
Jt-T
where a E R+ is a known constant, and B c. is a known symmetric, positive definite
constant gain matrix that satisfies the following inequality
1|BI |< b (5-38)
where b E R+ is a known constant. To facilitate the subsequent stability analysis, the error
between B and M-l(q) is defined by
TI(q) B M-(q),
(5-39)
where rI(q) E -.
satisfies the following inequality
II(q)|ll < T,
where rl e R+ denotes a known constant. The open-loop tracking error system can
be developed by multiplying the time derivative of (5-37) by M(q) and utilizing the
expressions in (5-1), (5-4), and (5-39) to obtain
M(q)e2 M(q)d + V(q, q)q + G(q) + F() + d + aM(q) u(t) M(q) [u u,]. (5-41)
Based on (5-41) and the subsequent stability analysis, the control input u(t) E R" is
designed as
U = kbe2,
(5-42)
where kb E R+ is a known control gain that can be expanded as
kb = kb, + kb, + kb3,
(5-43)
to facilitate the subsequent analysis, where kb,, kb2, and kb3 E R+ are known constants.
After adding and subtracting the auxiliary term Nd(qd, qd, id, t) e R" defined as
Nd = M (qd)qd + Vm(qd, qd) qd + G(qd) + F (qd) ,
(5-40)
and using (5-37) and (5-42), the expression in (5-41) can be rewritten as
M(q)e2 = M(q)e2 + N + S e1 kbe2 -1 (q)r [e2 eC2, (5-44)
where the auxiliary terms N(ei,e2,t), N(ei,e2,t), S(qdqd, d, t) E R" are defined as
N = N Nd, s =Nd + d, (5-45)
1.
N -lM(q)e2+M(q)qd+Vm(q, )q+G(q)+F(q)+aM(q)e2-a2M(q)ei+ei+aM(q)B u(O)dO,
where N(ei, e2, t) and S(qd, qdd, d, t) can be upper bounded as
N< P2(|I) II, S < < (5-46)
In (5-46), E2 E R+ is a known constant, the bounding function p2 ( II) E I is a positive
globally invertible nondecreasing function, and z E is defined as
z = eT eT C (5-47)
where ez e R' is defined as
e, u(O)dO.
t-T
Theorem 5. The controller given in (5-42) ensures SUUB tracking in the sense that
I||e(t)l| < coexp(-elt) + 2, (5-48)
where co, C1, C2 CE ]+ denote constants, provided the control gains a and kb introduced in
(5-37) and (5-42), ,' /.., /.: /; are selected according to the sufficient conditions:
Sb22 2m2 (kb, + kb2) + wk-r 2
a > kb3 > -- 12 y2 > 2r, (5-49)
4 1 2Tm2
where M2, b, E R+, y R+ are 1. ,,i 1. in (5-2), (5-38), and (5-40), "i.' /:, and 7,
wo E R+ are -;,I1-. ;. ,'/1.;i 1. f;,. constants.
Remark 1. The second sufficient gain condition indicates that w can be selected suffi-
i, ,;/ small and kb, can be selected suff.- i nil, 1. lag, provided 1 2rM2 > 0. The condition
that 1 2Trm2 > 0 indicates that the constant approximation matrix B must be chosen
suff. :, ./il, close to M-'(q) so that iB M-'(q) I| < 2 Experimental results illustrate
the performance/robustness of the developed controller with respect to the mismatch be-
tween B and M-l(q). S1 .. ..:I.'ll; results indicate an <.:,"'.:,,'.:i. ., amount of variation in
the performance even when each element of M-l(q) is overestimated by as much as 10r "
Different results i,,rn be obtained for different s;,il 1ii, but these results indicate that the
gain condition is reasonable.
Proof: Let y(t) E )D C T.' 2 be defined as
T
t( [e e (5-50)
where P (t), Q (t) E R denote LK functionals defined as [45]
P W ( t u(0)2 d ds, Q 2b T/= 1 e2 2d
t-7J s t ,-
where w e R+ is a known constant. A positive definite Lyapunov functional candidate
V (y, t) : x [0 oo) R is defined as
1 1
V(y, t) 2A eTe1 + Te/M(q)e2 + P + Q, (551)
and satisfies the following inequalities
A l112 < V < A 2 1 (5-52)
where Ai, A2 E R+ are defined in (5-23).
Taking the time derivative of (5-51) and using (5-37) and (5-44) yields
V = -aeTe + eBez + w-r u u2 + S+ N 2 ., j [(q)i ( 2 2)]
+ [|2ekb 2 2 2- 2 2 (O) 2 dO, (5-53)
Jt-r
where the Leibniz integral rule was applied to determine the time derivative of P(t) (see
the Appendix 7.2) and Q(t). Using (5-2), (5-38), and (5-46), the terms in (5-53) can be
upper bounded as
V < -a ||e111 kb Ie2 + '1/i2 I. I 112 + + T IIU112 + le211 \ + IC 2 ||11 ) I|| I I + b I|el| ||e,1 |
I' 21 + 1kb [k, I 211 2 -2 2] _- u(0)112 d0. (5-54)
The following terms in (5-54) can be upper bounded by utilizing Young's inequality:
4 7
lTm2/ b 621 62 62 I+ I 62 112
2 2
where 7 E R+ is a known constant. Further, by using the Cauchy Schwarz inequality, the
following term in (5-55) can be upper bounded as
| 2 j I ()2 d0. (5-56)
t-T
After adding and subtracting f_ IU (0)) 12 dO to (5-54), and utilizing (5-42), (5-43),
(5-55) and (5-56), the following expression is obtained:
b2 2 1 27
V < -(a ) 6e112 (kb3 wkI2r 2ram2kb) 62e12 -1 ( 2) 1e,112
47 72
-kb, 2 + 2 (1) l lkb, el2 + 22 ~2E I()( 2 d0.(5-57)
By completing the squares, the inequality in (5-57) can be upper bounded as
P2 -
V < 2- f2 I I(12 d + 2C, (5-58)
where 32 E R+ is denoted as
b b2 7 2) 2 1 2,T
32 min a (k- GL 2,
Since
Iu(o0) 12 d s < T sup [ t u(0)112 d] 0 r u(O) 12do,
t- 1 sE[t,t-T] s t-T
the expression in (5-58) can rewritten as
< {- 2 41 2; IiU(O) 2d0 I I U(O) 2d0 + 42
< 0- 2 2 Y k b
(5 59)
Using the definitions of z(t) in (5-47), y(t) in (5-50), and u(t) in (5-42), the expression in
(5-59) can be expressed as
< I02- {/C2 2 I}e2 + (5-60)
S 4kb, 11 4kb (560)
where /2( :11I) e R is defined as
P4kb2 2rm2 2wy2
/2 -min(02 kb,17 2)JJ] "
By further utilizing (5-52), the inequality in (5-60) can be written as
S< 32 +- (5-61)
A2 4kb2
Consider a set S defined as
SA z{(t) E1 | < 1 (2
In S, /32(: 1I|) can be lower bounded by a constant 2 c R+ as
62 < / 2( II). (5-63)
Based on (5-63), the linear differential equation in (5-61) can be solved as
V < V(0)e-- + -22 (5-64)
4kb, 62
provided 1||| I< p21 (2 /32kb) From (5-64), if z(0) E S then kb can be chosen according
to the sufficient conditions in (5-49) (i.e. a semi-global result) to yield result in (5-48).
Based on the definition of y(t), it can be concluded that el(t), e2(t) E L in S. Given
that el(t), e2(t), qd(t), d(t) in S, (5-4), (5-42), and (5-37) indicate that q(t),
q(t), u e in S.
5.4 Experimental Results and Discussion
Experiments for the developed controllers were conducted on a two-link robot
shown in Fig. 5-1. Each robot link is mounted on an NSK direct drive switched
Figure 5-1. Experimental testbed consiting of a 2-link robot. The input delay in the
system was artificially inserted in the control software.
reluctance motor (240.0 Nm Model YS5240-GN001, and 20.0 Nm Model YS2020-GN001,
respectively). The NSK motors are controlled through power electronics operating
in torque control mode. Rotor positions are measured through motor resolver with
a resolution of 614400 pulses/revolution. The control algorithms were executed on a
Pentium 2.8 GHz PC operating under QNX. Data acquisition and control implementation
were performed at a frequency of 1.0 kHz using the ServoToGo I/O board. Input delay
was artificially inserted in the system through the control software (i.e., the control
commands to the motors were d. 1 I, .1 by a value set by the user). The developed
controllers were tested for various values of input delay ranging from 1 ms to 200 ms.
The desired link trajectories for link 1 (qd (t)) and link 2 (qz(t)) were selected as (in
degrees):
qd (t) = qd,(t) = 20.0sin(1.5t)(l exp(-0.01t3)).
The controller developed in (5-8) (PID controller with 1/. ,.r; compensation) and the
controller developed in (5-42) (PD controller with /. ,.r; compensation) were compared
with traditional PID and PD controllers, respectively, in the presence of input delay in the
system. The input d, 1 .i1 two link robot dynamics are modeled as
Uir pi + 2p3 cos(q2) 2 + p3 cos(q2) i -3 sin(q2) 2 -p3 sin(q2)(g + q2)
2 P2 + p3 cos(q2) 2 ][2 P3 sin(q2)1 0
il fd, 0 i fsi 0 tanh(ql)
92 0 fd2 2 0 s2 tanh(q2)
where pi, P2, P3, fdi, fd2, fs, fs2 E R+ are unknown constants, and r E R+ is the
user-defined time delay value. However, the following values: pi = 3.473kg.m2, P2
0.196kg.m2, and ps = 0.242kg.m2 were used to calculate the inverse inertia matrix for
implementing the PID controller with 1/ /.,r; compensation but were not used to implement
the PD controller with 1/ /.,r; compensation.
The control gains for the experiments were obtained by choosing gains and then
adjusting based on performance (in particular, torque saturation). If the response
exhibited a prolonged transient response (compared with the response obtained with
other gains), the proportional gains were adjusted. If the response exhibited overshoot,
derivative gains were adjusted. At a particular input d. 1 iv value, the control gains were
first tuned for the PID/PD controllers with /1 /.,r; compensation and then compared with
traditional PID/PD controllers. Using the same control gains values as in the PID/PD
controllers with /. /.,;/ compensation, the control torques for the traditional PID/PD
controllers reached pre-set torque limits, leading to an incomplete experimental trial (e.g.,
if the control torque reaches 20 Nm, which is the set torque limit for the link-2 motor, the
control software aborts the experimental trial2 ). Therefore, for each case of input delay
(except at 1 ms), control gains for the traditional PID/PD controllers were retuned (i.e.,
lowered) to avoid torque saturation. In contrast to the above approach, the control gains
could potentially have been adjusted using more methodical approaches. For example, the
nonlinear system in [124] was linearized at several operating points and a linear controller
was designed for each point, and the gains were chosen by interpolating, or scheduling the
linear controllers. In [125], a neural network is used to tune the gains of a PID controller.
In [126] a genetic algorithm was used to fine tune the gains after initial guess were made
by the controller designer. The authors in [127] provide an extensive discussion on the use
of extremum seeking for tuning the gains of a PID controller. Additionally, in [128], the
tuning of a PID controller for robot manipulators is discussed.
The experimental results are summarized in Table 5-1. The error and torque plots
for the case when the input delay is 50 ms (as a representative example) are shown in
Figs. 5-3-5-4. The PD controller with /. /.,;/ compensation was also tested to observe
the sensitivity of the B gain matrix, defined in (5-37), where the input d.1 iv was
selected as 100 ms. Each element of the B gain matrix was incremented/decremented
by a certain percentage from the inverse inertia matrix (see Table 5-2). The purpose of
this set of experiments was to show that the gain condition discussed in Remark 1 is
a sufficient but not a necessary condition, and to explore the performance/robustness
of the controller in (5-42) given inexact approximations of the inertia matrix. The
controller exhibited no significant degradation, even when each element of the inertia
matrix is over-approximated by 1C(' However, underestimating the inverse inertia
matrix (particularly when deviation from the inverse inertia matrix was 75 percent),
2 Instead of aborting the experimental trial, the experiments could have also
been performed by utilizing the saturation torque as the control torque in case the
computed torque reaches or exceeds the torque limit; but for comparison purposes, the
aforementioned criterion was chosen.
yielded increased tracking errors. Different results may be obtained for different systems.
The third set of experiments, given in Table 5-3 were conducted to show that promising
results can be obtained even when the input delay value is not exactly known; however,
the tracking error performance degrades with increasing inaccuracy in delay value
approximation (e.g., in the case of PD + compensator, the tracking error increases
significantly when the d. 1 lv value is overestimated by 1 1'. or greater). For this set of
experiments the input delay was chosen to be 100 ms.
The experimental results clearly show that the PID/PD controllers with /. ,i ;
compensation perform better than the traditional PID/PD controllers. Both controllers
can be divided into respective PID/PD components and predictor (delay compensating)
terms. The better performance shown by the controllers can be attributed to the predictor
components in both the controllers. As an illustrative example, Fig. 5-2 shows the time
plots of the PD controller with /. /.I,; compensation and its control components. The
two components: PD component and delay compensating term are plotted to show
their behavior with respect to each other. The plot shows that the d. 1 li compensating
component is ah--iv-i following the PD component but is opposite in sign (like an mirror
image but less in magnitude). Thus, the net (actual) control torque is aliv-l less than
the PD control component. This implies that the delay compensating term tends to
correct the PD component (acts as a primary torque generator) which may have compiled
extraneous torque due to the input delay. The delay compensating term predicts the
correction term by finitely integrating control torque over the time interval ranging from
current time minus the time delay to current time.
5.5 Delay compensation in NMES through Predictor-based Control
The primary goal of the input delay research was to compensate for Electromechanical
delay (EMD) in NMES. EMD in muscle force generation is defined as the difference in
time from the arrival of action potential at the neuromuscular junction to the development
of tension in the muscle [8]. In NMES control, the EMD is modeled as an input delay
10. s .
S- I
S20
-10 ...... .
0 1 2 3 4 5
Time [see.]
Figure 5-2. The plot shows three torque terms: PD component shown in dotted line, delay
compensating term plotted in dashed line, and the net or actual control torque shown in
solid line. The PD component and the delay compensating term (finite integral term of
control values) are two components of the PD controller with /. 1,;/ compensation (actual
control torque). Note that the delay compensating term is alvb--, opposite in sign to the
PD component. Thus, the net control torque is alvb--, less than the PD controller. This
implies that the delay compensating term tends to correct the PD component which may
have compiled extraneous torque due to the input delay. The predictor term computes the
correction term by finitely integrating control torque over the time interval ranging from
current time minus the time delay to current time.
in the musculoskeletal dynamics [6] and occurs due to finite conduction velocities of the
chemical ions in the muscle in response to the external electrical input [36]. Input d.1 iv
can cause performance degradation as was observed during NMES experimental trials
on human subjects with RISE and NN+RISE controllers and has also been reported to
potentially cause instability during human stance experiments with NMES [40].
5.5.1 Experiments: Input Delay Characterization
Experiments were conducted to characterize input delay in healthy individuals during
NMES. The tested consisted of LEM (detailed in Section 3.4.4.1). The delay in NMES
RMS Error
Controller PID PID + CPTR PD PD + CPTR
Time Delay Linki Link2 Linki Link2 Linki Link2 Linki Link2
1 ms 0.1060 0.0890 0.1090 0.0870 0.0770 0.0830 0.0770 0.0760
2 ms 0.1070 0.1250 0.1130 0.0920 0.0650 0.1510 0.0690 0.0650
5 ms 0.1290 0.3700 0.1150 0.0770 0.061 0.291 0.0760 0.0820
10 ms 0.0890 0.2850 0.1310 0.091 0.0570 0.5050 0.0890 0.0880
50 ms 1.9540 1.2720 0.3700 0.3350 1.0370 1.6020 0.4070 0.3360
100 ms 3.1370 6.6050 1.0780 0.7260 3.1820 5.5950 1.1590 0.7290
200 ms 7.6290 6.7780 3.1180 3.6260 14.5320 17.5860 3.6250 2.3750
Maximum Absolute Peak Error
1 ms 0.1640 0.1730 0.1690 0.1780 0.1240 0.1580 0.1270 0.1500
2 ms 0.1720 0.2300 0.1790 0.180 0.1050 0.2750 0.1140 0.1250
5 ms 0.2040 0.6420 0.1790 0.1610 0.1080 0.5090 0.1270 0.1500
10 ms 0.1490 0.5120 0.2070 0.2110 0.1070 0.7070 0.1470 0.2000
50 ms 3.4300 2.0680 0.671 1.1960 1.7760 2.9980 0.7740 1.1930
100 ms 6.4840 11.6030 1.9640 2.4150 5.9300 11.551 1.9150 2.3330
200 ms 14.9600 12.5690 6.6000 10.4660 24.6290 32.7260 5.5200 6.8780
Table 5-1. Summarized experimental results of traditional PID/PD controllers and the
PID/PD controllers with d. 1 iv compensation. The controllers were tested for different
input delay values ranging from 1 ms to 200 ms. CPTR stands for compensator.
Elementwise percentage change RMS Error
in inverse inertia matrix Linki Link2
0 1.1720 1.0050
+10 1.2460 1.1680
-10 1.0780 0.9550
-50 1.5830 1.491
+50 1.5400 1.2490
+100 1.1910 1.0860
-75 2.9480 1.331
Table 5-2. Results compare performance of the PD controller with delay compensation,
when the B gain matrix is varied from the known inverse inertia matrix. The input delay
value was chosen to be 100 ms. The results indicate that large variations in the gain
matrix may be possible.
was measured as the difference between the time when voltage is applied to the muscle
and the time when the angle encoder detects the first leg movement. The input delay
values were measured for ten healthy individuals (9 male and 1 female). The tests on
each individual investigated the effect on input delay of three stimulation parameters:
frequency, pulsewidth, and voltage. Three different set of tests including: frequency vs
RMS Error
Percent uncertainty PD + Compensator PID + Compensator
in input delay Linki Link2 Linki Link2
(O'. 1.1590 0.7300 1.0780 0.7260
S(+)10' 1.2340 0.9660 0.9370 0.9100
(-)1(' 1.0790 1.2150 0.7560 0.4100
(+)21i'. 1.3380 1.5480 1.3040 1.8100
(-)2i'- 1.1920 1.7730 0.7820 0.6170
(+):l I 1.451 1.761 1.4980 0.6590
S(-)S -i'. 1.4520 1.3220 0.7680 0.6090
(+)50' 1.6290 2.5130 2.2420 1.1810
(- )5(' 1.1860 1.4500 0.9870 0.9070
S(+ i'. 3.5280 6.8190 3.0920 1.5100
(- i'. 1.2290 5.4080 0.9150 2.0530
(+)91' 4.0990 12.0200 3.3220 1.8360
(-) 91i' 3.2600 6.041 0.8740 2.461
(+)10'. 4.331 12.4450 4.2190 3.1010
(-)1O1 '. 3.1820 5.5950 3.1370 6.6050
Table 5-3. Experimental results when the input d 1li has uncertainty. The input delay
value was selected as 100 ms.
input delay, voltage vs input delay, and pulsewidth vs input delay were performed on
each individual. In each set of experiments, the other two stimulation parameters were
kept constant. Before the start of experiments, the subject was instructed to relax to
avoid voluntary leg motion. The threshold voltage was measured for each subject which
can be defined as the minimum voltage applied to the subject's muscle that produces
a movement large enough to be detected by the angle encoder. This measurement was
performed by applying a constant input voltage, beginning at 10 V and increasing the
voltage slightly until movement was detected. Once the threshold voltage was obtained,
the aforementioned three sets of experiments were performed for each individual.
The first set of experiments constituted varying frequency while keeping voltage and
pulsewidth constant. These tests consisted of measuring the input delay of the subject's
muscle for three 0.2 second impulses, each 5 seconds apart. Each impulse imparted a
constant voltage (threshold voltage + 10 V) to the muscle. The 5 second time separation
between the impulses allowed the subjects to voluntarily bring their leg back to the
Sr -
-3 -2
0 10 20 30 40 50 0 10 20 30 40 50
2 ^ I- i -f T -t
0i 0
-3
0 10 20 30 40 50 0 10 20 30 40 50
Time [sec.] Time [sec.]
Figure 5-3. The top-left and bottom-left plots show the errors of Link 1 and Link 2,
respectively, obtained from the PID controller with delay compensation and a traditional
PID controller. The top-right and bottom-right plots show the errors of Link 1 and Link
2, respectively, obtained from the PD controller with delay compensation and a traditional
PD controller. Errors obtained from the PID/PD + delay compensator are shown as
solid lines and the errors obtained from the traditional PID/PD controller are shown as
dash-dot lines. The input delay was chosen to be 50 ms.
rest position. Fig. 5-5 shows the typical EMD during NMES in a healthy individual.
Final input delay value was computed by averaging the measured d. 1 values over three
impulses. Eight experiments were performed for different frequencies, where the frequency
was chosen randomly from the range of 30 Hz and 100 Hz (intra range interval of 10
E 21
I
00
, -1
-ii
; -21
0 10 20 30 40 50 0 10 20 30 40 50
I. I .i l .
10 20 30
Time [sec]
0 10 20 30
Time [sec]
Figure 5-4. The top-left and bottom-left plots show the torques of Link 1 and Link 2,
respectively, obtained from the PID controller with delay compensation. The top-right and
bottom-right plots show the torques of Link 1 and Link 2, respectively, obtained from the
PD controller with delay compensation. The input delay was chosen to be 50 ms.
Hz). The pulse width for this type of the experiments was kept at 100ps. The second
type of experiments consisted of varying pulsewidth while keeping voltage and frequency
constant. Each experiment constituted three impulses as explained above for the frequency
tests. Nine experiments were performed for different pulsewidths, where pulsewidth was
randomly chosen from 100ps. to 1000ps (intra range interval of 100Ms). For this set
40 50
. .l i .
0 .5 .. : / .. ... ...
II. *
0.5 --- i
0 0.05 0.1 0.15 0.2 0.25 0.3
Time [sec.]
Figure 5-5. Typical input delay during NMES in a healthy individual. The desired
trajectory is shown in dashed line and the actual leg angle is shown in solid line. Note
that the actual leg angle starts rising around 70 ms.
of experiments, the frequency was kept constant at 30 Hz and the voltage consisted of
minimum threshold voltage + 10 V. The last set of experiments involved conducting
experiments with varying voltages. Same impulse program as used in the earlier set of
experiments was used, where pulsewidth and frequency were kept constant. The frequency
was kept at 30 Hz and the pulse width was kept at 100 mus. Three experiments were
performed for different voltages (threshold voltage + additional voltage, where additional
voltage was varied between 5 and 20 volts (intra range interval of 5 volts). Table 5-4 (as
a representative example) shows the summarized input delay variations with respect to
different stimulation parameters in a healthy individual.
ANOVA (Analysis of variance) tests were performed to determine the intraclass
correlations. An ANOVA test is generally employ, -1 to determine the statistical significance
between the means of data groups numbering more than two (using student t-test to
determine the statistical significance between more than two data groups can lead to
Type-I error (i.e., rejection of null hypothesis which in reality is true)). The results of
the stimulation frequency testing (see Fig. 5-6) showed that the difference in the means
of EMD was statistically significant (P-value = 1.50372E 10). Further, post-hoc test
utilizing Tukey's method showed that the EMD was longer for the lower frequencies than
for the higher frequencies. Particularly, the test showed that the average EMD of 76 ms
at a frequency of 30 Hz is statistically different from the average EMD of 51 ms at a
frequency of 100 Hz. However, the results of the stimulation pulse width (see Fig. 5-8) and
voltage experiments (see Fig. 5-7) showed no significant correlation between either varying
stimulation pulsewidth or stimulation voltage and electromechanical delay (P-value =
0.6870 and 0.072, respectively).
Frequency Vs. Time Delay
0.1
0.09
008 ISubject 1
-U-Subject2
-t-Subject4
2 \ -.m- Subject S
006 ..-Subject 6
Subject 7
005
SSubject9
--o- Subject 10
0.04
30 40 50 60 70 80 90 100
Frequency(Hz)
Figure 5-6. Average input delay values across different frequencies.
5.5.2 Experiments: PD Controller with Delay Compensation
The challenge in implementing the controllers in (5-8) and (5-42) is to measure
inertia and input delay in the muscle dynamics. Implementing the controller in (5-8)
becomes even more complicated due to the fact that it requires not only inertia of the
Time Delay vs Voltage
0.ii
0.09
S----Subject 1
0.08 "--"-- -----Subject 2
Subject
008 ---Subject 3
O -.," -*--Subject 4
S~-Subject S
NI- --^' ^- ~- -- --tSubject b
Subject 9
-@-Subject 10
0.05
0.04
5 7 9 11 13 15 17 19
Voltage
Figure 5-7. Average input delay values across different voltages.
musculoskeletal-LEM system to be measured but also the auxiliary function 6Q(q, c, t) E
R defined in (2-8), which consists of unmeasurable muscle force-velocity and muscle
force-length relationships to be known. However, the controller defined in (5-42) can be
implemented provided the following assumptions are made.
Assumption 1: The input delay is measurable and is constant. Although the input
delay for the NMES system is measurable but may not be constant due to v ,i i. I of
factors such as fatigue, non-isometric contractions, type of task, or stimulation parameters.
However, these variations are likely to be minimal in the duration of a single trial, and
the fact that the new controllers are shown to be robust to uncertainty in the input delay
value (see Table 5-3).
Assumption 2: The function JQ introduced in (3.3) can be upper bounded as
ai < JQ < 2, B J < a3 (5-65)
where al, a2, a3 E 2 are some known positive constants, and B is the control gain
introduced in (5-37).
Pulse Width vs. Time Delay
0.1
0.09
S Subject 1
0.08
S- Subject
0.07 /-- ---Subject4
SSubjecto5
0.06 -m--Subjtect6
SSubject7
0.05 Subject 9
SSubject 10
0.04
100 200 300 400 500 600 700 800 900 1000
Pulse Width (ps)
Figure 5-8. Average input delay values across different pulsewidths.
The tested for experiments consisted of LEM (detailed in Section 3.4.4.1). The
control objective was to track a continuous constant period (2 sec.) sinusoidal trajectory.
Three ble shmales (age: 21-28yrs) were chosen as the test subjects. After the protocol
(see section 3.4.4.1), the input delay value was measured for each subject. The measured
delay value was utilized for implementing the PD controllcr with I/ /,;/ compensation and
throughout the duration of trials, the same respective measured delay value was used
for each subject. The experiments compared the traditional PD controller with the PD
controller with Il/.,;I compensation. Each subject participated in two to four trials for each
controller 3 The experimental results obtained for each controller are summarized in Table
5-5. The table shows best two results (results with minimum RMS errors out of all trials)
obtained from each controller and subject.
3 maximum number of trials are limited due to increasing discomfort that arises due to
rapid muscle fatigue.
