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ANGULAR MOMENTUM DURING GAIT A COMPUTATIONAL SIMULATION By CAMERON NOTT A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010 2010 Cameron Nott To my beautiful wife ACKNOWLEDGMENTS First and foremost, I would like to thank my parents, sisters and grandparents who have been so supportive. I also ask their forgiveness for not being with them at the weddings, childbirths, surgeries, funerals and many other important events. Although my heart is heavy after the ten years spent more than fifteen thousand miles away from them, I am very grateful for the sacrifices they have made to allow me to obtain my doctoral degree. I especially thank my mother and father for remaining so positive and supportive despite all the tears shed at the airport every year or so. My education is as much their achievement as it is mine. Words cannot express the extent of my thanks. I would like to thank Dr. Kautz for always believing in my ability and being the prime example of professionalism, intelligence, and a super advisor in general. I thank Dr. Fregly for reestablishing my passion for exploring a small portion of a unified physical existence that God has created for us. Dr. Bowden and Dr. Gregory also deserve extended thanks for all the help and guidance along the way at a professional and personal level. My appreciation is also for Dr. Olcmen for helping me discover my potential, for the continuing advice, and for being a fantastic man and advisor. Finally and most importantly, I would like to thank my wife who has managed to make me laugh every day, supported me in every possible situation, and always been so exceptionally kind and warm hearted. TABLE OF CONTENTS page A C KN O W LED G M ENTS ............... ........................................... ............... 4 LIS T O F F IG U R E S .................................................................. 7 LIST OF ABBREVIATIONS..................... .......... .............................. 10 ABSTRACT ............... .................................... ....... ..... ...... ......... 11 CHAPTER 1 INTRODUCTION ........................... ............. .................. 13 Reduced Balance Capacity in the Stroke Population.................. ...... ........... 13 Dynamic Balance During Gait Can Be Studied Through Whole Body Angular Momentum, Foot Placement, and Joint Power Production.............. .......... 14 W hole Body Angular Momentum ......................... ................ ............. 14 Foot Placem ent D during G ait ................... .............. ............... ... ............... 16 Joint Power Production and Maintenance of Whole Body Angular Momentum ............ ..... ......... .. .. ...... ............ 17 Hemiparesis Adversely Affects Balance via Foot Placement and Power Production ............... ........ .. ... ........ ... .......... .. .......... ............ 17 Biomechanical Simulation a Means to Fully Understand Inadequate Power Generation as it Relates to Regulation of Angular Momentum.......................... 18 All Joint Moments Significantly Contribute to Whole Body Angular M om entum ........................ ... ..... .......... ..................... 19 Biomechanical Simulation Requires the Development of a Ground Contact M o d e l ............... ..... ...... ...... .... .... ... .......... ......................... 2 0 Biomechanical Simulation Optimization Will Reveal the Relation Between Joint Power Production and Whole Body Angular Momentum................... 21 C o nclusio n ......... ...... ............ .................................. ........................... 2 2 2 THE SENSITIVITY OF TRUNK ANGULAR ACCELERATION TO ALL JOINT MOMENTS ............... ......... ....... ....... .... ... ........ ....... ........ 23 Intro d uctio n ......... ...... ............ ................................. ........................... 2 3 M e th o d s .............. ..... ............ ................. ................................................. 2 4 S u b je c t .............. ..... ............ ................. ............................................ 2 4 Data Collection ...................................... ......... ......... ......... 24 Model........................................... 25 Results and Discussion.......................................... ............... 28 Sagittal Plane ................. ......... .......... ........28 F ronta l P lane ........................ ...... ..................... ..... ... .......... ........ .......... 29 Joint Moment Contributions to Trunk Angular Accelerations.......................... 29 Control of Trunk Angular Acceleration by the CNS ............... ............... 30 3 THE DEVELOPMENT OF A GROUND CONTACT MODEL FOR BIOMECHANICAL SIMULATION FROM TREADMILL DATA.............................. 38 Introduction ........................... ............... 38 Methods ............................ .................................... 39 Subjects................................................ 39 Data Collection .............. ......... .......... ........ 40 Pressure Sensitive Film ................................................... ... .. ............... 41 ThreePoint Calibration ......................... ......... .. ........ .......... ......... 43 Foot Model ........................................................................ ... ......... .................. 44 Full Body Model............................................. ............... 44 ViscoElastic Contact Elements ........................ .... ......... ...... ............... 46 F rictio n M o d e l ....................................................................... ................................. 4 9 Coulomb Friction ....... ....................... ........ ............... 49 V iscous Friction .............................................. .. .............................. 51 Torsional Friction ................................................... 51 Kinematic Optimization ............................................ 52 Results and Discussion.......................................... ............... 53 C o n c lu s io n ............. ........... .................. ................................... ............... 5 4 4 ANGULAR MOMENTUM CONTROL DURING GAIT ................ ................ 69 Introduction ....................... ...... ............... 69 M e th o d s ............... ....... .. ..................................................................... 7 1 Subjects ........... ............................ ........ ...... ............... 71 Data Collection ....................... ... ........................... 72 Model........................................... 72 Perturbation ............. ................ ......... ............... .... .... ............... 72 Whole Body Angular Momentum During the Gait Cycle ............................. .... 73 S a g itta l P la n e : ....... .................................................. ........... ............... 7 3 Angular Momentum in the Frontal plane ............... .. ............. .. ............... 74 The Relation Between Joint Power and Whole Body Angular Momentum.............. 75 Step Length and Moment Impulses Relation to Angular Momentum Correction..... 76 The Effects of Reduced Joint Power Production on Resisting Perturbation D u rin g G a it ............. ........... .. .......................... ............... 8 1 C o n c lu s io n ............. ........... .................. ................................... ............... 8 2 5 CONCLUSION AND DISCUSSION .......................... .......... 90 LIST OF REFERENCES ................... ............................... 92 BIOG RAPHICAL SKETCH ........................ ........ .......... .. .............................. 97 6 LIST OF FIGURES Figure page 21 Comparison of the current model (blue) to data from previously published w ork (red) (W inter 1991) ................................. ....... .. ............................. 31 22 The sensitivity of the trunk angular acceleration about the xaxis to the joint moments during gait ........................................... 32 23 The sensitivity of the trunk angular acceleration about the yaxis to the joint m om ents of the leg during gait.. ............. ............ ......... .......... ............... 33 24 The sensitivity of the trunk angular acceleration about the yaxis to the joint m om ents of the leg during gait.. ............. ............ ......... .......... ............... 34 25 Frontal and sagittal plane views illustrate how a sagittal moment can cause frontal plane accelerations by means of reaction forces along the kinematic c h a in ......... ............................................... .................................. 3 5 26 Contribution of each joint moment to trunk angular acceleration in the sagittal plane...................................... ......... ......... 36 27 Two methods of calculating the contact moments are illustrated by taking free bodies of the trunk and lower extremity ....................... .......................... 37 31 Alignment of the pressure film in the foot reference frame relative to the m option tracking m arkers .................. .... ............................. ................. 56 32 Calibration of the pressure film ........ ........ .... ................. ... ............... 57 33 Dimensionality reduction using kmeans clustering.. ................. ........ ......... 58 34 Anthroprometric model with (left) and without (right) segmental coordinate system definititions. ..................... ........ ....... ............... ............... 58 35 The convergence of an optimization to find the left shoulder joint center from a po o r in itia l g ue ss.................. ............................. .......... ...... 5 9 36 Comparison of the viscoelastic model's reproduction of the experimental data that w as used to train the m odel........................................... ... .................. 60 37 Comparison of an optimized viscoelastic element to experimentally collected data ............ .. .................................... ............ ........... ...... 6 1 38 The comparison of an example experimentally measured element penetration and a hypothetical penetration that would cause a viscoelastic element to exactly match its corresponding ground reaction force. The difference in the measurement falls well within the experimental error .............. 62 39 On an ADAL ................. .... ........................ ............ 63 310a The weight is at rest on the treadmill belt with no force being applied ............... 64 310b With a 9.7 kg force being applied along the yaxis (along the walking direction on the treadmill) no sliding motion of the belt or the weight on the belt occurs. ................................................... .... ........... 65 311 Kinematic alterations in joint angles produce foot motion that accommodates foot/ground contact modeling. ............................................... ............... 66 312 The effect of kinematic variations removes the nonphysical negative damping characteristic seen in the original experimental data .............. .......... 67 313 Resulting force and moment production of the foot contact model post kinematic optim ization ........... ............................. .................. 68 41 The angular momentum at the beginning of swing versus the angular momentum at midswing (left). The angular momentum at the beginning of swing versus the change in angular momentum through swing (right) .............. 84 42 AM profiles for normal (blue) and perturbed (red) states............... .............. 84 43 Joint powers from unperturbed (blue) and perturbed (red) states for a control and a hem iparetic subject............................................................. ......... 85 44 Calculated foot placement limits ..................... .................. ... ............... 86 45 Angular momentum of the model undergoing control via a simple low gain PID controller ........................................ ........... 86 46 The top row figures are from a simulation that reflects how a short step length for a given speed can result in a fall forward......................................... 87 47 The angular momentum profile for 1 gaticycle of the model under PID control for the model falling over due to a shortened step length (see figure 45, top) and the model walking normally (see figure 45, bottom). ............. ............... 87 48 The kinematic response of the subject to an anterior perturbation force of 10 Ibs applied just above the center of mass................................... ............ 88 49 The angular momentum profiles of the model undergoing perturbation with PID control gains. .......... ............ ......... ................ .... ............ 88 410 The perturbation force applied by the Force pod. The command was given to apply a 10b (44.49N) over a period of 0.5 seconds (100 Frames)................... 88 411 The response of the model to a 15% reduction in joint angular velocity in the left leg. The perturbation force of ten pounds caused a fall forward almost immediately. ............. .......... ...................... 89 LIST OF ABBREVIATIONS AM Angular Momentum COM Center of mass COP Center of pressure GRF Ground reaction force GRM Ground reaction moment acting at the COP 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 CONTROL OF ANGULAR MOMENTUM DURING HEMI PARETIC GAIT A COMPUTATIONAL SIMULATION By Cameron Nott August 2010 Chair: B. J. Fregly Major: Mechanical Engineering There is a 73% incidence of falls among individuals with mild to moderate stroke. 37% of patients that fall sustain injury that required medical treatment. 8% sustained fractures and the risk for hip fracture is ten times higher in the stroke population. General risk of fracture is two to seven times greater following a stroke. Due to the impact falls have on the health and well being of persons with stroke, there is a substantial need to assess dynamic balance during walking in order to ascertain who is at increased risk for falls. While little has been done to quantify balance during walking in the general population, much less has been done to quantify the effect of neurological impairment on balance during gait in persons with stroke. Approaches for determining who is at risk for falls have usually relied on clinical measurement scales as opposed to quantitative evaluation of joint power production ability. However, interpretations of these measures are based on statistical evidence and do not quantify a subject's balance performance during walking. To quantify and ultimately improve dynamic balance during neurologically impaired walking, it is necessary to quantify the biomechanical effects of altered or impaired joint power productions during paretic gait. These effects are relevant since a primary disability associated with poststroke hemiparesis is the failure to make rapid graded adjustment of muscle forces. Reduction in, or ill coordinated, muscle forces result in inefficient and ineffective net joint power production. In order to quantify the biomechanical effects related to a decrease in joint power production, a detailed biomechanical simulation of the human body during perturbed walking is necessary to elaborate on experimental findings. A simulation that can explore the effects of reduced power production on balance requires the development of a ground contact model that can explain the production of ground reaction forces and moments based on the deformation/kinematics of the foot. The following work develops the ground contact model and the biomechanical simulation that mathematically relates reduction in joint power production to deficits in balance. Furthermore, the work investigates the role of various joints' in maintaining balance and quantifies successful balance with a mathematical measure: whole body angular momentum. To improve understanding of the relation between joint power production and balance, the simulation is subjected to an optimization procedure that results in kinematic and kinetic solutions that return whole body angular momentum to a regular limit cycle following a perturbation. Studying the return of whole body angular momentum to a regular limit cycle after a perturbation assists in understanding balance since whole body angular momentum collectively quantifies rotational velocity of the entire body about the center of mass and in order to maintain balance, the rotational velocity of the entire body is necessarily bounded. CHAPTER 1 INTRODUCTION Reduced Balance Capacity in the Stroke Population The impact of stroke on walking is significant [1], with only 37% of stroke survivors able to walk after the first week. Even among those who achieve independent ambulation, significant residual deficits persist in balance [2, 3] and gait speed with a 73% incidence of falls among individuals with mild to moderate impairment 6 months poststroke[4]. A primary disability associated with poststroke hemiparesis is the failure to make rapid graded adjustment of muscle force and to sufficiently activate muscles [5]; consequently, a reduced ability to rapidly respond to perturbations that influence balance is inferred. Various compensatory walking patterns have been suggested to increase balance in hemiparetic gait (e.g., increased double support time); however, it is difficult to state whether kinematic or kinetic variations during paretic gait are adaptations done by choice to increase stability or the result of an impaired neural controller and decreased power production. Regardless, persons with hemiparesis do not have robust control over dynamic balance during gait and little is understood about how to quantify their ability to retain balance. Balance control is an interaction of the individual, the task, and the environment as actuated by the neuromuscular system. While various theoretical and clinical models exist for static balance, they are not directly applicable to all of the dynamic aspects of balance during walking [6]. Most falls occur during walking [7] but most clinical balance assessment tests do not include assessment of dynamic walking. Balance control mechanisms are specific to the task being performed and the environment in which the task is being performed [8] and recent walking recovery literature emphasizes the importance of task specificity during rehabilitation [9, 10]. None of the clinical assessments of balance control in gait allow for the assessment of reactive balance mechanisms, which are critical for both postural and equilibrium control. These reactive responses depend on rapid input from subcortical mechanisms depending on somatosensory and vestibular sources, both of which are often impaired in those with stroke [11]. There is a substantial need to assess dynamic balance and reactive responses during walking in order to ascertain who is at increased risk for falls and whether a targeted intervention has improved dynamic balance. Dynamic Balance During Gait Can Be Studied Through Whole Body Angular Momentum, Foot Placement, and Joint Power Production The principle of whole body angular momentum is the primary connection between joint power production and balance. Foot placement is a means of angular momentum regulation and a fall in any direction requires appropriate foot placement to regain balance. Whole Body Angular Momentum Whole body angular momentum (AM) is the summation of segmental moment of inertia multiplied by angular velocity. Thus, it collectively quantifies the rotational velocity of the entire body about the center of mass (COM) in a single measure. It can be calculated from kinematic data where the AM of the entire body is expressed as the superposition of individual segmental AM. To find the whole body AM about any arbitrary point q, a summation is performed [12]. Hq = ili Oi + (Pq Pi) X m(Vq Vi) (11) In the summation, Hq is the whole body AM about point q. Ii is the i'th segment's inertia tensor and oi is its angular velocity. pq and Vq are the position and velocity of point q respectively while pi and Vi are the position and velocity of the segment's center of mass. m is the segment's mass. To calculate whole body AM from kinematics, an anthropometric model is required to determine m and I and to measure segmental kinematics. A typical 3D model would consist of 13 rigid segments (two feet, two shanks, two thighs, pelvis, trunk, two forearms, two upper arms, and a head). Furthermore, a 3D model is imperative to ensure that interplanar coupling is taken into account. A 2D model will not reflect interplanar coupling [13]. To maintain upright walking, the whole body AM should maintain oscillation about zero, since a constant nonzero AM implies continuous rotation of all segments about an axis (other than the vertical axis) which would most likely represent falling [14]. Previously it has been noted that intersegmental AM cancellations are mostly responsible for a "close to zero" whole body AM and various successful control schemes have been developed based on this "zero moment assumption" in the robotics literature [15]. In highly sophisticated bipedal robots, control systems that maintain a constant zero AM, despite perturbation, result in stable and robust multisegmental bipedal gait. This has been described as the only reliable stability index of dynamic bipedal locomotion in robotics [16]. Despite the success of such control systems in robotics, it is noted from experimental results that humans do not follow this strategy completely. Humans tend to allow oscillation in AM about zero during gait [12, 17]. As opposed to the zero moment assumption, which assumes that the subjects strive to maintain a constant zero AM throughout the gait cycle, we hypothesize that inter segmental AM cancellations are not the primary mechanisms of AM stabilization. The hypothesis states that healthy and efficient walking requires fluctuation of the AM from zero, and it requires a mechanism whereby the AM can be directed to zero when necessary in normal gait or perturbed walking. We propose that this regulation occurs during normal and perturbed gait because AM is often manipulated purposefully during certain tasks [17]. Phases of normal walking have been described as a controlled fall, which can be thought of as a nonzero fluctuation in AM. To control this fall, it is proposed that foot placement is the primary mechanism by which AM is controlled [18, 19]. Foot Placement During Gait Little or no attention has been given to whole body AM in conjunction with foot placement and balance, especially during gait. Inverted pendulum models (which model the human body as a rigid strut with a point mass on the end) have been used to determine margin of stability (essentially the proximity of the center of mass to the base of support after taking into account the velocity of the center of mass) as a measure of dynamic balance [20]. Inverted pendulum models neglect the angular inertial properties of the human body amongst other assumptions. Furthermore, theoretically determined margin of stability have not been experimentally verified to relate to impaired subjects