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- Permanent Link:
- http://ufdc.ufl.edu/UFE0041961/00001
## Material Information- Title:
- Lyapunov-Based Control Methods for Neuromuscular Electrical Stimulation
- Creator:
- Sharma, Nitin
- Place of Publication:
- [Gainesville, Fla.]
- Publisher:
- University of Florida
- Publication Date:
- 2010
- Language:
- english
- Physical Description:
- 1 online resource (153 p.)
## Thesis/Dissertation Information- Degree:
- Doctorate ( Ph.D.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Mechanical Engineering
Mechanical and Aerospace Engineering - Committee Chair:
- Dixon, Warren E.
- Committee Members:
- Banks, Scott A.
Crane, Carl D. Hammer, Jacob Gregory, Chris - Graduation Date:
- 8/7/2010
## Subjects- Subjects / Keywords:
- Developmental delay ( jstor )
Electric potential ( jstor ) Electric stimulation therapy ( jstor ) Electrical stimulation ( jstor ) Experimental results ( jstor ) Fatigue ( jstor ) Inertia ( jstor ) Legs ( jstor ) Torque ( jstor ) Trajectories ( jstor ) Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF adaptive, electrical, lyapunov, neural, nmes, nonlinear, rise, time - Genre:
- Electronic Thesis or Dissertation
bibliography ( marcgt ) theses ( marcgt ) government publication (state, provincial, terriorial, dependent) ( marcgt ) Mechanical Engineering thesis, Ph.D.
## Notes- Abstract:
- 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 Chapter 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 Chapter 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 asymptotic 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 Chapter 3 focus on the use of a NN feedforward controller that is augmented with a continuous robust feedback term to yield an asymptotic result (in lieu of typical uniformly ultimately bounded (UUB) stability). Specifically, a NN-based controller and Lyapunov-based stability analysis are provided to enable semi-global asymptotic 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 Chapter 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, delays, changing load dynamics). Simulations are provided to illustrate the performance of the proposed controller. Continued efforts will focus on achieving asymptotic 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 Chapter 5 for an Euler-Lagrange system with time-delayed 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 healthy normal individuals to show the feasibility, performance, and robustness of the developed controller. In addition to efforts focussed on input delayed nonlinear systems, a parallel motivation exists to address another class of time delayed systems which consist of nonlinear systems with unknown state delays. A continuous robust adaptive control method is designed in Chapter 6 for a class of uncertain nonlinear systems with unknown constant time-delays 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-delays, 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 asymptotically tracks a desired time varying bounded trajectory. Chapter 7 concludes the dissertation with a discussion of the developed contributions and future efforts. ( en )
- General Note:
- In the series University of Florida Digital Collections.
- General Note:
- Includes vita.
- Bibliography:
- Includes bibliographical references.
- Source of Description:
- Description based on online resource; title from PDF title page.
- Source of Description:
- This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
- Thesis:
- Thesis (Ph.D.)--University of Florida, 2010.
- Local:
- Adviser: Dixon, Warren E.
- Statement of Responsibility:
- by Nitin Sharma.
## Record Information- Source Institution:
- UFRGP
- Rights Management:
- Applicable rights reserved.
- Embargo Date:
- 10/8/2010
- Resource Identifier:
- 004979722 ( ALEPH )
705932481 ( OCLC ) - Classification:
- LD1780 2010 ( lcc )
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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 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 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. 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), 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. 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 [121] R. Happee, lii', i-.- dynamic optimization including muscular dynamics, a new simulation method applied to goal directed movements," J. Biomech., vol. 27, no. 7, pp. 953-960, 1994. [122] F. L. Lewis, \X. i network control of robot manipulators," IEEE Expert, vol. 11, no. 3, pp. 64-75, 1996. [123] M. Ferrarin, F. Palazzo, R. Riener, and J. Quintern, "Model-based control of FES-induced single joint movements," IEEE Trans. Neural Syst. Rehabil. Eng., vol. 9, no. 3, pp. 245-257, Sep. 2001. [124] N. Stefanovic, M. Ding, and L. Pavel, "An application of L2 nonlinear control and gain scheduling to erbium doped fiber amplifiers," Control Eng. Pract., vol. 15, pp. 1107-1117, 2007. [125] T. Fuilii I1: Y. Kishida, M. Yoshioka, and S. Omatu, "Stabilization of double inverted pendulum with self-tuning neuro-PID," in Proc. IEEE-INNS-ENNS Int. Joint Conf. Neural Netw., vol. 4, 24-27 July 2000, pp. 345-348. [126] F. N I, ii i K. Kuril --Vi-hi K. Kiguchi, and K. Watanabe, "Simulation of fine gain tuning using genetic algorithms for model-based robotic servo controllers," in Proc. Int. Sym. Comput. Intell. Robot. Autom., 20-23 June 2007, pp. 196-201. [127] N. J. Killingsworth and M. Krstic, "PID tuning using extremum seeking: online, model-free performance optimization," IEEE Contr. Syst. Mag., vol. 26, no. 1, pp. 70-79, 2006. [128] R. Kelly, V. Santibanez, and A. Loria, Control of Robot Manipulators in Joint Space. Springer, 2005. [129] R. Anderson, M. Spong, and N. Sandia National Labs., Albuquerque, "Bilateral control of teleoperators with time delay," IEEE Trans. Autom. Control, vol. 34, no. 5, pp. 494-501, 1989. [130] G. Liu, J. Mu, D. Rees, and S. C'! n, "Design and stability analysis of networked control systems with random communication time delay using the modified MPC," Int. J. Control, vol. 79, no. 4, pp. 288-297, 2006. [131] T. Cameron, G. Loeb, R. Peck, J. Schulman, P. Strojnik, and P. Troyk, \ili romodular implants to provide electrical stimulation of paralyzed muscles and limbs," IEEE Trans. on Biomed. Eng., vol. 44, no. 9, pp. 781-790, 1997. 