Frequency [Hz] Pulsewidth [p sec.] Voltage [V] Ti 72 73 Avg. 7
30 100 10 0.069 0.053 0.073 0.065
40 100 10 0.076 0.064 0.077 0.072
50 100 10 0.073 0.069 0.075 0.072
60 100 10 0.062 0.074 0.06 0.065
70 100 10 0.064 0.066 0.051 0.060
80 100 10 0.062 0.059 0.077 0.066
90 100 10 0.062 0.057 0.048 0.056
100 100 10 0.055 0.061 0.059 0.058
30 200 10 0.065 0.066 0.094 0.075
30 300 10 0.07 0.072 0.079 0.074
30 400 10 0.065 0.065 0.09 0.073
30 500 10 0.058 0.056 0.071 0.062
30 600 10 0.05 0.073 0.064 0.062
30 700 10 0.065 0.077 0.058 0.067
30 800 10 0.065 0.067 0.061 0.064
30 900 10 0.071 0.053 0.055 0.060
30 1000 10 0.057 0.083 0.065 0.068
30 100 5 0.081 0.061 0.061 0.068
30 100 15 0.068 0.079 0.087 0.078
30 100 20 0.082 0.084 0.059 0.075
Table 5-4. Summarized input d.l 1 values of a healthy individual across different
stimulation parameters. Delay values (7) are shown in seconds. The voltages shown in
column 3 are the added voltages to the threshold voltage.
A Student's t-test was also performed to confirm statistical significance in the mean
differences of the RMS errors, maximum steady state errors (SSEs), RMS voltages, and
the maximum voltages. The statistical comparison was conducted on the averages of the
two best results obtained for each subject. The analysis shows that the mean differences
in the RMS errors, maximum SSEs, and maximum voltages are statistically significant
while the analysis shows no statistical difference in the RMS voltages. The mean RMS
error of 4.43 obtained with the PD controller with 1/ Ir.' compensation is lower than the
RMS error of 6.030 obtained with the PD controller. Also, the mean maximum SSE and
the mean maximum voltage obtained with the PD controller with /. /I,;/ compensation are
lower than the mean maximum SSE and the mean maximum voltage obtained with the
traditional PD controller. The respective p-values are given in the Table 5-5. The actual
leg angle, error, and voltage plots obtained from subject C (as a representative example)
are shown in Figs. 5-9 and 5-10.
-7
4 0 o ... .. .... .
S0 5 10 15 20
20
0
0 5 10 15 20
Time [sec.]
4
0 5 10 15 20
Time [sec.]
40
30 30
20
0 5 10 15 20
Time [sec.]
Figure 5-9. Top plot: Actual limb trajectory of a subject (solid line) versus the desired
trajectory (dashed line) input obtained with the PD controller with /. r,'; compensation.
Middle plot: The tracking error (desired angle minus actual angle) of a subject's leg,
tracking a constant (2 sec.) period desired trajectory. Bottom plot: The computed voltage
of the PD controller with 1/ ,r;i compensation during knee joint tracking.
5.6 Conclusion
Control methods are developed for a class of an unknown Euler-Lagrange systems
with input delay. The designed controllers have a predictor-based structure to compensate
for dl 1 i,- in the input. LK functionals are constructed to aid the stability analysis which
yields a semi global uniformly ultimately bounded result. The experimental results
show that the developed controllers have improved performance when compared to
0
0 5 10 15 20
Time [sec.]
P -10---
-20
0 5 10 15 20
Time [sec.]
50
0 5 10 15 20
Time [sec.]
Figure 5-10. Top plot: Actual limb trajectory of a subject (solid line) versus the desired
trajectory (dashed line) input, obtained with the traditional PD controller. Middle
plot: The tracking error (desired angle minus actual angle) of a subject's leg, tracking
a constant (2 sec.) period desired trajectory. Bottom plot: The computed PD voltage
during knee joint tracking. Note that the voltage saturates at the user-defined set lower
voltage threshold of 10 V
traditional PID/PD controllers in the presence of input delay. Additional experiments on
6. 11l!:1 individuals showed that the PD controller with delay compensation is capable to
compensate for input delay in NMES and also performs better than the traditional PD
controller. A key contribution is the development of the first ever controllers to address
delay in the input of an uncertain nonlinear system. The result has been heretofore an
open challenge because of the need to develop a stabilizing predictor for the dynamic
response of an uncertain nonlinear system. To develop the controllers, the time delay
RMS Error RMS Voltage [V] Max. SSE Max. Voltage [V]
Subject PD PD +CTR PD PD +CTR PD PD +CTR PD PD +CTR
A 4.480 5.260 31.49 33.18 11.840 11.51 42.95 42.02
A 7.630 3.520 29.30 32.26 20.41 9.040 50 44.38
B 8.480 6.350 20.93 22.93 25.780 9.6110 45.1 27.43
B 6.540 5.960 24.72 22.65 10.790 10.720 31.28 26.51
C 3.110 2.850 25.58 26.17 12.840 5.680 43.68 38.8
C 5.91 2.61 23.65 27.60 16.660 5.60 49.33 36.7
Mean 6.030 4.430 25.95 27.47 16.370 8.690 43.72 35.97
p value 0.003* 0.095 0.008* 0.040*
Table 5-5. Table compares the experimental results obtained from the traditional
PD controller and the PD controller with /. .',;/ compensation. indicates statistical
significance and CTR stands for compensator.
was required to be a known constant. While some applications have known d-.1 iv (e.g.,
teleoperation [129], some network d [-v.1l [130], time constants in biological systems
[6, 36]), the development of more generalized results (which have been developed for
some linear systems) with unknown time d.-1 i- remains an open challenge. However, the
experimental results with two-link robot illustrated some robustness with regard to the
uncertainty in the time delay.
CHAPTER 6
RISE-BASED ADAPTIVE CONTROL OF AN UNCERTAIN NONLINEAR SYSTEM
WITH UNKNOWN STATE DELAYS
6.1 Introduction
The development in this chapter is motivated by the lack of continuous robust
controllers that can achieve .,-i-i1l l ic stability for a class of uncertain time-d, 1 li'- 1
nonlinear systems with additive bounded disturbances. The approach described in the
current effort uses a continuous implicit learning [96] based Robust Integral of the Sign of
the Error (RISE) structure [11, 27]. Due to the added benefit of reduced control effort and
improved control performance, an adaptive controller in conjunction with RISE feedback
structure is designed. However, since the time d 1li value is not .liv -,i- known, it becomes
challenging to design a delay free adaptive control law. Through the use of a desired
compensation adaptive law (DCAL) based technique and segregating the appropriate
terms in the open loop error system, the dependence of parameter estimate laws on the
time d. 1 -, ,1 unknown regression matrix is removed. Contrary to previous results, there is
no singularity in the developed controller. A Lyapunov-based stability analysis is provided
that uses an LK functional along with Young's inequality to remove time d. 1 i. terms
and achieves .i-vmptotic tracking.
6.2 Problem Formulation
Consider a class of uncertain nonlinear systems with an unknown state delay as [87]
x = x2
x(t) = f(x(t)) + 61(x(t)) + g(x(t r)) + 62(x(t r)) + d(t) + bu(t)
y = x1 (6-1)
In (6-1), f(x(t)), 61(x(t)) e R2 are unknown functions, g(x(t r)), 2( (t r)) C RI
are unknown time-dl 1 i-, '1 functions, r E R+ is an unknown constant arbitrarily large
time delay, d(t) E R" is a bounded disturbance, b E R is an unknown positive constant,
u(t) C R" is the control input, and x(t) [xT X x... xT] T R"' denote system states,
where x(t) is assumed to be measurable. Also the following assumptions and notations will
be exploited in the subsequent development.
Notation: Throughout the paper, a time dependent d, 1 li- .1 function is denoted as
x(t r) or x,, and a time dependent function (without time delay) is denoted as x(t) or x.
Assumption 1: The unknown functions b- f(x), b-lg(x) are linearly parameterizable,
i.e., b-'f(x) Yi(x)01, b-'g(x) Y2x)02, where Yi(x) e R2 p, Y2x) C R"' are
regression matrices of known functions, 01 e RP21x, 02 e -.' -xl are constant unknown
parameter vectors, and pi, P2 are positive integers. The regression matrix Y2(xT) is not
computable due to the unknown time delay present in the state
Assumption 2: If x(t) E L, then g(x), 61(x), 62(x) are bounded. Moreover, the
first and second partial derivatives of g(x), 61(x), 62(x) with respect to x(t) exist and are
bounded (see [83, 87, 95]).
Assumption 3: The disturbance term and its first two time derivatives are bounded
(i.e., d(t), d(t), d(t) c L).
Assumption 4: The desired trajectory is designed such that yd(t), yd(t) cE where
yd)(t) denotes the ith time derivative for i = 1, 2,..., n + 2.
6.3 Error System Development
The control objective is to ensure that the output y(t) E R" tracks a desired
time-varying trajectory yd(t) E Rm despite uncertainties in the system and an unknown
time delay in the state. To quantify the objective, a tracking error, denoted by el(t) E R",
is defined as
el(t) Y (t) d(t). (6-2)
To facilitate the subsequent analysis, following filtered tracking errors are also defined as
e2(t) e1(t) + aleI(t), (6-3)
ei(t) A ei-(t) + ai- 1eCi(t) + ei-2, (6-4)
r(t) ,(t) + )aen(t), (6-5)
where ac, ..., aOn E R denote positive constant control gains. As defined in (6-5), the
filtered tracking error r(t) is not measurable since the expression depends on x,(t).
However, ei(t),...,en(t) E RI are measurable because (6-4) can be expressed in terms of
the tracking error el(t) as
i-1
ea(t) aije i 2,...,n, (6-6)
j=0
where aij E R are positive constants obtained from substituting (6-6) in (6-4) and
comparing coefficients [114]. It can be easily shown that
aij = 1, j =i 1. (6-7)
Using (6-2)-(6-7), the open loop error system can be written as
r= y() -y +) (6-8)
where l(el, ei,..., e 1)) e R'" is a function of known and measurable terms, defined as
n-2
S anj (+l) + anej) + (n-1)
j=0
The open-loop tracking error system can be developed by premultiplying (6-8) by b-1 and
utilizing the expressions in (6 1) and Assumption 1 to obtain the following expression:
b- r = Yl(x)01 + b-'1(x) + Y2(x)02 + b- 2(x,) + b-'d + u b- y" + b- l. (6-9)
In the subsequent development, a DCAL-based update law is developed in terms of Y2(')
without a state delay. After some algebraic manipulation, the expression in (6-9) can be
rewritten as
b-'r = b-' + S + S2 + W + b-d + Y,(d) + Y2(d)0 + u,
where the auxiliary functions Sl(xd, x), S2(Xdr, x ), W(xd, Xdr, ydj)) CE IR are defined as
S1 = YI()1 Y( (Xd)1 + b- l() b- 1(Xd), (6-10)
S2 Y2(x)2 Y2(Xd)2 + b-62(x,) b- 62(Xdr), (6-11)
W = b-61(Xd) + b- 2(Xd)- Y2(Xd)2 Y2(dr)2 b- ), (6-12)
where Xd [y yP y d ) T Rm" denotes a column vector containing the
desired trajectory and its derivatives. The grouping of terms and structure of (6-10) is
motivated by the subsequent stability analysis and the need to develop an adaptive update
law that is invariant to the unknown time delay. The auxiliary function Sl(xd, x) is defined
because these terms are not functions of the time-delay. The auxiliary function S2 (Xdr, Xr)
is introduced because the time-d'l 1' .1 states are isolated in this term, and W(xd, Xd,, ad)
is isolated because it only contains functions of the desired trajectory.
Based on the open-loop error system in (6-10), the control input u(t) E R" is
designed as
u = -YI(Xd)O1 Y2(xd)2 p. (6-13)
In (6-13), pE R"7 denotes the implicit learning-based [96] RISE term defined as the
generalized solution to
f = (kh + 1) r + 3sgn(en), p(O) = 0, (6-14)
where ks, 3E R are known positive constant gains. In (6-13), 01(t) E R1, 02(t)E -'
denote parameter estimate vectors defined as
01 = FlY (xd)r, 02 r2Y2 d)r, (6-15a)
where F1 e IRplxp1, F2 E -! ._P2 are known, constant, diagonal, positive definite adaptation
gain matrices. In (6-15a), Y2(xd) does not depend on the time d. 1 .1 desired state. This
delay free law is achieved by isolating the d, 1 li .1 term Y2(Xdr)02 in the auxiliary signal
W(xd, Xdr, y()) in (6 12). The adaptation laws in (6-15a) depend on the unmeasurable
signal r(t), but by using the fact that Y1(xd), Y2(Xd) are functions of the known time
varying desired trajectory, integration by parts can be used to implement 8 (t) for i = 1, 2
where only eT(t) is required as
0, 0(0) + PYIT(xd)eC.(7) 1 {Yxden a) x en } da.
The closed-loop error system can be developed by substituting (6-13) into (6-10) as
b-b-r = b + S +S2 + W +b-ld- + Yl(Xd) + Y2(Xd)O2, (6-16)
where Qi for i = 1, 2 are the parameter estimation error vectors defined as
0i = Oi 0i. (6-17)
To facilitate the subsequent stability analysis and to more clearly illustrate how the RISE
structure in (6-14) is used to reject the disturbance terms, the time derivative of (6-16) is
determined as
b-'r = N+Nd -en +Y((Xd)l +Y 2(Xd)2 -(ks +1) r sgn(e), (6-18)
where the auxiliary functions N(e, ...e, r, eli, ..., eT, r,), Nd(Xd, Xd, t) E Rm are defined as
N = b- i + S+ & S+ C, Y(xd) Fl(xd)r Y2(Xd2(Xd)r,
Nd W + b-'d. (6-19)
Using Assumptions 2, 3, and 4, Nd(xd, Xd, Xd, t) and its time derivative can be upper
bounded as
|Nd1 < (Nd, Nd < (NC (6-20)
where (Nd, (Nd E R are known positive constants. The expression defined in (6-19) can be
upper bounded using the Mean Value Theorem as [114]
N < pi(|| ||) 1|| 11 + p2(11 -- ) 1|| -| (6-21)
where z(t) E 1'1 m is defined as
z e e e r (6-22)
and the known bounding functions pi (|: ||), p2 (|| : I) E R are positive, globally invertible,
and nondecreasing functions. Note that the upper bound for the auxiliary function
N(el,e2, eC, e2.) in (6-21) is segregated into delay free and d. 1 i. '1 upper bound
functions. Motivation for this segregation of terms is to eliminate the delay dependent
term through the use of an LK functional in the stability analysis. Specifically, let
Q(t) E R denote an LK functional defined as
Q =- = p (||-(o ||) ||(::( )||2 ) da, (6-23)
where k E cR and p2a) are introduced in (6-14) and (6-21), respectively.
6.4 Stability Analysis
Theorem 6. The controller given in (6-13), (6-14), and (6-15a) ensures that all system
i.:l,.l are bounded under closed-loop operation. The tracking error is regulated in the
sense that
||ei(t)|| 0 as t oo,
provided the control gain ks introduced in (6-14) is selected suff .:. nill; 1I, and anl,
a0, and 3 are selected according to the following sufficient conditions:
3> (N, + N, a n-1, an > 2 (6-24)
( 1 ) 1 )
where a,_1, an are introduced in (6-4) and (6-5), ,. i. 1,:; /;3 is introduced in (6-14);
and (Nd and CNd are introduced in (6-20).
Proof: Let )D C )- p1) +p+p2+2 be a domain containing y(t) = 0, where y(t) e
R(n+ )mpl P2 +2 is defined as
y(t) A T P(t)T Q(t) ] (6-25)
where 6 (t) are defined in (6-17), z(t) and Q(t) are defined in (6-22) and (6-23),
respectively, and the auxiliary function P (t) E R is the generalized solution to the
differential equation
n
P (t) -L (t), P (0) =3 |e, (0) -e (0)TNd (0) (6 26)
i= 1
The auxiliary function L (t) E R in (6-26) is defined as
L (t) ^ rT (Nd (t) psgn (e,)). (6-27)
Provided the sufficient conditions stated in Theorem 6 are satisfied, then P (t) > 0 (see the
Appendix B).
Let VL (y, t) : D x [0, oc) IR denote a Lipschitz continuous regular positive definite
functional defined as
V( A 10 1A 22 1 1
V(y,t) ee+ ee2 ... + e + rb- + P + + O 0r1i (6 28)
+2 r2 12,
which satisfies the following inequalities
U (y) < V (y, t)< U2 (y), (6-29)
provided the sufficient conditions introduced in Theorem 6 are satisfied. In (6-29),
U1 (y) U2 (y) E R are continuous, positive definite functions defined as
Ui (y) = 71 Iy112 U2 (y) =7 I12 (6-30)
where 71, 72 E R are defined as
1
71 2- min(1, b- 7min {11}, F min {21}),
72 = max( -b-' l, rx {r1 (max { 1}),6 (6-31)
and 7min { }, 7max { } denote the minimum and maximum Eigenvalues, respectively. After
taking the time derivative of (6-28), VL (y, t) can be expressed as
VL (y, t) A e 2i + ee2 + + tb- +P+Q+ Q 1 + 0 0 + e1E02.
From (6-3), (6-4), (6-18), (6-26), (6-27), adaptation laws in (6-15a), and the time
derivative of Q(t) in (6-23), some of the differential equations describing the closed-loop
system for which the stability analysis is being performed have discontinuous right-hand
sides as
ei = e2 alel, (6-32a)
e2 = 63 a262 el, (6-32b)
(6-32c)
(6-32d)
(6-32e)
,n = r cien (6-32f)
b-'r 1 + Nd e, + Y(xd)1 + Y2(xd)02 (k, + 1) r sgn(en), (6-32g)
P (t) -rT (Nd (t) psgn (en)), (6-32h)
Q((t) 2 1(p((t))|() |() 2 ( T)) (t )2), (632i)
M (t ,\ I2(112 (6-32i)
2k, 2 2
0FI = -o Y (Xd)r, (6-32j)
oF2 1 2 2Y (Xd)r. (6-32k)
Let f(y, t) e I. 1)m+pi+p2+2 denote the right hand side of (6-32). f(y, t) is continuous
except in the set {(y,t)le2 = 0}. From [103-106], an absolute continuous Filippov solution
y(t) exists almost everywhere (a.e.) so that
e K[f](y,t) a.e.
The generalized time derivative of (6-28) exists a.e., and VL(y, t) a.e. VL (y, t) where
1
1 1 1 0 0
c C CT-1 2P 1 2Ql T F n 7 1- 1-1 F
1 2 .T T b- 2} 2 11 2F2IK 5 2 .' .
'-P Q O 2T.
For more details of the notations used in 6-32 to 6-33 and discussion, see Section 3.3.1.
After utilizing (6-3), (6-4), (6-18), (6-26), (6-27), adaptation laws in (6-15a) and the
time derivative of Q(t) in (6-23), the expression in (6-33) can be rewritten as
VL (y,t) C
-calele1 -
+r Y2 (d)2
- [T (Xd)r
- areee + e e -1 + eCr + r N + TNd rTe, + rYl(xd)l
(k, + 1) ||r|2 TK[sgn(en)]- rTNd(t) + OT K[sgn(e)]
022T(X)r + i) ) 2) (6-34)
02Y2Xd)F2hiP2H
Cancelling common terms yields and using (6-21)
n
(L {y> t) c ^ Q' o |e 1 112 + eT_ler -- r112 k, Ir12 + 112(11 ) :: -II I1r|1 + pl(1 11) 11 1,111 r 11
i=1
2-11 (11) 1 11 2(11 H11) 112
+ 22ks 2 2k (6-35)
2k, 2k,
After applying following Young's inequality to determine that
P2( -) 2 < l2 e llen 2 ei (6-36)
the ex2kn 2 n) cn be w n
the expression in (6-35) can be written as
n-2
-- |, 2 (an-1 1) en-1 2
i 1
2 1- I II k 11
2k8
(a,
Se 2- llr12
2} en F
After completing the squares, the expression in (6-37) can be written as
p2v I 1 I 1
(6-37)
where p2(1 11) IR is defined as
(6-38)
and 73 min [1i, a 2, o, n2, o.n- -. o 1 1] The bounding function p (| )
is a positive, globally invertible, and nondecreasing function that does not depend on
VL (y, t) C
2(11 ::(t)ll) P2(11 ::(t)l) + p2(11 :(t)lII),
the time-delay. The expression in (6-37) can be further upper bounded by a continuous,
positive semi-definite function
VL (Y,t) C -U(y) = -c :I- Vy CD (6-39)
for some positive constant c, where
-A y (t) e I I) m+pi+p +2 < -i1 2
Larger values of k, will expand the size of the domain D. The inequalities in (6-29)
and (6-39) can be used to show that V(y, t) E L, in D; hence, C1, 2,...,en, 01, 02 cE L
in D. The closed-loop error systems can now be used to conclude all remaining signals
are bounded in D, and the definitions for U(y) and z(t) can be used to prove that U(y) is
uniformly continuous in D. Let S C D denote a set defined as
S y(t) CD U2((t)) < 1 (1 )) (6-40)
The region of attraction in (6-40) can be made arbitrarily large to include any initial
conditions by increasing the control gain k, (i.e., a semi-global stability result), and hence
cll : '- 0 as t oo Vy(0) e S. (6-41)
Based on the definition of z(t), (6-41) can be used to show that
||ei(t)|| -0 as t oo Vy(0) e S. (6-42)
6.5 Simulations
To illustrate the performance of the RISE-based adaptive controller, we consider the
following first order scalar nonlinear plant [87]:
'i -= 2 (6-43)
2 = f(x) + g(x) + 61(x) + 62(x) + d + bu,
where f(x), g(x,) are linearly parameterizable functions, 61(x) is an unknown function,
62 (x) is an unknown d. 1 ',. 1 function, d(t) is a disturbance term, u(t) is the control
input, and b is an unknown coefficient. Since the time delay is unknown, the regression
matrix for g(x,) is unknown to the controller. However, the adaptive estimate laws
in the controller do not require the time delay value to be known. For the simulation
purposes, these functions and parameters are chosen: f(x) = 0.5sin(xl(t)); g(x,) =
0.2x(t 7r) + 2 cos(x2(t)); 61(X) = sin(5x2(t)); 62(xS) = 0.52(t- T) sin(2xi(t 7));
d = O.lsi(t); b = 1. The simulations are performed for the two cases of unknown time
d-.1 i-, namely, r = 3 s; r = 10 s. The desired trajectory is chosen as
Xd(t) = 0.5 [sin(t) + sin(0.5t)]. (6-44)
The following gains are chosen for r = 3 s and r = 10 s
ks = 10, aci 7, a2 6, /3 5, Fi = 0.5,
Fa = [2,0; 0,10].
From the results shown in Figs. 6-1-6-5, it is clear that the controller tracks the time
varying desired trajectory effectively. In both the cases, the steady state errors stay
between 0.003 radians and the control inputs are bounded. Also it can be seen that there
is a little variation in the control performances for time d,1 l' ,- = 3 s and r = 10 s.
6.6 Conclusion
A robust continuous RISE-based structure is utilized in conjunction with an adaptive
controller for stabilizing a class of uncertain nonlinear systems with unknown state d'-1 .,-
and bounded disturbances. By properly utilizing a DCAL-based method and segregating
the necessary terms, the controller and the adaptive estimate law do not depend on the
unknown time delay in the state. Appropriate LK functional is constructed to cancel
the time d,1 liv .1 terms in the stability analysis. Simulations are provided to show the
0.01
-0.01
-0.03
0 .03 '--------- --------
0 5 10 15 20
Time [sec]
Figure 6-1. Tracking error for the case r = 3 s.
-2
-4
0 5 10 15 20
Time [sec]
Figure 6-2. Control input for the case r = 3 s.
performance of the controller. A Lyapunov-based stability analysis proves .-i, ,!l I ic
stability for the closed loop nonlinear system.
1.5 .5. .- .. .. .. ... .
5 10 15
Time [sec]
Figure 6-3. Parameter estimates for the case = 3 s. Dashed line shows the parameter
estimate of 01. Solid line shows the parameter estimate of 02(1). Dash-dot line shows the
parameter estimate of 02(2).
5 10 15
Time [sec]
Figure 6-4. Tracking error for the case r
10 s.
0 5 10 15
Time [sec]
Figure 6-5. Control input for the case r
20
10 s.
.i..... -... .: ...- ; :;............:)..- .- -,- .....- .-. -. -.. .
1.5
I .... ......... .............. .
0.5..
Time [sec]
Figure 6-6. Parameter estimates for the case r = 10 s. Dashed line shows the parameter
estimate of 01. Solid line shows the parameter estimate of 02(1). Dash-dot line shows the
parameter estimate of 02(2).
CHAPTER 7
CONCLUSION AND FUTURE WORK
7.1 Conclusion
New nonlinear controllers are developed to tackle various technical challenges in
implementing NMES. These difficulties include unknown nonlinear mapping between
the applied voltage to the muscle and the force generated in the muscle, bounded
disturbances, muscle fatigue, and time delay. The first two controllers developed in
C'! lpter 3 deal with unknown nonlinear 1 IlpplH-ii bounded disturbances, and other
unknown nonlinearities and uncertainties. The Lyapunov-based stability analysis
is utilized to prove semi-global .I-vmptotic stability for the controllers. Extensive
experiments on healthy volunteers were conducted for both RISE and NN+RISE
controllers. Particularly, it was shown that the inclusion of neural network based
feedforward component in the RISE controller improves performance during NMES. Also,
preliminary experimental trials demonstrating sit-to-stand task depicted the feasibility of
the NN+RISE controller in a clinical-type scenario.
In C'! lpter 4, a NN-based controller is developed to compensate for fatigue. The
benefit of the controller is that it incorporates more muscle dynamics knowledge namely,
calcium and fatigue dynamics. The effectiveness of the controller to compensate fatigue is
shown through simulation results. Further simulations show that the controller performs
better than the RISE controller.
An important technical difficulty in NMES is input delay which becomes more
challenging due to the presence of unknown nonlinearities and disturbances. Lack of
input delay compensating controllers for uncertain nonlinear systems motivated to
develop predictor-based controllers for general Euler Lagrange system in C'! lpter 5.
The Lyapunov-based stability analysis utilizes LK functionals to prove semi-global UUB
tracking. Extensive experimental results show better performance of the controller in
comparison to the traditional PD/PID controller as well as their robustness to uncertainty
in input delay value and inertia matrix. Further, the feasibility of the predictor-based
controller for NMES is shown through experimental trials on healthy individuals. Also, a
study to characterize input delay in NMES is included in the chapter which shows that the
input delay is dependent on frequency.
The last chapter in the dissertation covers the development of RISE-based adaptive
controller for a class of nonlinear system with state d. 1 ,i- The significance of the result is
that a robust and continuous controller is developed for a nonlinear system with unknown
state d, 1 li- and additive disturbances. Lyapunov-based stability analysis aided with LK
functionals is utilized to show a semi-global .,-i-, !! II l ic tracking.
7.2 Future Work
The following points discuss future work that can be built on the current research
described in the dissertation.
* Current experiments focused extensively on testing controllers on healthy
volunteers. These experiments showed that the controllers hold potential for clinical
tasks. Also, a preliminary test with the NN+RISE controller showed a promising
sit-to-stand task performance. Therefore, extensive experiments can be performed
where controllers should be tested on patients for functional tasks such as walking
and sit-to-stand maneuvers.
Efforts in Chapter 3 showed that the RMS error difference (for both RISE and
NN+RISE controllers) between the flexion and extension phase of the leg movement
is statistically significant. These results -ii.;. -1 that the role of switching controllers
(hybrid control approach) can be investigated. Specifically, two different controllers
can be utilized where each controller is dedicated for a particular phase of the leg
movement.
The result developed in C(i lpter 4 has three main limitations: unmeasurable
calcium and fatigue dynamics, dependence on acceleration, and uniformly ultimately
bounded stability result. Efforts can be made to develop an observer-based controller
to remove the dependence on mathematical fatigue and calcium dynamics models.