ability to maintain balance during gait. Whether inverted pendulum assumptions are justified has been discussed by Herr et al. [12] with respect to human balance and Goswami [2123] with respect to bipedal robots. Regardless, it should be noted that whole body angular momentum carries vital information that allows a mathematical prediction of where a foot should be placed such that the subject does not fall [18, 19]. The maintenance of AM during gait therefore forms the theoretical basis of the this research utilizing the fact that foot placement is a means to ensure the return of AM to zero from a nonzero state, without disrupting the center of mass trajectory. Joint Power Production and Maintenance of Whole Body Angular Momentum Two factors are related to directing the angular momentum in the correct direction: the moment arm from the center of mass to the center or pressure, and the force acting at the center of pressure. These two factors can be related to whole body angular momentum in a simple fashion: S= M + Rx GRF (12) In this relation, H is the time rate of change of the angular momentum about the center of mass, GRM is the ground reaction moment and GRF is the ground reaction force. R is the vector from the center of mass to the center of pressure which is located on the ground surface. To substantiate a significant alteration in whole body angular momentum, the above relation shows that a significant contribution from a single foot is related to the distance of that foot from the center of mass and the magnitude of the ground reaction force that is produced by that foot. Since the ground reaction force can be related to any single joint moment by the standard inverse dynamics equations, joint power production is directly related to the change in angular momentum about the center of mass (as is foot placement). Hemiparesis Adversely Affects Balance via Foot Placement and Power Production Impaired subjects with an inability to rapidly respond to perturbation, do not have the power production capacity to place the foot fast enough to regain balance within the gait cycle [24]. Studying the response to perturbation will allow for the classification of balance recovery ability according to neurological deficit and also rate balance recovery strategies. Understanding the inability of paretic subjects to regain control of their angular momentum will further develop clinical assistance in the prevention of falls. When AM is perturbed, there is a time constraint that must be met to place the foot in order to correct the AM fluctuation. If the foot that is to correct AM is placed excessively late or not far enough from the COM, a fall may occur [18, 19]. Control subjects are expected to respond more rapidly than hemiparetic subjects and have foot placement that is more consistent with a mathematically viable solution for rectifying the postperturbation angular momentum [24]. Since hemiparetic subjects have poor coordination and reduced power production ability, they are likely less able to rapidly place a foot far enough forward to be in a position that can reestablish dynamic balance. Thus, hemiparetic subjects should require more time and steps to regain a steady AM profile and demonstrate more fluctuation in angular momentum for the same perturbation when compared to a control subject. Subjects who are more likely to fall are expected to be those who are less able to quickly regain their AM profile. Studying the dynamic balance response to perturbation may allow for the classification of dynamic balance performance according to specific neurological deficits. Biomechanical Simulation a Means to Fully Understand Inadequate Power Generation as it Relates to Regulation of Angular Momentum Biomechanical simulation is growing in popularity as a means to understand the mechanics of the human body. It is the application of the first principles of mechanics to living organisms. The first instance of the application of mechanics was written by Aristotle and was titled De Motu Animalium or On the Movement of Animals. In this work, Aristotle first performed "mind experiments" to determine how an animal could achieve a task without actually requiring the animal to do the task. Today the science ranges from learning about flight from the behavior of insects to making more comfortable mattresses. In this application biomechanical simulation is used to determine the joint power production requirements needed to overcome certain perturbing forces during gait. In order to complete the simulation, three topics are to be addressed. The first topic is the effect of joint moments on segments' angular momentum to which the joint does not directly apply. Since the trunk is more than one third the mass of the entire body, the effect of joint moments of the lower extremities on trunk angular acceleration is sufficient to establish whether whole body AM can be affected by all the lower extremity joints. The second is the generation of a suitable ground contact model. The third is the development of a biomechanical model to which optimization routines can be applied to determine suitable joint power productions that can stabilize whole body AM after a perturbation. All Joint Moments Significantly Contribute to Whole Body Angular Momentum Computationally advanced biomechanical analyses of gait demonstrate the often counterintuitive roles of joint moments on various aspects of gait such as propulsion, swing initiation, and balance [13, 2528]. Each joint moment can produce linear and angular acceleration of all body segments (including those on which the moment does not directly act) due to the dynamic coupling inherent in the interconnected musculoskeletal system. This study presents the quantitative relationships between individual joint moments and trunk control with respect to balance during gait to show that the ankle, knee, and hip joint moments all affect the angular acceleration of the trunk and consequently the angular momentum of the trunk which is a large portion of the whole body angular momentum which cannot be accounted for by intersegmental angular momentum cancellations. We show that trunk angular acceleration is affected by all the joints in the leg with varying degrees of dependence during the gait cycle. Furthermore, it is shown that interplanar coupling exists and a two dimensional analysis of trunk balance neglects important outofplane joint moments that affect trunk angular acceleration. Biomechanical Simulation Requires the Development of a Ground Contact Model Of the intricate modeling necessities specific to human gait, footground contact has been presented as challenging. The most common approach presented is a spring damper model in which springdamper elements dictate ground reaction forces [2934]. Another approach is to assume a welded joint between the ground the foot [3538] or to model the foot as a rigid rolling surface [39]. More complex methods such as finite element models from study on cadavers and magnetic resonance imaging have been proposed [40]. Viscoelastic sphere models have also been introduced to represent the plantar surface of the foot [41]. Viscoelastic spherical contact models have been suggested with increasing interest to model the foot [34, 41]. A contact model based on a viscoelastic sphere was first studied by the H. R. Hertz [42] and expresses the relationship between the sphere's deformation and the resulting force that causes the deformation. The viscoelastic sphere model presented by Guler et al relies on experimental data from heel pads and Guler et al conclude that viscoelastic spheres can be incorporated into foot models for simulation. More complex spherical models have been presented that replace constant exponent of 3/2 that exists in Hertz's model with a variable n which dictates the geometry of the model (Marhefka 1999). The primary procedure by which a contact element's parameters are derived is with forward simulation [2934]. In determining spring and damping constants in forward simulation, it is permissible to allow a certain amount of variation in kinematic response. This is typically the cost function of the optimization and the resulting kinematics of the foot does not match experimental kinematics exactly. The contact model developed herein uses neural networks to accommodate for experimental inaccuracies with respect to kinematic measurements. The input into the model is the kinematics of a set of penetrative elements that are fixed in a multi segmented foot. The positions of the elements are determined with a three point calibration using a pressure sensitive film. The result is a foot contact model that can accept measured kinematics and duplicate ground reaction force measurements. This is useful for inverse dynamic optimizations where the input is a set of simulated joint angles which, with the ground contact model, fully define the ground reaction forces and moments at the feet. Biomechanical Simulation Optimization Will Reveal the Relation Between Joint Power Production and Whole Body Angular Momentum To understand and quantify the regulation of whole body AM with reduced power production ability, an optimization will be run to determine the joint angle trajectories that will reestablish a biomechanical simulation's whole body AM to a normal trajectory as determined from experimental data. The experiential data is collected with a novel, stateoftheart perturbation system on an instrumented treadmill that fully records bilateral ground reaction forces and moments. Furthermore, a motion capture system measures the kinematics of the subject. With such a computational model in place, it will be possible to simulate perturbations applied to patients with hemiparesis without subjecting them to a dangerous situation or greater risk of falls. The simulation will also determine and quantify minimal joint power requirements needed to overcome various levels of perturbation forces. Conclusion The presented research is a means to quantify the balance capability of subjects with hemiparesis based on biomechanical principles and power production capability; something that is much needed in the clinical environment. The end result is a quantification of foot placement requirements to maintain balance during gait. In order to produce a quantification of a subjects' ability to withstand perturbation forces during gait, three tasks are to be achieved. The first is the establishment of the sensitivity of trunk angular acceleration on lower extremity joint moments. The second is the development of a ground contact model. The third is the application of the first and second tasks to a computational simulation using a biomechanical model that can determine the effects of reduced joint power production and it effect on overcoming perturbation. CHAPTER 2 THE SENSITIVITY OF TRUNK ANGULAR ACCELERATION TO ALL JOINT MOMENTS Introduction Dynamic balance during walking consists in large part of stabilization of the trunk. Trunk angular sway has been indicated as a reliable measure of balance stability [43, 44]; however, there are inconsistencies in the literature as to how each of the lower extremity joint moments affects trunk angular acceleration. Previous studies have argued that because the inertia of the trunk about the ankle joint is nearly eight times the inertia of the trunk about the hip joint, the ankle muscles would need nearly eight times the momentofforce relative to provide the same angular acceleration of the head, arms, and trunk segment as the hip muscles [45]. They concluded that the central nervous system (CNS) may recognize this and utilize the hip muscles as the primary actuator for trunk angular control with little involvement of the ankle muscles during gait. Alternatively, EMG studies have concluded that the role of distal musculature is as important as proximal hip musculature in maintaining balance using a movable platform to allow for perturbations [11, 46, 47]. Similar observations, using joint power and muscle EMG patterns, show that reactive balance control is more likely a synchronized effort of the lower extremity joint moments to prevent collapse during perturbations [48]. Although the contribution of each muscle (or joint moment) has been explored with regards to trunk propulsion and support and segmental power flow [2628, 4951], similar relationships to trunk angular acceleration have been relatively unexplored. In the present study, we use a three dimensional model and the resulting forward dynamic equations of motion to identify the relative contribution of each joint moment to the trunk angular acceleration and assess whether one joint more strongly accelerates the trunk than the others. We also analyze the contribution of sagittal plane moments on frontal plane trunk accelerations that are made possible through dynamic coupling [52], along with the contribution of frontal plane moments on sagittal plane accelerations, to assess whether caution is necessary when interpreting trunk angular acceleration with only a single plane analysis [53]. Methods Subject A single subject whose joint moments and joint angular velocities are consistent with published data used in similar trunk control research (e.g., Winter, 1991) (Figure 2 1) was used for demonstration purposes. The subject walked at selfselected speed on an instrumented treadmill (Techmachine, Andrezieux Boutheon, France) for three trials of thirty seconds each. Data Collection Kinematic data were collected using a 12 camera VICON system and sampled at 100Hz to capture spatial positions of markers placed on the subject in a modified Helen Hayes configuration. Rigid clusters of markers were placed on the feet, left and right shanks, left and right thighs, pelvis, trunk, head, and upper and lower arms to allow for six degree of freedom orientation measurement of each segment. Hip joint centers were determined using the CODA algorithm [54]. In this work we have assumed that treadmill and overground gait are similar enough to make this study relevant to overground walking. We believe this is justified because each muscle will still accelerate all segments in exactly the same way in each mode of walking. Model A 12 segment rigid body model consisted of two feet, shanks, thighs, forearms, and upper arms, a head, and a combined pelvis and trunk segment. Each segment's inertial characteristics were defined using Visual 3D (Cmotion, Germantown, Maryland). The equations of motion were derived using a standard d'Alembert approach and a sensitivity matrix relating the joint moments to trunk accelerations was derived. Each of the 12 segments of the subject was assumed to be a rigid segment and modeled accordingly. The equations that governed the system are shown in equation 1. Fe + F + F = contact + distance inerfa (21) M contact +M ds tan ce inertia = Here, F is a force vector acting on a segment and M is the moment vector acting on a segment. The subscripts denote the type of force or moment as being either contact, distance, or inertia which describes forces or moments due to contact, gravity, and inertial forces respectively. The inertial forces are further expanded in equation 2 as: asnerta = m a ne a(22) MCa = R (Ra I (R. )x(IR.)) In equation 2, the inertial moment is taken about the centerofmass (COM). R is the rotation matrix, which describes the segment in the lab reference frame, and I is the inertial matrix in the segment reference frame. Since the inertial matrix dotted with the rotation matrix generally results in a matrix with nonzero diagonal terms, rotational dynamic coupling is possible. The mass of the segment is defined by m and the angular acceleration and velocity of the segment with respect to the lab reference frame is defined by a and respectively. The inertial moments acting on the segment can be expressed about any point, 0, in the lab using equation 3. Mo po coM +MCOM (23) Inerrta Linertia Inertia In equation 3, po/cOM is the vector from point 0 in the lab to the COM of the segment, which is necessary to ensure that summation of the contact, distance, and inertial moments occur about the same point with respect to the same reference frame. Equations 1, 2, and 3 can be applied to any body or group of bodies which are selected as the free body in question. In order to establish the trunk angular acceleration as a function of the lower extremity joint moments, joint constraints are applied to the ankle, knee, and hip. The equation used to do so is given by equation 4. aD a +ad x P + a x P + 6d (pd x Pd)+ 0d (Cp Px) = 0 (24) Equation 4 relates the acceleration of the distal and proximal segment to one another since they are connected by a ball and socket joint. pd and pp are the vectors from a point 0 in the lab to the COM of the distal and proximal segments, respectively. It is assumed that the moment acting on the proximal end of the distal segment is equal and opposite to the moment acting on the distal end of the proximal segment. The moments acting at the proximal end of each segment (at each joint) are given by the equations (5). M'i(, = ( i x(m, .d)+ R, 1((R, ,).I, (R,. o,)x I .(R,. S,))) p co (m, g)COP xGRFGRMP M L (p =(1' x(m, as)+R 1((Rs ,) is(Rs i)xls.(Rs, )IP coi x(ms j)+ (25) Mo = x(m, 1 a,)+ R,1 ((Ra, *f,).I, (R, a,)x I (R, a,))) pivi" x(m, g) + 9M .... Similarly the forces acting on the proximal end of each segment are given by equation 6. contact mF F F g GRF c ntact = ms m S. + Ftact (26) (contact = T T mT g+ Fontact In the equations 5 and 6 the subscripts F, S, and T reflect the foot, shank, and thigh segments, respectively. The above equations of motion can be linearly parameterized into equation 7 which reflects the trunk angular acceleration as a function of the joint moments. s11 S12 S13 S14 S15 S16 S17 S18 S19 MA A aX S21 S22 S23 S24 S25 S26 S27 S28 S29 + MK +K (27) S31 S32 S33 S34 S35 S36 S37 S38 S39 MH CH _Z For the sake of example, the moment equations for a threedimensional inverted pendulum is derived and linearly parameterized to express the angular acceleration of the pendulum in terms of the moment at the pivot point. The moment equation for an inverted pendulum is given as: contact + contact This equation can be written in ~M,~ 0 P, I1 My +I1 P, O 0 P ... + I P, O P, P, a 6x+Pom X (Md) + Pxmg = 0 terms of its elements. P F I 0 o *I mo +... P. ma +1 P, P m L0 = a ma P P 0 mg a (28) (29) In equation 9, the angular acceleration's sensitivity to the contact moments are the inverse of the inertia matrix in the laboratory reference frame for a given configuration of the pendulum. The sensitivity matrix S (equation 7) is a time varying matrix which is dependent on the geometry of the model. The constant vector C is also dependant on the geometry of the model and the scalarsa,, ay, and a, in equation 7 are the angular accelerations of the trunk about the x, y and z axes, respectively. The joint constraints are defined by equation 4 in above. This equation assumes a ball and socket joint for all joints. The laboratory axes have the xaxis directed to the subject's right, the yaxis directed forward, and the zaxis directed vertically. Note, to express the threedimensional vectors in a sense that is most comparable with previously performed twodimensional analysis, moments are expressed with reference to axes that remain parallel with the laboratory axes but with an origin at the proximal end of the corresponding segment (c.f., anatomically defined axes). Results and Discussion Sagittal Plane The elements of the sensitivity matrix (angular acceleration per Nm of moment produced) which affect the sagittal plane trunk angular acceleration are shown in Figure 22. The sagittal plane trunk angular acceleration depends little on nonsagittal plane moments. The sagittal plane trunk angular acceleration has a negative sensitivity to the moment generated at the ankle for most of the gait cycle (Figure 22). Thus, an increase in plantar flexion moment (a more negative ankle moment about the xaxis) can cause the trunk to have an angular acceleration that leans the trunk posteriorly (shoulders back relative to hips, which is a positive angular acceleration about the xaxis). Similarly, an increase in a knee extension moment has the potential to cause an angular acceleration that leans the trunk posteriorly while an increase in hip flexion moment causes an angular acceleration that leans the trunk anteriorly. It should be noted that when the trunk angular acceleration is sensitive to a moment, it does not imply that the moment is influencing the trunk angular acceleration. If the moment is low, and the sensitivity is high, the joint may not influence the trunk angular acceleration. Thus, the sensitivity to a joint moment must always be considered in conjunction with the joint moment to assess the actual influence on the trunk angular acceleration. Frontal Plane The frontal plane trunk angular acceleration is similarly dependent on moments at all joints (Figure 23). In addition to the importance of the moment at the hip, an increase in left ankle plantar flexion moment or left knee extension moment each could possibly cause a trunk angular acceleration to the right (Figure 24). Joint Moment Contributions to Trunk Angular Accelerations Joint moments multiplied by their sensitivity yields results in units of acceleration with the result from each moment being superimposed to yield the total trunk angular acceleration. Note that each joint moment's contribution to the trunk angular acceleration in the sagittal and frontal planes are thus quantified and the extent and the magnitude of each moment's contribution varies throughout the gait cycle (Figure 25 and Figure 26, respectively). Note that at approximately 50% gait cycle: 1) sagittal plane plantarflexion and hip flexion moments approximately counteract one another with respect to their effect on trunk sagittal plane angular acceleration; and 2) sagittal plane plantarflexion moment approximately counteracts frontal plane hip moment with respect to their effect on trunk frontal plane angular acceleration. Control of Trunk Angular Acceleration by the CNS The contact moments acting on the trunk will almost be equal and opposite to the inertial torques since the gravitational torque is minimal due to the upright trunk causing the gravity vector to pass close to the centerofmass (Figure 27). Thus, the inertial torque is almost equal and opposite to the contact torques on the trunk as a result of the fundamental laws of mechanics (see Equation 1 in Appendix A). Furthermore, since distal joint moments can generate linear forces at the base of the trunk (Figure 24, Figure 27), it is possible that the CNS utilizes these forces to alter the angular acceleration of the trunk. An example of such a control system is the inverted pendulumcart problem in which linear forces at the base of the inverted pendulum can control pendulum balance (rotation) yet still progress the pendulum horizontally. If the trunk is modeled as an inverted pendulum with its base at the hip joint center, then the linear reaction forces (which are a function of distal joint moments) at the base of the trunk are an effective means to control balance of the trunk and maintain forward progression. Since both the forces and moments at the base of the human trunk can be used to control its angular acceleration, it is possible that the CNS uses both to control trunk balance. 