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. 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) 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) 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 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 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 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)). 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 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. 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 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) 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 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. 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 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 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 -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. 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) 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 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) 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 . 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. 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 -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 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 BIOGRAPHICAL SKETCH ................... .......... 153 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 To my loving wife, Deepti, my dear parents, Neena and Balwinder Sharma, and my affectionate sister, Nitika for their unwavering support 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 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. 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. 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. 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) 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 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). 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. 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. [67] L. Roca, J. Luis Guzman, J. E. Normey-Rico, M. Berenguel, and L. Yebra, "Robust constrained predictive feedback linearization controller in a solar desalination plant collector field," Control Eng. Pract., vol. 17, no. 9, pp. 1076-1088, Sep. 2009. [68] C. Xiang, L. Cao, Q. Wang, and T. Lee, "Design of predictor-based controllers for input-delay systems," in Proc. IEEE Int. Symp. Ind. Electron., 30 2008-July 2 2008, pp. 1009-1014. [69] H.-H. Wang, "Optimal vibration control for offshore structures subjected to wave loading with input d. 1 .i," in Int. Conf. Meas. Technol. Mechatron. 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Bliman, "Backstepping design for time-delay nonlinear systems," IEEE Trans. Autom. Control, vol. 51, no. 1, pp. 149-154, 2006. [77] M. Jankovic, "Control of cascade systems with time delay the integral cross-term approach," in Proc. IEEE Conf. Decis. Control, Dec. 2006, pp. 2547-2552. [78] A. Teel, "Connections between Razumikhin-type theorems and the ISS nonlinearsmall gain theorem," IEEE Trans. Autom. Control, vol. 43, no. 7, pp. 960-964, 1998. [79] M. Krstic, "On compensating long actuator d. 1 li- in nonlinear control," IEEE Trans. Autom. Control, vol. 53, no. 7, pp. 1684-1688, 2008. [80] Y. Xia and Y. Jia, "Robust sliding-mode control for uncertain time-delay systems: an LMI approach," IEEE Trans. Autom. Control, vol. 48, no. 6, pp. 1086-1091, June 2003. 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 [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 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) 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) 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. 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 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. 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 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) 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 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. REFERENCES [1] P. H. Peckham and D. B. Gray, "Functional neuromuscular stimulation," J. Rehabil. Res. Dev., vol. 33, pp. 9-11, 1996. [2] P. H. Peckham and J. S. Knutson, "Functional electrical stimulation for neuromuscular applications," Annu. Rev. Biomed. Eng., vol. 7, pp. 327-360, 2005. [3] J. J. Abbas and H. J. Chizeck, 1. I II ., 1 control of coronal plane hip angle in paraplegic subjects using functional neuromuscular stimulation," IEEE Trans. Biomed. Eng., vol. 38, no. 7, pp. 687-698, 1991. [4] N. Lan, P. E. Crago, and H. J. Chizeck, "Control of end-point forces of a multijoint limb by functional neuromuscular stimulation," IEEE Trans. Biomed. Eng., vol. 38, no. 10, pp. 953-965, 1991. [5] 1', I 1I I: control methods for task regulation by electrical stimulation of muscles," IEEE Trans. Biomed. Eng., vol. 38, no. 12, pp. 1213-1223, 1991. [6] T. Schauer, N. O. Negard, F. Previdi, K. J. Hunt, M. H. Fraser, E. Ferchland, and J. Raisch, "Online identification and nonlinear control of the electrically stimulated quadriceps muscle," Control Eng. Pract., vol. 13, pp. 1207-1219, 2005. [7] K. Stegath, N. Sharma, C. M. Gregory, and W. E. Dixon, "An extremum seeking method for non-isometric neuromuscular electrical stimulation," in Proc. IEEE Int. Conf. Syst. Man. C;,1,. ,,. 