Specifically, recurrent neural network based observer can be designed to identify
system states. Further, improvement in stability analysis can be achieved by
developing a controller with .-i- i!!ill ic tracking. An extensive investigation is
required to observe the effect of the controller in C(i lpter 4 on reducing fatigue.
Experiments should not only compare the result with an existing controller
for improved performance but should specifically study the effectiveness of the
included fatigue model for fatigue compensation. The results may (or may not)
point to a need for improved fatigue models that are more suitable for non-isometric
contractions and account for multiple factors affecting the fatigue onset in NMES.
Also, additional information can be gathered to predict fatigue onset through
incorporating Electromyogram (EMG) signals. Measuring surface EMG signals can
be used as an indicator or can be utilized to quantify the fatigue onset which can be
further incorporated in NMES control design.
S Currently most of the NMES control implementation utilize single modulation
methods (e.g., the experiments were performed with amplitude modulation
technique, where the frequency and pulsewidth were kept constant while voltage
is varied). Methods can be developed to modulate multiple stimulation parameters
simultaneously. However, more efforts will be required first to investigate the effects
of multiple modulation during NMES control. The benefits of this research may
manifest as improved control performance during fatigue onset (e.g., frequency i' '-
an important role in the fatigue onset. Modulating frequency along with amplitude
may delay the onset of fatigue during NMES.)
* One of the most important technical issue in NMES is the rapid onset of fatigue.
Numerous factors influence the early onset of fatigue during NMES control.
Overstimulation due to high gain controller is one of the factors that affects the
fatigue onset. Feedforward methods or using low gain control are alv-i- I -,'::-. -1. I
to avoid early onset of fatigue. However, high gain controllers are required to obtain
minimal tracking errors during functional tasks. A solution to optimize these two
conflicting strategies can be obtained by designing optimal controllers. Proper
mechanisms can be built into these controllers to provide a choice between better
error performance or delaying the fatigue onset.
* The focus of the current research was mainly on developing control techniques
for non-invasive surface electrical stimulation. The main disadvantage of surface
electrical stimulation is repetitive and non-selective recruitment of muscle fibres
which lie in the path of applied current. This type of muscle recruitment is the
main cause of rapid fatigue onset and is in contrast to the recruitment employ,
by the brain and central nervous system during voluntary contractions. In context
to this disadvantage, researchers have used invasive electrodes to stimulate specific
muscles or nerves in the paralyzed patients to produce desired functional movements.
The main benefit of these methods is selective and non-repetitive recruitment of
muscle fibres, thereby avoiding muscle fatigue. However, wires protruding out from
the skin and chances of infection have made this option unattractive. With the
advancement of technology, some researchers have developed micro-stimulators
called BIONs [131], which can be surgically implanted at specific sites in the muscle.
These microelectrodes which do not require wires are powered externally through
an inductive coil and a battery. Multiple BIONs to stimulate specific muscle sites
can not only be used to produce desired functional movements but also can be used
to eliminate muscle fatigue through utilizing non-repetitive and selective muscle
recruitment. In order to produce NMES control via BIONs, studies will be required
to imitate the strategies used by brain and central nervous system during voluntary
contractions. The real challenge will be to maintain stability and coordination of
multiple implanted BIONs in order to extract desired movements. Approaches from
hybrid control theory and co-operative control should be investigated to develop
NMES control via BIONs.
Development focused on input delay measurement in the C'! lpter 5 showed that
the input delay in NMES depends only on varying frequency. However, further
investigations are required to study the effect of fatigue and non-isometric
contractions on input delay. Also, results in C'! lpter 5 are only applicable with
known constant input delay values. Therefore, controllers need to be developed
to account for time-varying or unknown input delay. Other delay compensating
techniques such as model predictive control (\!PC) can also be investigated for
NMES. One of the advantages of MPC is that it inherently compensates for input
d- 1 i- Although the technique would require muscle dynamics to be known,
advantages such as performance and control optimization in addition to d. 1li
compensation makes MPC a worthy candidate for investigation.
APPENDIX A
(CHAPTER 5) PREDICTOR-BASED CONTROL FOR AN UNCERTAIN
EULER-LAGRANGE SYSTEM WITH INPUT DELAY
Lemma 1. D. fiu. Q(t) E R as
Q(t)= Y (1 (o (0) 11 ds.
The time derivative of Q(t) is
The time derivative of Q(t) is
Q(t=) = rL u(t)112
U (o)2 1dO..
t-T
Proof: The time derivative of Q(t)
dQ) [-T
on applying Leibniz integral rule can be written as
Q(t) t ( 11t(od 2dO dt
(/i'-T
d-T
dt ]-
(1-3)
The expression in (1-3) can be simplified as
t ()t ( ||/ a t
it(o) 11\2 do + L -
J-- -- s
Again applying Leibniz integral rule on second integral in (1-4)
t -t
(0o) \2 do + Ij j() 1 2 d
The expression in (15) can be simplified a
The expression in (1-5) can be simplified as
Further integrating the second integral in (1-6)
UrT Iu(t) 11
(1-1)
(1-2)
\s /
ds] I
jt(0)112 d0 ds.
(1 4)
ii s) 2 2) d ds.
11 d1 t+ at1()1 o
(1-5)
j I |(0)2 dO + a i(t)2 It ds.
Jt- 1-7
(1-6)
St(o) 112 de.
It-T
I(o)112 d6o) d(
Lemma 2. D. fiu P(t) E R as
P(t) = ( uIIu(e0) 2d) ds.
The time derivative of P(t) is
The time derivative of P(t) is
P(t) = r u(t) 2I
Proof: The proof is similar to the proof given for Lemma 1
(1-7)
rt
t_-T
(1-8)
APPENDIX B
(CHAPTER 6) RISE-BASED ADAPTIVE CONTROL OF AN UNCERTAIN
NONLINEAR SYSTEM WITH UNKNOWN STATE DELAYS
Lemma 3. D. fin, L(t) E R as
Then, if 0 -,/.:/,
then
L A rT(Nd psgn(e,)).
S> wNd
> Nd + -a
L (7) dT <
i= 1
(0) 1 eC (O)T Nd (0) ,
where ei (0) E R denotes the ith element of the vector en (0).
Proof: Integrating both sides of (2-9)
JL(a)da
0o
f [rT(Nd
Jo
Osgn(en))] da.
(2-12)
On substituting (6-5) in (2-12) yields
L L(o)du
It
n Nddu
Jon d
) ~.l
)t )t
0 en0sgne)d + j0 aeNd
Jo Jo
Osgn(en))da.
(2 13)
After utilizing integration by parts for the first integral and integrating the second integral
in (2-13), the following expression is obtained:
L(jo)d eTNd
0 n
Jot
e(N~s
n
e (0) Nd (0) + e(0)|
i 1
where the fact that sgn(en) can be denoted as
sgn(en) = [sgn(enl) sgn(en2)
(2-9)
(2-10)
(2-11)
d1 1)d l\
d- _psgn(en)) da,
a dao
i= 1
(2-14)
gn(e,,,)] ,
(2-15)
is utilized in the second integral. Using the bounds given in (6-20) and the fact that
-(t) || <
i 1
the expression in (2-14) can be upper bounded as
t
I L(,)du <
0n
(0 1 )
i 1
3 e~ )+jci eJ
((N, +
S-d da.
a
(2-17)
It is clear from (2-17) that if the following sufficient condition
/3> N +
a
is satisfied, then the following inequality holds
t n
L(a)d
0 i= 1
(t) ,
(2-16)
(2 18)
e,(0) N(0).
(2-19)
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BIOGRAPHICAL SKETCH
Nitin Sharma was born in November 1981 in Amritsar, India. He received his
Bachelor of Engineering degree in industrial engineering from Thapar University, India.
After his graduation in 2004, he was hired as a graduate engineer trainee from 2004 to
2005 and worked as an executive engineer from 2005 to 2006 in Maruti Suzuki India Ltd.
He then joined the Nonlinear Controls and Robotics (NCR) research group to pursue
his doctoral research under the advisement of Dr. Warren E. Dixon. He will be joining
as a postdoctoral fellow in Dr. Richard Stein's laboratory at the University of Alberta,
Edmonton, Canada.
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Iwouldliketoexpresssinceregratitudetomyadvisor,Dr.WarrenE.Dixon,forgivingmetheopportunitytoworkwithhim.IthankhimforexposingmetovastandexcitingresearchareaofnonlinearcontrolandmotivatingmetoworkonNeuromuscularElectricalStimulation(NMES)controlproblem.Ihavelearnttremendouslyfromhisexperienceandappreciatehissignicantroleindevelopingmyprofessionalskillsandcontributingtomyacademicsuccess.Iwouldalsoliketothankmyco-advisorDr.ChrisGregoryforansweringmyqueriesrelatedtomusclephysiologyandforguidingmeinbuildingcorrectprotocolsduringNMESexperiments.IalsoappreciatemycommitteemembersDr.ScottBanks,Dr.CarlD.CraneIIIandDr.JacobHammerforthetimeandhelptheyprovided.IwouldliketothankmycolleaguesfortheirsupportandappreciatetheirsteadfastvolunteeringinNMESexperiments.Iwouldliketothankmywifeforherloveandpatience.Also,Iwouldliketoattributemyoverallsuccesstomymotherwhotookhertimeandeorttoteachmeduringmychildhood.Finally,Iwouldliketothankmyfatherforhisbeliefinme. 4
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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 11 CHAPTER 1INTRODUCTION .................................. 14 1.1MotivationandProblemStatement ...................... 14 1.2Contributions .................................. 22 2MUSCLEACTIVATIONANDLIMBMODEL .................. 25 3NONLINEARNEUROMUSCULARELECTRICALSTIMULATION(NMES)TRACKINGCONTROLOFAHUMANLIMB .................. 30 3.1Introduction ................................... 30 3.2ControlDevelopment .............................. 31 3.3NonlinearNMESControlofaHumanLimbviaRobustIntegralofSignumofError(RISE)method ............................ 32 3.3.1StabilityAnalysis ............................ 34 3.3.2ExperimentalResults .......................... 38 3.3.2.1Testbedandprotocol ..................... 39 3.3.2.2Resultsanddiscussion .................... 40 3.3.3Conclusion ................................ 46 3.4ModiedNeuralNetwork-basedElectricalStimulationforHumanLimbTracking ..................................... 48 3.4.1Open-LoopErrorSystem ........................ 50 3.4.2Closed-LoopErrorSystem ....................... 51 3.4.3StabilityAnalysis ............................ 55 3.4.4ExperimentalResults .......................... 59 3.4.4.1Testbedandprotocol ..................... 59 3.4.4.2Resultsanddiscussion .................... 61 3.4.5Limitations ............................... 68 3.4.6Conclusion ................................ 71 4NONLINEARCONTROLOFNMES:INCORPORATINGFATIGUEANDCALCIUMDYNAMICS ............................... 73 4.1Introduction ................................... 73 4.2MuscleActivationandLimbModel ...................... 73 5
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.............................. 76 4.3.1Open-LoopErrorSystem ........................ 77 4.3.2Closed-LoopErrorSystem ....................... 79 4.3.3BacksteppingErrorSystem ....................... 81 4.4StabilityAnalysis ................................ 81 4.5Simulations ................................... 84 4.6Conclusion .................................... 85 5PREDICTOR-BASEDCONTROLFORANUNCERTAINEULER-LAGRANGESYSTEMWITHINPUTDELAY .......................... 89 5.1Introduction ................................... 89 5.2DynamicModelandProperties ........................ 90 5.3ControlDevelopment .............................. 91 5.3.1Objective ................................. 91 5.3.2ControldevelopmentgivenaKnownInertiaMatrix ......... 91 5.3.3ControldevelopmentwithanUnknownInertiaMatrix ........ 97 5.4ExperimentalResultsandDiscussion ..................... 103 5.5DelaycompensationinNMESthroughPredictor-basedControl ...... 106 5.5.1Experiments:InputDelayCharacterization .............. 107 5.5.2Experiments:PDControllerwithDelayCompensation ....... 113 5.6Conclusion .................................... 117 6RISE-BASEDADAPTIVECONTROLOFANUNCERTAINNONLINEARSYSTEMWITHUNKNOWNSTATEDELAYS .................. 120 6.1Introduction ................................... 120 6.2ProblemFormulation .............................. 120 6.3ErrorSystemDevelopment ........................... 121 6.4StabilityAnalysis ................................ 125 6.5Simulations ................................... 130 6.6Conclusion .................................... 131 7CONCLUSIONANDFUTUREWORK ...................... 135 7.1Conclusion .................................... 135 7.2FutureWork ................................... 136 APPENDIX APREDICTOR-BASEDCONTROLFORANUNCERTAINEULER-LAGRANGESYSTEMWITHINPUTDELAY .......................... 139 BRISE-BASEDADAPTIVECONTROLOFANUNCERTAINNONLINEARSYSTEMWITHUNKNOWNSTATEDELAYS .................. 141 REFERENCES ....................................... 143 6
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Table page 3-1TabulatedresultsindicatethatthetestsubjectwasnotlearningthedesiredtrajectorysincetheRMSerrorsarerelativelyequalforeachtrial. ........ 40 3-2Experimentalresultsfortwoperioddesiredtrajectory ............... 41 3-3Summarizedexperimentalresultsformultiple,higherfrequenciesandhigherrangeofmotion. ................................... 43 3-4SummarizedexperimentalresultsandPvaluesofonetailedpairedT-testfora1.5secondperioddesiredtrajectory. ........................ 63 3-5SummarizedexperimentalresultsandPvaluesofonetailedpairedT-testfordualperiodic(4-6second)desiredtrajectory. .................... 66 3-6Experimentalresultsforstepresponseandchangingloads ............ 66 3-7ThetableshowstheRMSerrorsduringextensionandexionphaseofthelegmovementacrossdierentsubjects, ......................... 71 5-1SummarizedexperimentalresultsoftraditionalPID/PDcontrollersandthePID/PDcontrollerswithdelaycompensation. ........................ 108 5-2ResultscompareperformanceofthePDcontrollerwithdelaycompensation,whentheBgainmatrixisvariedfromtheknowninverseinertiamatrix. .... 108 5-3Experimentalresultswhentheinputdelayhasuncertainty.Theinputdelayvaluewasselectedas100ms. ............................ 109 5-4Summarizedinputdelayvaluesofahealthyindividualacrossdierentstimulationparameters. ...................................... 116 5-5TablecomparestheexperimentalresultsobtainedfromthetraditionalPDcontrollerandthePDcontrollerwithdelaycompensation. .................. 119 8
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Figure page 2-1Muscleactivationandlimbmodel. ......................... 25 2-2Theleftimageillustratesaperson'sleftleginarelaxedstate. .......... 27 3-1Topplots:Actualleftlimbtrajectoryofasubject(solidline)versusthedesiredtwoperiodictrajectory(dashedline)input. .................... 42 3-2Topplot:Actuallimbtrajectory(solidline)versusthedesiredtripleperiodictrajectory(dashedline). ............................... 43 3-3Topplot:Actuallimbtrajectory(solidline)versusthedesiredconstantperiod(2sec)trajectory(dashedline). ........................... 44 3-4Topplot:Actuallimbtrajectory(solidline)versusthetripleperiodicdesiredtrajectorywithhigherrangeofmotion(dashedline). ............... 45 3-5Topplot:Actuallimbtrajectory(solidline)versusthedesiredconstantperiod(6sec)trajectory(dashedline). ........................... 46 3-6Topplot:Actuallimbtrajectory(solidline)versusdesiredsteptrajectory(dashedline). .......................................... 47 3-7Thetopplotshowstheactuallimbtrajectory(solidline)obtainedfromtheRISEcontrollerversusthedesired1.5secondperioddesiredtrajectory(dashedline). 61 3-8Thetopplotshowstheactuallimbtrajectory(solidline)obtainedfromtheNN+RISEcontrollerversusthedesired1.5secondperioddesiredtrajectory(dashedline). 62 3-9Thetopplotshowstheactuallimbtrajectory(solidline)obtainedfromtheRISEcontrollerversusthedualperiodicdesiredtrajectory(dashedline). ....... 64 3-10Thetopplotshowstheactuallimbtrajectory(solidline)obtainedfromtheNN+RISEcontrollerversusthedualperiodicdesiredtrajectory(dashedline). ....... 65 3-11ExperimentalplotsforstepchangeandloadadditionobtainedfromNN+RISEcontroller. ....................................... 67 3-12Initialsittingpositionduringsit-to-standexperiments.Theknee-anglewasmeasuredusingagoniometerattachedaroundtheknee-axisofthesubject'sleg. ...... 68 3-13Thetopplotshowstheactuallegangletrajectory(solidline)versusdesiredtrajectory(dottedline)obtainedduringthestandingexperiment. ........ 69 4-1Anuncertainfatiguemodelisincorporatedinthecontroldesigntoaddressmusclefatigue.Bestguessestimatesareusedforunknownmodelparameters. ..... 76 9
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.................................. 84 4-3Topplotshowsthekneeangleerrorfora2secondperiodtrajectoryusingtheproposedcontroller. .................................. 85 4-4Topplotshowsthekneeangleerrorfora6secondperiodtrajectoryusingtheRISEcontroller. .................................... 86 4-5Topplotshowsthekneeangleerrorfora2secondperiodtrajectoryusingtheRISEcontroller. .................................... 87 4-6RISEcontrollerwithfatigueinthedynamics .................... 87 4-7Performanceoftheproposedcontroller ....................... 88 4-8Fatiguevariable .................................... 88 5-1Experimentaltestbedconsitingofa2-linkrobot.Theinputdelayinthesystemwasarticiallyinsertedinthecontrolsoftware. .................. 103 5-2Theplotshowsthreetorqueterms ......................... 107 5-3Thetop-leftandbottom-leftplotsshowtheerrorsofLink1andLink2 ..... 110 5-4Thetop-leftandbottom-leftplotsshowthetorquesofLink1andLink2 .... 111 5-5TypicalinputdelayduringNMESinahealthyindividual. ............ 112 5-6Averageinputdelayvaluesacrossdierentfrequencies. .............. 113 5-7Averageinputdelayvaluesacrossdierentvoltages. ................ 114 5-8Averageinputdelayvaluesacrossdierentpulsewidths. .............. 115 5-9Topplot:Actuallimbtrajectoryofasubject(solidline)versusthedesiredtrajectory(dashedline)inputobtainedwiththePDcontrollerwithdelaycompensation. .. 117 5-10Topplot:Actuallimbtrajectoryofasubject(solidline)versusthedesiredtrajectory(dashedline)input .................................. 118 6-1Trackingerrorforthecase=3s: 132 6-2Controlinputforthecase=3s: 132 6-3Parameterestimatesforthecase=3s: 133 6-4Trackingerrorforthecase=10s: 133 6-5Controlinputforthecase=10s: 134 6-6Parameterestimatesforthecase=10s: 134 10
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Neuromuscularelectricalstimulation(NMES)istheapplicationofapotentialeldacrossamuscleinordertoproduceadesiredmusclecontraction.NMESisapromisingtreatmentthathasthepotentialtorestorefunctionaltasksinpersonswithmovementdisorders.Towardsthisgoal,theresearchobjectiveinthedissertationistodevelopNMEScontrollersthatwillenableaperson'slowershanktotrackacontinuousdesiredtrajectory(orconstantsetpoint). AnonlinearmusculoskeletalmodelisdevelopedinChapter 2 whichdescribesmuscleactivationandcontractiondynamicsandbodysegmentaldynamicsduringNMES.Thedenitionsofvariouscomponentsinthemusculoskeletaldynamicsareprovidedbutarenotrequiredforcontrolimplementation.Instead,thestructureoftherelationshipsisusedtodenepropertiesandmakeassumptionsforcontroldevelopment. AnonlinearcontrolmethodisdevelopedinChapter 3 tocontrolthehumanquadricepsfemorismuscleundergoingnon-isometriccontractions.Thedevelopedcontrollerdoesnotrequireamusclemodelandcanbeproventoyieldasymptoticstabilityforanonlinearmusclemodelinthepresenceofboundednonlineardisturbances.Theperformanceofthecontrollerisdemonstratedthroughaseriesofclosed-loopexperimentsonhealthynormalvolunteers.Theexperimentsillustratetheabilityofthecontrollertoenabletheshanktofollowtrajectorieswithdierentperiodsandrangesofmotion,andalsotrackdesiredstepchangeswithchangingloads. 11
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3 focusontheuseofaNNfeedforwardcontrollerthatisaugmentedwithacontinuousrobustfeedbacktermtoyieldanasymptoticresult(inlieuoftypicaluniformlyultimatelybounded(UUB)stability).Specically,aNN-basedcontrollerandLyapunov-basedstabilityanalysisareprovidedtoenablesemi-globalasymptotictrackingofadesiredtime-varyinglimbtrajectory(i.e.,non-isometriccontractions).TheaddedvalueofincorporatingaNNfeedforwardtermisillustratedthroughexperimentsonhealthynormalvolunteersthatcomparethedevelopedcontrollerwiththepureRISE-basedfeedbackcontroller. ApervasiveproblemwithcurrentNMEStechnologyistherapidonsetoftheunavoidablemusclefatigueduringNMES.Inclosed-loopNMEScontrol,disturbancessuchasmusclefatigueareoftentackledthroughhigh-gainfeedbackwhichcanoverstimulatethemusclewhichfurtherintensiesthefatigueonset.InChapter 4 ,aNMEScontrollerisdevelopedthatincorporatestheeectsofmusclefatiguethroughanuncertainfunctionofthecalciumdynamics.ANN-basedestimateofthefatiguemodelmismatchisincorporatedinanonlinearcontrollerthroughabacksteppingmethodtocontrolthehumanquadricepsfemorismuscleundergoingnon-isometriccontractions.ThedevelopedcontrollerisproventoyieldUUBstabilityforanuncertainnonlinearmusclemodelinthepresenceofboundednonlineardisturbances(e.g.,spasticity,delays,changingloaddynamics).Simulationsareprovidedtoillustratetheperformanceoftheproposedcontroller.ContinuedeortswillfocusonachievingasymptotictrackingversustheUUBresult,andonvalidatingthecontrollerthroughexperiments. AnotherimpedimentinNMEScontrolisthepresenceofinputoractuatordelay.Controlofnonlinearsystemswithactuatordelayisachallengingproblembecauseoftheneedtodevelopsomeformofpredictionofthenonlineardynamics.Theproblem 12
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5 foranEuler-Lagrangesystemwithtime-delayedactuation,parametricuncertainty,andadditiveboundeddisturbances.Onecontrollerisdevelopedundertheassumptionthattheinertiaisknown,andasecondcontrollerisdevelopedwhentheinertiaisunknown.Foreachcaseapredictor-likemethodisdevelopedtoaddressthetimedelayinthecontrolinput.Lyapunov-KrasovskiifunctionalsareusedwithinaLyapunov-basedstabilityanalysistoprovesemi-globalUUBtracking.ExtensiveexperimentsshowbetterperformancecomparedtotraditionalPD=PIDcontrolleraswellasrobustnesstouncertaintyintheinertiamatrixandtimedelayvalue.Experimentsareperformedonhealthynormalindividualstoshowthefeasibility,performance,androbustnessofthedevelopedcontroller. Inadditiontoeortsfocussedoninputdelayednonlinearsystems,aparallelmotivationexiststoaddressanotherclassoftimedelayedsystemswhichconsistofnonlinearsystemswithunknownstatedelays.AcontinuousrobustadaptivecontrolmethodisdesignedinChapter 6 foraclassofuncertainnonlinearsystemswithunknownconstanttime-delaysinthestates.Specically,therobustadaptivecontrolmethod,agradient-baseddesiredcompensationadaptationlaw(DCAL),andaLyapunov-Kravoskii(LK)functional-baseddelaycontroltermareutilizedtocompensateforunknowntime-delays,linearlyparameterizableuncertainties,andadditiveboundeddisturbancesforageneralnonlinearsystem.Despitethesedisturbances,aLyapunov-basedanalysisisusedtoconcludethatthesystemoutputasymptoticallytracksadesiredtimevaryingboundedtrajectory. Chapter 7 concludesthedissertationwithadiscussionofthedevelopedcontributionsandfutureeorts. 13