1 0.5 Winter .0" Current Model 0 .2 0.5 0 20 40 60 80 100 0 20 40 60 80 100 1 i i1.5 0U.5  40 61] 80  1 0.6. 0.4. 0 20 40 60 80 100 0 20 40 0 80 00 % Gait Cycle % Gait Cycle Figure 21. Comparison of the current model (blue) to data from previously published work (red) (Winter 1991). 0 2 2 0E >0 0) ^ (D sU) 2oE  o 0 > E CO) U _ Moment about xaxis Moment about yaxis Moment about zaxis 2 % Gait Cycle Figure 22. The sensitivity of the trunk angular acceleration about the xaxis to the joint moments during gait. A negative sensitivity implies that a positive moment about an axis causes a negative acceleration about the same axis and vice versa. a,& 0 0 1.5 00 (5 E 0 2.5 2.5 Moment about xaxis Moment about yaxis E Moment about zaxis wo . 0 1.5 0 100 % Gait Cycle Figure 23. The sensitivity of the trunk angular acceleration about the yaxis to the joint moments of the leg during gait. A negative sensitivity implies that a positive moment about an axis causes a negative acceleration about the same axis and vice versa. Frontal plane view Sagittal plane view of segmental coupling of segmental coupling Sagittal plane moment, can cause frontal plane angular accelerations Figure 24. The sensitivity of the trunk angular acceleration about the yaxis to the joint moments of the leg during gait. A negative sensitivity implies that a positive moment about an axis causes a negative acceleration about the same axis and vice versa. XAxis 50 100 YAxis 4001 ZAxis 4001 200 0o 400 200 0 200 400 400, 200[ 0 .... 200[ 400 50 100 400 200 0 200 200 400 50 100 50 100 Contribution to Trunk Ang. Accel (rad/s2) Moment (Nm) Trunk Angular Accel (rad/s2) 50 100 400 400 200 200 0. 0 200 200 400 V 400 50 100 % Gait Cycle  V 50 100 % Gait Cycle 4001 200 0OO 200 400 50 100 % Gait Cycle Figure 25. Frontal and sagittal plane views illustrate how a sagittal moment can cause frontal plane accelerations by means of reaction forces along the kinematic chain. 400 50 100 400, 200 0 200 400 200 YAxis 200 0 50 100 0 50 100 0 50 100 200 200 20C 20 20C 0 50 100 0 200 0 200 0 50 100 % Gait Cycle 200 0 P 0 50 100 Contribution to Trunk Angular Accel (rad/s2) )  Moment (Nm) Trunk Angular Accel (rad/s2) ) 50 100 0 50 100 % Gait Cycle 200 0 1200 0 50 100 % Gait Cycle Figure 26. Contribution of each joint moment to trunk angular acceleration in the sagittal plane. The summation of the contributions is equal to the total trunk angular acceleration in the sagittal plane. 200 200 200 0 200 XAxis ZAxis 0 Free body of lower extremities Forces and moments are equal and opposite Calculated from inertial property ic . inertial torques. and inertial force of trunk. Equal and opposite to contact forces ni, l moments by Newton's second law. 2; I.cpendent on knee and ankle joint forces and moments i"nertia Mdist Figure 27. Two methods of calculating the contact moments are illustrated by taking free bodies of the trunk and lower extremity. These methods illustrate why the contact moments on the trunk are mathematically equal and opposite to the inertial forces and moments acting on the trunk. i V K .^ isaw Free body of trunk ~ T CHAPTER 3 THE DEVELOPMENT OF A GROUND CONTACT MODEL FOR BIOMECHANICAL SIMULATION FROM TREADMILL DATA Introduction Simulation in biomechanics is a rapidly growing field with detailed computer models producing estimation of currently immeasurable parameters of human gait. Two primary types of simulation are typically performed: 1.) a forward simulation is one in which joint torques are provided to the model and the resulting kinematics can be calculated, and 2.) and inverse dynamic simulation calculates the joint torques from kinematics. Such simulations are invaluable to orthopedic sciences, rehabilitation, prosthetic design, and computer graphics amongst many other fields [5558]. Due to the biomechanical intricacies of the human body, numerous assumptions have to be made (mass properties, ground contact models, joint contact models, etc) to produce a solution in these simulations. Optimization of the mathematical elements of a simulation based on experimental data therefore becomes desirable. Of the intricate modeling necessities specific to human gait, footground contact has been presented as challenging. The most common approach presented is a spring damper model in which springdamper elements dictate ground reaction forces.[2934] Another approach is to assume a welded joint between the ground the foot [3538] or to model the foot as a rigid rolling surface [39]. More complex methods such as finite element models from study on cadavers and magnetic resonance imaging have been proposed [40]. Viscoelastic sphere models have also been introduced to represent the plantar surface of the foot [34, 41]. A contact model based on a viscoelastic sphere was first studied by the H. R. Hertz [42] and expresses the relationship between the sphere's deformation and the resulting force that causes the deformation. The visco elastic sphere model presented by Guler et al relies on experimental data from heel pads and Guler et al conclude that viscoelastic spheres can be incorporated into foot models for simulation [41]. More complex spherical models have been presented that replace constant exponent of 3/2 that exists in Hertz's model with a variable n which dictates the geometry of the model (Marhefka 1999). Despite the increasing investigation and complexity associated with foot contact modeling, four vital aspects of contact model design have not been addressed thoroughly. The first is the applicability of current models to treadmill walking, the second is the effect of kinematic measurement error on foot contact model development, the third is the effect of the discretinization of the continuous plantar surface of the foot on the frictional properties of the model, and the fourth is the reporting of the foot contact models ability to model all six of the ground reaction force and moment components. The following work address these four points by exploring the effects of treadmill belt movement in the medial lateral direction versus the anterior posterior direction, performing kinematic optimization within experimental measurement to improve model performance, introducing a novel elemental torsional friction to account for discretinizing the plantar surface of the foot to improve axial reaction moment modeling, and presenting the models ability to reconstruct all six components of the ground reaction force is presented. Methods Subjects A single subject was used for demonstration purposes. The subject walked at their self selected speed of 1.1 m/s on an instrumented treadmill (Techmachine, Andrezieux Boutheon, France) for three trials of thirty seconds each. Data Collection Kinematic data were collected using a 12 camera VICON system and sampled at 200Hz to capture spatial positions of markers placed on the subject's feet. The subject walked barefoot. Markers to capture the kinematics of the foot were placed on the medial, lateral, and posterior sides of the heel. A marker was placed on the medial and lateral malleoli, on the first and fifth metatarsal heads, on the big toe nail, on the second toe nail, and at the base of the second toe. A rigid cluster with three markers on was also placed on the mid foot (Figure 31). The subject was asked to perform numerous tasks on the treadmill. The first was to stand statically for a period of five seconds to determine the relative positions of markers to be used as tracking clusters (discussed in the "Model" section of this chapter). After the static trial, the subject was required to step on three marker balls securely attached to the ground level. This process was used in a three point calibration (discussed in "Pressure Sensitive Film" section of this chapter). Subsequently, the subject was asked to walk at various speeds (as naturally as possible) for forty seconds at a time. Once the subject reached steady state (typically within the first ten seconds) the data collection would commence. Two measurements systems were utilized to determine the forces and moments acting on the plantar surface of the feet. The first is a split belt instrumented treadmill (Techmachine, Andrezieux Boutheon, France) and the second is a pressure sensitive film (Tekscan, Inc. South Boston, USA). The instrumented treadmill has the capability to record horizontal and vertical forces while the pressure sensitive film records only vertical forces and pressure distribution. Pressure Sensitive Film The pressure sensitive film measures vertical ground reaction forces at 1200 discrete places on the plantar surface of the foot with a grid resolution of 5.08mm. The sensor film was taped to the plantar surface of the foot using strong double sided adhesive tape which effectively held the film in the correct position and acquired data at a rate of 100Hz. Since the film's measurement is contained in a three dimensional matrix (x position, yposition and time), the data requires more complex filtering procedures. The current work applied a 2 stage filter and a subsequent fifth order data interpolation. The first stage was to filter each frame of data (every point in time has 1200 readings at 1200 xy locations) such that the distribution of pressure measured on the plantar surface of the foot at a single point in time produced a smooth function. The second stage is to filter the data along time to remove the high frequency content of each individual sensor in the film. The data interpolation occurs along the time axis where every sensor's data is resampled at a rate to coincide with the data from the motion capture system. This interpolation utilized included fitting a quintic spline to every senor's time trajectory and resampling the spline at 200 Hz. Not every point on the grid proved necessary to include in the evaluation. Many sensors had a zero or insignificant contribution to the total pressure distribution. In order to filter out only the points that made a relevant contribution and determine the locations of clusters of elements that could be grouped into a segment, a breadthfirst search algorithm was used. The algorithm loops through every time frame of data. At a time frame, the data can be represented by a 60 X 20 matrix in which the rows indicate anterior/posterior direction (along the yaxis) and the columns represent medial/lateral direction (along the xaxis). In this matrix the algorithm selects the element with the maximum value. The next step is to check the 8 neighboring elements to see if their value is above a threshold that represents a significant contribution. In this work that threshold was set to be at 1% of the maximum value. If the neighbor's value is less than the threshold, the same process is not repeated for its neighbors. If the neighbor's element is above the threshold, then the position of this element is stored and its neighbor's are checked. The process is repeated until no more neighbors exist with values above the threshold. The result is a group of points that can be classified into a segment as a group (Figure 31). The algorithm effectively determined four clusters that represented the heel, the forefoot, the second toe and the big toe. The points of the heel were assumed to be at a fixed position in the foot segment; similarly, the midfoot points were assumed to be fixed in the foot reference frame, and the second toe and big toe clusters were assumed to be fixed in the toes segment. Upon completion of the filtering process, the pressure data from the remaining elements are calibrated at each point in time to exactly match the vertical force measurement from the treadmill. Thus, the pressure film is not utilized as a force measurement system as much as it is utilized as a pressure distribution measurement system. Prior to the calibration the pressure film according to the instrumented treadmill, a cross correlation is used to determine the time lag between the two signals since the signals were not collected in a synchronous fashion (Figure 32). In order to calibrate the film at a point in time, the 60 X 20 matrix is normalized to the summation of all elements' values at that value. The result is then multiplied by the value of the vertical force as measured by the instrumented treadmill at that point in time. After this procedure is performed, when a frame corresponding to a point in time is summated along its rows in columns, the result is a value that exactly corresponds to the value of the vertical force as measured from the instrumented treadmill at that point in time (Figure 32). ThreePoint Calibration The information of the pressure distribution is not effective unless its position is known within the foot's reference frame. In order to estimate the position of each one of the 1200 sensors in the foot reference frame, a three point calibration is performed. During this process the subject is required to step on three markers that are securely fastened to the floor's surface. The position of the markers fasted to the ground surface was recorded prior to this. The markers were seven millimeters in radius and planted on a two millimeter thick plastic base that was fastened to the ground. While the subject stepped on the markers, the kinematics of the subject's foot was recorded. The data that resulted from the three point calibration data collection was passed through the breadthfirst search algorithm described earlier in this section. The algorithm effectively determined the pressure regions that corresponded to the markers that the subject stepped on. Since the kinematic motion of the subject's foot was recorded during this process, a rotation matrix and reference frame origin could be determined such that the position of the pressure regions could in the pressure film reference frame could be translated and rotated to coincide with the marker positions as determined from the motion capture system. To determine the three Euler angles and three translations that successfully complete the three points calibration, an interiorreflective Newton method is used. The optimization involved the approximate solution of a large system using the method of preconditioned conjugate gradients at each iteration. The algorithm was implemented using Matlab's "fminunc" function with a largescale optimization. The algorithm successfully determined the three translations and three rotations that the pressure film points would have to undergo in the foot reference frame necessary to "lineup" the pressure regions with the positions of the markers fastened to the floor (Figure 31). Foot Model To model the foot, it was divided into two segments. The first segment is defined as the heel and its motion is tracked from the three markers placed on the medial, lateral, and posterior sides of the heel, and a cluster of three markers on the midfoot. The second segment is the toes segment which is tracked by using the markers on the hallux nail, index toe nail, and the marker at the base of the index toe. The kinematics of each segment is determined by singular value decomposition of a matrix derived from the positions of the land marks [59]. To reduce the complexity of the model eight viscoelastic elements were fixed in the foot reference frame and two elements were fixed in the toes reference frame. The locations of the elements were determined using the pressure sensitive film, the three point calibration routine, and kmeans clustering. Kmeans clustering was used to reduce the number of elements in each segment to a manageable amount. The number of elements in the midfoot and heel sections was reduced to four and a single representative element was to replace the elements in the index toe and a single representative element replaced the elements in the hallux (Figure 33). Full Body Model The model was created in Mathwork's (Natick, MA, USA) Matlab using the SimMechanics toolbox. The axis definition has the yaxis of the laboratory perpendicular to the treadmill surface. The zaxis is orthogonal to the yaxis and points in the opposite direction of the treadmill belts motion (i.e. the zaxis points forward). The xaxis is the cross product of the y and zaxes. The model and segmental coordinate systems are presented in Figure 34. Visualization of the model is done utilizing equivalent ellipsoids to represent each segment. The ellipsoid is determined from the segment's mass and mass distribution. The foot is visualized as a convex hull of the contact elements used to model foot ground contact. The model consists of thirteen rigid segments (toes, feet, legs, thighs, pelvis, trunk, head, upper arms, and lower arms) all interconnected with ball and socket joints. The joint locations were optimized. The optimization placed a fixed point in the proximal segment; it also placed another point fixed in the distal segment. The coincidence of these two points' trajectory was the cost function. When the optimizer placed two points fixed in the proximal and distal segment reference frames that had coincident trajectories through space due to the motion of the respective segments, the average position of the two points was considered to be the joint center (Figure 35). The optimization converged using a gradient descent approach (Matlab's "fminsearch"). After the joint center optimization the pelvis was constrained to move according to the experimental data. The thigh was then connected to the pelvis at the optimized joint center. This optimized joint center contains small residuals (stddev = 3.2 mm) and which results in a translational motion of the thigh that varies from the original experimental data within the residual of the joint center optimization. This joint center residual superimposes its effect after the knee and ankle joint centers. Prior to joint constraint, the maximum residual from least squares marker tracking on the foot was 2.8 mm. The radius of the markers used to track the system was 7.5 mm. Post joint constraint the origin of the foot had a mean error of 4.3 mm from the nonjoint constraint system. The maximum deviation of the origin of the foot in the joint constraint system was 8.5 mm from the origin of the foot in the nonjoint constrained system, and this residual occurred at heel strike. ViscoElastic Contact Elements As mentioned in the introduction of this chapter, the typical approach to modeling foot ground contact has been with viscoelastic elements. The typical form of such an element is given by an equation that represents a nonlinear damping element. Fz = c xn. (1 + b xn. i) (31) In this equation x is the penetration distance, i is the penetration velocity, and n is and exponent that can be included as a design variable since it is dependent on the hypothetical geometry of the contact element. The characteristic use of this equation allows the simulation of a hysteresis in the simulation of the vertical ground reaction force. Two more design variables are c and b which are the spring constant and damping constant respectively. Since the vertical force is known at 1200 locations on the plantar surface of the foot after utilizing the measurement capability of the pressure film, the viscoelastic element's capability as an appropriate element can be explored. In order to do this, an optimization procedure is completed in which the height of the element in the segment's reference frame, the spring constant, the damping coefficient, and the geometry exponent are all design variables. An element was placed at every point as dictated by the sensor positions in the pressure sensitive film. An optimization was performed on each individual element. The optimization was to determine the parameters of the viscoelastic element that cause the element to successfully reproduce the ground reaction force as determined from its respective sensor measurement of the ground reaction force in the pressure film. The cost function was selected as the leastsquares error between the vertical force from the simulation and vertical force from experimental data. The vertical force from the simulation is the superposition of all the elements vertical reaction force. A comparison of the two data sets is made in Figure 36. Early in the step, the viscoelastic elements had trouble trying to model the characteristics of the heel elements. This could be due to numerous reasons. The first reason is perhaps that the discrete natural of the elemental model does not contain a high enough resolution to accurately model the continuous surface of the heel. The second reason is that the compression of the heel pads may reach a maximum, beyond that, the calcaneous plays a significant role in the production of force. Compression of the calcaneous has a very high sensitivity to force production as is immeasurable with the equipment utilized in the current work. A third reason may simply be that the elemental penetration into the ground may not be measured accurately enough by the tracking markers. Researchers have successfully utilized the elements for accurate force simulation beyond that seen in Figure 36. The approach taken by these researchers is to determine reaction force and moments at the ankle through standard inverse dynamics. The parameters of the viscoelastic elements are then determined in a forward dynamics sense. That is, the kinematics of the foot are determined by driving the foot with the forces and moments at the ankle. During this process, the parameters of the viscoelastic elements are altered with an optimization until the motion of the foot best represents the experimental kinematics used in the initial inverse dynamic analysis, and also the corresponding ground reaction forces. Therefore, success has been due to the fact that the model is allowed to vary from the initial measured kinematics. Due to the high sensitivity of the ground reaction force production to kinematics, these slight variations from the measured kinematics can have profound effects on the ground reaction forces. To verify this hypothesis, a single element was selected which proved difficult for the viscoelastic element to model. To verify that slight variations in kinematics have profound effects on viscoelastic elements' potential to model ground reaction forces, a single element is selected with an especially complex force penetration pattern (Figure 37). After the optimization, the spring constant value on this particular element was 2372 N/m. To determine the appropriate parameters such that the penetration and penetration velocities model ground reaction forces at an element, the spring constant, damping coefficient, and exponent in equation 31 are adjusted; however, the optimizations rarely produce a perfect fit. If one reverses the process and solves the differential equation 31 for the penetration, x, by substituting in the true vertical forces from experimental data and the spring constant, damping coefficient, and exponent from the optimization, one will determine the penetrations that would produce the correct ground reaction force at that element. The differential equation which reveals the penetration trajectory that would