2007, pp. 2528-2532. [8] A. H. Vette, K. Masani, and M. R. Popovic, IpllI.!, i il I :ii' of a physiologically identified PD feedback controller for regulating the active ankle torque during quiet stance," IEEE Trans. Neural Syst. Rehabil. Eng., vol. 15, no. 2, pp. 235-243, June 2007. [9] G. Khang and F. E. Z i, 1 "Paraplegic standing controlled by functional neuromuscular stimulation: Part I computer model and contr .l--i-I. i1 design," IEEE Trans. Biomed. Eng., vol. 36, no. 9, pp. 873-884, 1989. [10] F. Previdi, M. Ferrarin, S. Savaresi, and S. Bittanti, "Gain scheduling control of functional electrical stimulation for assisted standing up and sitting down in paraplegia: a simulation study," Int. J. Adapt Control S.:i,,Il Process., vol. 19, pp. 327-338, 2005. [11] P. M. Patre, W. MacKunis, C. Makkar, and W. E. Dixon, "Asymptotic tracking for systems with structured and unstructured uncertainties," IEEE Trans. Control Syst. Technol., vol. 16, no. 2, pp. 373-379, 2008. [12] "Asymptotic tracking for systems with structured and unstructured uncertainties," in Proc. IEEE Conf. Decis. Control, San Diego, CA, Dec. 2006, pp. 441-446. 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 .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 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 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) 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 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) [13] N. Lan, H. F iw- and E. Crago, \. i, I network generation of muscle stimulation patterns for control of arm movements," IEEE Trans. Rehabil. Eng., vol. 2, no. 4, pp. 213-224, 1994. [14] J. J. Abbas and H. J. Chizeck, \N .i I! network control of functional neuromuscular stimulation systems: computer simulation studies," IEEE Trans. Biomed. Eng., vol. 42, no. 11, pp. 1117-1127, Nov. 1995. [15] D. Graupe and H. Kordylewski, "Artificial neural network control of FES in paraplegics for patient responsive ambulation," IEEE Trans. Biomed. Eng., vol. 42, no. 7, pp. 699-707, July 1995. [16] G.-C. C'!I(I,- J.-J. Lub, G.-D. Liao, J.-S. Lai, C.-K. C'I. 1, B.-L. Kuo, and T.-S. Kuo, "A neuro-control system for the knee joint position control with quadriceps stimulation," IEEE Trans. Rehabil. Eng., vol. 5, no. 1, pp. 2-11, Mar. 1997. [17] J. A. Riess and J. J. Abbas, "Adaptive neural network control of cyclic movements using functional neuromuscular stimulation," IEEE Trans. Neural Syst. Rehabil. Eng., vol. 8, pp. 42-52, 2000. [18] H. Kordylewski and D. Graupe, "Control of neuromuscular stimulation for ambulation by complete paraplegics via artificial neural networks," Neurol. Res., vol. 23, no. 5, pp. 472-481, 2001. [19] D. G. Zhang and K. Y. Zhu, "Simulation study of FES-assisted standing up with neural network control," in Proc. Annu. Int. Conf. IEEE Eng. Med. Biol. Soc., vol. 6, 2004, pp. 4118-4121. [20] J. P. Giuffrida and P. E. Crago, "Functional restoration of elbow extension after spinal-cord injury using a neural network-based synergistic FES controller," IEEE Trans. Neural Syst. Rehabil. Eng., vol. 13, no. 2, pp. 147-152, 2005. [21] Y.-L. Chen, W.-L. C('!, i C.-C. Hsiao, T.-S. Kuo, and J.-S. Lai, "Development of the FES system with neural network + PID controller for the stroke," in Proc. IEEE Int. Symp. Circuits Syst., May 23-26, 2005, pp. 5119-5121. [22] K. Kurosawa, R. Futami, T. Watanabe, and N. Hoshimiya, "Joint angle control by FES using a feedback error learning controller," IEEE Trans. Neural Syst. Rehabil. Eng., vol. 13, pp. 359-371, 2005. [23] A. Pedrocchi, S. Ferrante, E. De Momi, and G. Ferrigno, "Error mapping controller: a closed loop neuroprosthesis controlled by artificial neural networks," J. Neuroeng. Rehabil., vol. 3, no. 1, p. 25, 2006. [24] S. Kim, M. Fairchild, A. Iarkov, J. Abbas, and R. Jung, "Adaptive control for neuromuscular stimulation-assisted movement therapy in a rodent model," IEEE Trans. Biomed. Eng., vol. 56, pp. 452-461, 2008. 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 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. 