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1 2 ].AlthoughmostNMESproceduresinphysicaltherapyclinicsconsistoftabulatedopen-loopapplicationofelectricalstimulation,asignicantmarketexistsforthedevelopmentofnoninvasiveclosed-loopmethods.However,theapplicationanddevelopmentofNMEScontrolhavebeenstymiedbyseveraltechnicalchallenges.Specically,duetoavarietyofuncertaintiesinmusclephysiology(e.g.,temperature,pH,andarchitecture),predictingtheexactcontractionforceexertedbythemuscleisdicult.Onecauseofthisdicultyisthatthereisanunknownmappingbetweenthegeneratedmuscleforceandstimulationparameters.Thereareadditionalproblemswithdeliveringconsistentstimulationenergytothemuscleduetoavarietyoffactorsincluding:musclefatigue,inputdelay,electrodeplacement,hyperactivesomatosensoryreexes,inter-andintra-subjectvariabilityinmuscleproperties,changingmusclegeometryundertheelectrodesinnon-isometricconditions,percentageofsubcutaneousbodyfat,overallbodyhydration,etc. Giventheuncertaintiesinthestructureofthemusclemodelandtheparametricuncertaintyforspecicmuscles,someinvestigatorshaveexploredvariouslinearPID-basedmethods(cf.[ 3 { 8 ]andthereferencestherein).Typically,theseapproacheshaveonlybeenempiricallyinvestigatedandnoanalyticalstabilityanalysishasbeendevelopedthatprovidesanindicationoftheperformance,robustnessorstabilityofthesecontrolmethods.ThedevelopmentofastabilityanalysisforpreviousPID-basedNMEScontrollershasbeenevasivebecauseofthefactthatthegoverningequationsforamusclecontraction/limb 14
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6 9 10 ]);however,thegoverningequationsaretypicallylinearizedtoaccommodateagainschedulingorlinearoptimalcontrollerapproach. MotivatedtodevelopeectiveNMEScontrolinlightofthesechallenges,therstresultinChapter 3 developsanopen-looperrorsystemforageneraluncertainnonlinearmusclemodelbasedonavailableanalyticalandempiricaldata[ 11 12 ])thatfacilitatesthedevelopmentofanewcontinuousfeedbackmethod(coinedRISEforRobustIntegraloftheSignoftheError).Throughthiserror-systemdevelopment,thecontinuousRISEcontrollerisproven(throughaLyapunov-basedstabilityanalysis)toyieldanasymptoticstabilityresultdespitetheuncertainnonlinearmusclemodelandthepresenceofadditiveboundeddisturbances(e.g.,musclespasticity,fatigue,changingloadsinfunctionaltasks,andunmodeledmusclebehavior). Seminalworkin[ 13 { 18 ]continuetoinspirenewinvestigations(cf.[ 19 { 26 ]andthereferencestherein)inneuralnetwork(NN)-basedNMEScontroldevelopment.OnemotivationforNN-basedcontrollersisthedesiretoaugmentfeedbackmethodswithanadaptiveelementthatcanadjusttotheuncertainmusclemodel,ratherthanonlyrelyingonfeedbacktodominatetheuncertaintybasedonworsecasescenarios.NN-basedcontrolmethodshaveattractedmoreattentioninNMESthanotheradaptivefeedforwardmethodsbecauseofthenatureoftheunstructureduncertaintyandtheuniversalapproximationpropertyofNNs.However,sinceNNscanonlyapproximateafunctionwithinsomeresidualapproximationerror,allpreviousNN-basedcontrollersyielduniformlyultimatelyboundedstability(i.e.,theerrorsconvergetoaregionofboundedsteady-stateerror). TheresultinthethirdsectionofChapter 3 focusesonthedevelopmentofaRISE-basedNMEScontrollerandtheassociatedanalyticalstabilityanalysisthatyieldsasymptotictrackinginthepresenceofanonlinearuncertainmusclemodelwith 15
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27 ]indicatethattheRISE-basedfeedbackstructurecanbeaugmentedwithaNNfeedforwardtermtoyieldasymptotictrackingforsomeclassesofsystems.Basedonthesegeneralresults,anextensionisprovidedinthefourthsectionofChapter 3 wheretheRISE-basedmethodismodiedwithaNNtodevelopanewNMEScontrollerfortheuncertainmusclemodel. WhileeortsinChapter 3 ,provideaninroadtothedevelopmentofanalyticalNMEScontrollersforthenonlinearmusclemodel,theseresultsdonotaccountformusclefatigue,whichisaprimaryfactortoconsidertoyieldsomefunctionalresultsinmanyrehabilitationapplications.Heuristically,musclefatigueisadecreaseinthemuscleforceoutputforagiveninputandisacomplex,multifactorialphenomenon[ 28 { 30 ].Ingeneral,someofthefactorsassociatedwiththeonsetoffatiguearefailureofexcitationofmotorneurons,impairmentofactionpotentialpropagationinthemusclemembraneandconductivityofsarcoplasmicreticulumtoCa2+ionconcentration,andthechangeinconcentrationofcatabolitesandmetabolites[ 31 ].Factorssuchasthestimulationmethod,musclebrecomposition,stateoftrainingofthemuscle,andthedurationandtasktobeperformedhavebeennoticedtoaectfatigueduringNMES.GiventheimpactoffatigueeectsduringNMES,researchershaveproposeddierentstimulationstrategies[ 30 32 33 ]todelaytheonsetoffatiguesuchaschoosingdierentstimulationpatternsandparameters,improvingfatigueresistancethroughmuscleretraining,sequentialstimulation,andsizeorderrecruitment. Controllerscanbedesignedwithsomefeedforwardknowledgetoapproximatethefatigueonsetoremploysomeassumedmathematicalmodelofthefatigueinthecontroldesign.Researchersin[ 34 { 38 ]developedvariousmathematicalmodelsforfatigue.In 16
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34 ],amusculotendonmodelforaquadricepsmuscleundergoingisometriccontractionsduringfunctionalelectricalstimulation(FES)wasproposed.ThemodelincorporatedfatiguebasedontheintracellularpHlevelwherethefatigueparametersforatypicalsubjectwerefoundthroughmetabolicinformation,experimentationandcurvetting.Amoregeneralmathematicalmodelfordynamicfatiguedenedasafunctionofnormalizedmuscleactivationvariable(Ca2+dynamics)wasproposedin[ 35 36 ].Thefatiguewasintroducedasatnessfunctionthatvariesaccordingtotheincreaseordecreaseinmuscleactivationduringelectricalstimulation.Thefatiguetimeparameterswereestimatedfromstimulationexperiments.Modelsin[ 37 ]and[ 38 ]predictforceduetotheeectofstimulationpatternsandrestingtimeswithchangingphysiologicalconditions,wheremodelparameterizationrequiredinvestigatingexperimentalforcesgeneratedfromastandardizedstimulationprotocol.Althoughthesemathematicalmodelsforfatiguepredictionarepresentinliterature,fewresearchershaveutilizedtheseassumedfatiguemodelsinclosed-loopNMEScontrol.Resultsin[ 36 ]and[ 39 ]usethefatiguemodelproposedin[ 35 ]and[ 36 ]foraFEScontroller,wherepatientspecicparameters(e.g.fatiguetimeconstants)areassumedtobeknownalongwithexactmodelknowledgeofthecalciumdynamics.ThedicultyinvolvedinthecontroldesignusingcalciumdynamicsorintracellularpHlevelisthatthesestatescannotbemeasuredeasilyforreal-timecontrol.Therefore,thesestates(calciumdynamicsorpHlevel)aremodeledasarstorsecondorderordinarydierentialequation(cf.,[ 34 36 39 ])andtheparametersintheequationsareestimatedfromexperimentationorarebasedondatafrompaststudies. ThefocusofChapter 4 istoaddressmusclefatiguebyincorporatinganuncertainfatiguemodel(i.e.,themodeldevelopedin[ 35 ])intheNMEScontroller.Theuncertainfatiguemodelisdenedasafunctionofanormalizedmuscleactivationvariable.Thenormalizedmuscleactivationvariabledenotesthecalcium(Ca2+ion)dynamicswhichactasanintermediatevariablebetweencontractilemachineryandexternalstimulus.Thecalciumdynamicsaremodeledasarstorderdierentialequationbasedon[ 6 ]and 17
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39 ].AbacksteppingapproachisutilizedtodesignvirtualcontrolinputthatconsistsofNN-basedfeedforwardsignalandfeedbacksignal.Thedevelopedcontrolleryieldsauniformlyultimatelyboundedstabilityresultgivenanunknownnonlinearmusclemodelwithuncertainfatigueandcalciumdynamics. AnothertechnicalchallengethathampersthesatisfactoryNMEScontrolperformanceiselectromechanicaldelayinmuscleforcegenerationwhichisdenedasthedierenceintimefromthearrivalofactionpotentialattheneuromuscularjunctiontothedevelopmentoftensioninthemuscle[ 8 ].InNMEScontrol,theelectromechanicaldelayismodeledasaninputdelayinthemusculoskeletaldynamics[ 6 ]andoccursduetoniteconductionvelocitiesofthechemicalionsinthemuscleinresponsetotheexternalelectricalinput[ 36 ].InputdelaycancauseperformancedegradationaswasobservedduringNMESexperimentaltrialsonvolunteersubjectswithRISEandNN+RISEcontrollersandhasalsobeenreportedtopotentiallycauseinstabilityduringhumanstanceexperimentswithNMES[ 40 ].Timedelayinthecontrolinput(alsoknownasdeadtime,orinputdelay)isapervasiveproblemincontrolapplicationsotherthanNMEScontrol.Chemicalandcombustionprocesses,teleroboticsystems,vehicleplatoons,andcommunicationnetworks[ 41 { 44 ]oftenencounterdelaysinthecontrolinput.Suchdelaysareoftenattributedtosensormeasurementdelay,transportlags,communicationdelays,ortaskprioritization,andcanleadtopoorperformanceandpotentialinstability. Motivatedbyperformanceandstabilityproblems,variousmethodshavebeendevelopedforlinearsystemswithinputdelays(cf.[ 45 { 57 ]andthereferencestherein).Asdiscussedin[ 45 46 ],anoutcomeoftheseresultsisthedevelopmentanduseofpredictiontechniquessuchasArtsteinmodelreduction[ 48 ],nitespectrumassignment[ 51 ],andcontinuouspoleplacement[ 58 ].TheconceptofpredictivecontroloriginatedfromclassicSmithpredictormethods[ 59 ].TheSmithpredictorrequiresaplantmodelforoutputpredictionandhasbeenwidelystudiedandmodiedforcontrolpurposes(cf.[ 60 { 67 ]andreferencestherein).However,theSmithpredictordoesnotprovidegoodclosed-loop 18
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42 46 ].ContrarytotheSmithpredictor,nitespectrumassignmentorArtsteinmodelreductiontechniquesandtheirextensions[ 47 { 53 68 { 71 ]canbeappliedtounstableormultivariablelinearplants.Thesepredictor-basedmethodsutilizeniteintegralsoverpastcontrolvaluestoreducethedelayedsystemtoadelayfreesystem. Anotherapproachtodeveloppredictivecontrollersisbasedonthefactthatinputdelaysystemscanberepresentedbyhyperbolicpartialdierentialequations(cf.[ 45 46 ]andreferencestherein).Thisfactisexploitedin[ 54 { 57 ]todesigncontrollersforactuatordelayedlinearsystems.Thesenovelmethodsmodelthetimedelayedsystemasanordinarydierentialequation(ODE)-partialdierentialequation(PDE)cascadewherethenon-delayedinputactsatthePDEboundary.ThecontrolleristhendesignedbyemployingabacksteppingtypeapproachforPDEcontrol[ 72 ]. Predictortechniqueshavealsobeenextendedtoadaptivecontrolofunknownlinearplantsin[ 41 56 73 ].In[ 41 73 ]thecontrollerutilizesamodiedSmithpredictortypestructuretoachieveasemi-globalresult.In[ 56 ](andthecompanionpaper[ 55 ]),aglobaladaptivecontrollerisdevelopedthatcompensatesforuncertainplantparametersandapossiblylargeunknowndelay. Incomparisontoinputdelayedlinearsystems,fewerresultsareavailablefornonlinearsystems.Approachesforinputdelayednonlinearsystemssuchas[ 74 75 ]utilizeaSmithpredictor-basedgloballylinearizingcontrolmethodandrequireaknownnonlinearplantmodelfortimedelaycompensation.In[ 42 ],aspecictechniqueisdevelopedforateleroboticsystemwithconstantinputandfeedbackdelayswhereaSmithpredictorforalocallylinearizedsubsystemisusedincombinationwithaneuralnetworkcontrollerforaremotelylocateduncertainnonlinearplant.In[ 76 ],anapproachtoconstructLyapunov-Krasovskii(LK)functionalsforinputdelayednonlinearsysteminfeedbackformisprovided,andthecontrolmethodin[ 77 ]utilizesacompositeLyapunovfunctioncontaininganintegralcrosstermandLKfunctionalforstabilizingnonlinearcascade 19
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78 ]fornonlinearnite-dimensionalcontrolsystemsinpresenceofsmallinputdelaysbyutilizingaRazumikhin-typetheorem.In[ 79 ],thebacksteppingapproachthatutilizesODE-PDEcascadetransformationforinputdelayedsystemsisextendedtoascalarnonlinearsystemwithactuatordelayofunrestrictedlength.However,tothebestofourknowledge,noattempthasbeenmadetowardsstabilizinganinputdelayednonlinearsystemwithparametricuncertaintyand/oradditiveboundeddisturbances. MotivatedbythelackofNMEScontrollersthatcompensateforinputdelayandthedesiretodevelopnon-modelbasedcontrollersfornonlinearsystemswithinputdelayChapter 5 focusesonthedevelopmentofatrackingcontrollerforanuncertainnonlinearEuler-Lagrangesystemwithinputdelay.Theinputtimedelayisassumedtobeaknownconstantandcanbearbitrarylarge.Thedynamicsareassumedtocontainparametricuncertaintyandadditiveboundeddisturbances.Therstdevelopedcontrollerisbasedontheassumptionthatthemassinertiaisknown,whereasthesecondcontrollerisbasedontheassumptionthatthemassinertiaisunknown.Thekeycontributionsofthiseortisthedesignofadelaycompensatingauxiliarysignaltoobtainatimedelayfreeopen-looperrorsystemandtheconstructionofLKfunctionalstocanceltimedelayedterms.Theauxiliarysignalleadstothedevelopmentofapredictor-basedcontrollerthatcontainsaniteintegralofpastcontrolvalues.ThisdelayedstatetodelayfreetransformationisanalogoustotheArtsteinmodelreductionapproach,whereasimilarpredictor-basedcontrolisobtained.LKfunctionalscontainingniteintegralsofcontrolinputvaluesareusedinaLyapunov-basedanalysisthatprovesthetrackingerrorsaresemi-globaluniformlyultimatelybounded. Anotherclassoftime-delayedsystemswhicharealsoendemictoengineeringsystemsandcancausedegradedcontrolperformanceandmakeclosed-loopstabilizationdicultaresystemswithstatedelays.Intime-delayedsystems,thedynamicsnotonlydepends 20
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46 80 ].)AdesireparalleltoNMESresearchexistedtoaddressthisclassoftimedelaysystems.Variouscontrollershavebeendevelopedtoaddresstime-delayinducedperformanceandstabilityissuesasdescribedinthesurveypapers[ 45 46 ]andinrecentresultsthattargetcontrolofuncertainsystemswithstatedelays(cf.[ 80 { 86 ]andreferencestherein).Controlsynthesisandstabilityanalysismethodsfornonlineartime-delayedsystemsareoftenbasedonLyapunovtechniquesinconjunctionwithaLyapunov-Kravoskii(LK)functional(cf.[ 82 83 85 87 ]).Forexample,in[ 82 ],aniterativeprocedureutilizingLKfunctionalsforrobuststabilizationofaclassofnonlinearsystemswithtriangularstructureisdeveloped.However,asstatedin[ 88 ],thecontrollercannotbeconstructedfromthegiveniterativeprocedure.Semi-globaluniformlyultimatelybounded(SUUB)resultshavebeendevelopedfortime-delayednonlinearsystems[ 83 85 ]byutilizingneuralnetwork-basedcontrol,whereappropriateLKfunctionalsareutilizedtoremovetimedelayedstates.Adiscontinuousadaptivecontrollerwasrecentlydevelopedin[ 87 ]foranonlinearsystemwithanunknowntimedelaytoachieveaUUBresultwiththeaidofLKfunctionals.However,controllersdesignedin[ 83 87 ]canbecomesingularwhenthecontrolledstatereacheszeroandanadhoccontrolstrategyisproposedtoovercometheproblem.Moreover,asstatedin[ 89 ]and[ 90 ],thecontroldesignproceduredescribedin[ 85 ]cannotbegeneralizedfornthordernonlinearsystems. Slidingmodecontrol(SMC)hasalsobeenutilizedfortimedelayedsystemsin[ 80 91 { 94 ].However,utilizingSMCstillposesachallengingdesignandcomputationproblemwhendelaysarepresentinstates[ 45 46 ].Moreover,thediscontinuoussignfunctionpresentinSMCcontrolleroftengivesrisetotheundesirablechatteringphenomenonduringpracticalapplications.ToovercomethelimitationsofdiscontinuityinSMC,acontinuousadaptiveslidingmodestrategyisdesignedin[ 95 ]fornonlinearplantswith 21
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ThedevelopmentinChapter 6 ismotivatedbythelackofcontinuousrobustcontrollersthatcanachieveasymptoticstabilityforaclassofuncertaintime-delayednonlinearsystemswithadditiveboundeddisturbances.Theapproachdescribedinthecurrenteortusesacontinuousimplicitlearning[ 96 ]basedRobustIntegraloftheSignoftheError(RISE)structure[ 11 27 ].Duetotheaddedbenetofreducedcontroleortandimprovedcontrolperformance,anadaptivecontrollerinconjunctionwithRISEfeedbackstructureisdesigned.However,sincethetimedelayvalueisnotalwaysknown,itbecomeschallengingtodesignadelayfreeadaptivecontrollaw.Throughtheuseofadesiredcompensationadaptivelaw(DCAL)basedtechniqueandsegregatingtheappropriatetermsintheopenlooperrorsystem,thedependenceofparameterestimatelawsonthetimedelayedunknownregressionmatrixisremoved.Contrarytopreviousresults,thereisnosingularityinthedevelopedcontroller.ALyapunov-basedstabilityanalysisisprovidedthatusesanLKfunctionalalongwithYoung'sinequalitytoremovetimedelayedtermsandachievesasymptotictracking. 1. Chapter3,NonlinearNeuromuscularElectricalStimulationTrackingControlofaHumanLimb:Thecontributionofthischapteristoillustratehowarecentlydevelopedcontinuousfeedbackmethodcalledrobustintegralofsignumoftheerror(coinedasRISE)canbeappliedforNMESsystems.ThemusclemodeldevelopedinChapter 2 isrewritteninaformthatadherestoRISE-basedLyapunovstabilityanalysis.Throughthiserror-systemdevelopment,thecontinuousRISEcontrollerisproven(throughaLyapunov-basedstabilityanalysis)toyieldanasymptoticstabilityresultdespitetheuncertainnonlinearmusclemodelandthepresence 22
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ThesecondresultinthechapterfocussesonblendingNN-basedfeedforwardtechniquewithRISEbasedfeedbackmethodwhichwasshowntoyieldasymptotictrackinginthepresenceofanonlinearuncertainmusclemodelwithnonvanishingadditivedisturbances.Therstresultusesfeedbackandanimplicitlearningmechanismtodominateuncertaintyanddisturbances.Recentresultsfromgeneralcontrolsystemsliterature[ 27 ]indicatethattheRISE-basedfeedbackstructurecanbeaugmentedwithaNNfeedforwardtermtoyieldasymptotictrackingforsomeclassesofsystems.Basedonthesegeneralresults,theRISE-basedmethodismodiedwithamultilayeredNNtodevelopanewNMEScontrollerfortheuncertainmusclemodel.TheexperimentalresultsindicatethattheadditionoftheNNreducestherootmeansquared(RMS)trackingerrorforsimilarstimulationeortwhencomparedtotherstmethoddevelopedinthechapter(RISEmethodwithouttheNNfeedforwardcomponent).AdditionalexperimentswereconductedtodepictthattheNN-basedfeedforwardtechniqueholdspromiseinclinical-typetasks.Specically,apreliminarysit-to-standexperimentwasperformedtoshowcontroller'sfeasibilityforanyfunctionaltask. 2. Chapter4,NonlinearControlofNMES:IncorporatingFatigueandCalciumDy-namicsAnopen-looperrorsystemforanuncertainnonlinearmusclemodelisdevelopedthatincludesthefatigueandcalciumdynamics.AvirtualcontrolinputisdesignedusingnonlinearbacksteppingtechniquewhichiscomposedofaNNbasedfeedforwardsignalandanerrorbasedfeedbacksignal.TheNNbasedcontrolstructureisexploitednotonlytofeedforwardmuscledynamicsbutalsotoapproximatetheerrorgeneratedduetoparametricuncertaintiesintheassumedfatiguemodel.Theactualexternalcontrolinput(appliedvoltage)isdesignedbasedonthebacksteppingerror.Throughthiserror-systemdevelopment,thecontinuousNNbasedcontrollerisproven(throughaLyapunov-basedstabilityanalysis)toyieldanuniformlyultimatelyboundedstabilityresultdespitetheuncertainnonlinearmusclemodelandthepresenceofadditiveboundeddisturbances(e.g.,musclespasticity,changingloadsinfunctionaltasks,anddelays). 3. Chapter5,Predictor-BasedControlforanUncertainEuler-LagrangeSystemwithInputDelayThischapterfocusesonthedevelopmentofatrackingcontrollerfor 23
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4. Chapter6,RISE-BasedAdaptiveControlofanUncertainNonlinearSystemwithUnknownStateDelaysThedevelopmentinthischapterismotivatedbythelackofcontinuousrobustcontrollersthatcanachieveasymptoticstabilityforaclassofuncertaintime-delayednonlinearsystemswithadditiveboundeddisturbances.Theapproachdescribedinthecurrenteortusesacontinuousimplicitlearning[ 96 ]basedRobustIntegraloftheSignoftheError(RISE)structure[ 11 27 ].Duetotheaddedbenetofreducedcontroleortandimprovedcontrolperformance,anadaptivecontrollerinconjunctionwithRISEfeedbackstructureisdesigned.However,sincethetimedelayvalueisnotalwaysknown,itbecomeschallengingtodesignadelayfreeadaptivecontrollaw.Throughtheuseofadesiredcompensationadaptivelaw(DCAL)basedtechniqueandsegregatingtheappropriatetermsintheopenlooperrorsystem,thedependenceofparameterestimatelawsonthetimedelayedunknownregressionmatrixisremoved.Contrarytopreviousresults,thereisnosingularityinthedevelopedcontroller.ALyapunov-basedstabilityanalysisisprovidedthatusesanLKfunctionalalongwithYoung'sinequalitytoremovetimedelayedtermsandachievesasymptotictracking. 24
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Thefollowingmodeldevelopmentrepresentsthemusculoskeletaldynamicsduringneuromuscularelectricalstimulationperformedonhumanquadricepsmuscle.Themodelsimulateslimbdynamicswhenexternalvoltageisappliedonthemuscle.Thetotalmusclekneejointmodelcanbecategorizedintobodysegmentaldynamicsandmuscleactivationandcontractiondynamics.Themuscleactivationandcontractiondynamicsexplainstheforcegenerationinthemusclewhilethebodysegmentaldynamicsconsiderstheactivemomentandpassivejointmoments. Thetotalknee-jointdynamicscanbemodeledas[ 6 ] 2{1 ),MI(q)2Rdenotestheinertialeectsoftheshank-footcomplexaboutthe Figure2-1.Muscleactivationandlimbmodel.Theforcegeneratingcontractionandactivationdynamicsinthemuscleisdenotedbyanunknownnonlinearfunction(q;_q)2Rinthedynamics.ThedetailedcontractionandactivationdynamicsincludingfatigueandcalciumdynamicsareintroducedinChapter 4 25
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97 ],d(t)2Risconsideredasanunknownboundeddisturbancewhichrepresentsanunmodeledreexactivationofthemuscle(e.g.,musclespasticity)andotherunknownunmodeledphenomena(e.g.,dynamicfatigue,electromechanicaldelays),and(t)2Rdenotesthetorqueproducedatthekneejoint.Inthesubsequentdevelopment,theunknowndisturbanced(t)isassumedtobeboundedanditsrstandsecondtimederivativesareassumedtoexistandbebounded.Thesearereasonableassumptionsfortypicaldisturbancessuchasmusclespasticity,fatigue,andloadchangesduringfunctionaltasks.Forsimplicity,thepassivedampingandelasticforceofmuscleandjointsareconsideredtogether.Theinertialandgravitationaleectsin( 2{1 )canbemodeledas whereq(t),_q(t),q(t)2Rdenotetheangularposition,velocity,andaccelerationofthelowershankabouttheknee-joint,respectively(seeFig. 2-2 ),J2Rdenotestheunknowninertiaofthecombinedshankandfoot,m2Rdenotestheunknowncombinedmassoftheshankandfoot,l2Ristheunknowndistancebetweentheknee-jointandthelumpedcenterofmassoftheshankandfoot,andg2Rdenotesthegravitationalacceleration.TheelasticeectsaremodeledontheempiricalndingsbyFerrarinandPedottiin[ 97 ]as wherek1,k2,k32Rareunknownpositivecoecients.Asshownin[ 6 ],theviscousmomentMv(_q)canbemodelledas whereB1,B2,andB32Rareunknownpositiveconstants. 26
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ThetorqueproducedaboutthekneeiscontrolledthroughmuscleforcesthatareelicitedbyNMES.Forsimplicity(andwithoutlossofgenerality),thedevelopmentinthischapterfocusesonproducingkneetorquethroughmuscletendonforces,denotedbyFT(t)2R,generatedbyelectricalstimulationofthequadriceps(i.e.,antagonisticmuscleforcesarenotconsidered).Thekneetorqueisrelatedtothemuscletendonforceas where(q(t))2Rdenotesapositivemomentarmthatchangeswiththeextensionandexionofthelegasshowninstudiesby[ 98 ]and[ 99 ].ThetendonforceFT(t)in( 2{5 )isdenedas wherea(q(t))isdenedasthepennationanglebetweenthetendonandthemuscle.Thepennationangleofhumanquadricepsmusclechangesmonotonicallyduringquadricepscontractionandisacontinuouslydierentiable,positive,monotonic,andboundedfunctionwithaboundedrsttimederivative[ 100 ].Therelationshipbetweenmuscleforceandappliedvoltageisdenotedbytheunknownfunction(q;_q)2Ras 27