satisfy the equation is given by: Fk.xn X = kb(32) k~b~xn To solve this equation, Matlab's ODE45 is utilized which is a numerical integrator that uses the RungaKutta modus operandi. By performing this check, it is noticed that variation in kinematic measurement by less than 1mm sufficiently alters the penetrations to exactly match the experimental ground reaction forces (Figure 38). The reported accuracy of the motion capture system is on the order of 0.5 millimeters and the maximum residual on the segments tracking occurs on the toe segment and heel segment and is in the order of 0.75 millimeters. It can be concluded that the poor performance of the viscoelastic element to design a ground contact model is not due mathematical inconsistencies in the contact model's development, but rather due to insufficient measurement capability of the motion capture system. It is also concluded that in order to successfully apply a model containing a reduced number of elements, it is vital to alter the kinematics of the model to obtain the correct viscoelastic properties of each element. Friction Model Friction models are applied in various forms. In the current work a combination of viscous and Coulomb friction is utilized as well as the introduction of an elemental torsional friction associated with the rotational velocity of the element about the vertical axis. Coulomb Friction Coulomb friction is given as: 'h = IFy\ . (33) In equation 33, Fh is the horizontal frictional force (a two dimensional vector) which is pointed in the opposite direction of the two dimensional horizontal velocity, vh. The Coulomb coefficient of friction is ic and IFy is the magnitude of the vertical force (the force normal to the walking surface). This equation can be divided into its two scalar equations for the x and ycomponents by: Ff = F I (34) for the xdirection and: Ffz = IVc (35) For the zdirection.Typically the frictional properties of a surface are independent of direction; however, in the case of treadmill walking this is not always true. On a treadmill it is possible that the belt slides in the medial/lateral direction (x direction) while its motion in the anterior/posterior (zdirection) direction is dictated by the rollers of the treadmill. To verify this fact a 20kg mass was placed on the center of two different treadmills. An approximate 100 N force was applied to the mass in the xdirection and y direction separately. In the xdirection on treadmill 1 (ADAL treadmill; Techmachine, Andrezieux Boutheon, France) the weight and the belt moved 4mm in the direction of a 91.23 N force without the weight sliding on the belt (Figure 39). The belt slid on the smooth surface under the belt. In the ydirection the 95.15N force did not cause any motion as the belt would not slide over the rollers (which were locked from rolling) and the weight would not slide over the belt (Figure 310). Similar results were found on treadmill 2 (Bertec treadmill; Columbus, Ohio, USA). In the xdirection the weight and belt moved but no motion was seen in the ydirection. Again, flexibility in the belt allowed medial/lateral sliding in the xdirection of the belt over the smooth surface under the belt but the high coefficient of friction between the rollers and the belt did not allow motion of the belt in the fore/aft direction. It can be concluded that in order to model horizontal friction on a treadmill, different coefficients of friction must be selected for the x and ydirections. This is because there is generally a smooth surface under a treadmill belt and lateral belt flexibility allows for motion of the belt over this surface; however; the tensile flexibility of the belt is very high in the walking direction and the coefficient friction between the belt and the roller is very high; therefore, the belt will not slide in the ydirection. Since motion of the subject is measured relative to the lab and the medial lateral displacement of the belt is unknown, it is justifiable and necessary to use different coefficients of friction for the x and ydirections in equations 34 and 35 respectively. Viscous Friction Viscous friction is given by: 'FP = IFy 1 ,vh (36) This friction is also separable into two components and different coefficients of friction are required for treadmill walking for the x and ydirections as shown for Coulomb friction. Both Coulomb and viscous friction can be superimposed to determine linear friction in the model. Torsional Friction If a single element is to represent a group of elements then rotation of that group of elements about the representative element has the ability to produce linear frictional forces that oppose the rotation. The frictional linear forces produce a pure moment about the representative element. The cumulative frictional moment about the representative point can then be modeled by a form analogous to linear Coulomb and viscous friction: Mf= FyI 9,y I Wy IFy\ (.C WY In this equation Mfis the moment about the element that opposes rotation. 9 is the rotational coefficient of friction and wy is the angular velocity of the segment. Typically a set of forces acting on a rigid body can be replaced with the resultant force acting on the body and a pure moment. If the single element is to replace an infinite amount of individual elements on a continuous surface, then a means for producing the resulting pure moment must be made available. Equation 39 completes the frictional model of the representative element. Kinematic Optimization Since it has been shown that kinematic variation within experimental measurement tolerances affect the ability of the model to accurately represent the ground reaction forces and moments, a joint angle optimization was performed on the whole body model with the foot model attached. The optimization allowed for small variations in hip, knee, and ankle angles (less than 2 degrees) using particle swarm optimization but assumed that the hip joint center moved with the pelvis in a fixed fashion. The optimization utilized bsplines with fifteen control points for each joint angle trajectory. Within the joint angle optimization, two secondary optimizations are performed to determine the viscoelastic element parameters and the frictional parameters that achieve the measured ground reaction forces and moments. The optimizations were performed with and without the torsional compensation at each element and with and without kinematic adjustment. The elemental forces in the ydirection (vertical) are known experimentally from the pressure sensitive film; therefore the viscoelastic element parameters to determine (39) vertical force are first performed using optimization to solve for n in equation 33 and nonnegative regression to find c and b. Since the vertical force (normal force) is now modeled at each element, the results can be used in an optimization to find the frictional parameters to match the remaining ground reaction forces and moments. Results and Discussion A kinematic adjustment was made to the joints of the full body model. Three degrees of freedom were allowed to vary at the hip, one (flexion/extension) at the knee, three at the ankle joint, and three at the toes joint. No angle was allowed to vary more than two degrees in any direction. The optimization was performed with two optimizers in parallel. A gradient descent method (Mathworks fmincon) and a particle swarm optimizer. Periodically (every hour on a Pentium 4 processor with 1GB DDR RAM) the best result from each optimizer was compared and the best result applied to the optimizer with the worse result. Initially the particle swarm algorithm provided the best solutions and as the algorithms began to converge the gradient descent method provided better results. The optimization continued for 6 hours and 43 before the minimum was reached. The resulting joint angles are presented in Figure 311. In Figure 312 the change in the force penetration curve for an element at the heel is presented as an example. Prior to the kinematic optimization, the force penetration curve exhibits a nonphysical damping characteristic. The penetration reaches a certain point and the force increases drastically with no change in the element penetration. This is not a function that can be modeled with a viscoelastic element. After slight variations in the model kinematics, a more physically viable force penetration curve is noticed which lends itself to being modeled with a viscoelastic element. The final results of the ground reaction forces and moments of the model are presented in Figure 313. The gains ranged from 5.43x103 to 3.29x106 N/m for the viscoelastic element. After the kinematic optimization the mean squared error of the knee joint from its original position was 7.3 mm. The position of the ankle joint center had a mean squared error of 22.4mm from its original measured trajectory. Conclusion A foot contact model for simulation has been developed herein. Viscoelastic elements were utilized to create the dynamic relationship between kinematics and the vertical ground reaction forces. With the utilization of a pressure sensitive film, it has been shown that the success of a foot ground contact model has a high dependency on measurement accuracy and that small kinematic variations within measurement tolerances improve the ability of a viscoelastic element to model the reaction forces. It has been shown that a foot contact model that is developed from data collected on a treadmill will most likely have to utilize directionally dependant coefficients of friction since high quality, high performance instrumented treadmills still contain medial/lateral belt movement that is not present in the walking direction of the treadmill. A torsional frictional model has been proposed to account for the pure moment that is produced about a representative element from continuous surface area in the immediate vicinity of the representative element. The model was deemed necessary due to a dimensionality reduction using kmeans clustering and a breadth first search algorithm. The clustering relied on a three point calibration that aligned the elements of the pressure sensitive film accurately within the foot reference frame. In combing all of the above aspects, a model is developed that reflects all six components of ground reaction force during gait. 0.55  00 00880 000 0...... O O 0.5h 0.45 0.45 0.4 0.35 0.3 uv I  1 0.42 0.46 "0.5 XPosition (m) Figure 31. Alignment of the pressure film in the foot reference frame relative to the motion tracking markers (using a three point calibration) and also the results from the breadthfirst algorithm search which found only the relevant force elements and clustered the elements into anatomically relevant groups Midfoot o Heel o Second Toe o Big Toe X Heel Marker Medial Heel Marker * Cluster Marker 1 * Cluster Marker 2 * Cluster Marker 3 * Medial Met Head * Lateral Met Head O Hallux Marker Index Toe Marker O Base Marker X Lateral Heel Marker . . .. . PreCalibration and Time Syncronization 1o.. , 2300 2400 2500 2600 2700 2800 2900 Post Time Syncronization 2300 2400 2500 2600 2700 2800 2900 aTime F 1 ram) 0I I \ S I % 2300 2400 2500 2600 2700 2800 2900 weight. Middle: Time synchronized data using a cross correlation to determine lag. Bottom: Final result from calibration. 0 2300 2400 2500 2600 2700 2800 2900 Posl Calibratnon I_ 400 200 2300 2400 25()0 2600 2700 2800 29010 Time framess) Figure 32. Calibration of the pressure film. Top: Original data normalized by body weight. Middle: Time synchronized data using a cross correlation to determine lag. Bottom: Final result from calibration. Left Foot Right Foot 0.2 0.15 0.1 0.05 0.02 0.02 0.02 0.02 Figure 33. Dimensionality reduction using kmeans clustering. All the elements in the segment are replaced with the points reflected by the large solid dots. Figure 34. Anthroprometric model with (left) and without (right) segmental coordinate system definititions. The large number of coordinate systems at the feet are associated with the contact elements in the foot and toes reference frames. Left Shoulder  Initial Guess 200 40 60 80 0. 200 400 600 800 1000 C 0 1.39 (.37 g 1.37 >1 200 400 600 800 1000 E 0.55 S0.45 V 200 400 600 800 1000 Time (Frames) Left Shoulder  After Joint Center Optimization 35 J.4 45 200 400 600 800 1000 Point Fixed in Arm Point Fixed in Trunk 1.38 1.36 200 400 600 800 1000 0.55 0.5 0.45 200 400 600 800 1000 Time (Frames) Figure 35. The convergence of an optimization to find the left shoulder joint center from a poor initial guess. z 500 0 04 a I 50 050 0 10 IL 02 n Lw. 800 H 100 150 200 250 300 100 150 200 250 300 I ' I I 50 100 150 200 250 300 z S500 0 o LL 0 . 400 200 50 100 150 200 250 300 Time (Frames) Model ,,^ ,. Acmuall I 1L I I F I j 1 vn I I i I I ,/ *! 1 i II i I I I I: I I 1 I I t / f :i t l 50 100 150 200 Time (rames) Figure 36. Comparison of the viscoelastic model's reproduction of the experimental data that was used to train the model 250 300 "' Measured =Optimized ViscoElastic Element Model \7 I 6 6 o II o 4 LL 2 00 850 I. 900 Time 950 1000 1050 (Frames) 2 Penetration (m) 4 x 103 Figure 37. Comparison of an optimized viscoelastic element to experimentally collected data. 1100 5 8. Z o o 4 LL 5x 103 Penetration needed Sto exactly match force E 4\ with previously determined S/ \ gains C / Original Penetrali,:n CIL i O 3 %  0 50 100 150 200 250 Time (Frames) 10 Z I. 0 4 0* 0 1 2 3 4 5 Penetration (m) x lo' Figure 38. The comparison of an example experimentally measured element penetration and a hypothetical penetration that would cause a viscoelastic element to exactly match its corresponding ground reaction force. The difference in the measurement falls well within the experimental error. I ii 'C2 Figure 39. On an ADAL treadmill, the weight with no force acting on it is at rest at 50cm (left). With a 9.3kg force along the xaxis (normal to the walking direction of the treadmill), the belt slides 4mm to the right (right). The weight does not slide on the belt. BOE i a Figure 310a. The weight is at rest on the treadmill belt with no force being applied. I I BaI .11 I '4 * Figure 310b. With a 9.7 kg force being applied along the yaxis (along the walking direction on the treadmill) no sliding motion of the belt or the weight on the belt occurs. l_i 5  0.4 S0.2 / \ 0.2 0.4 0.2 0.2 " N 0 1 0 1 X>~ m 0.4 0.4 Altere Sw 0; w 0.2 Origin, 0 1 0 1 0.4 0 S0. 0.5  0 1 0 1 S0.4 0.2 S0.2 c 0 S0.2 0.4 0 1 0 1 Time (sec) Time (sec) Figure 311. Kinematic alterations in joint angles produce foot motion that accommodates foot/ground contact modeling. Pre Kinematic Optimization 0.02 Penetration (m) Post Kinematic Optimization 250 + Actual o Best Fit Model 0.04 0.02 Penetration (m) 0.04 Figure 312. The effect of kinematic variations removes the nonphysical negative damping characteristic seen in the original experimental data. 60  7  Q I I d I l I i , I i eg E SC   II 0 0.5 1 1.5 O) i IC 5 LL i l I n n I I I 1 llj i 0i 0" 1 1 5 T Ia , m l1 lIi C 0 05 1 15 Timli. (ia I v 10 V c I 1500 0 0.5 1 1.5 E100 0 051> .5 0 05 1 15 a (I Tirre [(g Figure 313. Resulting force and moment production of the foot contact model post kinematic optimization. A torsional friction model is utilized at each element and differing linear coefficients of friction are used in the x and z directions due to medial/lateral treadmill belt flexibility. 68 CHAPTER 4 ANGULAR MOMENTUM CONTROL DURING GAIT Introduction The impact of stroke on walking is significant [1], with only 37% of stroke survivors able to walk after the first week. Even among those who achieve independent ambulation, significant residual deficits persist in balance [2, 3] and gait speed with a 73% incidence of falls among individuals with mild to moderate impairment 6 months poststroke[4]. A primary disability associated with poststroke hemiparesis is the failure to make rapid graded adjustment of muscle force and to sufficiently activate muscles [5]; consequently, a reduced ability to rapidly respond to perturbations that influence balance is inferred. Various compensatory walking patterns have been suggested to increase balance in hemiparetic gait (e.g., increased double support time); however, it is difficult to state whether kinematic or kinetic variations during paretic gait are adaptations done by choice to increase stability or the result of an impaired neural controller and decreased power production. Regardless, persons with hemiparesis do not have robust control over dynamic balance during gait and little is understood about how to quantify their ability to retain balance. There is a substantial need to assess dynamic balance during walking in order to ascertain who is at increased risk for falls and whether a targeted intervention has improved dynamic balance. While little has been done to quantify balance during walking in the general population, much less has been done to quantify balance during gait in persons with neurological deficits such as stroke. Approaches for determining who is at risk for falls have usually relied on clinical measurement scales as opposed to quantitative evaluation of either stability of the walking pattern or balance mechanisms active during walking. Numerous clinically inspired tests have been used to predict falls and classify subjects into "fallers" and "nonfallers" [60]. For example, the Berg Balance Scale was designed to provide an indication of a subject's static balance [61, 62]. However, interpretations of these measures are based on statistical evidence and do not quantify a subject's balance performance during walking. Furthermore, these assessments are not based on balance control mechanisms present in healthy subjects, which would likely be a more helpful approach to improve the efficiency of balance rehabilitation. To explain and ultimately improve dynamic balance during walking, it is necessary to interrogate balancespecific neuralmuscular systems during gait. In order to understand the complexities of balance during gait, we believe that it will prove necessary to quantify destabilization of gait following perturbation and determine the biomechanical mechanisms by which the consequent stabilization is achieved. To maintain upright walking, the whole body AM should maintain oscillation about zero, since a constant nonzero AM implies continuous rotation of all segments about an axis (other than the vertical axis) which would most likely represent falling [14]. Previously it has been noted that intersegmental AM cancellations are mostly responsible for a "close to zero" whole body AM and various successful control schemes have been developed based on this "zero moment assumption" in the robotics literature [15]. In highly sophisticated bipedal robots, control systems that maintain a constant zero AM, despite perturbation, result in stable and robust multisegmental bipedal gait. This has been described as the only reliable stability index of dynamic bipedal locomotion in robotics [16]. Despite the success of such control systems in robotics, it is noted from experimental results that humans do not follow this strategy completely. Humans tend to allow oscillation in AM about zero during gait [12, 17]; however, it has been proposed that angular momentum is controlled rigorously by the central nervous system and can be assumed to be zero to make predictions on COP location [12, 17]. As opposed to the zero moment assumption, which assumes that the subjects strive to maintain a constant zero AM throughout the gait cycle, we hypothesize that intersegmental AM cancellations are not the primary mechanisms of AM stabilization. The hypothesis states that healthy and efficient walking requires fluctuation of the AM from zero, and it requires a mechanism whereby the AM can be directed to zero when necessary in normal gait or perturbed walking. We propose that this regulation occurs during normal and perturbed gait because AM is often manipulated purposefully during certain tasks [17]. Phases of normal walking have been described as a controlled fall, which can be thought of as a nonzero fluctuation in AM. To control this fall, well timed joint power production is necessary and specific foot placement and it is the goal of this work to investigate both characteristics. Methods Subjects Twentyone healthy subjects were used for to demonstrate theoretical step limits and a single healthy subject was used to in the perturbation study for demonstration purposes. A single subject six months poststroke or longer also participated in the perturbation study. The subjects walked at self selected speed on an instrumented treadmill (Techmachine, Andrezieux Boutheon, France) for three trials of thirty seconds each. The instrumented treadmill is equipped to measure 3 orthogonal components of ground reaction forces at the subjects' feet and three orthogonal components of ground reaction moments. These six measurements allow for the calculation of the center of pressure (COP) under the foot. This can be done under both feet since the treadmill is a split belt treadmill. Data Collection Kinematic data were collected using a 12 camera VICON system and sampled at 200Hz to capture spatial positions of markers placed on the subject. Rigid clusters of markers were placed on 13 of segments assumed to be rigid bodies. Clusters where placed on the feet, legs, thighs, pelvis, trunk, head, upper arms, and lower arms. A linear algebra method based on the singular value decomposition of a matrix constructed from the positions of markers in the clusters is used to track segment motion in a least squares sense [59]. Subjects' joint power productions were determined from standard inverse dynamics procedures. Model A description of the model utilized on the subject for the perturbation study in this chapter is provided in detail in Chapter 2. Perturbation Force perturbations have rarely been applied during gait because their application requires the design of significantly complex custom hardware. The accurate application of a force is required in which direction, magnitude and point of application are all known. In order to apply a quantified force to a subject walking on a treadmill, a sophisticated and rapidly responding feedback control system is needed in conjunction with a powerful and speedy actuator. Our research facility has recently had built (Aretec, LLC, Ashburn, VA) a custom piece of hardware that solved the problems associated with impulse response tests on a walking subject. The Force Pod is based on a master/ slave configuration that maintains tension in a cable between the device and the subject without applying a net force until commanded to do so. The configuration winds up cable slack and releases cable tension in an appropriate fashion to not affect the subject. At a commanded time, the system winds up the cable and measures the force being applied to the subject. The system then alters the length of the cable to ensure that the movement of the subject does not influence the magnitude or direction of the force the cable is inducing on the subject. In conjunction with an advanced safety harness system, state of the art motion capture hardware, and a split belt instrumented treadmill already in place, the Force Pod can determine the impulse response of a