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, 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 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 [25] A. Ajoudani and A. Erfanian, "A neuro-sliding-mode control with adaptive modeling of uncertainty for control of movement in paralyzed limbs using functional electrical stimulation," IEEE Trans. Biomed. Eng., vol. 56, no. 7, pp. 1771-1780, Jul. 2009. [26] J. Lujan and P. Crago, "Automated optimal coordination of multiple-DOF neuromuscular actions in feedforward neuroprostheses," IEEE Trans. Biomed. Eng., vol. 56, no. 1, pp. 179-187, Jan. 2009. [27] P. M. Patre, W. MacKunis, K. Kaiser, and W. E. Dixon, "Asymptotic tracking for uncertain dynamic systems via a rmiiltili, r neural network feedforward and RISE feedback control structure," IEEE Trans. Autom. Control, vol. 53, no. 9, pp. 2180-2185, 2008. [28] M. J. Levy, M. and Z. Susak, "Recruitment, force and fatigue characteristics of quadriceps muscles of paraplegics, isometrically activated by surface FES," J. Biomed. Eng., vol. 12, pp. 150-156, 1990. [29] D. Russ, K. Vandenborne, and S. Binder-Macleod, I-i'. in fatigue during intermittent electrical stimulation of human skeletal muscle," J. Appl. Ph; -.,1 vol. 93, no. 2, pp. 469-478, 2002. [30] J. Mizmhi, "Fatigue in muscles activated by functional electrical stimulation," Crit. Rev. Phys. Rehabil. Med., vol. 9, no. 2, pp. 93-129, 1997. [31] E. Asmussen, \Iiiucle fatigue," Med. Sci. Sports. Exerc., vol. 11, no. 4, pp. 313-321, 1979. [32] R. Maladen, R. Perumal, A. Wexler, and S. Binder-Macleod, "Effects of activation pattern on nonisometric human skeletal muscle performance," J. Appl. Phi;i-./ vol. 102, no. 5, pp. 1985-91, 2007. [33] S. Binder-Macleod, J. Dean, and J. Ding, "Electrical stimulation factors in potentiation of human quadriceps femoris," Muscle Nerve, vol. 25, no. 2, pp. 271-9, 2002. [34] Y. Giat, J. Mizrahi, and M. Levy, "A musculotendon model of the fatigue profiles of paralyzed quadriceps muscle under FES," IEEE Trans. Biomed. Eng., vol. 40, no. 7, pp. 664-674, 1993. [35] R. Riener, J. Quintern, and G. Schmidt, "Biomechanical model of the human knee evaluated by neuromuscular stimulation," J. Biomech., vol. 29, pp. 1157-1167, 1996. [36] R. Riener and T. Fuhr, "Patient-driven control of FES-supported standing up: A simulation study," IEEE Trans. Rehabil. Eng., vol. 6, pp. 113-124, 1998. [37] J. Ding, A. Wexler, and S. Binder-Macleod, "A predictive fatigue model. I. predicting the effect of stimulation frequency and pattern on fatigue," IEEE Trans. Rehabil. Eng., vol. 10, no. 1, pp. 48-58, 2002. 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. 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) 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. [108] B. Paden and S. Sastry, "A calculus for computing Filippov's differential inclusion with application to the variable structure control of robot manipulators," IEEE Trans. Circuits Syst., vol. 34, pp. 73-82, 1987. [109] D. Shevitz and B. 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Lecture Notes in Control and Information Sciences. Heidelberg, Germany: Springer, 2004, vol. 301, pp. 251-264. [115] C. Makkar, G. Hu, W. G. Sawyer, and W. E. Dixon, "Lyapunov-based tracking control in the presence of uncertain nonlinear parameterizable friction," IEEE Trans. Autom. Control, vol. 52, no. 10, pp. 1988-1994, 2007. [116] N. Sharma, K. Stegath, C. M. Gregory, and W. E. Dixon, "Nonlinear neuromuscular electrical stimulation tracking control of a human limb," IEEE Trans. Neural Syst. Rehabil. Eng., vol. 17, no. 6, pp. 576-584, Dec. 2009. [117] N. Sharma, C. M. Gregory, M. Johnson, and W. E. Dixon, "Modified neural network-based electrical stimulation for human limb tracking," in Proc. IEEE Int. Symp. Intell. Control, Sep. 2008, pp. 1320-1325. [118] C. M. Gregory, W. Dixon, and C. S. Bickel, I1p I of varying pulse frequency and duration on muscle torque production and fatigue," Muscle and Nerve, vol. 35, no. 4, pp. 504-509, 2007. [119] F. Z, ii '\! ,ucle and tendon: properties, models, scaling, and application to biomechanics and motor control," Crit. Rev. Biomed. Eng., vol. 17, no. 4, pp. 359-411, 1989. [120] H. Hatze, "A myocybernetic control model of skeletal muscle," B':. 1.'.. .'1 C,1., .,I /- ics, vol. 25, no. 2, pp. 103-119, 1977. 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. [81] X.-J. Jing, D.-L. Tan, and Y.-C. Wang, "An LMI approach to stability of systems with severe time-delay," IEEE Trans. Autom. Control, vol. 49, no. 7, pp. 1192-1195, July 2004. [82] S. Nguang, "Robust stabilization of a class of time-delay nonlinear systems," IEEE Trans. Autom. Control, vol. 45, no. 4, pp. 756-762, 2000. [83] S. Ge, F. Hong, and T. H. Lee, "Adaptive neural network control of nonlinear systems with unknown time d. 1 ,- IEEE Trans. Autom. Control, vol. 48, no. 11, pp. 2004-2010, Nov. 2003. [84] S. Mondie and V. Kharitonov, "Exponential estimates for retarded time-delay systems: an LMI approach," IEEE Trans. Autom. Control, vol. 50, no. 2, pp. 268-273, Feb. 2005. [85] D. Ho, L. Junmin, and Y. Niu, "Adaptive neural control for a class of nonlinearly parametric time-delay systems," IEEE Trans. Neural Networks, vol. 16, pp. 625-635, 2005. [86] X. Li and C. de Souza, "Delay-dependent robust stability and stabilization of uncertain linear delay systems: a linear matrix inequality approach," IEEE Trans. Autom. Control, vol. 42, no. 8, pp. 1144-1148, Aug 1997. [87] S. Ge, F. Hong, and T. Lee, "Robust adaptive control of nonlinear systems with unknown time d-1iv -," Automatica, vol. 41, no. 7, pp. 1181-1190, Jul. 2005. [88] S. Zhou, G. Feng, and S. Nguang, "Comments on "robust stabilization of a class of time-delay nonlinear systems," IEEE Trans. Autom. Control, vol. 47, no. 9, 2002. [89] S. J. Yoo, J. B. Park, and Y. H. Choi, "Comments on "adaptive neural control for a class of nonlinearly parametric time-d, 1li systems"," IEEE Trans. Neural Networks, vol. 19, no. 8, pp. 1496-1498, Aug. 2008. [90] D. Ho, J. Li, and Y. Niu, "Reply to "comments on "adaptive neural control for a class of nonlinearly parametric time-delay systems""," IEEE Trans. Neural Networks, vol. 19, no. 8, pp. 1498-1498, Aug. 2008. [91] K. Shyu and J. Yan, "Robust stability of uncertain time-delay systems and its stabilization by variable structure control," Int. J. Control, vol. 57, no. 1, pp. 237-246, 1993. [92] F. Gouaisbaut, W. Perruquetti, and J. P. Richard, "A sliding mode control for linear systems with input and state d-.1 -," in Proc. IEEE Conf. Decis. Control, vol. 4, Dec. 1999, pp. 4234-4239. [93] F. Gouaisbaut, M. Dambrine, and J. P. Richard, "Sliding mode control of TDS via functional surfaces," in Proc. IEEE Conf. Decis. Control, vol. 5, Dec. 2001, pp. 4630-4634. 