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101 102 ],orpulsewidth)iswellestablished.Theempiricaldatain[ 101 ]and[ 102 ]indicatesthefunction(q;_q)isacontinuouslydierentiable,non-zero,positive,monotonic,andboundedfunction,anditsrsttimederivativeisbounded. Thetotalforcegeneratedatthetendoncouldbeconsideredasthesumofforcesgeneratedbyanactiveelement(oftendenotedbyFCE),thetensiongeneratedbyapassiveelasticelement(oftendenotedbyFPE),andforcesgeneratedbyviscousuids(oftendenotedbyFVE).Theseforceshavedynamiccharacteristics.Forexample,thepassiveelementincreaseswithincreasingmusclelength,andthemusclestinesshasbeenreportedtochangebygreaterthantwoordersofmagnitude[ 34 ]underdynamiccontractions.Themusclemodelinthechapterconsidersthetotalmuscleforcecomposedofthesumoftheseelementsasthefunctionofanunknownnonlinearfunction(q;_q)andanappliedvoltageV(t):Theintroductionoftheunknownnonlinearfunction(q;_q)enablesthemusclecontractiontobeconsideredundergeneraldynamicconditionsinthesubsequentcontroldevelopment.Expressingthemusclecontractionforcesinthismannerenablesthedevelopmentofacontrolmethodthatisrobusttochangesintheforces,becausetheseeectsareincludedintheuncertainnonlinearmusclemodelthatisincorporatedinthestabilityanalysis.Themodeldevelopedin( 2{1 )-( 2{7 )isusedtoexaminethestabilityofthesubsequentlydevelopedcontroller,butthecontrollerdoesnotexplicitlydependonthesemodels.Thefollowingassumptionsareusedtofacilitatethesubsequentcontroldevelopmentandstabilityanalysis. 98 99 ]whosersttwotimederivativesexist,andbasedontheempiricaldata[ 101 102 ],thefunction(q;_q)isassumedtobeanon-zero,positive,andboundedfunctionwithaboundedrstandsecondtimederivatives. 28
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=cosa;(2{8) wheretherstandsecondtimederivativesof(q;_q)areassumedtoexistandbebounded(seeAssumption1). 29
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11 12 ])anditsextensionthroughcombiningNN-basedfeedforwardmethod.Throughthiserror-systemdevelopment,thecontinuousRISEcontrolleranditsmodicationisproven(throughaLyapunov-basedstabilityanalysis)toyieldanasymptoticstabilityresultdespitetheuncertainnonlinearmusclemodelandthepresenceofadditiveboundeddisturbances(e.g.,musclespasticity,fatigue,changingloadsinfunctionaltasks).Theperformanceofthetwononlinearcontrollersisexperimentallyveriedforhumanlegtrackingbyapplyingthecontrollerasavoltagepotentialacrossexternalelectrodesattachedtothedistal-medialandproximal-lateralportionofthequadricepsfemorismusclegroup.TheRISEandNN+RISEcontrollersareimplementedbyavoltagemodulationschemewithaxedfrequencyandaxedpulsewidth.Othermodulationstrategies(e.g.,frequencyorpulse-widthmodulation)couldhavealsobeenimplemented(andappliedtootherskeletalmusclegroups)withoutlossofgenerality. ThirdsectionofthechapterdiscussesthedevelopmentofRISEcontrollerforuncertainnonlinearmusclemodel.Theexperimentsillustratetheabilityofthecontrollertoenablethelegshanktotracksingleandmultipleperiodtrajectorieswithdierentperiodsandrangesofmotion,andalsotrackdesiredstepchangeswithchangingloads.InfourthsectiontheRISE-basedmethodismodiedwithaNNtodevelopanewNMEScontrollerfortheuncertainmusclemodel.TheexperimentalresultsindicatethattheadditionoftheNNreducestherootmeansquared(RMS)trackingerrorforsimilarstimulationeortwhencomparedtotherstresult(RISEmethodwithouttheNN 30
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Toquantifytheobjective,apositiontrackingerror,denotedbye1(t)2R,isdenedas whereqd(t)isanaprioritrajectorywhichisdesignedsuchthatqd(t),qid(t)2L1,whereqid(t)denotestheithderivativefori=1;2;3;4.Tofacilitatethesubsequentanalysis,lteredtrackingerrors,denotedbye2(t)andr(t)2R,aredenedas 31
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3{3 )byJandutilizingtheexpressionsin( 2{1 )and( 2{5 )-( 3{2 ),thefollowingexpressioncanbeobtained: whereW(_e1;e2;t)2Risanauxiliarysignaldenedas andthecontinuous,positive,monotonic,andboundedauxiliaryfunction(q;t)2Risdenedin( 2{8 ).Aftermultiplying( 3{4 )by1(q;t)2R,thefollowingexpressionisobtained: whereJ(q;t)2R;d(q;t)2R,andW(_e1;e2;t)2Raredenedas Tofacilitatethesubsequentstabilityanalysis,theopen-looperrorsystemfor( 3{6 )canbedeterminedas 2_Jr+N_Ve2;(3{9) whereN(e1;e2;r;t)2Rdenotestheunmeasurableauxiliaryterm 2_Jr+_d(q;t):(3{10) 32
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Afteraddingandsubtracting( 3{11 )to( 3{9 ),theopen-looperrorsystemcanbeexpressedas 2_Jr;(3{12) wheretheunmeasurableauxiliaryterm~N(e1;e2;r;t)2Risdenedas ~N(t)=NNd:(3{13) Motivationforexpressingtheopen-looperrorsystemasin( 3{12 )isgivenbythedesiretosegregatetheuncertainnonlinearitiesanddisturbancesfromthemodelintotermsthatareboundedbystate-dependentboundsandtermsthatareupperboundedbyconstants.Specically,theMeanValueTheoremcanbeappliedtoupperbound~N(e1;e2;r;t)bystate-dependenttermsas wherez(t)2R3isdenedas andtheboundingfunction(kzk)isapositive,globallyinvertible,nondecreasingfunction.Thefactthatqd(t),qid(t)2L18i=1;2;3;4canbeusedtoupperboundNd(qd;_qd;qd;...qd;t)as whereNdand_Nd2Rareknownpositiveconstants. 33
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2{1 )-( 2{7 ),theRISE-basedvoltagecontrolinputV(t)isdesignedas whereks2Rdenotespositiveconstantadjustablecontrolgain,and(t)2Risthegeneralizedsolutionto _(t)=(ks+1)2e2(t)+sgn(e2(t));(0)=0;(3{18) where2Rdenotespositiveconstantadjustablecontrolgain,andsgn()denotesthesignumfunction.Althoughthecontrolinputispresentintheopen-looperrorsystemin( 3{4 ),anextraderivativeisusedtodeveloptheopen-looperrorsystemin( 3{12 )tofacilitatethedesignoftheRISE-basedcontroller.Specically,thetime-derivativeoftheRISEinputin( 3{17 )lookslikeadiscontinuousslidingmodecontroller.Slidingmodecontrolisdesirablebecauseitisamethodthatcanbeusedtorejecttheadditiveboundeddisturbancespresentinthemuscledynamics(e.g.,musclespasticity,loadchanges,electromechanicaldelays)whilestillobtaininganasymptoticstabilityresult.Thedisadvantageofaslidingmodecontrolleristhatitisdiscontinuous.Bystructuringtheopen-looperrorsystemasin( 3{12 ),theRISEcontrollerin( 3{17 )canbeimplementedasacontinuouscontroller(i.e.,theuniqueintegralofthesignoftheerror)andstillyieldanasymptoticstabilityresult.Withoutlossofgenerality,thedevelopedvoltagecontrolinputcanbeimplementedthroughvariousmodulationmethods(i.e.,voltage,frequency,orpulsewidthmodulation). Theorem1. 3{17 )ensuresthatallsystemsignalsareboundedunderclosed-loopoperation.Thepositiontrackingerrorisregulatedinthesensethat 34
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3{17 )isselectedsucientlylarge,andisselectedaccordingtothefollowingsucientcondition: 3{16 ). 1 andtheauxiliaryfunctionP(t)2Risthegeneralizedsolutiontothedierentialequation _P(t)=L(t);P(0)=je2(0)je2(0)Nd(0):(3{22) TheauxiliaryfunctionL(t)2Rin( 3{22 )isdenedas ProvidedthesucientconditionsstatedinTheorem 1 aresatised,thenP(t)0. LetVL(y;t):D[0;1)!RdenoteaLipschitzcontinuousregularpositivedenitefunctionaldenedas 2eT2e2+1 2rTJr+P;(3{24) whichsatisfytheinequalities providedthesucientconditionintroducedTheorem 1 issatised,whereU1(y);U2(y)2Rarecontinuous,positivedenitefunctions.Aftertakingthetimederivativeof( 3{24 ),_VL(y;t)canbeexpressedas _VL(y;t),2e1_e1+1 2e2_e2+Jr_r+1 2_Jr2+_P:(3{26) 35
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3{2 ),( 3{3 ),( 3{12 ),( 3{22 ),and( 3{23 ),someofthedierentialequationsdescribingtheclosed-loopsystemforwhichthestabilityanalysisisbeingperformedhavediscontinuousright-handsidesas _e1=e21e1; _e2=r2e2; 2_Jr+~N+Nde2(ks+1)rsgn(e2); _P(t)=r(Nd(t)sgn(e2)): Letf(y;t)2R4denotetherighthandsideof( 3{27 ).Sincethesubsequentanalysisrequiresthatasolutionexistsfor_y=f(y;t),itisimportanttoshowtheexistenceofthesolutionto( 3{27 ).Asdescribedin[ 103 { 106 ],theexistenceofFilippov'sgeneralizedsolutioncanbeestablishedfor( 3{27 ).First,notethatf(y;t)iscontinuousexceptinthesetf(y;t)je2=0g.From[ 103 { 106 ],anabsolutecontinuousFilippovsolutiony(t)existsalmosteverywhere(a.e.)sothat _y2K[f](y;t)a:e:(3{28) Exceptthepointsonthediscontinuoussurfacef(y;t)je2=0g,theFilippovset-valuedmapincludesuniquesolution.UnderFilippov'sframework,ageneralizedLyapunovstabilitytheorycanbeused(see[ 106 { 109 ]forfurtherdetails)toestablishstrongstabilityoftheclosed-loopsystem.Thegeneralizedtimederivativeof( 3{24 )existsa.e.,and_VL(y;t)2a:e:~VL(y;t)where~VL=2@VL(y;t)TK_e1_e2_r1 2P1 2_P1T:=rVTLK_e1_e2_r1 2P1 2_P1T2e1e2rJ2P1 21 2_Jr2K_e1_e2_r1 2P1 2_P1T;
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107 ],andK[]isdenedas[ 108 109 ]K[f](y),\>0\N=0 3{2 ),( 3{3 ),( 3{12 ),( 3{17 ),( 3{18 ),( 3{22 ),and( 3{23 ) 2_Jr2+r~N+rNdre2(ks+1)r2rK[sgn(e2)]1 2_Jr2rNd(t)+rK[sgn(e2)]; where[ 108 ] suchthat Cancellingcommontermsandbasedonthefactthat 2e1e2ke2k2+ke1k2;(3{32) ( 3{29 )canbewrittenas Asshownin( 3{29 )-( 3{33 ),theuniqueintegralsignumtermintheRISEcontrollerisusedtocompensateforthedisturbancetermsincludedinNd(qd;_qd;qd;...qd;t),providedthecontrolgainisselectedaccordingto( 3{20 ).Using( 3{14 ),thetermr(t)~N(e1;e2;r;t),canbeupperboundedbyfollowinginequality: 37
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Completingthesquaresforthebracketedtermsin( 3{35 )yields Thefollowingexpressioncanbeobtainedfrom( 3{36 ): whereU(y)isacontinuouspositivedenitefunction,providedksisselectedsucientlylargebasedontheinitialconditionsofthesystem.Thatis,theregionofattractioncanbemadearbitrarilylargetoincludeanyinitialconditionsbyincreasingthecontrolgainks(i.e.,asemi-globaltypeofstabilityresult),andhence Basedonthedenitionofz(t)in( 3{15 ),( 3{38 )canbeusedtoshowthat 3{17 ).Thevoltagecontrollerwasimplementedthroughanamplitudemodulationschemecomposedofavariableamplitudepositivesquarewavewithaxedpulsewidthof100secandxedfrequencyof30Hz.The100secpulsewidthandthe30Hzstimulationfrequencywerechosena-prioriandrepresentparametricsettingsthatarewithintherangestypicallyreportedduringNMESstudies.Duringstimulationat100secpulsewidths,humanskeletalmuscleresponsetochangesinstimulationamplitude(force-amplituderelationship)andfrequency(force-frequencyrelationship)arehighlypredictableandthusdeemed 38
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110 ]whichshowthatasstimulationfrequencyisincreasedmuscleforceincreasestoasaturationlimit.Higherfrequenciescanbechosentogeneratemoreforceuptoasaturationlimit,butmusclestendtofatiguefasterathigherfrequencies.The30Hzpulsewaveyieldsreducedfatigueincomparisontohigherfrequenciesbutlowerfrequenciestendtoproducerippledkneemotion[ 35 110 ].Thereforestimulationfrequenciesintherangeof30-40Hzisanoptimalchoiceforconductingexternalelectricalstimulation.ThefollowingresultsindicatethattheRISEalgorithmwasabletominimizethekneeangleerrorwhiledynamicallytrackingadesiredtrajectory. Intheexperiment,bipolarself-adhesiveneuromuscularstimulationelectrodeswereplacedoverthedistal-medialandproximal-lateralportionofthequadricepsfemorismusclegroupandconnectedtothecustomstimulationcircuitry.Priortoparticipatinginthestudy,writteninformedconsentwasobtainedfromallthesubjects,asapprovedbytheInstitutionalReviewBoardattheUniversityofFlorida.Trackingexperimentsforatwoperioddesiredtrajectorywereconductedonbothlegsofvesubjects.Thesubjectsincludedtwohealthyfemalesandthreehealthymalesintheagegroupof22to26years.Theelectricalstimulationresponsesofhealthysubjectshavebeenreportedassimilartoparaplegicsubjects'responses[ 16 22 39 111 ].ThereforehealthysubjectswereusedinNMESexperimentsassubstituteforparaplegicpatientswhichwerenotavailable.AsdescribedinSection 3.3.2.2 ,theresultswereapproximatelyequalacrossthesubjects(i.e.,astandarddeviationof0.53degreesofRootMeanSquared(RMS)trackingerror). 39
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Duringtheexperimentseachsubjectwasinstructedtorelaxandtoallowthestimulationtocontrolthelimbmotion(i.e.,thesubjectswerenotsupposedtoinuencethelegmotionvoluntarilyandwerenotallowedtoobservethedesiredtrajectory).Varyingthetimeperiodandrangeofmotionmayalsohelptoreduceanypossibletrajectorylearningandanticipationbyahealthysubject.Toexperimentallyexamineifanytrajectorylearningoccurred,foursuccessivetestswereconductedonahealthysubjectwithatwominuteintervalbetweentrials.Theexperimentswereconductedfor15secondsonadualperiodtrajectoryof4and6seconds.TheresultingRMSerrorsaregiveninTable 3-1 .TheresultsinTable 3-1 illustratethattrajectorylearningbythesubjectisnotapparentsincethestandarddeviationbetweenthesuccessivetrialsis0:039degrees. RMSerror(indeg.) 4:35 2 4:28 3 4:26 4 4:29 Table3-1.TabulatedresultsindicatethatthetestsubjectwasnotlearningthedesiredtrajectorysincetheRMSerrorsarerelativelyequalforeachtrial. 3-1 ,aresummarizedinTable 3-2 .InTable 3-2 ,themaximumsteady-stateerrorisdenedasthemaximumabsolutevalueoferrorthatoccursafter4secondsofthetrial.Themaximumsteady-stateerrorsrangefrom4.25to7.55degreeswithameanof6.32degreesandastandarddeviationof1.18degrees.TheRMStrackingerrorsrangefrom2to3:47withameanRMSerrorof2.75degreesandastandarddeviationof0.53degrees.ThetrackingerrorresultsforSubjectBandthecorrespondingoutputvoltagescomputedbytheRISEmethod(priortovoltagemodulation)areshowninFig. 3-1 .The 40
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Leg RMSError Max.SteadyStateError Left 2:89 Right 2:36 Left 2:00 Right 2:35 Left 2:07 Right 2:94 Left 3:47 Right 2:89 Left 3:11 Right 3:45 2:75 0:53 Tofurtherillustratetheperformanceofthecontroller,experimentswerealsoconductedfortrajectorieswithfasterandslowerperiodsandlargerrangesofmotion.Specically,thecontroller'sperformancewastestedforadesiredtrajectorywithaconstant2secondperiod,aconstant6secondperiod,atripleperiodictrajectorywithcyclesof2,4;and6secondsandforahigherrangeofmotionof65degrees.AsindicatedinTable 3-1 ,theresultsforthetwoperiodtrajectoryyieldedsimilarresultsforallthesubjects.Hence,theseadditionaltestswereperformedonasingleindividualsimplytoillustratethecapabilitiesofthecontroller,withtheunderstandingthatsomevariationswouldbeapparentwhenimplementedondierentindividuals.TheRMStrackingerrorsandmaximumsteady-stateerrorsareprovidedinTable 3-3 .TheRMSerrorandthemaximumsteadystateerrorsarelowestforaconstant6secondperioddesiredtrajectoryandhigherforfastertrajectoriesandhigherrangeofmotion.Theseresultsareanexpectedoutcomesincetrackingmoreaggressivetrajectoriesgenerallyyieldsmoreerror.Thetripleperiodictrajectoryconsistsofamixofslowerandfasterperiod 41
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trajectories,thereforetheRMSandthemaximumsteadystateerrorsareinbetweentherespectiveerrorsobtainedformoreaggressive2secondperiodandhigherrangeofmotiondesiredtrajectories.Figs. 3-2 3-5 depicttheerrorsfortheexperimentssummarizedinTable 3-3 Additionalexperimentswerealsoconductedtoexaminetheperformanceofthecontrollerinresponsetostepchangesandchangingloads.Specically,adesiredtrajectoryofastepinputwascommandedwitha10poundloadattachedtotheLEM.Anadditional 42
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A B 2:88 6.13 Constant2sec. 4:11 10.67 Tripleperiodic(6;4;2)sec. 3:27 7.82 Tripleperiodic(6;4;2)secwithhigherrangeofmotion 5:46 12.48 Table3-3.Summarizedexperimentalresultsformultiple,higherfrequenciesandhigherrangeofmotion.Column(A)indicatesRMSerrorindegrees,andcolumn(B)indicatesmaximumsteadystateerrorindegrees. Figure3-2.Topplot:Actuallimbtrajectory(solidline)versusthedesiredtripleperiodictrajectory(dashedline).Bottomplot:Thelimbtrackingerror(desiredangleminusactualangle)ofasubjecttrackingatripleperiodicdesiredtrajectory. 10poundloadwasaddedoncethelimbstabilizedafterastepdownof15degrees.Thelimbwasagaincommandedtoperformastepresponsetoraisethelimbbackupanadditional15degreeswiththetotalloadof20pounds.TheresultsareshowninFig. 3-6 .Thesteadystateerrorwaswithin1degree.Amaximumerrorof3degreeswasobservedwhentheexternalloadwasadded.Theresultsgivesomeindicationofthecontroller's 43
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abilitytoadapttochangesinloadandstepinputsandmotivatepossiblefuturecasestudieswithneurologicallyimpairedindividualsthatexpressmusclespasticity. Foreachexperiment,thecomputedvoltageinputwasmodulatedbyaxedpulsewidthof100secandxedfrequencyof30Hz.Thestimulationfrequencywasselectedbasedonsubjectcomfortandtominimizefatigue.Duringpreliminaryexperimentswithstimulationfrequenciesof100Hz,thesubjectsfatiguedapproximatelytwotimesfasterthaninthecurrentresults.Theresultsalsoindicatethata100secpulsewidthwasacceptable,thoughfuturestudieswillinvestigatehigherpulsewidthsintherangeof300350secwhichrecruitmoreslowfatiguingmotorunits[ 110 ].Ourpreviouspreliminaryexperimentsindicatedthatlongerpulsewidths(e.g.,1msec)producedsimilareectsasadirectcurrentvoltage. 44
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TheuseoftheRISEcontrolstructureismotivatedbyitsimplicitlearningcharacteristics[ 96 ]anditsabilitytocompensateforadditivesystemdisturbancesandparametricuncertaintiesinthesystem.TheadvantageoftheRISEcontrolleristhatitdoesnotrequiremusclemodelknowledgeandguaranteesasymptoticstabilityofthenonlinearsystem.Theexperimentalresultsindicatethatthisfeedbackmethodmayhavepromiseinsomeclinicalapplications. AlthoughtheRISEcontrollerwassuccessfullyimplemented,theperformanceofthecontrollermaybeimprovedbyincludingafeedforwardcontrolstructuresuchasneuralnetworks(ablackboxfunctionapproximationtechnique)orphysiological/phenomenologicalmusclemodels.SincetheRISEcontrollerisahighgainfeedbackcontrollerthatyieldsasymptoticperformance,addingafeedforwardcontrolelementmayimprovetransientand 45
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steadystateperformanceandreducetheoverallcontroleort,therebyreducingmusclefatigue.Anotherpossibleimprovementtothecontrolleristoaccountforfatigue.Fatiguecanbereducedforshortdurationsbyselectingoptimalstimulationparameters,butfunctionalelectricalstimulation(FES)mayrequireacontrollerthatadaptswithfatiguetoyieldperformancegainsforlongertimedurations.Thereforeourfuturegoalwillbetoincludeafatiguemodelinthesystemtoenhancethecontrollerperformance. 46
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changes,andchangesintheload.Specically,theexperimentalresultsindicatedthatwithnomusclemodel(andonlyvoltageamplitudemodulation),theRISEalgorithmcoulddeterminetheappropriatestimulationvoltageforthetrackingobjective.Forthefastesttestedtrajectorythemaximumsteady-statetrackingerrorswereapproximately10degrees,whereasthemaximumsteady-stateerrorinslowertrajectorieswereaslittleasapproximately4degrees.Anadvantageofthiscontrolleristhatitcanbeappliedwithoutknowledgeofpatientspecicparameterslikelimbmassorinertia,limbcenterofgravitylocation,parametersthatmodelpassiveandelasticforceelements.Thus,itsapplicationwouldnotrequirespecicexpertiseorextensivetestingpriortouse.Thecontroldevelopmentalsoaccountsforunmodeleddisturbance(e.g.musclespasticity)thatarecommonlyobservedinclinicalpopulations.Theproposedstrategyholdspromisefor 47
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2{1 ).Duetotheuniversalapproximationproperty,NN-basedestimationmethodscanbeusedtorepresenttheunknownnonlinearmusclemodelbyathree-layerNNas[ 112 ] 48
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3{40 ),U2R(N1+1)N2andW2R(N2+1)nareboundedconstantidealweightmatricesfortherst-to-secondandsecond-to-thirdlayersrespectively,whereN1isthenumberofneuronsintheinputlayer,N2isthenumberofneuronsinthehiddenlayer,andnisthenumberofneuronsintheoutputlayer.Thesigmoidactivationfunctionin( 3{40 )isdenotedby():RN1+1!RN2+1;and(x):RN1+1!Rnisthefunctionalreconstructionerror.Theadditionalterm"1"intheinputvectorx(t)andactivationterm()allowsforthresholdstobeincludedastherstcolumnsoftheweightmatrices[ 112 ].Thus,anytuningofWandUthenincludestuningofthresholds.Basedon( 3{40 ),thetypicalthreelayerNNapproximationforf(x)isgivenas[ 112 ] ^f(x)=^WT(^UTx);(3{41) where^U2R(N1+1)N2and^W2R(N2+1)naresubsequentlydesignedestimatesoftheidealweightmatrices.Theestimatemismatchfortheidealweightmatrices,denotedby~U(t)2R(N1+1)N2and~W(t)2R(N2+1)n,aredenedas ~U=U^U;~W=W^W;(3{42) andthemismatchforthehidden-layeroutputerrorforagivenx(t),denotedby~(x)2RN2+1,isdenedas ~=^=(UTx)(^UTx):(3{43) TheNNestimatehasceratinpropertiesandassumptionsthatfacilitatethesubsequentdevelopment. 112 ] 49
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3{44 )into( 3{43 )thefollowingexpressioncanbeobtained: ~=^0~UTy+O(~UTy)2;(3{45) where^0=0(^UTy): wherekkFistheFrobeniusnormofamatrix,tr()isthetraceofamatrix.TheidealweightsinaNNarebounded,butknowledgeofthisboundisanon-standardassumptionintypicalNNliterature(althoughthisassumptionisalsousedintextbookssuchas[ 112 113 ]).Iftheidealweightsareconstrainedtostaywithinsomepredenedthreshold,thenthefunctionreconstructionerrorwillbelarger.Typically,thiswouldyieldalargerultimatesteady-statebound.Yet,inthecurrentresult,themismatchresultingfromlimitingthemagnitudeoftheweightsiscompensatedthroughtheRISEfeedbackstructure(i.e.,theRISEstructureeliminatesthedisturbanceduetothefunctionreconstructionerror). 3{3 )byJandbyutilizingtheexpressionsin( 2{1 )and( 2{5 ){( 3{2 )as 50
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2{8 ).Thedynamicsin( 3{48 )canberewrittenas wheretheauxiliaryfunctionsfd(qd;_qd;qd)2RandS(q;qd;_q;_qd;qd)2Raredenedas andJ(q;_q)2R;L(q;_q)2R,andd(q;t)2Raredenedas Theexpressionin( 3{50 )canberepresentedbyathree-layerNNas wherexd(t)2R4isdenedasxd(t)=[1qd(t)_qd(t)qd(t)].Basedontheassumptionthatthedesiredtrajectoryisbounded,thefollowinginequalitieshold whereb1;b2andb32Rareknownpositiveconstants. 51
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3{49 )andthesubsequentstabilityanalysis,thecontroltorqueinputisdesignedas[ 27 ] where^fd(t)2Risthethree-layerNNfeedforwardestimatedesignedas ^fd=^WT(^UTxd)(3{55) and(t)2RistheRISEfeedbacktermdesignedas[ 11 96 114 115 ] TheestimatesfortheNNweightsin( 3{55 )aregeneratedon-lineusingaprojectionalgorithmas where12R(N2+1)(N2+1)and22R44areconstant,positivedenite,symmetricgainmatrices.In( 3{56 ),ks2Rdenotespositiveconstantadjustablecontrolgain,and(t)2Risthegeneralizedsolutionto _(t)=(ks+1)2e2(t)+1sgn(e2(t));(0)=0;(3{58) where12Rdenotespositiveconstantadjustablecontrolgain,andsgn()denotesthesignumfunction.Theclosed-looptrackingerrorsystemcanbedevelopedbysubstituting( 3{54 )into( 3{49 )as where ~fd(xd)=fd^fd:(3{60) 52