human subject to perturbation in a controlled and safe fashion. The Force Pod was utilized to apply a perturbation to the subject just after toe off. The force as applied at should height pulling the subject forward. At this point it should be noted that the subject was walking on the treadmill and a forward pulling force could bring the subject to the front of the treadmill. Trying to avoid coming off the front of the treadmill may have played a part in the subject's response, but this didn't seem to be the case here. Whole Body Angular Momentum During the Gait Cycle The gait cycle can be divided into 6 regions: 1) initial double support, 2&3) first and second half of single limb stance respectively, 4) second double support and 5&6) first and second half of ipsilateral swing respectively. Sagittal Plane: In healthy subjects, each of the regions shows characteristic fluctuations in sagittal plane angular momentum. Just prior to each heel strike, there is a rapid change in angular momentum, which reflects a forward freefall [63] during the second half of swing phase (either region 3 or 6). This forward freefall is an intentional fluctuation in angular momentum. Subjects tend to land on the swinging leg which is aptly placed such that the ground reaction forces acting on the foot create a moment about the COM which results in the angular momentum returning to a value above or close to zero at the end of double support (regions 1 and 4). If the foot is not placed in the correct position, the moment will not be sufficiently large enough and the whole body angular momentum will not be corrected. In the middle of swing (end of regions 2 and 5) the angular momentum reaches a maximum and begins its descent throughout the second half of swing (regions 3 and 6) to complete the limit cycle for each step. Thus, there are two angular momentum cycles in the sagittal plane for every gait cycle and the transition from one rotational state to the next occurs during double support. Angular Momentum in the Frontal plane In order to regulate angular momentum in the frontal plane (lateral balance) during swing, the stance leg is responsible for generating ground reaction forces that will redirect the angular momentum. An initial nonzero angular momentum (at the beginning of swing) is normally zero at the end of the first half of swing and changes sign in the second half of swing, and vice versa for an initial positive angular momentum. In either case, the angular momentum is required to go from an initial value at the beginning of swing to a zero value at a point close to midswing (since the sign of the angular momentum cannot change at midswing without the angular momentum first becoming zero). The initial angular momentum in swing is highly inversely correlated (r = 0.857; p<0.0001) with the change in angular momentum during the first half of swing; however, the initial angular momentum is not correlated (r = 0.096, p<0.0001) to the magnitude of the angular momentum at midpoint of swing (Figure 41). This shows that angular momentum in the frontal plane is a controlled value and generally driven to a value close to zero during the first half of swing regardless of the initial conditions of swing. The Relation Between Joint Power and Whole Body Angular Momentum Whole body angular momentum (AM) is the summation of segmental moment of inertia multiplied by angular velocity. Thus, it collectively quantifies the rotational velocity of the entire body about the center of mass (COM) in a single measure. It can be calculated from kinematic data where the AM of the entire body is expressed as the superposition of individual segmental AM. To find the whole body AM about any arbitrary point q, a summation is performed [12]. Hq = Zili i ti + (Pq Pi) x mi(Vq Vi) (41) In the summation, Hq is the whole body AM about point q. Ii is the i'th segment's inertia tensor and oi is its angular velocity. pq and Vq are the position and velocity of point q respectively while Pi and Vi are the position and velocity of the segment's center of mass. mi is the segment's mass. To calculate whole body AM from kinematics, an anthropometric model is required to determine m and I and to measure segmental kinematics. The change in whole body AM about point q can be related to the sum of the moments acting on the subject. Mq = dH+ ( pq)xm a (42) In equation 42, the summation on the left side represents the summations of all moments acting on the subject about point q. The acceleration of point q is aq and m is the mass of the subject. If point q is selected as the COM of the subjects the second term on the right hand side becomes zero since f is the subject's COM. The moments acting on the subject can be expressed as the ground reaction moment and the moment arm from the COM to the COP crossed with the ground reaction forces. GRM + (COP COM) x GRF = dH (43) dt From the equation, it can be noticed that in order to change the whole body angular momentum about the COM the subject is required to alter the COP location (which is related to foot placement) and the GRF (which is related to joint power production). Thus, subjects with reduced power production and reduced coordination to accurately place the foot will have difficulty in correcting whole body AM to reestablish it to a regular limit cycle. When comparing the AM profile of a patient with hemiparesis to a control subject, it is noticed that the control subject needed fewer gait cycles to return the AM profile to within two standard deviations of the regular AM trajectory (Figure 4 2). Although it can be experimentally verified that hemi paretic subjects require more time to reestablish AM to a regular limit cycle, it is unknown to what extent power production inhibits the reestablishment of AM to its original trajectory. Experimental data does show the difference in power production in the response to a perturbation from paretic subjects and from controls (Figure 43). Step Length and Moment Impulses Relation to Angular Momentum Correction An external moment acting about a subjects COM will cause a linear change in angular momentum. The only way to counteract the change is to place a foot in the appropriate position and generate a ground reaction force that develops a near equal and opposite moment about the COM. In fact, the angular momentum is given by the impulse of the moment acting about the COM: J Mq dt = Hq (44) if the moments are summed about the subjects COM. Thus an angular momentum perturbation is quantifiable in units of Newtonmeterseconds. The only external moment available to the subject to counteract the change in angular momentum due to a perturbation is the ground reaction forces acting at the COP. The ground reaction moments at the COP are, by definition of COP, zero about the horizontal axes and rotation about the vertical axis does not imply falling. It is only the linear ground reaction forces that can generate moments which oppose angular momentum perturbations which could result in falling. Two aspects affect the production of a responsive moment about the subjects COM: the lever arm from the COP to the COM and the force at the COP. A subject has to essentially produce a counter impulse to correct unwanted changes in angular momentum. An impulse of lower magnitude is to be applied over a longer period of time to have the same effect as a shorter impulse of high magnitude. In order to generate a sufficient impulse in a finite amount of time (i.e. before it's too late to avoid catastrophic failure), the foot must be placed in a manner such that the angular momentum about the horizontal axis can be taken from a nonzero value to a close to zero value. Consider the sagittal plane and consider falling forward to be an increase in negative angular momentum about the xaxis. That is, the xaxis points to the subjects right if a right hand coordinate system is used. Then assume that R = COP COM. Let F, = COP(tHS) and F2 = COP(tTo) be the position of the COP at ipsilateral heel strike (HS) and contralateral toe off (TO) respectively. Assume there exists a function G(t) such that R(t) = F, COM(t) + (F2 F1) G(t) (45) Where G(t) = when t = tHS and 0 < G(t) < 1 tH < < tTO 1 when t = tTO) G(t) is a function that describes the weight shift from the trailing leg to the leading leg in double support which starts at tHS and ends at tro. To determine the foot placement in the medial direction it will be assumed that G(t) = G(tro) 9 tro < t < tMS where tMS is the time at the end of the first half of swing or midswing (MS). The moment equation about the subjects COM can be written in general as: M, + R x m (a g) = H (46) where H is the time rate of change of the angular momentum about the COM, M is the ground reaction moment, m is the total mass of the subject, a is the acceleration of the subjects COM, and g is the gravitational constant. Expanding equation 2 and writing the equations that relate the angular momentum about the x and yaxes to COM accelerations and the elements of R, it can be shown that: IH = m (az g) Ry m ay, Rz (47) Hy = m (g az) Rx m ax Rz (48) In equations 47 and 48, the subscripts denote the scalar elements of the corresponding vectors and g = 9.81m/s2. Foot placement and sagittal plane angular momentum: The integral of Hj over double support is: S, dt = f m (a, g) Ry m ay Rzdt (49) Substituting the equations and expanding the integrals, it can be shown that the change in angular momentum over double support can be expressed in terms of F2y, which is the position of the COP at toe off, a constant and an indicator of the placement of the foot at heel strike. StTO tTO HxI t = m(az ) ( COMy)dt + F2y m(a g)(t)dt tHS HS Fy r m(a g)G(t)dt fro mayRzdt (410) Note in equation 410, all terms are dependent on the trajectory of the COM. Rz is the vertical distance between the COM and the ground. The criterion for taking the angular momentum from a negative to zero or a positive value through double support (to rectify a forward freefall by placing a foot sufficiently far in front of the COM) is that Hx (tTo) 2 0. In order for this to be true, F2y should have a minimum value as described in equation 411 tTO m(az) (F yCOMy)dt+Fy JTO m(azg)G(t)dt+f TomayRzdH (to) F2 HS HS HS (411) 2y> TO m(azg)G(t)dt tHS For foot placement and frontal plane angular momentum, the integral of Hk over swing is: Jt HIy dt = tT m (g az) Rx m ax, Rzdt (412) Similarly to the sagittal plane angular momentum analysis, tMS tMS H tms = m(g az) (Fx COMx)dt + F2x m(g az)G(t)dt tTO tiTO Fx tS m(g a)G(t)dt fttmaxRzdt (413) In order to maintain stability in the frontal plane, the angular momentum in the frontal plane is required to go from a positive to a negative value and a negative to positive value during left and right leg swing respectively. Only the case for right leg swing will be presented as the left leg swing is analogous in symmetric gait with a sign change. Going from a negative to positive angular momentum during a period of time requires that Hy(tMs) 2 0. Using this relationship and equation (413), a bound can be placed on F2x to ensure the change in angular momentum that is needed for stable gait. Similarly to the derivation for the limits in the sagittal plane: ;m(SL a)(F1xCOM,)dt+Fi xtmsm(gaz)G(t)dt+fttM maxRzdtHx(tt 2x TO TO TO (414) 2zx > o777^  ,. (414) ftMS m(gaz)G(t)dt For any given COM trajectory, a minimum step length/width required to be able to rectify angular momentum during the first half of swing. A drawback of this approach is that a mixing function is assumed for a weight shift pattern which is fair for regular gait; however, this function may not apply if a subject is undergoing perturbation. Subjects adhere to these limits (Figure 44). In the frontal plane, subjects' average limit was at 65.2611.52% of the average step width in relation to a step width that fluctuated by 4.82%. Subjects' step width was approximately 35% wider than the limit. In the sagittal plane, subjects did not overextend their step length far beyond the limit. This limit was 42.766.11% of the stride length from the COM of the contralateral leg at the previous midswing of the ipsilateral leg and subjects placed their foot at 49.44.0% of the stride length, which reflects symmetric gait. No subject fell significantly or consistently short of the determined limit. In simulation the biomechanical model was used to verify that a shortened step consequent to a perturbation places excess strain on the neural control system. Since biomechanical models tend to fall over anyway despite joint center optimization, COM/Inertial property optimization, and a highly advanced finely tuned foot contact model, a controller was applied to the rotational dynamics of the mathematical model to maintain stable gait without any perturbation. A simple low gain PID controller was applied to the COM of the subject. The controller applied a torque that opposed any change in angular momentum about the COM. No linear controller for translational forces was needed maintain stable upright gait on the simulation due to the ground contact model, only the angular momentum controller. The controller gains were set such that it the angular momentum of the walking model reflected that of the subject which the model was based on (Figure 45). The assumption was that the controller accommodates and model residuals or errors. Once the controller was in place to regulate angular momentum to a certain degree, the joint angles were defined such that the foot fell short of the above calculated limits by a foot length. The low gain controller could not control the angular momentum, the system became unstable, and the model fell forward (Figure 46). The angular momentum profile also increased without bound until the model struck the ground (Figure 47). In order to shorten the step length in the simulation, the joint angular velocity of the joint Euler angles was multiplied a 0.75. The result was then integrated to determine the new joint Euler angle and the first value in time of the initial Euler angle was used as an initial condition for the integration. Doing this for all four angles in the leg (hip, knee, ankle, and toes) fully defined the new step length. This reduction in joint angular velocity is synonymous with reduced joint power production. The Effects of Reduced Joint Power Production on Resisting Perturbation During Gait The response of a 1321b subject undergoing a 101b perturbation while walking on an instrumented treadmill is illustrated in figure 48. The subject lowered the COM, leaned backward, and rapidly placed the lead foot relatively far in front of the COM. The perturbation force lasted for 0.5 seconds. By leaning back the subject generates a negative moment about the xaxis to counter act the perturbation force. The PID controller gains were increased so that the controlled model could withstand the perturbation force. The model's joint angles were constrained to be the same as the experimental data initially. Figure 49 shows the angular momentum profile for the model and for the subject while figure 410 shows the perturbation force that was applied by the Force POD. To investigate the effect of reduced power production, the models joint angular velocity was reduced in the stepping leg (since joint power is torque multiplied by angular velocity) at varying increments to note the effect on angular momentum and on the stability of the entire system. All other joint angles were constrained to be the same as the experimentally measured angles. With a 15% reduction in joint angular velocity in the leg that was to be placed the model fell over from the perturbation almost immediately. The PID no longer had sufficient gains to accommodate the 101b perturbation force (Figure 411). Conclusion Persons with a reduction in joint power are more likely to fall from a perturbation than those without joint power production. The complexities and redundancies of the human locomotors system make investigation into these principles difficult. An increasing application of biomechanical models is assisting in the understanding of the complex phenomena associated with human bipedal gait and the control thereof. Angular momentum is a gaining rapid recognition in the exploration of stable bipedal locomotion. In this work angular momentum successfully predicts foot placement preference in control subjects and should be developed further in the clinical setting to improve the versatility and robustness of gait in persons with stroke. Although a simple PID controller cannot be compared to the complexity of the human brain (even one with neurological inconsistencies), the work herein does develop an understanding for the taxation on a control system from reduced power production. A simulation utilizing a model with foot/ground contact interaction, which only required angular momentum control to achieve stable linear gait, explored and validated the necessity for rapid foot placement and power production for the prevention of fall. I 1. R = 0.09605 0 1 2 !r1ir lm EPjlUr MCHrtum R .0 65739 I 1 M 2 WialrlA lbl, MO"nL Figure 41. The angular momentum at the beginning of swing versus the angular momentum at midswing (left). The angular momentum at the beginning of swing versus the change in angular momentum through swing (right). Data is from a right stance leg which is why the initial angular momentum is primarily positive. S20 6z0r ' 0 50 100 Unperturbed Step JO  o 50 Ifo Perturbed Step 1 B 100 0 0 1Im Unperturbed Perturbed Step Step1 0 o Ica Perturbed Step 2 40 Perturbe2 Step 2 Perturbed Step 3 s4 e 3 60  Perturbed Step 3 a 5o ID Perturbed Step 4 Perturbed Step 5 Perturbed Perturbed Step 4 Step 5 Figure 42. AM profiles for normal (blue) and perturbed (red) states. The shaded region represents three standard deviations from the mean trajectory. The perturbation occurs at the black line in the frames labeled "Perturbed Step 1". 84 0 50 10H 0 0 6 i Step Step 1 Step Time = 1 0155 Step Time = 1 Perturbation Perturbed Step Step 1 Step Time = 10155 Step Time = 81 Perturbation Perturbed Step Step 1 Step Time = 1.0155 Step Time = 1 C14, X0 ;  Perturbation Perturbed Step Step 1 Step Time = 1.1135 Step Time = 81 Step Time = 1.113 Step Time = 0.81 0 1W Perturbed Step 2 Step Time = 1.063 0o 10 Perturbed Step 2 Step Time = 0779 ^A.. Perturbed Step 2 Step Time = 1 0'o& 3 0 j  Step Time = 0.779 PeouJ~ Step 2 50 0l o o ,BO [ sd I 0 0 1W 0 W PM D W 1M Perturbed Perturbed Perturbed Step 3 Step 4 Step 5 Step Time = 1D08 Step Time = 1.0565 Step Time = 1.0655 0 100 0 4 e 0 4o 1i0 Perturbed Perturbed Perturbed Step 3 Step 4 Step 5 Step Time =1.0165 Step Time =1.0705 Step Time = 1 0745 Perturbed Perbtrbed Perturbed Step 3 Step 4 Step 5 Step Time = 1.08 Step Time = 1.0565 Step Time = 1.0655 14 14, 14, 0 I 10, Perturbed Perturbed Perturbed Step 3 Step 4 Step 5 StepTime =1.0165 Step Time = 10705 StepTime = 10745 Figure 43. Joint powers from unperturbed (blue) and perturbed (red) states for a control and a hemiparetic subject. The perturbation occurs at the vertical black line in the window entitled "Perturbation step". i E'iensenr i Function) SRijl.i FooJt Placement I F r ;iT, Function) si Tc. Wialr i..mrr,. COM S' M Jiiid D: ir, Stdev COM ,, .,StdevCOM 2 0.2 0.4 0.6 0 .8 1 1.2 XDistance (Normalized to average stride length) Figure 44: Calculated foot placement limits, COP trajectory, COM trajectory, and the foot placement density functions relative to the previous foot placement of the contra lateral leg for subjects walking at self selected speeds. The xdistance is in the walking direction. E / Actual Z  S\ ^ N S4  0.2 ............ 0,2 0.4 0.6 0.8 1 1.2 50 100 150 200 250 300 350 Di Time (Normalized to average stde length) Figure 45. Angular momentum of the model undergoing control via a simple low gained the foot placement density functions relative to the previous foot placement of the PID controller. The controller opposed changes in angular momentum enough to cause stable gait but did not hinder the model from achieving a natural is in the walking direction. 'E Actual If ,2 Model F 0 :;4 50 100 150 200 250 300 350 Time (Frames) Figure 45. Angular momentum of the model undergoing control via a simple low gain PID controller. The controller opposed changes in angular momentum enough to cause stable gait but did not hinder the model from achieving a natural motion. Figure 46. The top row figures are from a simulation that reflects how a short step length for a given speed can result in a fall forward. The lower is the simulation in normal gait. A PID controller is applied in both cases. 30 Normal Profile of Model under PID control 25 Profile of Model under PID control with a step that falls short of the step length 20 minumum z 10 0S / ' E 5 5 0 50 100 150 200 Time (Frames) Figure 47. The angular momentum profile for 1 gaticycle of the model under PID control for the model falling over due to a shortened step length (see figure 4 5, top) and the model walking normally (see figure 45, bottom). Figure 48. The kinematic response of the subject to an anterior perturbation force of 10 Ibs applied just above the center of mass. ' 300 400 Time (Frames) \.9 #/ "it 500 Figure 49. The angular momentum profiles of the model undergoing perturbation with PID control gains tuned to give a similar response to experimental data and the experimental angular momentum profile of a subject undergoing the same perturbation. 40  S20[ o 0 100 200 300 400 500 600 700 Time (Frames) Figure 410. The perturbation force applied by the Force pod. The command was given to apply a 101b (44.49N) over a period of 0.5 seconds (100 Frames). E4 z li2 7 E E 2 E 2 0c ~I It * I I /I 9. 100 100 ,, / 0 i0 200 Actual Model /I a,/' 600 700 Figure 411. The response of the model to a 15% reduction in joint angular velocity in the left leg. The perturbation force of ten pounds caused a fall forward almost immediately. CHAPTER 5 CONCLUSION AND DISCUSSION There is a 73% incidence of falls among individuals with mild to moderate stroke. 37% of patients that fall sustain injury that required medical treatment. 8% sustained fractures and the risk for hip fracture is ten times higher in the stroke population. General risk of fracture is two to seven times greater following a stroke [4]. Due to the impact falls have on the health and well being of persons with stroke, there is a substantial need to assess dynamic balance during walking in order to ascertain who is at increased risk for falls. While little has been done to quantify balance during walking in the general population, much less has been done to quantify the effect of neurological impairment on balance during gait in persons with stroke. Approaches for determining who is at risk for falls have usually relied on clinical measurement scales as opposed to quantitative evaluation of joint power production ability. However, interpretations of these measures are based on statistical evidence and do not quantify a subject's balance performance during walking. To quantify and ultimately improve dynamic balance during neurologically impaired walking, it is necessary to quantify the biomechanical effects of altered or impaired joint power productions during paretic gait. These effects are relevant since a primary disability associated with poststroke hemiparesis is the failure to make rapid graded adjustment of muscle forces [5, 64]. Reduction in, or ill coordinated, muscle forces result in inefficient and ineffective net joint power production. In order to quantify the biomechanical effects related to a decrease in joint power production, a detailed biomechanical simulation of the human body during perturbed walking is necessary to elaborate on experimental findings. A simulation that can explore the effects of reduced power production on balance requires the development of a ground contact model that can explain the production of ground reaction forces and moments based on the deformation/kinematics of the foot. The work developed the ground contact model and the biomechanical simulation that mathematically relates reduction in joint power production to deficits in balance. Furthermore, the work investigates the role of various joints' in maintaining balance and quantifies successful balance with a mathematical measure: whole body angular momentum. LIST OF REFERENCES 1. AHCPR, Clinical practice guideline: poststroke rehabilitation. 