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 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). 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 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 - 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 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) 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 S2010 Nitin Sharma [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 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 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 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 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 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( 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 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. 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 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 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. 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) 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. |

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PAGE 1 1 PAGE 2 2 PAGE 3 3 PAGE 4 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 PAGE 5 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 PAGE 6 .............................. 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 PAGE 7 ................................ 153 7 PAGE 8 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 PAGE 9 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 PAGE 10 .................................. 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 PAGE 11 Neuromuscularelectricalstimulation(NMES)istheapplicationofapotentialeldacrossamuscleinordertoproduceadesiredmusclecontraction.NMESisapromisingtreatmentthathasthepotentialtorestorefunctionaltasksinpersonswithmovementdisorders.Towardsthisgoal,theresearchobjectiveinthedissertationistodevelopNMEScontrollersthatwillenableaperson'slowershanktotrackacontinuousdesiredtrajectory(orconstantsetpoint). AnonlinearmusculoskeletalmodelisdevelopedinChapter 2 whichdescribesmuscleactivationandcontractiondynamicsandbodysegmentaldynamicsduringNMES.Thedenitionsofvariouscomponentsinthemusculoskeletaldynamicsareprovidedbutarenotrequiredforcontrolimplementation.Instead,thestructureoftherelationshipsisusedtodenepropertiesandmakeassumptionsforcontroldevelopment. AnonlinearcontrolmethodisdevelopedinChapter 3 tocontrolthehumanquadricepsfemorismuscleundergoingnon-isometriccontractions.Thedevelopedcontrollerdoesnotrequireamusclemodelandcanbeproventoyieldasymptoticstabilityforanonlinearmusclemodelinthepresenceofboundednonlineardisturbances.Theperformanceofthecontrollerisdemonstratedthroughaseriesofclosed-loopexperimentsonhealthynormalvolunteers.Theexperimentsillustratetheabilityofthecontrollertoenabletheshanktofollowtrajectorieswithdierentperiodsandrangesofmotion,andalsotrackdesiredstepchangeswithchangingloads. 11 PAGE 12 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 PAGE 13 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 PAGE 14 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 PAGE 15 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 PAGE 16 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 PAGE 17 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 PAGE 18 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 PAGE 19 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 PAGE 20 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 PAGE 21 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 PAGE 22 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 PAGE 23 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 PAGE 24 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 PAGE 25 Thefollowingmodeldevelopmentrepresentsthemusculoskeletaldynamicsduringneuromuscularelectricalstimulationperformedonhumanquadricepsmuscle.Themodelsimulateslimbdynamicswhenexternalvoltageisappliedonthemuscle.Thetotalmusclekneejointmodelcanbecategorizedintobodysegmentaldynamicsandmuscleactivationandcontractiondynamics.Themuscleactivationandcontractiondynamicsexplainstheforcegenerationinthemusclewhilethebodysegmentaldynamicsconsiderstheactivemomentandpassivejointmoments. Thetotalknee-jointdynamicscanbemodeledas[ 6 ] 2{1 ),MI(q)2Rdenotestheinertialeectsoftheshank-footcomplexaboutthe Figure2-1.Muscleactivationandlimbmodel.Theforcegeneratingcontractionandactivationdynamicsinthemuscleisdenotedbyanunknownnonlinearfunction(q;_q)2Rinthedynamics.ThedetailedcontractionandactivationdynamicsincludingfatigueandcalciumdynamicsareintroducedinChapter 4 25 PAGE 26 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 PAGE 27 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 PAGE 28 101 102 ],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 PAGE 29 =cosa;(2{8) wheretherstandsecondtimederivativesof(q;_q)areassumedtoexistandbebounded(seeAssumption1). 