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3{59 )canbedeterminedas AlthoughthevoltagecontrolinputV(t)ispresentintheopenlooperrorsystemin( 3{49 ),anadditionalderivativeistakentofacilitatethedesignoftheRISE-basedfeedbackcontroller.Aftersubstitutingthetimederivativeof( 3{60 )into( 3{61 )byusing( 3{52 )and( 3{55 ),theclosedloopsystemcanbeexpressedasJ_r=_Jr+WT0(UTxd)UT_xd^WT(^UTxd)^WT0(^UTxd)^UT_xd^WT0(^UTxd)^UTxd+_(xd)+_S_+_d; where0(^UTx)=d(UTx)=d(UTx)jUTx=^UTx:AfteraddingandsubtractingthetermsWT^0^VT_xd+^WT^0~VT_xdto( 3{62 ),thefollowingexpressioncanbeobtained:J_r=_Jr+^WT^0~VT_xd+~WT^0^VT_xd^WT^0~VT_xdWT^0^VT_xd 3{43 ).UsingtheNNweighttuninglawsdescribedin( 3{57 ),theexpressionin( 3{63 )canberewrittenas 2_Jr+~N+Ne2(ks+1)rsgn(e2);(3{64) wheretheunmeasurableauxiliaryterms~N(e1;e2;r;t)andN(^W;^U;xd;t)2Rgivenin( 3{64 )aredenedas ~N(t)=1 2_Jr+_S+e2proj1^0^UT_xdeT2T^^WT^0proj2_xd^0T^We2TTxd(3{65) 53
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3{66 ),Nd(q;_q;xd;_xd;t)2Risdenedas whileNB(^W;^U;xd;_xd;t)2Risdenedas whereNB1(^W;^U;xd;_xd;t)andNB2(^W;^U;xd;_xd;t)2Raredenedas and Motivationforthedenitionsin( 3{65 )-( 3{67 )arebasedontheneedtosegregatetermsthatareboundedbystate-dependentboundsandtermsthatareupperboundedbyconstantsforthedevelopmentoftheNNweightupdatelawsandthesubsequentstabilityanalysis.Theauxiliarytermin( 3{68 )isfurthersegregatedtodevelopgainconditionsinthestabilityanalysis.Basedonthesegregationoftermsin( 3{65 ),theMeanValueTheoremcanbeappliedtoupperbound~N(e1;e2;r;t)as wherez(t)2R3isdenedas andtheboundingfunction(kzk)2Risapositivegloballyinvertiblenondecreasingfunction.BasedonAssumption3inChapter 2 ,( 3{46 ),( 3{47 ),( 3{53 ),and( 3{68 )-( 3{70 ),thefollowinginequalitiescanbedeveloped[ 27 ]:kNdk1kNBk2_Nd3
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Theorem2. 3{54 )-( 3{58 )ensuresthatallsystemsignalsareboundedunderclosed-loopoperationandthatthepositiontrackingerrorisregulatedinthesensethat 3{56 )and( 3{58 )areselectedsucientlylarge. 2 wheretheauxiliaryfunctionQ(t)2Risdenedas andP(t)2Risthegeneralizedsolutiontothedierentialequation _P(t)=L(t);P(0)=1je2(0)je2(0)N(0):(3{77) Since1and2in( 3{76 )areconstant,symmetric,andpositivedenitematrices,and2>0;itisstraightforwardthatQ(t)0:TheauxiliaryfunctionL(t)2Rin( 3{77 )isdenedas where1;22Rintroducedin( 3{58 )and( 3{78 )respectively,arepositiveconstantschosenaccordingtothefollowingsucientconditions 55
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3{73 ).Providedthesucientconditionsin( 3{79 )aresatised,thenP(t)0. LetVL(y;t):D[0;1)!RdenoteaLipschitzcontinuousregularpositivedenitefunctionaldenedas 2e22+1 2Jr2+P+Q;(3{80) whichsatisestheinequalities providedthesucientconditionsin( 3{79 )aresatised,whereU1(y);U2(y)2Rarecontinuous,positivedenitefunctionsdenedas where1;22Rareknownpositivefunctionsorconstants.From( 3{2 ),( 3{3 ),( 3{64 ),( 3{77 ),( 3{78 ),andaftertakingthetimederivativeof( 3{76 ),someofthedierentialequationsdescribingtheclosed-loopsystemforwhichthestabilityanalysisisbeingperformedhavediscontinuousright-handsidesas _e1=e21e1; _e2=r2e2; 2_Jr+~N+Ne2(ks+1)rsgn(e2); _P(t)=r(NB1(t)+Nd(t)1sgn(e2))_e2NB2(t)+2e2(t)2; _Q(t)=tr2~WT11~W+tr2~UT12~U: Letf(y;t)2R5denotetherighthandsideof( 3{83 ).f(y;t)iscontinuousexceptinthesetf(y;t)je2=0g.From[ 103 { 106 ],anabsolutecontinuousFilippovsolutiony(t)existsalmosteverywhere(a.e.)sothat_y2K[f](y;t)a:e:
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3{80 )existsa.e.,and_VL(y;t)2a:e:~VL(y;t)where 2P1 2_P1 2Q1 2_Q1T =rVTLK_e1_e2_r1 2P1 2_P1 2Q1 2_Q1T;2e1e2rJ2P1 22Q1 21 2_Jr2K_e1_e2_r1 2P1 2_P1 2Q1 2_Q1T: 3{83 to 3{84 anddiscussion,seeSection 3.3.1 .Afterutilizing( 3{2 ),( 3{3 ),( 3{64 ),( 3{77 ),( 3{78 ),theexpressionin 3{84 canberewrittenas 2_Jr2+r~N+rNre2(ks+1)r2rK[sgn(e2)]1 2_Jr2rNB1rNd(t)+rK[sgn(e2)]_e2NB2(t)+2e22+tr2~WT11~W+tr2~UT12~U: Using( 3{57 ),( 3{66 ),( 3{68 ),( 3{70 ),cancellingcommonterms,andbasedonthefactthat2e1e2ke2k2+ke1k2; 3{85 )canbewrittenas Asshownin( 3{85 )-( 3{86 ),theuniqueintegralsignumtermintheRISEcontrollerisusedtocompensateforthedisturbancetermsincludedinNd(qd;_qd;qd;...qd;t)andNB1(^W;^U;xd;_xd;t);providedthecontrolgain1and2areselectedaccordingto( 3{79 ).FurtherthetermNB2(^W;^U;xd;_xd;t)ispartiallyrejectedbytheuniqueintegralsignumtermandpartiallycancelledbyadaptiveupdatelaw.Using( 3{71 ),theterm 57
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Completingthesquaresforthebracketedtermsin( 3{87 )yields Thefollowingexpressioncanbeobtainedfrom( 3{88 ): whereU(y)=ckzk2,forsomepositiveconstantc2R,isacontinuouspositivesemi-denitefunctionthatisdenedonthefollowingdomain:D4=ny2R5jkyk12p whereSDisintroducedinTheorem 2 .Theregionofattractionin( 3{90 )canbemadearbitrarilylargetoincludeanyinitialconditionsbyincreasingthecontrolgainks(i.e.,asemi-globaltypeofstabilityresult),andhence Basedonthedenitionofz(t)in( 3{72 ),( 3{91 )canbeusedtoshowthat 58
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3{54 )-( 3{58 )inexperimentswithvolunteersubjects.Theseresultswerecomparedwiththepreviousresultsin[ 116 ]thatusedtheRISEfeedbackstructurewithouttheNNfeedforwardterm.TheNMEScontrollerwasimplementedasanamplitudemodulatedvoltagecomposedofapositiverectangularpulsewithaxedwidthof400secandxedfrequencyof30Hz.TheapriorichosenstimulationparametersarewithintherangestypicallyreportedduringNMESstudies[ 110 116 ].Withoutlossofgenerality,thecontrollerisapplicabletodierentstimulationprotocols(i.e.,voltage,frequency,orpulsewidthmodulation).Thefollowingresultsindicatethatthedevelopedcontroller(henceforthdenotedasNN+RISE)wasabletominimizethekneeangleerrorwhiledynamicallytrackingadesiredtrajectory. Theobjectiveinonesetofexperimentswastoenablethekneeandlowerlegtofollowanangulartrajectory,whereas,theobjectiveofasecondsetofexperimentswastoregulatethekneeandlowerlegtoaconstantdesiredsetpoint.Anadditionalpreliminarytestwasalsoperformedtotestthecapabilityofthecontrollerforasit-to-standtask.Foreachsetofexperiments,bipolarself-adhesiveneuromuscularstimulationelectrodeswereplacedoverthedistal-medialandproximal-lateralportionofthequadricepsfemorismusclegroupofvolunteersandconnectedtocustomstimulationcircuitry.Theexperimentswereconductedonnon-impairedmaleandfemalesubjects(asinourpreviousstudyin[ 116 ])withagerangesof20to35years,withwritteninformedconsentasapprovedby 59
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16 22 39 111 ].Thevolunteerswereinstructedtorelaxasmuchaspossibleandtoallowthestimulationtocontrolthelimbmotion(i.e.,thesubjectwasnotsupposedtoinuencethelegmotionvoluntarilyandwasnotallowedtoseethedesiredtrajectory). TheNN+RISEcontrollerwasimplementedwithathreeinputlayerneurons,twenty-vehiddenlayerneurons,andoneoutputlayerneuron.Theneuralnetworkweightswereestimatedon-lineaccordingtotheadaptivealgorithmin( 3{57 ).Foreachexperiment,thecomputedvoltageinputwasmodulatedbyaxedpulsewidthof400secandxedfrequencyof30Hz.Thestimulationfrequencywasselectedbasedonsubjectcomfortandtominimizefatigue.Ninesubjects(8males,1female)wereincludedinthestudy.Thestudywasconductedfordierenttypesofdesiredtrajectoriesincluding:a1.5secondperiodictrajectory,adualperiodictrajectory(4-6second),andasteptrajectory.Forthe1.5secondperiodictrajectory,controllerswereimplementedonbothlegsoffoursubjects,whiletherestofthetestswereperformedononlyonelegoftheotherthreesubjectssincetheywerenotavailableforfurthertesting.Threesubjects(1male,1female(bothlegs);1male(oneleg))wereaskedtovolunteerforthedualperiodicdesiredtrajectorytestswhileregulationtestswereperformedononeofthelegsoftwosubjects.Eachsubjectparticipatedinonetrialpercriteria(e.g.,oneresultwasobtainedinasessionforagivendesiredtrajectory).Foreachsession,apre-trialtestwasperformedoneachvolunteertondtheappropriateinitialvoltageforthecontrollertoreducetheinitialtransienterror.Afterthepre-trialtest,theRISEcontrollerwasimplementedoneachsubjectforathirtyseconddurationanditsperformancewasrecorded.ArestperiodofveminuteswasprovidedbeforetheNN+RISEcontrollerwasimplementedforanadditionalthirtysecondduration. 60
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3-7 3-8 andaresummarizedinTable 3-4 .InTable 3-4 ,themaximumsteadystatevoltage(SSV)andmaximumsteadystateerror(SSE)aredenedasthecomputedvoltageandabsolutevalueoferrorrespectively,thatoccurafter1:5secondsofthetrial.Pairedonetailedt-tests(acrossthesubjectgroup)wereperformedwithalevelofsignicancesetat=0:05.Theresultsindicatethatthedevelopedcontrollerdemonstratestheabilityofthekneeangletotrackadesiredtrajectorywithamean(foreleventests)RMSerrorof2.92degreeswithameanmaximumsteadystateerrorof7.01degrees.CombiningtheNNwiththeRISEfeedbackstructurein[ 116 ]yields(statisticallysignicant)reducedmeanRMSerror 61
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forapproximatelythesameinputstimulus.ThemaximumsteadystatevoltagesfortheRISEandNN+RISEcontrollersrevealednostatisticaldierences.ToillustratethattheperformanceofNN+RISEcontroller(incomparisontotheRISEcontrolleralone)canbemoresignicantfordierentdesiredtrajectories,bothcontrollerswereimplementedonthreesubjects(2male,1female)withthecontrolobjectivetotrackadualperiodic(46second)desiredtrajectorywithahigherrangeofmotion.ThestimulationresultsfromtheRISEandtheNN+RISEcontrollersareshowninFigs. 3-9 and 3-10 andaresummarizedinTable 3-5 .InTable 3-5 ,themaximumSSVandSSEwereobservedafter4secondsofthetrial.TheresultsillustrateNN+RISEcontrolleryieldsreducedmeanRMSerror(acrossthegroup)andreducedmeanmaximumSSE(acrossthegroup)for 62
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Leg RMSError MaxSSE RMSVoltage[V] MaxSSV[V] RISE NNR RISE NNR RISE NNR RISE NNR Left 3:59 23:98 29:5 31 A Right 2:60 25:40 32:95 31:5 B Left 2:47 22:81 28:1 26:95 B Right 2:83 23:03 29:8 30:5 C Left 3:18 6:17 40:14 48:9 44:8 C Right 2:97 35:15 46:4 42:3 D Left 3:23 28:24 30 34:1 D Right 3:53 14:95 24:2 23:4 E Left 3:92 31:46 45 40:5 F Left 3:38 28:13 31:8 34:1 G Left 3:52 43:44 49:8 50 3:20 28:79 36:04 35:38 Std.Dev. 0:45 8:29 9:44 8:08 p-value 0:02 0:08 0:28 0:22 approximatelythesameinputstimulus.Pairedonetailedt-tests(acrossthesubjectgroup)wereperformedwithalevelofsignicancesetat=0:05.TheresultsshowthatthedierenceinmeanRMSerrorandmeanmaximumSSEwerestatisticallysignicant.ThePvalueforthemeanRMSerror(0:00043)andmeanmaximumSSE(0:0033)t-testobtainedinthecaseofdualperiodictrajectoryissmallerwhencomparedtothePvalues(0:02and0:08,respectively)obtainedforthe1.5secondtrajectory.ThisdierenceindicatestheincreasedroleoftheNNforslowertrajectories(wheretheadaptationgainscanbeincreased). Asin[ 117 ],additionalexperimentswerealsoconductedtoexaminetheperformanceoftheNN+RISEcontrollerinresponsetostepchangesandchangingloads.Specically,adesiredtrajectoryofastepinputwascommandedwitha10poundloadattachedtotheLEM.Anadditional10poundloadwasaddedoncethelimbstabilizedat15degrees.Thelimbwasagaincommandedtoperformastepresponsetoraisethelimbbackupan 63
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additional15degreeswiththetotalloadof20pounds.TheresultsfromarepresentativesubjectusingNN+RISEcontrollerareshowninFig. 3-11 .TheexperimentalresultsforthestepresponseandloadadditionaregiveninTable 3-6 .Theresultsgivesomeindicationofthecontroller'sabilitytoadapttochangesinloadandstepinputsandmotivatepossiblefuturecasestudies. ExperimentswerealsoperformedtotesttheNN+RISEcontrollerforasit-to-standtask.Thesetestswereconductedonahealthyindividualinitiallyseatedonachair(seeFig. 3-12 ).Thekneeanglewasmeasuredusingagoniometer(manufacturedbyBiometrics 64
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Ltd.)attachedtobothsidesofthesubject'sknee,wheretheinitialkneeangleissettozero(sittingposition).Thegoniometerwasinterfacedwiththecustomcomputercontrolledstimulationcircuitviaanangledisplayunit(ADU301).Theobjectivewastocontroltheangularkneetrajectorythatresultedinthevolunteerrisingfromaseatedposition,withanaldesiredangleof90(standingposition).Theerror,voltage,anddesiredversusactualkneeangleplotsareshowninFig. 3-13 .TheRMSerrorandvoltageduringthisexperimentwereobtainedas2:92and26:88V;respectively.Thenalsteadystateerrorreachedwithin0:5,themaximumtransienterrorwasobservedas8:23;
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Leg RMSError MaxSSE RMSVoltage[V] MaxSSV[V] RISE NNR RISE NNR RISE NNR RISE NNR Left 2:35 29:19 34:10 34:09 A Right 1:73 29:67 35:75 34:62 B Left 3:52 36:34 44:04 43:47 B Right 3:39 38:57 45:30 46:19 C Right 3:84 24:09 29:67 29:68 2:97 31:57 37:77 37:61 Std.Dev. 0:89 5:85 6:69 6:93 p-value 0:0033 0:43 Leg Max.SSE(afterstepinput) Max.Tran-sientError Max.Error(duringdis-turbance) Max.SSV(af-terstepinput)[Volts] Left 0:7 B Right 0:6 Table3-6.Experimentalresultsforstepresponseandchangingloads andthemaximumvoltagewasobtainedas35:1V:Thesignicanceofthesetestsistodepicttheapplicabilityofthecontrolleronclinicaltaskssuchassittostandmaneuvers.Althoughtheexperimentswereconductedonahealthyindividual,thesepreliminaryresultsshowthatthecontrollerholdspromisetoprovidesatisfactoryperformanceonpatientsinaclinical-typescenario. TheNN+RISEstructureismotivatedbythedesiretoblendaNN-basedfeedforwardmethodwithacontinuousfeedbackRISEstructuretoobtainasymptoticlimbtrackingdespiteanuncertainnonlinearmuscleresponse.TheabilityoftheneuralnetworkstolearnuncertainandunknownmuscledynamicsiscomplementedbytheabilityofRISEtocompensateforadditivesystemdisturbances(hyperactivesomatosensoryreexesthatmaybepresentinimpairedindividuals)andNNapproximationerror.AlthoughtheNN+RISEcontrollerwassuccessfullyimplementedandcomparedtoRISEcontrollerinthepresentwork,theperformanceofthecontrollermaybefurtherimprovedineortstoreducethe 66
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eectsofmusclefatigueinfuturestudies.Fatiguecanbereducedforshortdurationsbyselectingoptimalstimulationparameters,butfunctionalelectricalstimulation(FES)mayrequireacontrollerthatadaptswithfatiguetoyieldperformancegainsforlongertimedurations.Thereforeourfuturegoalwillbetoincludeafatiguemodelandincorporatingcalciumdynamicsinthemuscledynamicstoenhancethecontrollerperformance. 67
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116 ].However,severallimitationsexistintheexperimentalstudy.ThecontributionfromtheNNcomponentwasobservedtoincreasebuttheRISEcontributiondidnotdeclineproportionally.Apossiblereasonforthisobservationisthatthe1.5secondperioddesiredtrajectoryhasalargedesiredaccelerationqd(t),whichisaninputtotheNNthatcanleadtolargevoltageswingsduringthetransientstage.Toreducelargevoltagevariantsduringthetransientduetoqd(t),theupdatelawgainsarereducedincomparisontogainsthatcouldbeemployedduringlessaggressivetrajectories.Theexperimentalresultswithslowertrajectories(dualperiodic-4-6secondperiod)illustratethattheNNcomponentcanplayalargerroledependingonthetrajectory.Specically,thedualperiodictrajectoryresultsindicatethattheRMSerrorobtainedwiththeNN+RISEcontrollerislowerthantheRMSerrorobtainedwiththeRISEcontrollerwithalowerPvalue(0:00043)comparedtothePvalue(0:02)obtainedwiththe1.5secondperiodtrajectory. 68
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Sinceatrajectoryforaspecicfunctionaltaskwasnotprovided,thedesiredtrajectoryusedintherstsetofexperimentswassimplyselectedasacontinuoussinusoidwithaconstant1:5secondperiod.Thedesiredtrajectorywasarbitrarilyselected,buttheperiodofthesinusoidisinspiredbyatypicalwalkinggaittrajectory.Astheworktransitionstoapplicationswhereaspecicfunctionaltrajectoryisgenerated,thecontrol 69
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AnanalysisofRMSerrorsduringextensionandexionphaseofthelegmovementsacrossdierentsubjects,trajectories(1.5secondanddualperiodic),andbothcontrollersshowedthatthemeanRMSerrorismorewhenlegismovingupwards(extensionphase)comparedtoperiodswhenlegismovingdownwards(exionphase).At-testanalysisshowedthattheresultsarestatisticallysignicantwithpvaluesof0:00013and0:0014obtainedfromRISEandNN+RISEcontrollers,respectively.ThemeanRMSerrorsduringextensionphaseforRISEandNN+RISEcontrollerswere3:49and2:68,respectivelywhilemeanRMSerrorsduringexionphaseforRISEandNN+RISEcontrollerswere2:96and2:42;respectively.SummarizedRMSerrorsforbothphasesareshowninTable 3-7 .Anincreasederrorduringextensionphasecanbeattributedtohighercontroleortrequiredduringextension.Theperformanceduringtheextensionphasecanalsobeaggravatedbyincreasedtimedelayandmusclefatigueduetotherequirementforhighermuscleforcecomparedtotheexionphase.Thisanalysisindicatesapossibleneedforseparatecontrolstrategiesduringextensionandexionphaseofthelegmovement.Particularly,futureeortswillinvestigateahybridcontrolapproachforeachphaseofmotion. Currentlytheexperimentswereperformedonnon-impairedpersons.Infuturestudieswithimpairedindividuals,ouruntestedhypothesisisthattheaddedvalueoftheNNfeedforwardcomponentwillbeevenmorepronounced(andthatthecontrollerwillremainstable)asdisturbancesduetomorerapidfatigueandmoresensitivesomatosensoryreexesmaybepresentinimpairedindividuals.Todelaytheonsetoffatigue,dierentresearchershaveproposeddierentstimulationstrategies[ 32 33 118 ]suchaschoosingdierentstimulationpatternsandparameters.TheNMEScontrollerinthisstudywas 70
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Leg Trajectory RMSError(RISE) RMSError(NN+RISE) Extension Flexion Extension Flexion A Left Dualperiod Right Dualperiod Left Dualperiod Right Dualperiod Right Dualperiod Left 1.5second Right 1.5second Left 1.5second Right 1.5second Left 1.5second Right 1.5second 2:96 Left 1.5second Right 1.5second Left 1.5second Left 1.5second Right 1.5second 3:49 0:00013 0:0014 implementedusingconstantpulsewidthamplitudemodulationofthevoltage.However,thecontrollercanbeimplementedusingothermodulationschemessuchaspulsewidthandfrequencymodulationwithoutanyimplicationsonthestabilityanalysis,buttheeectsofusingfrequencymodulationorvaryingpulsetrains(e.g.apulsetraincontainingdoublets)remaintobeinvestigatedclinically. 71
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72
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35 ])intheNMEScontroller.Thecontributionofthemethodisthatonlybestguessestimatesofpatientspecicfatiguetimeconstantsandnaturalfrequencyofcalciumdynamicsarerequiredandthemismatchbetweentheestimatedparametersandactualparametersisincludedinastabilityanalysis.Thefatiguemodelisdenedasafunctionofanormalizedmuscleactivationvariable.Thenormalizedmuscleactivationvariabledenotesthecalcium(Ca2+ion)dynamicswhichactasanintermediatevariablebetweencontractilemachineryandexternalstimulus.Thecalciumdynamicsaremodeledasarstorderdierentialequationbasedon[ 6 ]and[ 39 ].Anopen-looperrorsystemforanuncertainnonlinearmusclemodelisdevelopedthatincludesthefatigueandcalciumdynamics.AvirtualcontrolinputisdesignedusingnonlinearbacksteppingtechniquewhichiscomposedofaNNbasedfeedforwardsignalandanerrorbasedfeedbacksignal.TheNNbasedcontrolstructureisexploitednotonlytofeedforwardmuscledynamicsbutalsotoapproximatetheerrorgeneratedduetoparametricuncertaintiesintheassumedfatiguemodel.Theactualexternalcontrolinput(appliedvoltage)isdesignedbasedonthebacksteppingerror.Throughthiserror-systemdevelopment,thecontinuousNNbasedcontrollerisproven(throughaLyapunov-basedstabilityanalysis)toyieldanuniformlyultimatelyboundedstabilityresultdespitetheuncertainnonlinearmusclemodelandthepresenceofadditiveboundeddisturbances(e.g.,musclespasticity,changingloadsinfunctionaltasks,anddelays). 2 ismodiedtoconsidercalciumandfatiguedynamicsduringneuromuscularelectricalstimulation.Theadditionaldynamics 73
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2 ThetorqueproducedaboutthekneeisgeneratedthroughmuscleforcesthatareelicitedbyNMES.TheactivemomentgeneratingforceatthekneejointisthetendonforceFT(t)2Rdenedas[ 119 ] wherea(q(t))2Risdenedasthepennationanglebetweenthetendonandthemuscle,whereq(t),_q(t)2Rdenotetheangularpositionandvelocityofthelowershankabouttheknee-joint,respectively(seeFig. 2-2 ).Thepennationangleofthehumanquadricepsmusclechangesmonotonicallyduringquadricepscontractionandisacontinuouslydierentiable,positive,monotonic,andboundedfunctionwithaboundedrsttimederivative[ 100 ].ThemuscleforceF(t)2Rin( 4{1 )isdenedas[ 36 ] whereFm2Risthemaximumisometricforcegeneratedbythemuscle.Theuncertainnonlinearfunctions1(q);2(q;_q)2Rin( 4{2 )areforce-lengthandforce-velocityrelationships,respectively,denedas[ 36 120 121 ] whereb,l(q)2Rin( 4{3 )denotetheunknownshapefactorandthenormalizedlengthwithrespecttotheoptimalmusclelength,respectively,andv(q;_q)2Risanunknownnon-negativenormalizedvelocitywithrespecttothemaximalcontractionvelocityofthemuscle,andc1;c2;c3;c4areunknown,bounded,positiveconstants. 74
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Thedenitionsin( 4{3 )and( 4{4 )arenotdirectlyusedinthecontroldevelopment.Instead,thestructureoftherelationshipsin( 4{3 )and( 4{4 )isusedtoconcludethat1(q)and2(q;_q)arecontinuouslydierentiable,non-zero,positive,monotonic,andboundedfunctions,withboundedrsttimederivatives.Themuscleforcein( 4{2 )iscoupledtotheactualexternalvoltagecontrolinputV(t)2Rthroughanintermediatenormalizedmuscleactivationvariablex(t)2R.Themuscleactivationvariableisgovernedbyfollowingdierentialequation[ 34 119 ] 2_x=wx+wsat[V(t)];(4{5) wherew2Ristheconstantnaturalfrequencyofthecalciumdynamics.Thefunctionsat[V(t)]2R(i.e.,recruitmentcurve)isdenotedbyapiecewiselinearfunctionas whereVmin2Ristheminimumvoltagerequiredtogeneratenoticeablemovementorforceproductioninamuscle,andVmax2Risthevoltageofthemuscleatwhichnoconsiderableincreaseinforceormovementisobserved.Basedon( 4{5 )and( 4{6 ),alineardierentialinequalitycanbedevelopedtoshowthatx(t)2[0;1]:Musclefatigueisincludedin( 4{2 )throughtheinvertible,positive,boundedfatiguefunction'(x)2Rthatisgeneratedfromtherstorderdierentialequation[ 35 36 ] _'=1 where'ministheunknownminimumfatigueconstantofthemuscle,andTf,Trareunknowntimeconstantsforfatigueandrecoveryinthemuscle,respectively. 75
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whereqd(t)isanaprioritrajectorywhichisdesignedsuchthatqd(t),qid(t)2L1,whereq(i)d(t)denotestheithderivativefori=1;2;3;4.Tofacilitatethesubsequentanalysis,alteredtrackingerror,denotedbyr(t);isdenedas where2Rdenotesapositiveconstant. 76
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4{9 ),multiplyingtheresultingexpressionbyJ,andthenutilizingtheexpressionsin( 4{1 ),( 4{2 ),( 2{1 ),( 2{5 )and( 4{8 )as wheretheauxiliaryfunction(q;_q)2Risdenedas Aftermultiplying( 4{10 )by1(q;_q)2R,thefollowingexpressionisobtained: whereJ(q;t);d(q;t),L(q;_q)2RaredenedasJ=1J;d=1d;L=1(Me+Mg+Mv): 4.2 ),(q;_q)iscontinuouslydierentiable,positive,monotonic,andbounded.Alsothefunction1(q;_q)isbounded.Thersttimederivativesof(q;_q)and1(q;_q)existandarebounded.TheinertiafunctionJispositivedeniteandcanbeupperandlowerboundedas wherea1;a22Raresomeknownpositiveconstants.Alsousingtheboundednessof(q;_q);_(q;_q);1(q;_q) wherej;2Raresomeknownpositiveconstants. 77