1995, Rockville, MD: AHCPR. 2. Horak, F.B., D.M. Wrisley, and J. Frank, The Balance Evaluation Systems Test (BESTest) to differentiate balance deficits. Physical therapy, 2009. 89(5): p. 48498. 3. 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Kautz, Muscle force redistributes segmental power for body progression during walking. Gait Posture, 2003(in press). 32. Ogihara, N. and N. Yamazaki, Generation of human bipedal locomotion by a bio mimetic neuromusculoskeletal model. Biol Cybern, 2001. 84(1): p. 111. 33. Miyashita, K., S. Ok, and K. Hase, Evolutionary generation of humanlike bipedal locomotion. Mechatronics, 2003. 13(89): p. 791807. 34. Gilchrist, L.A. and D.A. Winter, A twopart, viscoelastic foot model for use in gait simulations. Journal of Biomechanics, 1996. 29(6): p. 795798. 35. Amirouche, F.M.L., S.K. Ider, and J. Trimble, Analytical Method for the Analysis and Simulation of Human Locomotion. Journal of Biomechanical Engineering, 1990. 112(4): p. 379386. 36. Mochon, S. and T.A. McMahon, Ballistic walking. J Biomech, 1980. 13(1): p. 4957. 37. Siegler, S., R. Seliktar, and W. Hyman, Simulation of human gait with the aid of a simple mechanical model. Journal of Biomechanics, 1982. 15(6): p. 415425. 38. Onyshko, S. and D.A. Winter, A mathematical model for the dynamics of human locomotion. Journal of Biomechanics, 1980. 13(4): p. 361368. 39. Ju, M.S. and J.M. Mansour, Simulation of the Double Limb Support Phase of Human Gait. Journal of Biomechanical Engineering, 1988. 110(3): p. 223229. 40. Gefen, A., et al., Biomechanical Analysis of the ThreeDimensional Foot Structure During Gait: A Basic Tool for Clinical Applications. Journal of Biomechanical Engineering, 2000. 122(6): p. 630639. 41. Cenk Guler, H., N. Berme, and S.R. Simon, A viscoelastic sphere model for the representation of plantar soft tissue during simulations. Journal of Biomechanics, 1998. 31(9): p. 847853. 42. Hertz, H., Aoeber die BerAhrung fester elastischer KArper. Journal fAA4r die reine und angewandte Mathematik, 1826. 92: p. 156171. 43. Allum, J.H.J., et al., Trunk sway measures of postural stability during clinical balance tests: effects of a unilateral vestibular deficit. Gait & posture, 2001. 14(3): p. 227237. 44. Adkin, A.L., B.R. Bloem, and J.H.J. Allum, Trunk sway measurements during stance and gait tasks in Parkinson's disease. Gait & posture, 2005. 22(3): p. 240 249. 45. Winter, D.A., Human balance and posture control during standing and walking. Gait & posture, 1995. 3(4): p. 193214. 46. Tang, A. and W. Rymer, Abnormal forceEMG relations in paretic limbs of hemiparetic human subjects. J Neurol Neurosurg Psychiatry, 1981. 44(8): p. 690 8. 47. Tang, P.F., M.H. Woollacott, and R.K.Y. Chong, Control of reactive balance adjustments in perturbed human walking: roles of proximal and distal postural muscle activity. Experimental brain research, 1998. 119(2): p. 141152. 48. Ferber, R., et al., Reactive balance adjustments to unexpected perturbations during human walking. Gait & posture, 2002. 16(3): p. 238248. 49. Neptune, R.R. and S.A. Kautz, Muscle activation and deactivation dynamics: the governing properties in fast cyclical human movement performance? Exerc Sport Sci Rev, 2001. 29(2): p. 7680. 50. Neptune, R.R., S.A. Kautz, and F.E. Zajac, Contributions of the individual ankle plantar flexors to support, forward progression and swing initiation during walking. J Biomech, 2001. 34(11): p. 138798. 51. Neptune, R.R., S.A. Kautz, and F.E. Zajac, Comments on "Propulsive adaptation to changing gait speed". Journal of Biomechanics, 2001. 34(12): p. 166770. 52. Zajac, F. and M. Gordon, Determining muscle's force and action in multiarticular movement. Exerc Sport Sci Rev, 1989. 17: p. 187230. 53. MacKinnon, C.D. and D.A. Winter, Control of whole body balance in the frontal plane during human walking. Journal of Biomechanics, 1993. 26(6): p. 633644. 54. Bell, A.L., R.A. Brand, and D.R. Pedersen, Prediction of hip joint centre location from external landmarks. Human Movement Science, 1989. 8(1): p. 316. 55. Chao, E.Y.S., Graphicbased musculoskeletal model for biomechanical analyses and animation. Medical engineering & physics, 2003. 25(3): p. 201212. 56. O'Toole, R.V., et al., Biomechanics for preoperative planning and surgical simulations in orthopaedics. Computers in Biology and Medicine, 1995. 25(2): p. 183191. 57. Korioth, T.W.P. and A. Versluis, Modeling the Mechanical Behavior of the Jaws and Their Related Structures By Finite Element (Fe) Analysis. Critical Reviews in Oral Biology & Medicine, 1997. 8(1): p. 90104. 58. Multon, F., et al., Computer animation of human walking: a survey. The Journal of Visualization and Computer Animation, 1999. 10: p. 3954. 59. Soderkvist, I. and P.A. Wedin, Determining the movements of the skeleton using wellconfigured markers. Journal of Biomechanics, 1993. 26(12): p. 14731477. 60. ShumwayCook, A., et al., Predicting the probability for falls in community dwelling older adults. Phys Ther, 1997. 77(8): p. 8129. 61. Berg, K.O., et al., Measuring balance in the elderly: validation of an instrument. Can.J.Public Health, 1992. 83 Suppl 2: p. S711. 62. Steffen, T.M., T.A. Hacker, and L. Mollinger, Age and genderrelated test performance in communitydwelling elderly people: SixMinute Walk Test, Berg Balance Scale, Timed Up & Go Test, and gait speeds. Physical therapy, 2002. 82(2): p. 12837. 63. Perry, J., Gait Analysis: Normal and Pathological Function. 1992, Thorofare, NJ: Slack, Inc. 64. Mulroy, S., et al., Use of cluster analysis for gait pattern classification of patients in the early and late recovery phases following stroke. Gait & posture, 2003. 18(1): p. 11425. BIOGRAPHICAL SKETCH Cameron Nott was born in South Africa in 1981. He started his academic career in the United States of America in 2001 as a student athlete at McNeese State University in Louisiana. There, he specialized in jet engine control systems as a mechanical engineering student. Consequently, Cameron was recruited to the University of Alabama as a track athlete where his research during his master's in aerospace engineering involved supersonic wind tunnel control systems, jet engine health monitoring systems, and laser Doppler velocometery with application of artificial intelligence. During this period, Cameron began to become interested in neural prosthesis and decided to pursue his doctoral degree at the Brain Rehabilitation Research Center in the Malcolm Randall VA Medical Center in Gainesville, Florida. There, he obtained his PhD and went on to be an assistant research professor at the Medical University of South Carolina, South Carolina. PAGE 1 1 ANGULAR MOMENTUM DURING GAIT A COMPUTATIONAL SIMULATION By CAMERON NOTT A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010 PAGE 2 2 2010 Cameron Nott PAGE 3 3 To my beautiful wife PAGE 4 4 ACKNOWLEDGMENTS First and foremost, I would like to thank my parents, sisters and grandparents who have been so supportive. I also ask their forgiveness for not being with them at the weddings, childbirths, surgeries, funerals and many other important events. Although my heart is heavy after the ten years spent more than fifteen thousand miles away from them, I am very grateful for the sacrifices they have made to allow me to obtain my doctoral degree. I especially thank my mother and father for remaining so positive and supportive despite all the tears shed at the airport every year or so. My education is as much their achievement as it is mine. Words cann ot express the extent of my thanks. I would like to thank Dr. Kautz for always believing in my ability and being the prime example of professionalism, intelligence, and a super advisor in general. I thank Dr. Fregly for re establishing my passion for exploring a small portion of a unified physical existence that God has created for us. Dr. Bowden and Dr. Gregory also deserve extended thanks for all the help and guidance along the way at a professional and personal level. My appreciation is also for Dr. Olcm en for helping me discover my potential, for the continuing advice, and for being a fantastic man and advisor. Finally and most importantly, I would like to thank my wife who has managed to make me laugh every day, supported me in every possible situation, and always been so exceptionally kind and warm hearted. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS .................................................................................................. 4 LIST OF FIGURES .......................................................................................................... 7 LIST OF ABBREVIATIONS ........................................................................................... 10 ABSTRACT ................................................................................................................... 11 CHAPTER 1 INTRODUCTION .................................................................................................... 13 Reduced Balance Capacity in the Stroke Population .............................................. 13 Dynamic Balance During Gait Can Be Studied Through Whole Body Angular Momentum, Foot P lacement, and Joint Power Production .................................. 14 Whole Body Angular Momentum ...................................................................... 14 Foot Placement During Gait ............................................................................. 16 Joint Power Production and Maintenance of Whole Body Angular Momentum .................................................................................................... 17 Hemiparesis Adversely Affects Balance via Foot Placement and Pow er Production ..................................................................................................... 17 Biomechanical Simulation a Means to Fully Understand Inadequate Power Generation as it Relates to Regulation of Angular Momentum ............................ 18 All Joint Moments Significantly Contribute to Whole Body Angular Momentum .................................................................................................... 19 Biomechanical Simulation Requires the Development of a Ground Contact Model ............................................................................................................ 20 Biomechanical Simulation Optimization Will Reveal the Relation Between Joint Power Production and Whole Body Angular Momentum ...................... 21 Conclusion .............................................................................................................. 22 2 THE SENSITIVITY OF TRUNK ANGULAR ACCELERATION TO ALL JOINT MOMENTS ............................................................................................................. 23 Introduction ............................................................................................................. 23 Methods .................................................................................................................. 24 Subject ............................................................................................................. 24 Data Collection ................................................................................................. 24 Model ................................................................................................................ 25 Results and Discussion ........................................................................................... 28 Sagittal Plane ................................................................................................... 28 Frontal Plane .................................................................................................... 29 Joint Moment Contributions to Trunk Angular Accelerations ............................ 29 Control of Trunk Angular Acceleration by the CNS .......................................... 30 PAGE 6 6 3 THE DEVELOPEMENT OF A GROUND CONTACT MODEL FOR BIOMECHANICAL SIMULATION FROM TREADMILL DATA ................................ 38 Introduction ............................................................................................................. 38 Methods .................................................................................................................. 39 Subjects ............................................................................................................ 39 Data Collection ................................................................................................. 40 Pressure Sensitive Film .................................................................................... 41 Three Point Calibration .................................................................................... 43 Foot Model ....................................................................................................... 44 Full Body Model ................................................................................................ 44 Visco Elastic Co ntact Elements .............................................................................. 46 Friction Model ......................................................................................................... 49 Coulomb Friction .............................................................................................. 49 Viscous Friction ................................................................................................ 51 Torsional Friction .............................................................................................. 51 Kinematic Optimization ........................................................................................... 52 Results and Discussion ........................................................................................... 53 Conclusion .............................................................................................................. 54 4 ANGULAR MOMENTUM CONTROL DURING GAIT ............................................. 69 Introduction ............................................................................................................. 69 Methods .................................................................................................................. 71 Subjects ............................................................................................................ 71 Data Collection ................................................................................................. 72 Model ................................................................................................................ 72 Pe rturbation ...................................................................................................... 72 Whole Body Angular Momentum During the Gait Cycle ......................................... 73 Sagittal Plane: .................................................................................................. 73 Angular Momentum in the Frontal plane .......................................................... 74 The Relation Between Joint Power and Whole Body Angular Momentum .............. 75 Step Length and Moment Impulses Relation to Angular Momentum Correction ..... 76 The Effects of Reduced Joint Power Production on Resisting Perturbation During Gait .......................................................................................................... 81 Conclusion .............................................................................................................. 82 5 CONCLUSION AND DISCUSSION ........................................................................ 90 LIST OF REFERENCES ............................................................................................... 92 BIOGRAPHICAL SKETCH ............................................................................................ 97 PAGE 7 7 LIST OF FIGURES Figure page 2 1 Comparison of the current model (blue) to data from previously published work (red) (Winter 1991). .................................................................................... 31 2 2 The sensitivity of the trunk angular acceleration about the x axis to the joint moments during gait. .......................................................................................... 32 2 3 The sensitivity of the trunk angular acceleration about the y axis to the joint moments of the leg during gait.. ......................................................................... 33 2 4 The sensitivity of the trunk angular acceleration about the y axis to the joint moments of the leg during gait.. ......................................................................... 34 2 5 Frontal and sagittal plane views illustrate how a sagittal moment can cause frontal plane accelerations by means of reaction forces along the kinematic chai n. .................................................................................................................. 35 2 6 Contribution of each joint moment to trunk angular acceleration in the sagittal plane. .................................................................................................................. 36 2 7 Two methods of calculating the contact moments are illustrated by taking free bodies of the trunk and lower extremity. ...................................................... 37 3 1 Alignment of the pressure film in the foot reference frame relative to the motion tracking markers ..................................................................................... 5 6 3 2 Calibration of the pressure film. .......................................................................... 57 3 3 Dimensionality reduction using kmeans clustering.. ........................................... 58 3 4 Anthroprometric model with (left) and without (right) segmental coordinate system definititions.. ........................................................................................... 58 3 5 The convergence of an optimization to find the left shoulder joint center from a poor initial guess. ............................................................................................. 59 3 6 Comparison of the viscoelastic models reproduction of the experimental data that was used to train the model ................................................................. 60 3 7 Comparison of an optimized viscoelastic element to experimentally collected data. ................................................................................................................... 61 PAGE 8 8 3 8 The comparison of an example experimentally measured element penetration and a hypothetical penetration that would cause a viscoelastic element to exactly match its corresponding ground reaction force. The difference in the measurement falls well within the experimental error. .............. 62 3 9 On an ADAL ....................................................................................................... 63 3 10a The weight is at rest on the treadmill belt with no force being applied. ............... 64 3 10b With a 9.7 kg force being applied along the y axis (along the walking direction on the treadmill) no sliding motion of the belt or the weight on the belt occurs. ......................................................................................................... 65 3 11 Kinematic alterations in joint angles produce foot motion that accommodates foot/ground contact modeling. ............................................................................ 66 3 12 The effect of kinematic variations removes the nonphysical negative damping characteristic seen in the original experimental data. .......................... 67 3 13 Resulting force and moment production of the foot contact model post kinematic optimization.. ...................................................................................... 68 4 1 The angular momentum at the beginning of swing versus the angular momentum at mid swing (left). The angular momentum at the beginning of swing versus the change in angular momentum through swing (right).. ............. 84 4 2 AM profiles for normal (blue) and perturbed (red) states. ................................... 84 4 3 Joint powers from unperturbed (blue) and perturbed (red) states for a control and a hemiparetic subject.. ................................................................................. 85 4 4 Calculated foot placement limits. ........................................................................ 86 4 5 Angular momentum of the model undergoing control via a simple low gain PID controller ...................................................................................................... 86 4 6 The top row figures are from a simulation that reflects how a short step length for a given speed can result in a fall forward. ........................................... 87 4 7 Th e angular momentum profile for 1 gaticycle of the model under PID control for the model falling over due to a shortened step length (see figure 45, top) and the model walking normally (see figure 45, bottom). .................................. 87 4 8 The kinematic response of the subject to an anterior perturbation force of 10 lbs applied just above the center of mass. .......................................................... 88 4 9 The angular momentum profiles of the model undergoing perturbation with PID control gains ............................................................................................... 88 PAGE 9 9 4 10 The perturbation force applied by the Force pod. The command was given to apply a 10lb (44.49N) over a period of 0.5 seconds (100 Frames). .................... 88 4 11 The response of the model t o a 15% reduction in joint angular velocity in the left leg. The perturbation force of ten pounds caused a fall forward almost immediately. ....................................................................................................... 