29 PAGE 30 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 PAGE 31 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 PAGE 32 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 PAGE 33 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 PAGE 34 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 PAGE 35 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 PAGE 36 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; PAGE 37 107 ],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 PAGE 38 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 PAGE 39 110 ]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 PAGE 40 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 PAGE 41 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 PAGE 42 trajectories,thereforetheRMSandthemaximumsteadystateerrorsareinbetweentherespectiveerrorsobtainedformoreaggressive2secondperiodandhigherrangeofmotiondesiredtrajectories.Figs. 3-2 3-5 depicttheerrorsfortheexperimentssummarizedinTable 3-3 Additionalexperimentswerealsoconductedtoexaminetheperformanceofthecontrollerinresponsetostepchangesandchangingloads.Specically,adesiredtrajectoryofastepinputwascommandedwitha10poundloadattachedtotheLEM.Anadditional 42 PAGE 43 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 PAGE 44 abilitytoadapttochangesinloadandstepinputsandmotivatepossiblefuturecasestudieswithneurologicallyimpairedindividualsthatexpressmusclespasticity. Foreachexperiment,thecomputedvoltageinputwasmodulatedbyaxedpulsewidthof100secandxedfrequencyof30Hz.Thestimulationfrequencywasselectedbasedonsubjectcomfortandtominimizefatigue.Duringpreliminaryexperimentswithstimulationfrequenciesof100Hz,thesubjectsfatiguedapproximatelytwotimesfasterthaninthecurrentresults.Theresultsalsoindicatethata100secpulsewidthwasacceptable,thoughfuturestudieswillinvestigatehigherpulsewidthsintherangeof300350secwhichrecruitmoreslowfatiguingmotorunits[ 110 ].Ourpreviouspreliminaryexperimentsindicatedthatlongerpulsewidths(e.g.,1msec)producedsimilareectsasadirectcurrentvoltage. 44 PAGE 45 TheuseoftheRISEcontrolstructureismotivatedbyitsimplicitlearningcharacteristics[ 96 ]anditsabilitytocompensateforadditivesystemdisturbancesandparametricuncertaintiesinthesystem.TheadvantageoftheRISEcontrolleristhatitdoesnotrequiremusclemodelknowledgeandguaranteesasymptoticstabilityofthenonlinearsystem.Theexperimentalresultsindicatethatthisfeedbackmethodmayhavepromiseinsomeclinicalapplications. AlthoughtheRISEcontrollerwassuccessfullyimplemented,theperformanceofthecontrollermaybeimprovedbyincludingafeedforwardcontrolstructuresuchasneuralnetworks(ablackboxfunctionapproximationtechnique)orphysiological/phenomenologicalmusclemodels.SincetheRISEcontrollerisahighgainfeedbackcontrollerthatyieldsasymptoticperformance,addingafeedforwardcontrolelementmayimprovetransientand 45 PAGE 46 steadystateperformanceandreducetheoverallcontroleort,therebyreducingmusclefatigue.Anotherpossibleimprovementtothecontrolleristoaccountforfatigue.Fatiguecanbereducedforshortdurationsbyselectingoptimalstimulationparameters,butfunctionalelectricalstimulation(FES)mayrequireacontrollerthatadaptswithfatiguetoyieldperformancegainsforlongertimedurations.Thereforeourfuturegoalwillbetoincludeafatiguemodelinthesystemtoenhancethecontrollerperformance. 46 PAGE 47 changes,andchangesintheload.Specically,theexperimentalresultsindicatedthatwithnomusclemodel(andonlyvoltageamplitudemodulation),theRISEalgorithmcoulddeterminetheappropriatestimulationvoltageforthetrackingobjective.Forthefastesttestedtrajectorythemaximumsteady-statetrackingerrorswereapproximately10degrees,whereasthemaximumsteady-stateerrorinslowertrajectorieswereaslittleasapproximately4degrees.Anadvantageofthiscontrolleristhatitcanbeappliedwithoutknowledgeofpatientspecicparameterslikelimbmassorinertia,limbcenterofgravitylocation,parametersthatmodelpassiveandelasticforceelements.Thus,itsapplicationwouldnotrequirespecicexpertiseorextensivetestingpriortouse.Thecontroldevelopmentalsoaccountsforunmodeleddisturbance(e.g.musclespasticity)thatarecommonlyobservedinclinicalpopulations.Theproposedstrategyholdspromisefor 47 PAGE 48 2{1 ).Duetotheuniversalapproximationproperty,NN-basedestimationmethodscanbeusedtorepresenttheunknownnonlinearmusclemodelbyathree-layerNNas[ 112 ] 48 PAGE 49 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 PAGE 50 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 PAGE 51 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 PAGE 52 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 PAGE 53 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 PAGE 54 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 PAGE 55 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 PAGE 56 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: PAGE 57 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 PAGE 58 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 PAGE 59 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 PAGE 60 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 PAGE 61 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 PAGE 62 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 PAGE 63 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 PAGE 64 additional15degreeswiththetotalloadof20pounds.TheresultsfromarepresentativesubjectusingNN+RISEcontrollerareshowninFig. 3-11 .TheexperimentalresultsforthestepresponseandloadadditionaregiveninTable 3-6 .Theresultsgivesomeindicationofthecontroller'sabilitytoadapttochangesinloadandstepinputsandmotivatepossiblefuturecasestudies. ExperimentswerealsoperformedtotesttheNN+RISEcontrollerforasit-to-standtask.Thesetestswereconductedonahealthyindividualinitiallyseatedonachair(seeFig. 