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4{7 )apositiveestimate^'(^x)isgeneratedas^'=1 ^Tf('min^')^x+1 ^Tr(1^')(1^x), (4{15)1^'(0)>0; 4{5 )as 2^x=^w^x+^wsat[V(t)];(4{16) where^w2Rdenotestheconstantbestguessestimateofnaturalfrequencyofcalciumdynamicsw:Theestimatedfunction^'(^x)isupperboundedbyapositiveconstant'2R.Specically,'canbedeterminedas '=^'(0)+1+^Tr Thealgorithmusedin( 4{15 )ensuresthat^'(^x)remainsstrictlypositive.Basedon( 4{6 )and( 4{16 ),alineardierentialinequalitycanbedevelopedtoshowthat^x(t)2[0;1]: 4{12 )toyield 2jre'~x'e^x^'^x;(4{18) wheretheauxiliaryfunctionS(q;_q;qd;e;r;^x)2Risdenedas 2jr+e~'^x(4{19) andtheerrorfunctions'e(x;^x);~'(^x);~x(t)2Raredenotedas ~'(^x)='(^x)^'(^x);(4{20) 78
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Since'(^x)and'(x)areboundedfunctions,theerrorfunction'e(t)canbeupperboundedas where'2Rissomeknownpositiveconstant.TheauxiliaryfunctionS(q;_q;qd;e;r;^x)canberepresentedbyathree-layerNNas wherey(t)2R7isdenedas and(y)isafunctionalreconstructionerrorthatisboundedbyaconstantas 4{18 ),abackstepping-basedapproachisusedtoinjectavirtualcontrolinputxd(t)2R(i.e.,desiredcalciumdynamics)as 2jre^'^x+^'xd^'xd:(4{27) Basedon( 4{27 ),thevirtualcontrolinputisdesignedasathreelayerNNfeedforwardtermplusafeedbacktermas whereks2Rdenotesapositiveconstantadjustablecontrolgain.ThefeedforwardNNcomponentin( 4{28 ),denotedby^S(t)2Risgeneratedas ^S=^WT(^UTy):(4{29) 79
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4{29 )aregeneratedon-lineusingprojectionalgorithmas[ 27 ] where12R(N2+1)(N2+1)and22R(N1+1)(N1+1)areconstant,positivedenite,symmetricgainmatrices.Theclosed-looptrackingerrorsystemcanbedevelopedbysubstituting( 4{28 )into( 4{27 )as 2jre+~S+d'~x'e^xksr^'ex;(4{31) where~S(y)2Risdenedas ~S(y)=S^S;(4{32) andex(t)2Risthebacksteppingerrordenedas TheclosedloopsystemcanbeexpressedasJ_r=1 2jre+WT(UTy)^WT(^UTy)+(y)+d'~x'e^xksr^'ex: AfteraddingandsubtractingthetermsWT^+^WT~to( 4{34 ),thefollowingexpressioncanbeobtained:J_r=1 2jre+~WT^+^WT~+~WT~+(y)+d'~x'e^xksr^'ex; wherethenotations^()and~()areintroducedin( 3{43 ).TheTaylorseriesapproximationdescribedin( 3{44 )and( 3{45 )cannowbeusedtorewrite( 4{35 )as 2jre+N+~WT^+^WT^0~UTyksr^'ex;(4{36) 80
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Basedon( 4{14 ),( 4{23 ),( 4{26 ),( 4{30 ),thefactthatx(t);^x(t)2[0;1],andtheassumptionthatdesiredtrajectoriesarebounded,thefollowinginequalitycanbedeveloped[ 122 ]: wherei2R,(i=1;2)areknownpositiveconstantsandz2R2isdenedas 4{33 )canbedeterminedbyusing( 4{16 )as _ex=^w Basedon( 4{6 )and( 4{40 ),andassumptionthatcontrolinputremainsbelowthesaturationvoltageVmax,thecontrolinput(Voltageinput)V(t)2RisdesignedasV(t)=(VmaxVmin)(^w wherek2Rdenotesapositiveconstantadjustablecontrolgain.Substituting( 4{41 )into( 4{40 ),yields _ex=^'rkex:(4{42) Theorem3. 4{28 )and( 4{41 )ensuresthatallsystemsignalsareboundedunderclosed-loopoperationandthatthepositiontrackingerrorisregulatedinthe
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4{9 ),( 4{49 ),( 4{50 )areselectedaccordingtothefollowingsucientcondition: 4{38 ). 2eTe+1 2rTJr+1 2eTxex+1 2tr(~WT11~W)+1 2tr(~UT12~U): Byusing( 4{13 )andtypicalNNproperties[ 112 ],VL(t)canbeupperandlowerboundedas where1;2;2Rareknownpositiveconstants,andX(t)2R3isdenedas Takingthetimederivativeof( 4{45 ),utilizing( 4{9 ),( 4{36 ),( 4{42 ),andcancelingsimilartermsyields_VL=eTe+rTNrTksr+rT~WT^+rT^WT^0~UTyeTxkexrT(j_J)rtr(~WT11~W)tr(~UT12~U): Using( 4{14 )and( 4{38 ),theexpressionin( 4{48 )canbeupperboundedas_VLe2ks1r2+2kzkjrj+[jrj1ks2r2]ke2x+rT~WT^+rT^WT^0~UTytr(~WT11^W)tr(~UT12^U); 82
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Completingthesquaresforthebracketedtermin( 4{49 )andusingtheupdatelawsin( 4{30 )yields _VL[min(1;ks1)2]kzk2ke2x+21 Theinequalityin( 4{46 )canbeusedtorewrite( 4{51 )as _VL 2VL+";(4{52) where"2Risapositiveconstantdenedas 2;(4{53) and2Risdenedas Thelineardierentialinequalityin( 4{52 )canbesolvedas 2t+"2 2ti:(4{55) Providedthesucientconditionin( 4{44 )issatised,theexpressionsin( 4{45 )and( 4{55 )indicatethate(t);r(t);ex(t);~W(t);~U(t)2L1.Giventhate(t);r(t);qd(t);_qd(t)2L1;( 4{8 )and( 4{9 )indicatethatq(t);_q(t)2L1:Since~W(t);~U(t)2L1;( 3{42 )andAssumption1( 3.4 )canbeusedtoconcludethat^W(t);^U(t)2L1:Basedon( 4{5 ),itcanbeshownthat^x(t)2[0;1]:Giventhatqd(t),e(t);r(t);q(t);_q(t);^x(t)2L1;theNNinputvectory(t)2L1from( 4{25 ):Sinceex(t);^x(t)2L1;( 4{33 )canbeusedtoshowthatxd(t)2L1:Giventhatr(t);^W(t);^U(t);xd(t)2L1,( 4{28 )and( 4{29 )indicatethat^S(t),^'1(t)2L1:Sincee(t);r(t);^W(t);~W(t);~U(t);ex(t)^'(t)2L1,( 4{36 )and( 4{38 )indicatethat_r(t)2L1:Asr(t);y(t);^W(t)2L1;theupdatelaws^W(t);^U(t)2L1:Since^'(t);^x(t)2L1,itcanbeshownthat^'(t)2L1:Giventhatthe^'(t);^'1(t);_r(t);
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6 36 123 ].TheRISEandtheproposedcontrolleraretestedfortwodierentdesiredtrajectories:1)slowtrajectorywith6secondperiod,2)fasttrajectorywith2secondperiod. Figure4-2.Topplotshowsthekneeangleerrorfora6secondperiodtrajectoryusingtheproposedcontroller.Middleplotshowsthepulsewidthcomputedbytheproposedcontroller.Bottomplotshowstheactuallegangle(dashedline)vsdesiredtrajectory(solidline). FromtheresultsshowninFigs. 4-2 4-8 ,itisclearthattheproposedcontrollertracksbothtimevaryingdesiredtrajectoriesbetterthantheRISEcontroller.Figs. 4-4 and 4-5 illustratetheperformanceoftheRISEcontrollerwhenimplementedonmuscledynamicswithoutincludingthefatiguedynamics.ThesteadystateerrorfromtheRISEcontrollerisbetween8fordesiredtrajectorywithperiod6seconds.ThesteadystateerrorinthecaseofRISEcontrollerincreasesto14whenfastertrajectorywithperiod2secondsisused.Fig. 4-6 depictsthatthecontrolperformancedegradeslaterintimewhenRISE 84
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controllerisimplementedonmuscledynamicswithfatiguemodelincluded.Theproposedcontrollerwasimplementedonthecompletemuscledynamicsthatincludedthefatiguedynamics.Figs. 4-2 4-3 and 4-7 showthatthesteadystateerrorinthecaseofproposedcontrollerremainswithin0:5forbothslowandfasttrajectories.Fig. 4-8 showshowthefatiguevariableevolveswithtimeasadeceasinginputgain.Theproposedcontrollerisabletocompensateforthedecreasingcontrolgain,andtheperformancedoesnotdegradeovertimeasshowninFig. 4-7 85
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performedtoproveuniformlyultimatelyboundedresultinthepresenceofboundeddisturbances(e.gmusclespasticity),parametricuncertainties.SimulationresultsclearlyillustratethattheproposedcontrollerperformsbetterintermsofreducederrorincomparisontotheRISEcontroller.However,theperformanceofthecontrolleronvolunteersorpatientsremainstobeseen.Thecontroller'sdependenceonaccelerationandmathematicalfatigueandcalciummodelshinderitsimplementationonvolunteers.Themathematicalcalciumandfatiguemodelswereincorporatedduetothefactthatthemeasurementofactualfatiguestateandcalciumvariableisdicult.Futureeortscanbemadetoincorporateanobserver-baseddesigninthecontrollerinordertoestimatethefatigueandcalciumstates. 86
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Figure4-6.RISEcontrollerwithfatigueinthedynamics:Topplotshowsthekneeangleerrorfora6secondperiodtrajectoryusingtheRISEcontroller.MiddleplotshowsthepulsewidthcomputedbytheRISEcontroller.Bottomplotshowstheactuallegangle(dashedline)vsdesiredtrajectory(solidline). 87
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Figure4-8.Fatiguevariable 88
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Theprimarymotiveofthisresearchistodevelopandimplementacontrollerthatcompensatesforelectromechanicaldelay(EMD)inNMES.ThelastsectionofthechapterfocussesoncharacterizingEMDduringNMES.ExperimentsresultsobtainedfromhealthyvolunteersareprovidedwhichdescribetheeectofstimulationparametersontheEMDduringNMES.Finally,aPDcontrollerwithanaugmentedpredictorcomponent 89
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In( 5{1 ),M(q)2Rnndenotesageneralizedinertiamatrix,Vm(q;_q)2Rnndenotesageneralizedcentripetal-Coriolismatrix,G(q)2Rndenotesageneralizedgravityvector,F(_q)2Rndenotesgeneralizedfriction,d(t)2Rndenotesanexogenousdisturbance(e.g.,unmodeledeects),u(t)2Rnrepresentsthegeneralizeddelayedinputcontrolvector,where2Risaconstanttimedelay,andq(t);_q(t);q(t)2Rndenotethegeneralizedstates.Thesubsequentdevelopmentisbasedontheassumptionsthatq(t)and_q(t)aremeasurable,Vm(q;_q);G(q);F(_q);d(t)areunknown,thetimedelayconstant2Risknown 5.3.2 ,M(q)isassumedtobeknowntoillustratethedevelopmentofaPID-likecontroller.InSection 5.3.3 ,thisassumptionisremovedandaPD-likecontrollerisdeveloped.Throughoutthepaper,atimedependentdelayedfunctionisdenotedasx(t)(orasx)andatimedependentfunction(withouttimedelay)isdenotedasx(t)(orasx):Thefollowingassumptionsareusedinthesubsequentdevelopment. 90
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wherem1;m22R+areknownconstantsandkkdenotesthestandardEuclideannorm. where1;22R+areknownconstants. 5.3.1Objective 5{1 )totrackadesiredtrajectory,denotedbyqd(t)2Rn.Toquantifytheobjective,apositiontrackingerror,denotedbye1(t)2Rn,isdenedas 91
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5{1 )toaninputdelayfreesystem,anauxiliarysignaldenotedbyr(t)2Rn;isalsodenedas where22R+denotesaconstant.Theauxiliarysignalr(t)isonlyintroducedtofacilitatethesubsequentanalysis,andisnotusedinthecontroldesignsincetheexpressionin( 5{6 )dependsontheunmeasurablegeneralizedstateq(t): 5{6 )byM(q)andutilizingtheexpressionsin( 5{1 ),( 5{4 ),and( 5{5 ),thetransformedopen-looptrackingerrorsystemcanbeexpressedinaninputdelayfreeformas Basedon( 5{7 )andthesubsequentstabilityanalysis,thecontrolinputu(t)2Rnisdesignedas whereka2R+isaknownconstantthatcanbeexpandedas tofacilitatethesubsequentstabilityanalysis,whereka1;ka22R+areknownconstants.Thecontrolleru(t)in( 5{8 )isaproportionalintegralderivative(PID)controllermodiedbyapredictorlikefeedbacktermfortimedelaycompensation.Althoughthecontrolinputu(t)ispresentintheopenlooperrorsystemin( 5{7 ),anadditionalderivativeistakentofacilitatethesubsequentstabilityanalysis.Thetimederivativeof( 5{7 )canbeexpressedas 2_M(q)r+N+_dkar;(5{10) 92
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2_M(q)r+M(q)...qd+_M(q)qd+_Vm(q;_q)_q+Vm(q;_q)q+_G(q)+_F(_q) (5{11) +(1+2)M(q)r12M(q)e221M(q)e122M(q)e2+1_M(q)_e1+2_M(q)e2(1+2)(uu)21M(q)_e1; 5{6 )isusedtowritethetimederivativeof( 5{8 )as_u=kar: 5{10 ),thefollowingexpressionisobtained: 2_M(q)r+~N+Se2kar;(5{12) wheretheauxiliaryfunctions~N(e1;e2;r;t)2RnandS(qd;_qd;qd;...qd;t)2Rnaredenedas ~N=NNd+e2;S=Nd+_d:(5{13) Sometermsintheclosed-loopdynamicsin( 5{12 )aresegregatedintoauxiliarytermsin( 5{13 )becauseofdierencesinhowthetermscanbeupperbounded.Forexample,Assumptions2;3and4,canbeusedtoupperboundS(qd;_qd;qd;...qd;t)as where"12R+isaknownconstantandtheMeanValueTheoremcanbeusedtoupperbound~N(e1;e2;r;t)as ~N1(kzk)kzk;(5{15) 93
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andtheboundingfunction1(kzk)2Risaknownpositivegloballyinvertiblenondecreasingfunction.In( 5{16 ),ez2Rnisdenedasez4=uu=Ztt_u()d; 5{8 )ensuressemi-globallyuniformlyultimatelybounded(SUUB)trackinginthesensethat 5{5 ),( 5{6 ),and( 5{8 ),respectivelyareselectedaccordingtothefollowingsucientconditions: 2;2>1+222 whereQ(t)2Risdenedas[ 45 76 ] where!2R+isaknownconstant.ApositivedeniteLyapunovfunctionalcandidateV(y;t):D[01)!Risdenedas 2eT2e2+1 2rTM(q)r+Q;(5{21) 94
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where1;22R+areknownconstantsdenedas 2min[m1;1];2=max[1 2m2;1];(5{23) wherem1andm2aredenedin( 5{2 ). Afterutilizing( 5{5 ),( 5{6 ),and( 5{12 )andcancellingthesimilarterms,thetimederivativeof( 5{21 )is _V=2eT1e221eT1e12eT2e2karTr+eT2M1(q)ez+rTS+rT~N+!k_uk2!Zttk_u()k2d;(5{24) wheretheLeibnizintegralrulewasappliedtodeterminethetimederivativeofQ(t)in( 5{20 )(seetheAppendix 7.2 ).Theexpressionin( 5{24 )canbeupperboundedbyusing( 5{3 ),( 5{14 )and( 5{15 )as _V(211)ke1k2(21)ke2k2kakrk2+2ke2kkezk +!k_uk2+"1krk+1(kzk)kzkkrk!Zttk_u()k2d: 5{25 )canbeupperboundedbyusingYoung'sinequality: where2R+isaknownconstant.Further,byusingtheCauchySchwarzinequality,thefollowingtermin( 5{26 )canbeupperboundedas 95
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2Rttk_u()k2din( 5{25 )yields _V(211)ke1k2(21)ke2k2kakrk2+2ke2kkezk+!k_uk2 +"1krk+1(kzk)kzkkrk! 2Zttk_u()k2d 2Zttk_u()k2d: 5{9 )andtheboundsgivenin( 5{26 )and( 5{27 ),theinequalityin( 5{28 )canbeupperboundedas _V(211)ke1k2(21222 2kezk2+"1krk+1(kzk)kzkkrkka2krk2ka1krk2 2Zttk_u()k2d: Aftercompletingthesquares,theinequalityin( 5{29 )canbeupperboundedas _V1kzk2 2Zttk_u()k2d+21(kzk) 4ka1kzk2+"21 where12R+isdenedas1=min(21222 2: 5{30 )canberewrittenas _V121(kzk) 4ka1kzk21 Usingthedenitionofz(t)in( 5{16 )andy(t)in( 5{19 ),theexpressionin( 5{31 )canbeexpressedas _V1kyk2121(kzk) 4ka1kezk2+"21 where1(kzk)2R+isdenedas1=min121(kzk) 4ka1;1
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5{22 ),theinequalityin( 5{32 )canbeupperboundedas _V1 ConsiderasetSdenedas InS,1(kzk)canbelowerboundedbyaconstant12R+as Basedon( 5{35 ),thelineardierentialequationin( 5{33 )canbesolvedas providedkzk112p 5{36 ),ifz(0)2Sthenkacanbechosenaccordingtothesucientconditionsin( 5{18 )(i.e.asemi-globalresult)toyieldtheresultin( 5{17 ).Basedondenitionofy(t),itcanbeconcludedthate1(t);e2(t);r(t)2L1inS.Giventhate1(t);e2(t);qd(t);_qd(t)2L1inS;( 5{4 )and( 5{5 )indicatethatq(t);_q(t)2L1inS:Sincer(t);e2(t);q(t);_q(t);_qd(t);qd(t)2L1inS,andu(t)u(t)=Rtt_u()d=kaRttr()d(byLeibnitz-Newtonformula)2L1inS,then( 5{6 )andAssumption3indicatethatq(t)2L1inS:Giventhatr(t);e2(t);q(t);_q(t);_qd(t)qd(t)2L1inS,( 5{7 )andAssumptions3and4indicatethatu(t)2L1inS. 97
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whereb2R+isaknownconstant.Tofacilitatethesubsequentstabilityanalysis,theerrorbetweenBandM1(q)isdenedby where(q)2Rnnsatisesthefollowinginequality where2R+denotesaknownconstant.Theopen-looptrackingerrorsystemcanbedevelopedbymultiplyingthetimederivativeof( 5{37 )byM(q)andutilizingtheexpressionsin( 5{1 ),( 5{4 ),and( 5{39 )toobtain Basedon( 5{41 )andthesubsequentstabilityanalysis,thecontrolinputu(t)2Rnisdesignedas wherekb2R+isaknowncontrolgainthatcanbeexpandedas tofacilitatethesubsequentanalysis,wherekb1;kb2;andkb32R+areknownconstants:AfteraddingandsubtractingtheauxiliarytermNd(qd;_qd;qd;t)2RndenedasNd=M(qd)qd+Vm(qd;_qd)_qd+G(qd)+F(_qd);
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5{37 )and( 5{42 ),theexpressionin( 5{41 )canberewrittenas 2_M(q)e2+~N+Se1kbe2kbM(q)[e2e2];(5{44) wheretheauxiliaryterms~N(e1;e2;t);N(e1;e2;t);S(qd;_qd;qd;t)2Rnaredenedas ~N=NNd;S=Nd+d;(5{45)N=1 2_M(q)e2+M(q)qd+Vm(q;_q)_q+G(q)+F(q)+M(q)e22M(q)e1+e1+M(q)BZttu()d; ~N2(kzk)kzk;kSk"2:(5{46) In( 5{46 ),"22R+isaknownconstant,theboundingfunction2(kzk)2Risapositivegloballyinvertiblenondecreasingfunction,andz2R3nisdenedas whereez2Rnisdenedasez=Zttu()d: 5{42 )ensuresSUUBtrackinginthesensethat 5{37 )and( 5{42 ),respectivelyareselectedaccordingtothesucientconditions: 5{2 ),( 5{38 ),and( 5{40 ),respectively,and;!2R+aresubsequentlydenedconstants.
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2m2:Experimentalresultsillustratetheperformance/robustnessofthedevelopedcontrollerwithrespecttothemismatchbe-tweenBandM1(q).Specically,resultsindicateaninsignicantamountofvariationintheperformanceevenwheneachelementofM1(q)isoverestimatedbyasmuchas100%.Dierentresultsmaybeobtainedfordierentsystems,buttheseresultsindicatethatthegainconditionisreasonable. whereP(t),Q(t)2RdenoteLKfunctionalsdenedas[ 45 ]P=!ZttZtsku()k2dds;Q=m2kb 2eT1e1+1 2eT2M(q)e2+P+Q;(5{51) andsatisesthefollowinginequalities where1;22R+aredenedin( 5{23 ). Takingthetimederivativeof( 5{51 )andusing( 5{37 )and( 5{44 )yields _V=eT1e1+eT1Bez+!kuk2+eT2hS+~Nkbe2kbM(q)(e2e2)i+m2kb wheretheLeibnizintegralrulewasappliedtodeterminethetimederivativeofP(t)(seetheAppendix 7.2 )andQ(t).Using( 5{2 ),( 5{38 ),and( 5{46 ),thetermsin( 5{53 )canbe 100
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_Vke1k2kbke2k2+m2kbke2k2+!kuk2+ke2k"2+ke2k2(kzk)kzk+bke1kkezk+m2kbke2kke2k+m2kb Thefollowingtermsin( 5{54 )canbeupperboundedbyutilizingYoung'sinequality: m2kbke2kke2km2kb 5{55 )canbeupperboundedas Afteraddingandsubtracting 2Rttku()k2dto( 5{54 ),andutilizing( 5{42 ),( 5{43 ),( 5{55 )and( 5{56 ),thefollowingexpressionisobtained: _V(b22 2)kezk2kb1ke2k2+2(kzk)kzkke2kkb2ke2k2+ke2k"2 2Zttku()k2d: Bycompletingthesquares,theinequalityin( 5{57 )canbeupperboundedas _V222(kzk) 4kb1kzk2 2Zttku()k2d+"22 where22R+isdenotedas2=minb22 2):
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5{58 )canrewrittenas _V222(kzk) 4kb1kzk2 22ZttZtsku()k2d+"22 Usingthedenitionsofz(t)in( 5{47 ),y(t)in( 5{50 ),andu(t)in( 5{42 ),theexpressionin( 5{59 )canbeexpressedas _V2kyk2222(kzk) 4kb1kezk2+"22 where2(kzk)2R+isdenedas2=min222(kzk) 4kb1;kb 2m2;1 2!2: 5{52 ),theinequalityin( 5{60 )canbewrittenas _V2 ConsiderasetSdenedas InS,2(kzk)canbelowerboundedbyaconstant22R+as Basedon( 5{63 ),thelineardierentialequationin( 5{61 )canbesolvedas providedkzk<122p 5{64 ),ifz(0)2Sthenkbcanbechosenaccordingtothesucientconditionsin( 5{49 )(i.e.asemi-globalresult)toyieldresultin( 5{48 ).Basedonthedenitionofy(t),itcanbeconcludedthate1(t);e2(t)2L1inS.Giventhate1(t);e2(t);qd(t);_qd(t)2L1inS;( 5{4 ),( 5{42 ),and( 5{37 )indicatethatq(t);_q(t);u2L1inS:
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5-1 .EachrobotlinkismountedonanNSKdirectdriveswitched Figure5-1.Experimentaltestbedconsitingofa2-linkrobot.Theinputdelayinthesystemwasarticiallyinsertedinthecontrolsoftware. reluctancemotor(240.0NmModelYS5240-GN001,and20.0NmModelYS2020-GN001,respectively).TheNSKmotorsarecontrolledthroughpowerelectronicsoperatingintorquecontrolmode.Rotorpositionsaremeasuredthroughmotorresolverwitharesolutionof614400pulses/revolution.ThecontrolalgorithmswereexecutedonaPentium2.8GHzPCoperatingunderQNX.Dataacquisitionandcontrolimplementationwereperformedatafrequencyof1.0kHzusingtheServoToGoI/Oboard.Inputdelaywasarticiallyinsertedinthesystemthroughthecontrolsoftware(i.e.,thecontrolcommandstothemotorsweredelayedbyavaluesetbytheuser).Thedevelopedcontrollersweretestedforvariousvaluesofinputdelayrangingfrom1msto200ms.Thedesiredlinktrajectoriesforlink1(qd1(t))andlink2(qd2(t))wereselectedas(indegrees):qd1(t)=qd2(t)=20:0sin(1:5t)(1exp(0:01t3)):
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5{8 )(PIDcontrollerwithdelaycompensation)andthecontrollerdevelopedin( 5{42 )(PDcontrollerwithdelaycompensation)werecomparedwithtraditionalPIDandPDcontrollers,respectively,inthepresenceofinputdelayinthesystem.Theinputdelayedtwolinkrobotdynamicsaremodeledas264u1u2375=264p1+2p3cos(q2)p2+p3cos(q2)p2+p3cos(q2)p2375264q1q2375+264p3sin(q2)_q2p3sin(q2)(_q1+_q2)p3sin(q2)_q10375264_q1_q2375+264fd100fd2375264_q1_q2375+264fs100fs2375264tanh(_q1)tanh(_q2)375; 104
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124 ]waslinearizedatseveraloperatingpointsandalinearcontrollerwasdesignedforeachpoint,andthegainswerechosenbyinterpolating,orschedulingthelinearcontrollers.In[ 125 ],aneuralnetworkisusedtotunethegainsofaPIDcontroller.In[ 126 ]ageneticalgorithmwasusedtonetunethegainsafterinitialguessweremadebythecontrollerdesigner.Theauthorsin[ 127 ]provideanextensivediscussionontheuseofextremumseekingfortuningthegainsofaPIDcontroller.Additionally,in[ 128 ],thetuningofaPIDcontrollerforrobotmanipulatorsisdiscussed. TheexperimentalresultsaresummarizedinTable 5-1 .Theerrorandtorqueplotsforthecasewhentheinputdelayis50ms(asarepresentativeexample)areshowninFigs. 5-3 5-4 .ThePDcontrollerwithdelaycompensationwasalsotestedtoobservethesensitivityoftheBgainmatrix,denedin( 5{37 ),wheretheinputdelaywasselectedas100ms.EachelementoftheBgainmatrixwasincremented/decrementedbyacertainpercentagefromtheinverseinertiamatrix(seeTable 5-2 ).ThepurposeofthissetofexperimentswastoshowthatthegainconditiondiscussedinRemark 1 isasucientbutnotanecessarycondition,andtoexploretheperformance/robustnessofthecontrollerin( 5{42 )giveninexactapproximationsoftheinertiamatrix.Thecontrollerexhibitednosignicantdegradation,evenwheneachelementoftheinertiamatrixisover-approximatedby100%.However,underestimatingtheinverseinertiamatrix(particularlywhendeviationfromtheinverseinertiamatrixwas75percent), 105
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5-3 wereconductedtoshowthatpromisingresultscanbeobtainedevenwhentheinputdelayvalueisnotexactlyknown;however,thetrackingerrorperformancedegradeswithincreasinginaccuracyindelayvalueapproximation(e.g.,inthecaseofPD+compensator,thetrackingerrorincreasessignicantlywhenthedelayvalueisoverestimatedby80%orgreater).Forthissetofexperimentstheinputdelaywaschosentobe100ms. TheexperimentalresultsclearlyshowthatthePID/PDcontrollerswithdelaycompensationperformbetterthanthetraditionalPID/PDcontrollers.BothcontrollerscanbedividedintorespectivePID/PDcomponentsandpredictor(delaycompensating)terms.Thebetterperformanceshownbythecontrollerscanbeattributedtothepredictorcomponentsinboththecontrollers.Asanillustrativeexample,Fig. 5-2 showsthetimeplotsofthePDcontrollerwithdelaycompensationanditscontrolcomponents.Thetwocomponents:PDcomponentanddelaycompensatingtermareplottedtoshowtheirbehaviorwithrespecttoeachother.TheplotshowsthatthedelaycompensatingcomponentisalwaysfollowingthePDcomponentbutisoppositeinsign(likeanmirrorimagebutlessinmagnitude).Thus,thenet(actual)controltorqueisalwayslessthanthePDcontrolcomponent.ThisimpliesthatthedelaycompensatingtermtendstocorrectthePDcomponent(actsasaprimarytorquegenerator)whichmayhavecompiledextraneoustorqueduetotheinputdelay.Thedelaycompensatingtermpredictsthecorrectiontermbynitelyintegratingcontroltorqueoverthetimeintervalrangingfromcurrenttimeminusthetimedelaytocurrenttime. 8 ].InNMEScontrol,theEMDismodeledasaninputdelay 106
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inthemusculoskeletaldynamics[ 6 ]andoccursduetoniteconductionvelocitiesofthechemicalionsinthemuscleinresponsetotheexternalelectricalinput[ 36 ].InputdelaycancauseperformancedegradationaswasobservedduringNMESexperimentaltrialsonhumansubjectswithRISEandNN+RISEcontrollersandhasalsobeenreportedtopotentiallycauseinstabilityduringhumanstanceexperimentswithNMES[ 40 ]. 3.4.4.1 ).ThedelayinNMES 107
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Controller PID PID+CPTR PD PD+CPTR TimeDelay Link1 Link2 Link1 Link2 Link1 Link2 Link1 Link2 0:106 0:107 0:129 0:089 1:954 3:137 7:629 0:164 0:172 0:204 0:149 3:430 6:484 14:960 RMSError ininverseinertiamatrix Link1 Link2 1:172 1:246 1:078 1:583 1:540 1:191 2:948 wasmeasuredasthedierencebetweenthetimewhenvoltageisappliedtothemuscleandthetimewhentheangleencoderdetectstherstlegmovement.Theinputdelayvaluesweremeasuredfortenhealthyindividuals(9maleand1female).Thetestsoneachindividualinvestigatedtheeectoninputdelayofthreestimulationparameters:frequency,pulsewidth,andvoltage.Threedierentsetoftestsincluding:frequencyvs 108