89 PAGE 10 10 LIST OF ABBREVIATIONS AM Angular Momentum COM Center of mass COP Center of pressure GRF Ground reaction force GRM Ground reaction moment acting at the COP PAGE 11 11 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 CONTROL OF ANGULAR MOMENTUM DURING HEMI PARETIC GAIT A COMPUTATIONAL SIMULATION By Cameron Nott August 2010 C hair: B. J. Fregly Major: Mechanical Engineering There is a 73% incidence of falls among individuals with mild to moderate stroke. 37% of patients that fall sustain injury that required medical treatment. 8% sustained fractures and the risk for hip fracture is ten times higher in the stroke population. G eneral risk of fracture is two to seven times greater following a stroke. Due to the impact falls have on the health and well being of persons with stroke, there is a substantial need to assess dynamic balance during walking in order to ascertain who is at increased risk for falls. While little has been done to quantify balance during walking in the general population, much less has been done to quantify the effect of neurological impairment on balance during gait in persons with stroke. Approaches for det ermining who is at risk for falls have usually relied on clinical measurement scales as opposed to quantitative evaluation of joint power production ability However, interpretations of these measures are based on statistical evidence and do not quantify a subjects balance performance during walking To quantify and ultimately improve dynamic balance during neurologically impaired walking, it is necessary to quantify the biomechanical effects of altered or impaired joint PAGE 12 12 power productions during paretic gait. These effects are relevant since a primary disability associated with post stroke hemiparesis is the failure to make rapid graded adjustment of muscle forces. Reduction in, or ill coordinated, muscle forces result in inefficient and ineffective net joint power production. In order to quantify the biomechanical effects related to a decrease in joint power production, a detailed biomechanical simulation of the human body during perturbed walking is necessary to elaborate on experimental findings. A simulation that can explore the effects of reduced power production on balance requires the development of a ground contact model that can explain the production of ground reaction forces and moments based on the deformation/k inematics of the foot. The following work develops the ground contact model and the biomechanical simulation that mathematically relates reduction in joint power production to deficits in balance. Furthermore, the work investigates the role of various join ts in maintaining balance and quantifies successful balance with a mathematical measure: whole body angular momentum. To improve understanding of the relation between joint power production and balance, the simulation is subjected to an optimization procedure that results in kinematic and kinetic solutions that return whole body angular momentum to a regular limit cycle following a perturbation. Studying the return of whole body angular momentum to a regular limit cycle after a perturbation assists in understanding balance since whole body angular momentum collectively quantifies rotational velocity of the entire body about the center of mass and in order to maintain balance, the rotational velocity of the entire body is necessarily bounded. PAGE 13 13 CHAPTER 1 INTRODUCTION Reduced Balance Capacity in the Stroke Population The impact of stroke on walking is significant [ 1] with only 37% of stroke survivors able to walk after the first week. Even among those who achieve independent ambulation, significant residual deficits persist in balance [2, 3] and gait speed with a 73% incidence of falls among individuals with mild to moderate impairment 6 months post stroke [4]. A primary disability associated with post stroke hemiparesis is the failure to make rapid graded adjustment of muscle force and to sufficiently activate muscles [5]; consequently, a reduced ability to rapidly respond to perturbations that influence balance is inferred. Various compensatory walking patterns have been suggested to increase balance in hemipar etic gait (e.g., increased double support time); however, it is difficult to state whether kinematic or kinetic variations during paretic gait are adaptations done by choice to increase stability or the result of an impaired neural controller and decreased power production. Regardless, persons with hemiparesis do not have robust control over dynamic balance during gait and little is understood about how to quantify their ability to retain balance. Balance control is an interaction of the individual, the tas k, and the environment as actuated by the neuromuscular system. While various theoretical and clinical models exist for static balance, they are not directly applicable to all of the dynamic aspects of balance during walking [6]. Most falls occur during walking [7] but most clinical balance assessment tests do not include assessment of dynamic walking. Balance control mechanisms are specific to the task being performed and the environment in which the task is being performed [8] and recent walking recovery literature emphasizes the PAGE 14 14 importance of task specificity during rehabilitation [9, 10] None of the clinical assessments of balance control in gait allow for the assessment of reactive balance mechanisms, which are critical for both postural and equilibr ium control. These reactive responses depend on rapid input from subcortical mechanisms depending on somatosensory and vestibular sources, both of which are often impaired in those with stroke [11] There is a substantial need to assess dynamic balance and reactive responses during walking in order to ascertain who is at increased risk for falls and whether a targeted intervention has improved dynamic balance. Dynamic Balance During Gait Can Be Studied Through Whole Body Angular Momentum, Foot Placement, and Joint Power Production The principle of whole body angular momentum is the primary connection between joint power production and balance. F oot placement is a means of angular momentum regulation and a fall in any direction requires appropriate foot placement to regain balance. Whole Body Angular Momentum Whole body angular momentum (AM) is the summation of segmental moment of i nertia multiplied by angular velocity. Thus, it collectively quantifies the rotational velocity of the entire body about the center of mass (COM) in a single measure. It can be calculated from kinematic data where the AM of the entire body is expressed as the superposition of individual segmental AM To find the whole body AM about any arbitrary point q, a summation is performed [12] = + p p ( v v ) (1 1) PAGE 15 15 In the summation, is the whole body AM about point q is the i th segments inertia tensor and is its angular velocity. p and v are the position and velocity of point q respectively while p and v are the position and velocity of the segments center of mass. m is the segments mass. To calculate whole body AM from kinematics, an anthropometric model is required to determine m and I and to measure segmental kinematics. A typical 3D model would consist of 13 rigid segments (two feet, two shanks, two thighs, pelvis, trunk, two forearms, two upper arms, and a head). Furthermore, a 3D model is imperative to ensure that interplan a r coupling is taken into account. A 2D model w ill not reflect interplanar coupling [13] To maintain uprig ht walking, the whole body AM should maintain oscillation about zero, since a constant nonzero AM implies continuous rotation of all segments about an axis (other than the vertical axis) which would most likely represent falling [14] Previously it has been noted that inter segmental AM cancellations are mostly responsible for a close to zero whole body AM and various successful control schemes have been developed based on this zero moment assumption in the robotics literature [15] In highly sophisticated bipedal robots, control systems that maintain a constant zero AM, despite perturbation, result in stable and robust multi segmental bipedal gait. This has been described as the only reliable stability index of dynamic bipedal locomotion in robotics [16] Despite the success of such control systems in robotics, it is noted from experimental results that humans do not follow this strategy completely. Humans tend to allow oscillation in AM about zero during gait [12, 17] As opposed to the zero moment assumption, which assumes that the subjects strive to maintain a constant zero AM throughout the gait cy cle, we hypothesize that inter  PAGE 16 16 segmental AM cancellations are not the primary mechanisms of AM stabilization. The hypothesis states that healthy and efficient walking requires fluctuation of the AM from zero, and it requires a mechanism whereby the AM can be directed to zero when necessary in normal gait or perturbed walking. We propose that this regulation occurs during normal and perturbed gait because AM is often manipulated purposefully during certain tasks [17] Phases of normal walking have been described as a controlled fall, which can be thought of as a nonzero fluc tuation in AM. To control this fall, it is proposed that foot placement is the primary mechanism by which AM is controlled [18, 19] Foot Placement During Gait Little or no attention has been given to whole body AM in conjunction with foot placement and balance, especially during gait. Inverted pendulum models (which model the human body as a rigid strut with a point mass on the end) have been used to determine margin of stability (essentially the proximity of the center of mass to the base of support after taking into account the velocity of the center of mass ) as a measure of dynamic balance [20] Inverted pendulum models neglect the angular inertial properties of the human body amongst other assumptions. Furthermore, theoretically determined margin of stability have not been experimentally verified to relate to im paired subjects ability to maintain balance during gait. Whether inverted pendulum assumptions are justified has been discussed by Herr et al. [12] with respect to human balance and Goswami [21 23] with respect to bipedal robots. Regardless, it should be noted that whole body angular momentum carries vital information that allows a mathematical prediction of where a foot should be placed such that the subject does not fall [18, 19] The maintenance of AM during gait therefore forms the theoretical basis of the this PAGE 17 17 research utilizing t he fact that foot placement is a means to ensure the return of AM to zero from a nonzero state, without disrupting the center of mass trajectory Joint Power Production and Maintenance of Whole Body A ngular M omentum Two factors are related to directing the angular momentum in the correct direction: the moment arm from the center of mass to the center or pressure, and the force acting at the center of pressure. These two factors can be related to whole body angular momentum in a simple fashion: = + (1 2) In this relation is the time rate of change of the angular momentum about the center of mass is the ground reaction moment and is the ground reaction force. is the vector from the center of mass to the center of pressure which is located on the ground surface. To substantiate a significant alteration in whole body angular momentum, the above relation shows that a significant contribution from a single foot is r elated to the distance of that foot from the center of mass and the magnitude of the ground reaction force that is produced by that foot. Since the ground reaction force can be related to any single joint moment by the standard inverse dynamics equations, joint power production is directly related to the change in angular momentum about the center of mass (as is foot placement). Hemiparesis Adversely Affects Balance via Foot Placement and Power Production I mpaired subjects with an inability to rapidly respond to perturbation, do not have the power production capacity to place the foot fast enough to regain balance within the gait cycle [24] Studying the response to perturbation will allow for the classification of PAGE 18 18 balance recovery ability according to neurological deficit and also rate balance recovery strategies. Understan ding the inability of paretic subjects to regain control of their angular momentum will further develop clinical assistance in the prevention of falls. When AM is perturbed, there is a time constraint that must be met to place the foot in order to correct the AM fluctuation. If the foot that is to correct AM is placed excessively late or not far enough from the COM a fall may occur [18, 19] Control subjects are expected to respond more rapidly than hemiparetic subjects and have foot placement that is more consistent with a mathematically viable solution for rectifying the post perturbation angular mom entum [24] Since hemiparetic subjects have poor coordination and reduced power production ability, they are likely less able to rapidly place a foot far enough forward to be in a position that can reestablish dynamic balance. Thus, hemiparetic subjects should require more time and steps to regain a s teady AM profile and demonstrate more fluctuation in angular momentum for the same perturbation when compared to a control subject. Subjects who are more likely to fall are expected to be those who are less able to quickly regain their AM profile. Studying the dynamic balance response to perturbation may allow for the classification of dynamic balance performance according to specific neurological deficits. Biomechanical Simulation a Means to Fully Understand Inadequate Power Generation as it Relates to Regulation of Angular Momentum Biomechanical simulation is growing in popularity as a means to understand the mechanics of the human body. It is the application of the first principles of mechanics to living organisms. The first instance of the application of mechanics was written by Aristotle and was titled De Motu Animalium or On the Movement of Animals. In this work, Aristotle first performed mind experiments to determine how an animal could PAGE 19 19 achieve a task without actually requiring the animal to do the task. Today the science ranges from learning about flight from the behavior of insects to making more comfortable mattresses. In this application biomechanical simulation is used to determine the joint power production requirements needed to overcome cer tain perturbing forces during gait. In order to complete the simulation, three topics are to be addressed. The first topic is the effect of joint moments on segments angular momentum to which the joint does not directly apply. Since the trunk is more than one third the mass of the entire body, the effect of joint moments of the lower extremities on trunk angular acceleration is sufficient to establish whether whole body AM can be affected by all the lower extremity joints. The second is the generation of a suitable ground contact model. The third is the development of a biomechanical model to which optimization routines can be applied to determine suitable joint power productions that can stabilize whole body AM after a perturbation. All Joint Moments Signi ficantly Contribute to Whole Body Angular Momentum Computationally advanced biomechanical analyses of gait demonstrate the often counter intuitive roles of joint moments on various aspects of gait such as propulsion, swing initiation, and balance [13, 2528] Each joint m oment can produce linear and angular acceleration of all body segments (including those on which the moment does not directly act) due to the dynamic coupling inherent in the interconnected musculoskeletal system. This study presents the quantitative relationships between individual joint moments and trunk control with respect to balance during gait to show that the ankle, knee, a nd hip joint moments all affect the angular acceleration of the trunk and consequently the angular momentum of the trunk which is a large portion of the whole body angular momentum which cannot be accounted for by inter segmental PAGE 20 20 angular momentum cancelati ons We show that trunk angular acceleration is affected by all the joints in the leg with varying degrees of dependence during the gait cycle. Furthermore, it is shown that inter planar coupling exists and a two dimensional analysis of trunk balance negle cts important out of plane joint moments that affect trunk angular acceleration. Biomechanical Simulation Requires the Development of a Ground Contact Model Of the intricate modeling necessities specific to human gait, foot ground contact has been presented as challenging. The most common approach presented is a spring damper model in which spring damper elements dictate ground reaction forces [29 34] Another approach is to assume a welded joint between the ground the foot [35 38] or to model the foot as a rigid rolling surface [39] More complex methods such as finite element models from study on cadavers and magnetic resonance imaging have been proposed [40] Visco elastic sphere models have also been introduced to represent the plantar surface of the foot [41] Visco elastic spherical contact models have been suggested with increasing interest to model the foot [34, 41] A contact model based on a visco elastic sphere was first studied by the H. R. Hertz [42] and expresses the relationship between the spheres deformation and the resulting force that causes the deformation. The viscoelastic sphere model presented by Guler et al relies on experimental data from heel pads and Guler et al conclude that viscoelastic spheres can be incorporated into foot models for simulation. More complex spherical models have been presented that replace constant exponent of 3/2 that exists in Hertzs model with a variable n which dictates the geometry of the model (Marhefka 1999). The primary procedure by which a c ontact elements parameters are derived is with forward simulation [29 34] In determining spring and damping constants in forward PAGE 21 21 simulation, it is permissible to allow a certain amount of variation in kinematic response. This is typically the cost function of the optimization and the resulting kinematics of the foot does not match experimental kinematics exactly. The contact model developed herein uses neural networks to accommodate for experimental inaccuracies with respect to kinematic measurements. The input into the model is the kinematics of a set of penetrative elements that are fixed in a multi segmented foot. The positions of the elements are determined with a three point calibration using a pressure sensitive film. The result is a foot contact model that can accept measured kinematics and duplicate ground reaction force measurements. This is useful for inverse dynamic optimizations where the input is a set of simulated joint angles which, with the ground contact model, fully define the ground reaction forces and moments at the feet. Biom echanical Simulation Optimization Will Reveal the Relation Between Joint Power Production and Whole Body Angular Momentum To understand and quantify the regulation of whole body AM with reduced power production ability, an optimization will be run to deter mine the joint angle trajectories that will reestablish a biomechanical simulations whole body AM to a normal trajectory as determined from experimental data. The experiential data is collected with a novel, state of the art perturbation system on an inst rumented treadmill that fully records bilateral ground reaction forces and moments. Furthermore, a motion capture system measures the kinematics of the subject. With such a computational model in place, it will be possible to simulate perturbations applied to patients with hemiparesis without subjecting them to a dangerous situation or greater risk of falls. The simulation will also determine and PAGE 22 22 quantify minimal joint power requirements needed to overcome various levels of perturbation forces. Conclusion The presented research is a means to quantify the balance capability of subjects with hemiparesis based on biomechanical principles and power production capability; something that is much needed in the clinical environment. The end result is a quantificati on of foot placement requirements to maintain balance during gait. In order to produce a quantification of a subjects ability to withstand perturbation forces during gait, three tasks are to be achieved. The first is the establishment of the sensiti vity of trunk angular acceleration on lower extremity joint moments. The second is the development of a ground contact model. The third is the application of th e first and second tasks to a computational simulation using a biomechanical model that can determine the effects of reduced joint power production and it effect on overcoming perturbation. PAGE 23 23 CHAPTER 2 THE SENSITIVITY OF T RUNK ANGULAR ACCELERATION TO ALL JOINT MOMENTS Introduction Dynamic balance during walking consists in large part of stabilization o f the trunk. Trunk angular sway has been indicated as a reliable measure of balance stability [43, 44] ; however, there are inconsistencies in the literature as to how each of the lower extremity joint moments affects trunk angular acceleration. Previous studies have argued that b ecause the inertia of the trunk about the ankle joint is nearly eight times the iner tia of the trunk about the hip joint, the ankle muscles would need nearly ei ght times the moment of force relative to provide the same angular acce leration of the head, arms, and trunk segment as the hip muscles [45] They concluded that the central nervous system (CNS) may recognize this and utilize the hip muscles as the primary ac tuator for trunk angular control with little involvement of the ankle muscles during gait. Alternatively, EMG studies have concluded that the role of distal musculature is as important as proximal hip musculature in maintaining balance using a movable plat form to allow for perturbations [11, 46, 47] Similar observations, using joint power and muscle EMG patterns, show that reactive balance control is more likely a synchronized effort of the lower extremity joint moments to prevent collapse during perturbations [48] Although the contribution of each muscle (or joint moment) has been explored with regards to trunk propulsion and support and segmental power flow [26 28, 4951] similar relationships to trunk angular acceleration have been relatively unexplored. In the present study, we use a three dimensional model and the resulting forward dynamic equations of motion to identify the relative contribution of each joint moment t o the trunk PAGE 24 24 angular acceleration and assess whether one joint more strongly accelerates the trunk than the others. We also analyze the contribution of sagittal plane moments on frontal plane trunk accelerations that are made possible through dynamic coupli ng [52] along with the contribution of frontal plane moments on sagittal plane accelerations, to assess whether caution is necessary when interpreting trunk angular acceleration with only a single plane analysis [53] Methods Subject A single subject whose joint moments and joint angular velocities are consistent with published data used in similar trunk control research (e.g., Wint er, 1991) ( Fig ure 21) was used for demonstration purposes. The subject walked at self selected speed on an instrumented treadmill ( Techmachine, Andrezieux Boutheon, France) for three trials of thirty seconds each. Data Collection Kinematic data were collected using a 12 camera VICON system and sampled at 100Hz to capture spatial positions of markers placed on the subject in a modified Helen Hayes configuration. Rigid clusters of markers were placed on the feet, left and right shanks, left and right thighs, pelvis, trunk, head, and upper and lower arms to allow for six degree of freedom orientation measurement of each segment. Hip joint centers were determined using the CODA algorithm [54] In this work we have assumed that treadmill and overground gait are similar enough to make this study relevant to overground walking. We believe this is justified because each muscle will still accelerate all segments in exactly the same way in each mode of walking. PAGE 25 25 Model A 12 segment rigid body model consisted of two feet, shanks, thighs, forearms, and upper arms, a head, and a combined pelvis and trunk segment. Each segments inertial characteristics were defined using Visual 3D (C motion, Germantown, Maryland). The equations of motion were derived using a standard dAlembert approach and a sensitivity matrix relating the joint moments to trunk accelerations was derived. Each of the 12 segments of the subject was assumed to be a rigid segment and modeled accordingly. The equations that governed the system are shown in equation 1. 