3-12 ).Thekneeanglewasmeasuredusingagoniometer(manufacturedbyBiometrics 64 PAGE 65 Ltd.)attachedtobothsidesofthesubject'sknee,wheretheinitialkneeangleissettozero(sittingposition).Thegoniometerwasinterfacedwiththecustomcomputercontrolledstimulationcircuitviaanangledisplayunit(ADU301).Theobjectivewastocontroltheangularkneetrajectorythatresultedinthevolunteerrisingfromaseatedposition,withanaldesiredangleof90(standingposition).Theerror,voltage,anddesiredversusactualkneeangleplotsareshowninFig. 3-13 .TheRMSerrorandvoltageduringthisexperimentwereobtainedas2:92and26:88V;respectively.Thenalsteadystateerrorreachedwithin0:5,themaximumtransienterrorwasobservedas8:23; PAGE 66 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 PAGE 67 eectsofmusclefatigueinfuturestudies.Fatiguecanbereducedforshortdurationsbyselectingoptimalstimulationparameters,butfunctionalelectricalstimulation(FES)mayrequireacontrollerthatadaptswithfatiguetoyieldperformancegainsforlongertimedurations.Thereforeourfuturegoalwillbetoincludeafatiguemodelandincorporatingcalciumdynamicsinthemuscledynamicstoenhancethecontrollerperformance. 67 PAGE 68 116 ].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 PAGE 69 Sinceatrajectoryforaspecicfunctionaltaskwasnotprovided,thedesiredtrajectoryusedintherstsetofexperimentswassimplyselectedasacontinuoussinusoidwithaconstant1:5secondperiod.Thedesiredtrajectorywasarbitrarilyselected,buttheperiodofthesinusoidisinspiredbyatypicalwalkinggaittrajectory.Astheworktransitionstoapplicationswhereaspecicfunctionaltrajectoryisgenerated,thecontrol 69 PAGE 70 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 PAGE 71 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 PAGE 72 72 PAGE 73 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 PAGE 74 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 PAGE 75 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 PAGE 76 whereqd(t)isanaprioritrajectorywhichisdesignedsuchthatqd(t),qid(t)2L1,whereq(i)d(t)denotestheithderivativefori=1;2;3;4.Tofacilitatethesubsequentanalysis,alteredtrackingerror,denotedbyr(t);isdenedas where2Rdenotesapositiveconstant. 76 PAGE 77 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 PAGE 78 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 PAGE 79 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 PAGE 80 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 PAGE 81 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 PAGE 82 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 PAGE 83 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); PAGE 84 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 PAGE 85 controllerisimplementedonmuscledynamicswithfatiguemodelincluded.Theproposedcontrollerwasimplementedonthecompletemuscledynamicsthatincludedthefatiguedynamics.Figs. 4-2 4-3 and 4-7 showthatthesteadystateerrorinthecaseofproposedcontrollerremainswithin0:5forbothslowandfasttrajectories.Fig. 4-8 showshowthefatiguevariableevolveswithtimeasadeceasinginputgain.Theproposedcontrollerisabletocompensateforthedecreasingcontrolgain,andtheperformancedoesnotdegradeovertimeasshowninFig. 4-7 85 PAGE 86 performedtoproveuniformlyultimatelyboundedresultinthepresenceofboundeddisturbances(e.gmusclespasticity),parametricuncertainties.SimulationresultsclearlyillustratethattheproposedcontrollerperformsbetterintermsofreducederrorincomparisontotheRISEcontroller.However,theperformanceofthecontrolleronvolunteersorpatientsremainstobeseen.Thecontroller'sdependenceonaccelerationandmathematicalfatigueandcalciummodelshinderitsimplementationonvolunteers.Themathematicalcalciumandfatiguemodelswereincorporatedduetothefactthatthemeasurementofactualfatiguestateandcalciumvariableisdicult.Futureeortscanbemadetoincorporateanobserver-baseddesigninthecontrollerinordertoestimatethefatigueandcalciumstates. 86 PAGE 87 Figure4-6.RISEcontrollerwithfatigueinthedynamics:Topplotshowsthekneeangleerrorfora6secondperiodtrajectoryusingtheRISEcontroller.MiddleplotshowsthepulsewidthcomputedbytheRISEcontroller.Bottomplotshowstheactuallegangle(dashedline)vsdesiredtrajectory(solidline). 87 PAGE 88 Figure4-8.Fatiguevariable 88 PAGE 89 Theprimarymotiveofthisresearchistodevelopandimplementacontrollerthatcompensatesforelectromechanicaldelay(EMD)inNMES.ThelastsectionofthechapterfocussesoncharacterizingEMDduringNMES.ExperimentsresultsobtainedfromhealthyvolunteersareprovidedwhichdescribetheeectofstimulationparametersontheEMDduringNMES.Finally,aPDcontrollerwithanaugmentedpredictorcomponent 89 PAGE 90 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 PAGE 91 wherem1;m22R+areknownconstantsandkkdenotesthestandardEuclideannorm. where1;22R+areknownconstants. 5.3.1Objective 5{1 )totrackadesiredtrajectory,denotedbyqd(t)2Rn.Toquantifytheobjective,apositiontrackingerror,denotedbye1(t)2Rn,isdenedas 91 PAGE 92 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 PAGE 93 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 PAGE 94 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 PAGE 95 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 PAGE 96 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 PAGE 97 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 PAGE 98 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); PAGE 99 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. PAGE 100 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 PAGE 101 _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): PAGE 102 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: PAGE 103 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)): PAGE 104 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 PAGE 105 124 ]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 PAGE 106 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 PAGE 107 inthemusculoskeletaldynamics[ 6 ]andoccursduetoniteconductionvelocitiesofthechemicalionsinthemuscleinresponsetotheexternalelectricalinput[ 36 ].InputdelaycancauseperformancedegradationaswasobservedduringNMESexperimentaltrialsonhumansubjectswithRISEandNN+RISEcontrollersandhasalsobeenreportedtopotentiallycauseinstabilityduringhumanstanceexperimentswithNMES[ 40 ]. 