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Percentuncertainty PD+Compensator PID+Compensator ininputdelay Link1 Link2 Link1 Link2 1:159 1:234 1:079 1:338 1:192 1:451 1:452 1:629 1:186 3:528 1:229 4:099 3:260 4:331 3:182 inputdelay,voltagevsinputdelay,andpulsewidthvsinputdelaywereperformedoneachindividual.Ineachsetofexperiments,theothertwostimulationparameterswerekeptconstant.Beforethestartofexperiments,thesubjectwasinstructedtorelaxtoavoidvoluntarylegmotion.Thethresholdvoltagewasmeasuredforeachsubjectwhichcanbedenedastheminimumvoltageappliedtothesubject'smusclethatproducesamovementlargeenoughtobedetectedbytheangleencoder.Thismeasurementwasperformedbyapplyingaconstantinputvoltage,beginningat10Vandincreasingthevoltageslightlyuntilmovementwasdetected.Oncethethresholdvoltagewasobtained,theaforementionedthreesetsofexperimentswereperformedforeachindividual. Therstsetofexperimentsconstitutedvaryingfrequencywhilekeepingvoltageandpulsewidthconstant.Thesetestsconsistedofmeasuringtheinputdelayofthesubject'smuscleforthree0.2secondimpulses,each5secondsapart.Eachimpulseimpartedaconstantvoltage(thresholdvoltage+10V)tothemuscle.The5secondtimeseparationbetweentheimpulsesallowedthesubjectstovoluntarilybringtheirlegbacktothe 109
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restposition.Fig. 5-5 showsthetypicalEMDduringNMESinahealthyindividual.Finalinputdelayvaluewascomputedbyaveragingthemeasureddelayvaluesoverthreeimpulses.Eightexperimentswereperformedfordierentfrequencies,wherethefrequencywaschosenrandomlyfromtherangeof30Hzand100Hz(intrarangeintervalof10 110
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Hz).Thepulsewidthforthistypeoftheexperimentswaskeptat100s.Thesecondtypeofexperimentsconsistedofvaryingpulsewidthwhilekeepingvoltageandfrequencyconstant.Eachexperimentconstitutedthreeimpulsesasexplainedaboveforthefrequencytests.Nineexperimentswereperformedfordierentpulsewidths,wherepulsewidthwasrandomlychosenfrom100s.to1000s(intrarangeintervalof100s).Forthisset 111
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ofexperiments,thefrequencywaskeptconstantat30Hzandthevoltageconsistedofminimumthresholdvoltage+10V.Thelastsetofexperimentsinvolvedconductingexperimentswithvaryingvoltages.Sameimpulseprogramasusedintheearliersetofexperimentswasused,wherepulsewidthandfrequencywerekeptconstant.Thefrequencywaskeptat30Hzandthepulsewidthwaskeptat100mus.Threeexperimentswereperformedfordierentvoltages(thresholdvoltage+additionalvoltage,whereadditionalvoltagewasvariedbetween5and20volts(intrarangeintervalof5volts).Table 5-4 (asarepresentativeexample)showsthesummarizedinputdelayvariationswithrespecttodierentstimulationparametersinahealthyindividual. ANOVA(Analysisofvariance)testswereperformedtodeterminetheintraclasscorrelations.AnANOVAtestisgenerallyemployedtodeterminethestatisticalsignicancebetweenthemeansofdatagroupsnumberingmorethantwo(usingstudentt-testto 112
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5-6 )showedthatthedierenceinthemeansofEMDwasstatisticallysignicant(P-value=1:50372E10).Further,post-hoctestutilizingTukey'smethodshowedthattheEMDwaslongerforthelowerfrequenciesthanforthehigherfrequencies.Particularly,thetestshowedthattheaverageEMDof76msatafrequencyof30HzisstatisticallydierentfromtheaverageEMDof51msatafrequencyof100Hz.However,theresultsofthestimulationpulsewidth(seeFig. 5-8 )andvoltageexperiments(seeFig. 5-7 )showednosignicantcorrelationbetweeneithervaryingstimulationpulsewidthorstimulationvoltageandelectromechanicaldelay(P-value=0:6870and0:072,respectively). Figure5-6.Averageinputdelayvaluesacrossdierentfrequencies. 5{8 )and( 5{42 )istomeasureinertiaandinputdelayinthemuscledynamics.Implementingthecontrollerin( 5{8 )becomesevenmorecomplicatedduetothefactthatitrequiresnotonlyinertiaofthe 113
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musculoskeletal-LEMsystemtobemeasuredbutalsotheauxiliaryfunction(q;_q;t)2Rdenedin( 2{8 ),whichconsistsofunmeasurablemuscleforce-velocityandmuscleforce-lengthrelationshipstobeknown.However,thecontrollerdenedin( 5{42 )canbeimplementedprovidedthefollowingassumptionsaremade. 5-3 ). 3.3 )canbeupperboundedas wherea1;a2;a32Raresomeknownpositiveconstants,andBisthecontrolgainintroducedin( 5{37 ). 114
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ThetestbedforexperimentsconsistedofLEM(detailedinSection 3.4.4.1 ).Thecontrolobjectivewastotrackacontinuousconstantperiod(2sec.)sinusoidaltrajectory.Threehealthymales(age:21-28yrs)werechosenasthetestsubjects.Aftertheprotocol(seesection 3.4.4.1 ),theinputdelayvaluewasmeasuredforeachsubject.ThemeasureddelayvaluewasutilizedforimplementingthePDcontrollerwithdelaycompensationandthroughoutthedurationoftrials,thesamerespectivemeasureddelayvaluewasusedforeachsubject.TheexperimentscomparedthetraditionalPDcontrollerwiththePDcontrollerwithdelaycompensation.Eachsubjectparticipatedintwotofourtrialsforeachcontroller 5-5 .Thetableshowsbesttworesults(resultswithminimumRMSerrorsoutofalltrials)obtainedfromeachcontrollerandsubject. 115
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Pulsewidth[sec.] Voltage[V] 100 10 0.069 0.053 0.073 0.065 40 100 10 0.076 0.064 0.077 0.072 50 100 10 0.073 0.069 0.075 0.072 60 100 10 0.062 0.074 0.06 0.065 70 100 10 0.064 0.066 0.051 0.060 80 100 10 0.062 0.059 0.077 0.066 90 100 10 0.062 0.057 0.048 0.056 100 100 10 0.055 0.061 0.059 0.058 30 200 10 0.065 0.066 0.094 0.075 30 300 10 0.07 0.072 0.079 0.074 30 400 10 0.065 0.065 0.09 0.073 30 500 10 0.058 0.056 0.071 0.062 30 600 10 0.05 0.073 0.064 0.062 30 700 10 0.065 0.077 0.058 0.067 30 800 10 0.065 0.067 0.061 0.064 30 900 10 0.071 0.053 0.055 0.060 30 1000 10 0.057 0.083 0.065 0.068 30 100 5 0.081 0.061 0.061 0.068 30 100 15 0.068 0.079 0.087 0.078 30 100 20 0.082 0.084 0.059 0.075 Table5-4.Summarizedinputdelayvaluesofahealthyindividualacrossdierentstimulationparameters.Delayvalues()areshowninseconds.Thevoltagesshownincolumn3aretheaddedvoltagestothethresholdvoltage. AStudent'st-testwasalsoperformedtoconrmstatisticalsignicanceinthemeandierencesoftheRMSerrors,maximumsteadystateerrors(SSEs),RMSvoltages,andthemaximumvoltages.Thestatisticalcomparisonwasconductedontheaveragesofthetwobestresultsobtainedforeachsubject.TheanalysisshowsthatthemeandierencesintheRMSerrors,maximumSSEs,andmaximumvoltagesarestatisticallysignicantwhiletheanalysisshowsnostatisticaldierenceintheRMSvoltages.ThemeanRMSerrorof4:43obtainedwiththePDcontrollerwithdelaycompensationislowerthantheRMSerrorof6:03obtainedwiththePDcontroller.Also,themeanmaximumSSEandthemeanmaximumvoltageobtainedwiththePDcontrollerwithdelaycompensationarelowerthanthemeanmaximumSSEandthemeanmaximumvoltageobtainedwiththetraditionalPDcontroller.Therespectivep-valuesaregivenintheTable 5-5 .Theactual 116
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5-9 and 5-10 Figure5-9.Topplot:Actuallimbtrajectoryofasubject(solidline)versusthedesiredtrajectory(dashedline)inputobtainedwiththePDcontrollerwithdelaycompensation.Middleplot:Thetrackingerror(desiredangleminusactualangle)ofasubject'sleg,trackingaconstant(2sec.)perioddesiredtrajectory.Bottomplot:ThecomputedvoltageofthePDcontrollerwithdelaycompensationduringkneejointtracking. 117
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118
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RMSVoltage[V] Max.SSE Max.Voltage[V] Subject PD PD+CTR PD PD+CTR PD PD+CTR PD PD+CTR A 4:48 33:18 11:84 42:02 A 7:63 32:26 20:41 44:38 B 8:48 22:93 25:78 27:43 B 6:54 22:65 10:79 26:51 C 3:11 26:17 12:84 38:8 C 5:91 27:60 16:66 36:7 Mean 6:03 27:47 16:37 35:97 pvalue 0:003 0:008 wasrequiredtobeaknownconstant.Whilesomeapplicationshaveknowndelays(e.g.,teleoperation[ 129 ],somenetworkdelays[ 130 ],timeconstantsinbiologicalsystems[ 6 36 ]),thedevelopmentofmoregeneralizedresults(whichhavebeendevelopedforsomelinearsystems)withunknowntimedelaysremainsanopenchallenge.However,theexperimentalresultswithtwo-linkrobotillustratedsomerobustnesswithregardtotheuncertaintyinthetimedelay. 119
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96 ]basedRobustIntegraloftheSignoftheError(RISE)structure[ 11 27 ].Duetotheaddedbenetofreducedcontroleortandimprovedcontrolperformance,anadaptivecontrollerinconjunctionwithRISEfeedbackstructureisdesigned.However,sincethetimedelayvalueisnotalwaysknown,itbecomeschallengingtodesignadelayfreeadaptivecontrollaw.Throughtheuseofadesiredcompensationadaptivelaw(DCAL)basedtechniqueandsegregatingtheappropriatetermsintheopenlooperrorsystem,thedependenceofparameterestimatelawsonthetimedelayedunknownregressionmatrixisremoved.Contrarytopreviousresults,thereisnosingularityinthedevelopedcontroller.ALyapunov-basedstabilityanalysisisprovidedthatusesanLKfunctionalalongwithYoung'sinequalitytoremovetimedelayedtermsandachievesasymptotictracking. 87 ] _x1=x2_xn1(t)=xn_xn(t)=f(x(t))+1(x(t))+g(x(t))+2(x(t))+d(t)+bu(t)y=x1 120
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6{1 ),f(x(t));1(x(t))2Rmareunknownfunctions,g(x(t));2(x(t))2Rmareunknowntime-delayedfunctions,2R+isanunknownconstantarbitrarilylargetimedelay,d(t)2Rmisaboundeddisturbance,b2Risanunknownpositiveconstant,u(t)2Rmisthecontrolinput,andx(t)=xT1xT2:::xTnT2Rmndenotesystemstates,wherex(t)isassumedtobemeasurable.Alsothefollowingassumptionsandnotationswillbeexploitedinthesubsequentdevelopment. 83 87 95 ]). 121
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where1;:::;n2Rdenotepositiveconstantcontrolgains.Asdenedin( 6{5 ),thelteredtrackingerrorr(t)isnotmeasurablesincetheexpressiondependson_xn(t):However,e1(t);:::;en(t)2Rmaremeasurablebecause( 6{4 )canbeexpressedintermsofthetrackingerrore1(t)as whereaij2Rarepositiveconstantsobtainedfromsubstituting( 6{6 )in( 6{4 )andcomparingcoecients[ 114 ].Itcanbeeasilyshownthat Using( 6{2 )-( 6{7 ),theopenlooperrorsystemcanbewrittenas wherel(e1;_e1;:::;e(n1)1)2Rmisafunctionofknownandmeasurableterms,denedasl=n2Xj=0anje(j+1)1+ne(j)1+ne(n1)1: 6{8 )byb1andutilizingtheexpressionsin( 6{1 )andAssumption1toobtainthefollowingexpression: 122
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6{9 )canberewrittenas wherexd=hyTd_yTd:::y(n1)TdiT2Rmndenotesacolumnvectorcontainingthedesiredtrajectoryanditsderivatives.Thegroupingoftermsandstructureof( 6{10 )ismotivatedbythesubsequentstabilityanalysisandtheneedtodevelopanadaptiveupdatelawthatisinvarianttotheunknowntimedelay.TheauxiliaryfunctionS1(xd;x)isdenedbecausethesetermsarenotfunctionsofthetime-delay.TheauxiliaryfunctionS2(xd;x)isintroducedbecausethetime-delayedstatesareisolatedinthisterm,andW(xd;xd;_xd)isisolatedbecauseitonlycontainsfunctionsofthedesiredtrajectory. Basedontheopen-looperrorsystemin( 6{10 ),thecontrolinputu(t)2Rmisdesignedas In( 6{13 ),2Rmdenotestheimplicitlearning-based[ 96 ]RISEtermdenedasthegeneralizedsolutionto _=(ks+1)r+sgn(en);(0)=0;(6{14) 123
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6{13 ),^1(t)2Rp1;^2(t)2Rp2denoteparameterestimatevectorsdenedas _^1=1_YT1(xd)r;_^2=2_YT2(xd)r;(6{15a) where12Rp1p1,22Rp2p2areknown,constant,diagonal,positivedeniteadaptationgainmatrices.In( 6{15a );_YT2(xd)doesnotdependonthetimedelayeddesiredstate.ThisdelayfreelawisachievedbyisolatingthedelayedtermY2(xd)2intheauxiliarysignalW(xd;xd;y(n)d)in( 6{12 ).Theadaptationlawsin( 6{15a )dependontheunmeasurablesignalr(t);butbyusingthefactthat_Y1(xd),_Y2(xd)arefunctionsoftheknowntimevaryingdesiredtrajectory,integrationbypartscanbeusedtoimplement^i(t)fori=1;2whereonlyen(t)isrequiredas ^i=^i(0)+i_YTi(xd)en()jt0it0nYTi(xd)en()n_YTi(xd)en()od: 6{13 )into( 6{10 )as where~ifori=1;2aretheparameterestimationerrorvectorsdenedas ~i=i^i:(6{17) TofacilitatethesubsequentstabilityanalysisandtomoreclearlyillustratehowtheRISEstructurein( 6{14 )isusedtorejectthedisturbanceterms,thetimederivativeof( 6{16 )isdeterminedas 124
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~N=b1_l+_S1+_S2+enY1(xd)1_Y1(xd)rY2(xd)2_Y2(xd)r;Nd=_W+b1_d: UsingAssumptions2,3;and4,Nd(xd;_xd;xd;t)anditstimederivativecanbeupperboundedas whereNd;_Nd2Rareknownpositiveconstants.Theexpressiondenedin( 6{19 )canbeupperboundedusingtheMeanValueTheoremas[ 114 ] wherez(t)2R(n+1)misdenedas andtheknownboundingfunctions1(kzk);2(kzk)2Rarepositive,globallyinvertible,andnondecreasingfunctions.Notethattheupperboundfortheauxiliaryfunction~N(e1;e2;e1;e2)in( 6{21 )issegregatedintodelayfreeanddelayedupperboundfunctions.MotivationforthissegregationoftermsistoeliminatethedelaydependenttermthroughtheuseofanLKfunctionalinthestabilityanalysis.Specically,letQ(t)2RdenoteanLKfunctionaldenedas 2ksZtt22(kz()k)kz()k2d;(6{23) whereks2Rand2()areintroducedin( 6{14 )and( 6{21 ),respectively. Theorem6. 6{13 ),( 6{14 ),and( 6{15a )ensuresthatallsystemsignalsareboundedunderclosed-loopoperation.Thetrackingerrorisregulatedinthe
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6{14 )isselectedsucientlylarge,andn1;n;andareselectedaccordingtothefollowingsucientconditions: 2(6{24) 6{4 )and( 6{5 ),respectively;isintroducedin( 6{14 );andNdand_Ndareintroducedin( 6{20 ). where~i(t)aredenedin( 6{17 ),z(t)andQ(t)aredenedin( 6{22 )and( 6{23 ),respectively,andtheauxiliaryfunctionP(t)2Risthegeneralizedsolutiontothedierentialequation _P(t)=L(t);P(0)=nXi=1jeni(0)jen(0)TNd(0)(6{26) TheauxiliaryfunctionL(t)2Rin( 6{26 )isdenedas ProvidedthesucientconditionsstatedinTheorem 6 aresatised,thenP(t)0(seetheAppendixB). LetVL(y;t):D[0;1)!RdenoteaLipschitzcontinuousregularpositivedenitefunctionaldenedas 2eT1e1+1 2eT2e2+:::+1 2eTnen+1 2rTb1r+P+Q+1 2~T111~1 +1 2~T212~2;
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providedthesucientconditionsintroducedinTheorem 6 aresatised.In( 6{29 ),U1(y);U2(y)2Rarecontinuous,positivedenitefunctionsdenedas where1;22Raredenedas 2min(1;b1;min11;min12);2=max(1 2b1;1;max11;max12); andminfg;maxfgdenotetheminimumandmaximumEigenvalues,respectively.Aftertakingthetimederivativeof( 6{28 ),_VL(y;t)canbeexpressedas_VL(y;t),eT1_e1+eT2_e2+:::+eTn_en+rTb1_r+_P+_Q+~T111~1+~T212~2: 6{3 ),( 6{4 ),( 6{18 ),( 6{26 ),( 6{27 ),adaptationlawsin( 6{15a ),andthetimederivativeofQ(t)in( 6{23 ),someofthedierentialequationsdescribingtheclosed-loopsystemforwhichthestabilityanalysisisbeingperformedhavediscontinuousright-hand 127
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_e1=e21e1; _e2=e32e2e1; _en=rnen _P(t)=rT(Nd(t)sgn(en)); _Q(t)=1 2ks22(kz(t)k)kz(t)k222(kz(t)k)kz(t)k2; ~T111~1=~T1_YT1(xd)r; ~T212~2=~T2_YT2(xd)r: Letf(y;t)2R(n+1)m+p1+p2+2denotetherighthandsideof( 6{32 ).f(y;t)iscontinuousexceptinthesetf(y;t)je2=0g.From[ 103 { 106 ],anabsolutecontinuousFilippovsolutiony(t)existsalmosteverywhere(a.e.)sothat_y2K[f](y;t)a:e: 6{28 )existsa.e.,and_VL(y;t)2a:e:~VL(y;t)where 2P1 2_P1 2Q1 2_Q~1~2T; =rVTLK_e1_e2_en_r1 2P1 2_P1 2Q1 2_Q~1~2T;eT1eT2eTnrTb12P1 22Q1 2~T111~T212K_e1_e2_en_r1 2P1 2_P1 2Q1 2_Q~1~2T:
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6{32 to 6{33 anddiscussion,seeSection 3.3.1 .Afterutilizing( 6{3 ),( 6{4 ),( 6{18 ),( 6{26 ),( 6{27 ),adaptationlawsin( 6{15a )andthetimederivativeofQ(t)in( 6{23 ),theexpressionin( 6{33 )canberewrittenas 2ks22(kzk)kzk222(kzk)kzk2: Cancellingcommontermsyieldsandusing( 6{21 ) AfterapplyingfollowingYoung'sinequalitytodeterminethat 2ken1k2+kenk2;(6{36) theexpressionin( 6{35 )canbewrittenas~VL(y;t)n2Xi=1ikeik2n11 2ken1k2n1 2kenk2krk2ks 6{37 )canbewrittenas 2kskzk2(6{37) where2(kzk)2Risdenedas and34=min1;2;:::;n2;n11 2;n1 2;1:Theboundingfunction(kzk)isapositive,globallyinvertible,andnondecreasingfunctionthatdoesnotdependon 129
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6{37 )canbefurtherupperboundedbyacontinuous,positivesemi-denitefunction forsomepositiveconstantc,whereD,ny(t)2R(n+1)m+p1+p2+2jkyk1p 6{29 )and( 6{39 )canbeusedtoshowthatV(y;t)2L1inD;hence,e1;e2;:::;en;~1;~22L1inD.Theclosed-looperrorsystemscannowbeusedtoconcludeallremainingsignalsareboundedinD,andthedenitionsforU(y)andz(t)canbeusedtoprovethatU(y)isuniformlycontinuousinD.LetSDdenoteasetdenedas Theregionofattractionin( 6{40 )canbemadearbitrarilylargetoincludeanyinitialconditionsbyincreasingthecontrolgainks(i.e.,asemi-globalstabilityresult),andhence Basedonthedenitionofz(t),( 6{41 )canbeusedtoshowthat 87 ]: _x1=x2 _x2=f(x)+g(x)+1(x)+2(x)+d+bu;
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Thefollowinggainsarechosenfor=3sand=10sks=10;1=7;2=6;=5;1=0:5;2=[2;0;0;10]. FromtheresultsshowninFigs. 6-1 6-5 ,itisclearthatthecontrollertracksthetimevaryingdesiredtrajectoryeectively.Inboththecases,thesteadystateerrorsstaybetween0:003radiansandthecontrolinputsarebounded:Alsoitcanbeseenthatthereisalittlevariationinthecontrolperformancesfortimedelays=3sand=10s: 131
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132
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Figure6-4.Trackingerrorforthecase=10s:
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134
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3 dealwithunknownnonlinearmapping,boundeddisturbances,andotherunknownnonlinearitiesanduncertainties.TheLyapunov-basedstabilityanalysisisutilizedtoprovesemi-globalasymptoticstabilityforthecontrollers.ExtensiveexperimentsonhealthyvolunteerswereconductedforbothRISEandNN+RISEcontrollers.Particularly,itwasshownthattheinclusionofneuralnetworkbasedfeedforwardcomponentintheRISEcontrollerimprovesperformanceduringNMES.Also,preliminaryexperimentaltrialsdemonstratingsit-to-standtaskdepictedthefeasibilityoftheNN+RISEcontrollerinaclinical-typescenario. InChapter 4 ,aNN-basedcontrollerisdevelopedtocompensateforfatigue.Thebenetofthecontrolleristhatitincorporatesmoremuscledynamicsknowledgenamely,calciumandfatiguedynamics.Theeectivenessofthecontrollertocompensatefatigueisshownthroughsimulationresults.FurthersimulationsshowthatthecontrollerperformsbetterthantheRISEcontroller. AnimportanttechnicaldicultyinNMESisinputdelaywhichbecomesmorechallengingduetothepresenceofunknownnonlinearitiesanddisturbances.Lackofinputdelaycompensatingcontrollersforuncertainnonlinearsystemsmotivatedtodeveloppredictor-basedcontrollersforgeneralEulerLagrangesysteminChapter 5 .TheLyapunov-basedstabilityanalysisutilizesLKfunctionalstoprovesemi-globalUUBtracking.ExtensiveexperimentalresultsshowbetterperformanceofthecontrollerincomparisontothetraditionalPD/PIDcontrolleraswellastheirrobustnesstouncertainty 135
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ThelastchapterinthedissertationcoversthedevelopmentofRISE-basedadaptivecontrollerforaclassofnonlinearsystemwithstatedelays.Thesignicanceoftheresultisthatarobustandcontinuouscontrollerisdevelopedforanonlinearsystemwithunknownstatedelaysandadditivedisturbances.Lyapunov-basedstabilityanalysisaidedwithLKfunctionalsisutilizedtoshowasemi-globalasymptotictracking. 3 showedthattheRMSerrordierence(forbothRISEandNN+RISEcontrollers)betweentheexionandextensionphaseofthelegmovementisstatisticallysignicant.Theseresultssuggestthattheroleofswitchingcontrollers(hybridcontrolapproach)canbeinvestigated.Specically,twodierentcontrollerscanbeutilizedwhereeachcontrollerisdedicatedforaparticularphaseofthelegmovement. 4 hasthreemainlimitations:unmeasurablecalciumandfatiguedynamics,dependenceonacceleration,anduniformlyultimatelyboundedstabilityresult.Eortscanbemadetodevelopanobserver-basedcontrollertoremovethedependenceonmathematicalfatigueandcalciumdynamicsmodels.Specically,recurrentneuralnetworkbasedobservercanbedesignedtoidentifysystemstates.Further,improvementinstabilityanalysiscanbeachievedbydevelopingacontrollerwithasymptotictracking.AnextensiveinvestigationisrequiredtoobservetheeectofthecontrollerinChapter 4 onreducingfatigue.Experimentsshouldnotonlycomparetheresultwithanexistingcontrollerforimprovedperformancebutshouldspecicallystudytheeectivenessoftheincludedfatiguemodelforfatiguecompensation.Theresultsmay(ormaynot) 136
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131 ],whichcanbesurgicallyimplantedatspecicsitesinthemuscle.Thesemicroelectrodeswhichdonotrequirewiresarepoweredexternallythroughaninductivecoilandabattery.MultipleBIONstostimulatespecicmusclesitescannotonlybeusedtoproducedesiredfunctionalmovementsbutalsocanbeusedtoeliminatemusclefatiguethroughutilizingnon-repetitiveandselectivemusclerecruitment.InordertoproduceNMEScontrolviaBIONs,studieswillberequired 137
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5 showedthattheinputdelayinNMESdependsonlyonvaryingfrequency.However,furtherinvestigationsarerequiredtostudytheeectoffatigueandnon-isometriccontractionsoninputdelay.Also,resultsinChapter 5 areonlyapplicablewithknownconstantinputdelayvalues.Therefore,controllersneedtobedevelopedtoaccountfortime-varyingorunknowninputdelay.Otherdelaycompensatingtechniquessuchasmodelpredictivecontrol(MPC)canalsobeinvestigatedforNMES.OneoftheadvantagesofMPCisthatitinherentlycompensatesforinputdelays.Althoughthetechniquewouldrequiremuscledynamicstobeknown,advantagessuchasperformanceandcontroloptimizationinadditiontodelaycompensationmakesMPCaworthycandidateforinvestigation. 138
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_Q(t)=!d dtZttZtsk_u()k2dds; _Q(t)=!Zttk_u()k2ddt dt!Zttk_u()k2dd(t) @tZtsk_u()k2dds:(1{3) Theexpressionin( 1{3 )canbesimpliedas @tZtsk_u()k2dds:(1{4) AgainapplyingLeibnizintegralruleonsecondintegralin( 1{4 ) dtk_u(s)k2ds dt+Zts@ @tk_u()k2dds:(1{5) Theexpressionin( 1{5 )canbesimpliedas Furtherintegratingthesecondintegralin( 1{6 )
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1 140
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Proof:Integratingbothsidesof( 2{9 ) Onsubstituting( 6{5 )in( 2{12 )yieldsZt0L()d=Zt0_eTnNddZt0_eTnsgn(en)d+Zt0eTn(Ndsgn(en))d: Afterutilizingintegrationbypartsfortherstintegralandintegratingthesecondintegralin( 2{13 ),thefollowingexpressionisobtained:Zt0L()d=eTnNdeTn(0)Nd(0)+nXi=1jeni(0)jnXi=1jeni(t)j+Zt0eTn(Nd1 wherethefactthatsgn(en)canbedenotedas 141
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6{20 )andthefactthat theexpressionin( 2{14 )canbeupperboundedasZt0L()dnXi=1jeni(0)jeTn(0)Nd(0)+(Ndkenk)+Zt0kenkNd+_Nd Itisclearfrom( 2{17 )thatifthefollowingsucientcondition issatised,thenthefollowinginequalityholds 142
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NitinSharmawasborninNovember1981inAmritsar,India.HereceivedhisBachelorofEngineeringdegreeinindustrialengineeringfromThaparUniversity,India.Afterhisgraduationin2004,hewashiredasagraduateengineertraineefrom2004to2005andworkedasanexecutiveengineerfrom2005to2006inMarutiSuzukiIndiaLtd.HethenjoinedtheNonlinearControlsandRobotics(NCR)researchgrouptopursuehisdoctoralresearchundertheadvisementofDr.WarrenE.Dixon.HewillbejoiningasapostdoctoralfellowinDr.RichardStein'slaboratoryattheUniversityofAlberta,Edmonton,Canada. 153