3.4.4.1 ).ThedelayinNMES 107 PAGE 108 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 PAGE 109 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 PAGE 110 restposition.Fig. 5-5 showsthetypicalEMDduringNMESinahealthyindividual.Finalinputdelayvaluewascomputedbyaveragingthemeasureddelayvaluesoverthreeimpulses.Eightexperimentswereperformedfordierentfrequencies,wherethefrequencywaschosenrandomlyfromtherangeof30Hzand100Hz(intrarangeintervalof10 110 PAGE 111 Hz).Thepulsewidthforthistypeoftheexperimentswaskeptat100s.Thesecondtypeofexperimentsconsistedofvaryingpulsewidthwhilekeepingvoltageandfrequencyconstant.Eachexperimentconstitutedthreeimpulsesasexplainedaboveforthefrequencytests.Nineexperimentswereperformedfordierentpulsewidths,wherepulsewidthwasrandomlychosenfrom100s.to1000s(intrarangeintervalof100s).Forthisset 111 PAGE 112 ofexperiments,thefrequencywaskeptconstantat30Hzandthevoltageconsistedofminimumthresholdvoltage+10V.Thelastsetofexperimentsinvolvedconductingexperimentswithvaryingvoltages.Sameimpulseprogramasusedintheearliersetofexperimentswasused,wherepulsewidthandfrequencywerekeptconstant.Thefrequencywaskeptat30Hzandthepulsewidthwaskeptat100mus.Threeexperimentswereperformedfordierentvoltages(thresholdvoltage+additionalvoltage,whereadditionalvoltagewasvariedbetween5and20volts(intrarangeintervalof5volts).Table 5-4 (asarepresentativeexample)showsthesummarizedinputdelayvariationswithrespecttodierentstimulationparametersinahealthyindividual. ANOVA(Analysisofvariance)testswereperformedtodeterminetheintraclasscorrelations.AnANOVAtestisgenerallyemployedtodeterminethestatisticalsignicancebetweenthemeansofdatagroupsnumberingmorethantwo(usingstudentt-testto 112 PAGE 113 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 PAGE 114 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 PAGE 115 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 PAGE 116 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 PAGE 117 5-9 and 5-10 Figure5-9.Topplot:Actuallimbtrajectoryofasubject(solidline)versusthedesiredtrajectory(dashedline)inputobtainedwiththePDcontrollerwithdelaycompensation.Middleplot:Thetrackingerror(desiredangleminusactualangle)ofasubject'sleg,trackingaconstant(2sec.)perioddesiredtrajectory.Bottomplot:ThecomputedvoltageofthePDcontrollerwithdelaycompensationduringkneejointtracking. 117 PAGE 118 118 PAGE 119 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 PAGE 120 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 PAGE 121 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 PAGE 122 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 PAGE 123 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 PAGE 124 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 PAGE 125 ~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 PAGE 126 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; PAGE 127 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 PAGE 128 _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: PAGE 129 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 PAGE 130 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; PAGE 131 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 PAGE 132 132 PAGE 133 Figure6-4.Trackingerrorforthecase=10s: PAGE 134 134 PAGE 135 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 PAGE 136 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 PAGE 137 131 ],whichcanbesurgicallyimplantedatspecicsitesinthemuscle.Thesemicroelectrodeswhichdonotrequirewiresarepoweredexternallythroughaninductivecoilandabattery.MultipleBIONstostimulatespecicmusclesitescannotonlybeusedtoproducedesiredfunctionalmovementsbutalsocanbeusedtoeliminatemusclefatiguethroughutilizingnon-repetitiveandselectivemusclerecruitment.InordertoproduceNMEScontrolviaBIONs,studieswillberequired 137 PAGE 138 5 showedthattheinputdelayinNMESdependsonlyonvaryingfrequency.However,furtherinvestigationsarerequiredtostudytheeectoffatigueandnon-isometriccontractionsoninputdelay.Also,resultsinChapter 5 areonlyapplicablewithknownconstantinputdelayvalues.Therefore,controllersneedtobedevelopedtoaccountfortime-varyingorunknowninputdelay.Otherdelaycompensatingtechniquessuchasmodelpredictivecontrol(MPC)canalsobeinvestigatedforNMES.OneoftheadvantagesofMPCisthatitinherentlycompensatesforinputdelays.Althoughthetechniquewouldrequiremuscledynamicstobeknown,advantagessuchasperformanceandcontroloptimizationinadditiontodelaycompensationmakesMPCaworthycandidateforinvestigation. 138 PAGE 139 _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 ) PAGE 140 1 140 PAGE 141 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 PAGE 142 6{20 )andthefactthat theexpressionin( 2{14 )canbeupperboundedasZt0L()dnXi=1jeni(0)jeTn(0)Nd(0)+(Ndkenk)+Zt0kenkNd+_Nd Itisclearfrom( 2{17 )thatifthefollowingsucientcondition issatised,thenthefollowinginequalityholds 142 PAGE 143 [1] P.H.PeckhamandD.B.Gray,\Functionalneuromuscularstimulation,"J.Rehabil.Res.Dev.,vol.33,pp.9{11,1996. 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