0 0tan tan inertia ce dis contact inertia ce dis contactM M M F F F ( 2 1 ) Here, F is a force vector acting on a segment and M is the moment vector acting on a segment. The subscripts denote the type of force or moment as being either contact, distance, or inertia which describes forces or moments due to contact, grav ity, and inertial forces respectively. The inertial forces are further expanded in equation 2 as: )) ( ) ( (1 R I R I R RCOM inertia inertiaM a m F ( 2 2 ) In equation 2, the inertial moment is taken about the center of mass (COM). R is the rotation matrix, which describes the segment in the lab reference frame, and I is the inertial matrix in the segment reference frame. Since the inertial matrix dotted with the rotation matrix generally results in a matrix with nonzero diagonal terms, rotational dynamic coupling is possible. The mass of the segment is defined by m and the angular acceleration and velocity of the segment with respect to the lab reference frame is PAGE 26 26 defined by and respectively. The inertial moments acting on the segment can be expressed about any point, O in the lab using equation 3. COM inertia inertia COM O O inerrtiaM F p M / ( 2 3 ) In equation 3, COM Op/ is the vector from point O in the lab to the COM of the segment, which is necessary to ensure that summation of the contact, distance, and inertial moments occur about the same point with respect to the same reference frame. Equations 1, 2, and 3 can be applied to any body or group of bodies which are selected as the free body in question. In order to establish the trunk angular acceleration as a function of the lower extremity joint moments, joint constraints are applied to the ankle, knee, and hip. The equation used to do so is given by equation 4. 0 ) ( ) ( p p p d d d p p d d P Dp p p p a a ( 2 4 ) Equation 4 relates the acceleration of the distal and proximal segment to one another since they are connected by a ball and socket joint. dp and pp are the vectors from a point O in the lab to the COM of the distal and proximal segments, respectively. It is assumed that the moment acting on the proximal end of the distal segment is equal and opposite to the moment acting on the distal end of the proxi mal segment. The moments acting at the proximal end of each segment (at each joint) are given by the equations (5). S contact T T T T T T T T T T T T T T T contact F contact S S S S S S S S S S S S S S S contact F F F F F F F F F F F F F F F contactM g m p a m p M M g m p a m p M GRM GRF COP g m p a m p MCOM COM COM COM COM COM ) ( )) ( ) ( ) ( ( ) ( ) ( )) ( ) ( ) ( ( ) ( ) ( )) ( ) ( ) ( ( ) (1 1 1 R I R I R R R I R I R R R I R I R R ( 2 5 ) PAGE 27 27 Similarly the forces acting on the proximal end of each segment are given by equation 6. S contact T T T T contact F contact S S S S contact F F F F contactF g m a m F F g m a m F GRF g m a m F ( 2 6 ) In the equations 5 and 6 the subscripts F S and T reflect the foot, shank, and thigh segments, respectively. The above equations of motion can be linearly parameterized into equation 7 which reflects the trunk angular acceleration as a function of the joint moments. Z Y X H K A H K AC C C M M M S S S S S S S S S S S S S S S S S S S S S S S S S S S 39 38 37 36 35 34 33 32 31 29 28 27 26 25 24 23 22 21 19 18 17 16 15 14 13 12 11 ( 2 7 ) For the sake of example, the moment equations for a threedimensional inverted pendulum is derived and linearly parameterized to express the angular acceleration of the pendulum in terms of the moment at the pivot point. The moment equation for an inverted pendulum is given as: 0 ) ( g m P a m P F P Mcom com contact contact I I ( 2 8 ) This equation can be written in terms of its elements. z y x com x y x z y z z y x com x y x z y z z y x x y x z y z z y x x y x z y z z y xmg P P P P P P a m a m a m P P P P P P F F F P P P P P P M M M 0 0 0 0 0 0 0 0 ... ... 0 0 0 0 0 01 1 1 1 1I I I I I I ( 2 9 ) PAGE 28 28 In equation 9, the angular accelerations sensitivity to the cont act moments are the inverse of the inertia matrix in the laboratory reference frame for a given configuration of the pendulum. The sensitivity matrix S (equation 7) is a time varying matrix which is dependent on the geometry of the model. The constant vec tor C is also dependant on the geometry of the model and the scalars x y and z in equation 7 are the angular accelerations of the trunk about the x, y and z axes, respectively. The joint constraints are defined by equation 4 in above. This equation assumes a ball and socket joint for all joints. The laboratory axes have the x axis directed to the subjects right, the y axis directed forward, and the z axis directed vertically. Note, to exp ress the threedimensional vectors in a sense that is most comparable with previously performed twodimensional analysis, moments are expressed with reference to axes that remain parallel with the laboratory axes but with an origin at the proximal end of t he corresponding segment (c.f., anatomically defined axes). Results and Discussion Sagittal P lane The elements of the sensitivity matrix (angular acceleration per N m of moment produced) which affect the sagittal plane trunk angular acceleration are shown in Figure 2 2. The sagittal plane trunk angular acceleration depends little on nonsagittal plane moments. The sagittal plane trunk angular acceleration has a negative sensitivity to the moment generated at the ankle for most of the gait cycle (Figure 22 ). Thus, an increase in plantar flexion moment (a more negative ankle moment about the x axis) can cause PAGE 29 29 the trunk to have an angular acceleration that leans the trunk posteriorly (shoulders back relative to hips, which is a positive angular acceleration about the x axis). Similarly, an increase in a knee extension moment has the potential to cause an angular acceleration that leans the trunk posteriorly while an increase in hip flexion moment causes an angular acceleration that leans the trunk anteriorly. It should be noted that when the trunk angular acceleration is sensitive to a moment, it does not imply that the moment is influencing the trunk angular acceleration. If the moment is low, and the sensitivity is high, the joint may not influence the trunk angular acceleration. Thus, the sensitivity to a joint moment must always be considered in conjunction with the joint moment to assess the actual influence on the trunk angular acceleration. Frontal P lane T he frontal plane trunk angular acceleration is sim ilarly dependent on moments at all joints (Figure 2 3). In addition to the importance of the moment at the hip, an increase in left ankle plantar flexion moment or left knee extension moment each could possibly cause a trunk angular acceleration to the rig ht (Figure 24). Joint Moment Contributions to Trunk Angular Accelerations Joint moments multiplied by their sensitivity yields results in units of acceleration with the result from each moment being superimposed to yield the total trunk angular accelerati on. Note that each joint moments contribution to the trunk angular acceleration in the sagittal and frontal planes are thus quantified and the extent and the magnitude of each moments contribution varies throughout the gait cycle (Figure 25 and Figure 26, respectively). Note that at approximately 50% gait cycle: 1) sagittal plane plantarflexion and hip flexion moments approximately counteract one another with respect to their effect on trunk sagittal plane angular acceleration; and 2) sagittal plane PAGE 30 30 pla ntarflexion moment approximately counteracts frontal plane hip moment with respect to their effect on trunk frontal plane angular acceleration. Control of Trunk Angular Acceleration by the CNS The contact moments acting on the trunk will almost be equal and opposite to the inertial torques since the gravitational torque is minimal due to the upright trunk causing the gravity vector to pass close to the center of mass (Figure 27). Thus, the inertial torque is almost equal and opposite to the contact torques on the trunk as a result of the fundamental laws of mechanics (see Equation 1 in Appendix A). Furthermore, since distal joint moments can generate linear forces at the base of the trunk (Figure 24, Figure 27), it is possible that the CNS utilizes these forces to alter the angular acceleration of the trunk. An example of such a control system is the invertedpendulum cart problem in which linear forces at the base of the inverted pendulum can control pendulum balance (rotation) yet still progress the pend ulum horizontally. If the trunk is modeled as an inverted pendulum with its base at the hip joint center, then the linear reaction forces (which are a function of distal joint moments) at the base of the trunk are an effective means to control balance of t he trunk and maintain forward progression. Since both the forces and moments at the base of the human trunk can be used to control its angular acceleration, it is possible that the CNS uses both to control trunk balance. PAGE 31 31 Fig ure 21 Comparison of the current model (blue) to data from previously published work (red) (Winter 1991). PAGE 32 32 Figure 22. The sensitivity of the trunk angular acceleration about the x axis to the joint moments during gait. A negative sensitivity implies that a posi tive moment about an axis causes a negative acceleration about the same axis and vice versa. PAGE 33 33 Figure 23. The sensitivity of the trunk angular acceleration about the y axis to the joint moments of the leg during gait. A negative sensitivity implies that a positive moment about an axis causes a negative acceleration about the same axis and vice versa. PAGE 34 34 Figure 24. The sensitivity of the trunk angular acceleration about the y axis to the joint moments of the leg during gait. A negative sensitivity implies that a positive moment about an axis causes a negative acceleration about the same axis and vice versa. PAGE 35 35 Figure 25. Frontal and sagittal plane views illustrate how a sagittal moment can cause frontal plane accelerations by means of reaction forces along the kinematic chain. PAGE 36 36 Figure 26. Contribution of each joint moment to trunk angular acceleration in the sagittal plane. The summation of the contributions is equal to the total trunk angular acceleration in the sagittal plane. PAGE 37 37 Figure 27. Two methods of calculating the contact moments are illustrated by taking free bodies of the trunk and lower extremity. These methods illustrate why the contact moments on the trunk are mathematical ly equal and opposite to the inertial forces and moments acting on the trunk PAGE 38 38 CHAPTER 3 THE DEVELOPEMENT OF A GROUND CONTACT MODEL FOR BIOMECHANICAL SIMULATION FROM TREADMILL DATA Introduction Simulation in biomechanics is a rapidly growing field with detailed computer models producing estimation of currently immeasurable parameters of human gait. Two primary types of simulation are typically performed: 1.) a forward simulation is one in which j oint torques are provided to the model and the resulting kinematics can be calculated, and 2.) and inverse dynamic simulation calculates the joint torques from kinematics. Such simulations are invaluable to orthopedic sciences, rehabilitation, prosthetic design, and computer graphics amongst many other fields [55 58] Due to the biomechanical intricacies of the human body, numerous assumptions have to be made (mass properties, ground contact models, joint contact models, etc) to produce a solution in these simulations. Optimization of the mathematical elements of a simulation based on experimental data therefore becomes desirable. Of the intricate modeling necessities specific to human gait, foot ground contact has been presented as challenging. The most common approach presented is a spring damper model in which spring damper elements dictate ground reaction forces. [29 34] Another approach is to assume a welded joint between the ground the foot [35 38] or to model the foot as a rigid rolling surface [39] More complex methods such as finite element models from study on cadavers and magnetic resonance imaging have been proposed [40] Visco elastic sphere models have also been introduced to represent the plantar surface of the foot [34, 41] A contact model based on a viscoelastic sphere was first studied by the H. R. Hertz [42] and expresses the relationship between the spheres deformation and the resulting force that causes the deformation. The visco PAGE 39 39 elastic sphere model presented by Guler et al relies on experimental data from heel pads and Guler et al conclude that viscoelastic spheres can be incorporated into foot models for simulation [41] More complex spherical models have been presented that replace constant exponent of 3/2 that exists in Hertzs model with a variable n which dictates the geometry of the model (Marhefka 1999). Despite the increasing investigation and complexity associated with foot contact modeling, four vital aspects of contact model design have not been addressed thoroughly The first is the applicability of current models to treadmill walking, the second is the effect of kinematic measurement error on foot contact model development, the third is the effect of the discretinization of the continuous plantar surface of the foot on the frictional properties of the model and the fourth is the reporting of the foot contact models ability to model all six of the ground reaction force and moment components. The following work address these four points by exploring the effects of treadmill belt movement in the medial lateral direction versus the anterior posterior direction, performing kinem atic optimization within experimental measurement to improve model performance, introducing a novel elemental torsional friction to account for discretinizing the plantar surface of the foot to improve axial reaction moment modeling, and presenting the models ability to reconstruct all six components of the ground reaction force is presented. Methods Subjects A single subject was used for demonstration purposes. The subject walked at their self selected speed of 1.1 m/s on an instrumented treadmill ( Techmachine, Andrezieux Boutheon, France) for three trials of thirty seconds each. PAGE 40 40 Data Collection Kinematic data were collected using a 12 camera VICON system and sampled at 200Hz to capture spatial positions of markers placed on the subject s feet. The s ubject walked barefoot. Markers to capture the kinematics of the foot were placed on the medial, lateral, and posterior sides of the heel. A marker was placed on the medial and lateral malleoli, on the first and fifth metatarsal heads, on the big toe nail, on the second toe nail, and at the base of the second toe. A rigid cluster with three markers on was also placed on the mid foot (Figure 31). The subject was asked to perform numerous tasks on the treadmill. The first was to stand statically for a period of five seconds to determine the relative positions of markers to be used as tracking clusters (discussed in the Model section of this chapter). After the static trial, the subject was required to step on three marker balls securely attached to the ground level. This process was used in a three point calibration (discussed in Pressure Sensitive Film section of this chapter). Subsequently, the subject was asked to walk at various speeds (as naturally as possible) for forty seconds at a time. Once the s ubject reached steady state (typically within the first ten seconds) the data collection would commence. Two measurements systems were utilized to determine the forces and moments acting on the plantar surface of the feet. The first is a split belt instrum ented treadmill ( Techmachine, Andrezieux Boutheon, France ) and the second is a pressure sensitive film (Tekscan, Inc. South Boston, USA). The instrumented treadmill has the capability to record horizontal and vertical forces while the pressure sensitive fi lm records only vertical forces and pressure distribution. PAGE 41 41 Pressure Sensitive Film The pressure sensitive film measures vertical ground reaction forces at 1200 discrete places on the plantar surface of the foot with a grid resolution of 5.08 mm The sensor film was taped to the plantar surface of the foot using strong double sided adhesive tape which effectively held the film in the correct position and acquired data at a rate of 100Hz. Since the films measurement is contained in a three dimensional matrix (x position, y position and time), the data requires more complex filtering procedures. The current work applied a 2 stage filter and a subsequent fifth order data interpolation. The first stage was to filter each frame of data (every point in time has 1 200 readings at 1200 x y locations) such that the distribution of pressure measured on the plantar surface of the foot at a single point in time produced a smooth function. The second stage is to filter the data along time to remove the high frequency cont ent of each individual sensor in the film. The data interpolation occurs along the time axis where every sensors data is resampled at a rate to coincide with the data from the motion capture system. This interpolation utilized included fitting a quintic s pline to every senors time trajectory and resampling the spline at 200 Hz. Not every point on the grid proved necessary to include in the evaluation. Many sensors had a zero or insignificant contribution to the total pressure distribution. In order to filter out only the points that made a relevant contribution and determine the locations of clusters of elements that could be grouped into a segment, a breadthfirst search algorithm was used. The algorithm loops through every time frame of data. At a time f rame, the data can be represented by a 60 X 20 matrix in which the rows indicate anterior/posterior direction (along the y axis) and the columns represent medial/lateral PAGE 42 42 direction (along the x axis). In this matrix the algorithm selects the element with th e maximum value. The next step is to check the 8 neighboring elements to see if their value is above a threshold that represents a significant contribution. In this work that threshold was set to be at 1% of the maximum value. If the neighbors value is less than the threshold, the same process is not repeated for its neighbors. If the neighbors element is above the threshold, then the position of this element is stored and its neighbors are checked. The process is repeated until no more neighbors exist w ith values above the threshold. The result is a group of points that can be classified into a segment as a group (Figure 31). The algorithm effectively determined four clusters that represented the heel, the forefoot, the second toe and the big toe. The p oints of the heel were assumed to be at a fixed position in the foot segment; similarly, the midfoot points were assumed to be fixed in the foot reference frame, and the second toe and big toe clusters were assumed to be fixed in the toes segment. Upon completion of the filtering process, the pressure data from the remaining elements are calibrated at each point in time to exactly match the vertical force measurement from the treadmill. Thus, the pressure film is not utilized as a force measurement system as much as it is utilized as a pressure distribution measurement system. Prior to the calibration the pressure film according to the instrumented treadmill, a cross correlation is used to determine the time lag between the two signals since the signals wer e not collected in a synchronous fashion (Figure 32). In order to calibrate the film at a point in time, the 60 X 20 matrix is normalized to the summation of all elements values at that value. The result is then multiplied by the value of the vertical f orce as measured by the instrumented treadmill at that point in PAGE 43 43 time. After this procedure is performed, when a frame corresponding to a point in time is summated along its rows in columns, the result is a value that exactly corresponds to the value of the vertical force as measured from the instrumented treadmill at that point in time (Figure 3 2). Three P oint C alibration The information of the pressure distribution is not effective unless its position is known within the foots reference frame. In order to estimate the position of each one of the 1200 sensors in the foot reference frame, a three point calibration is performed. During this process the subject is required to step on three markers that are securely fastened to the floors surface. The positi on of the markers fasted to the ground surface was recorded prior to this. The markers were seven millimeters in radius and planted on a two millimeter thick plastic base that was fastened to the ground. While the subject stepped on the markers, the kinematics of the subjects foot was recorded. The data that resulted from the three point calibration data collection was passed through the breadthfirst search algorithm described earlier in this section. The algorithm effectively determined the pressure regi ons that corresponded to the markers that the subject stepped on. Since the kinematic motion of the subjects foot was recorded during this process, a rotation matrix and reference frame origin could be determined such that the position of the pressure reg ions could in the pressure film reference frame could be translated and rotated to coincide with the marker positions as determined from the motion capture system. To determine the three Euler angles and three translations that successfully complete the t hree points calibration, an interior reflective Newton method is used. The optimization involved the approximate solution of a large system using the method of PAGE 44 44 preconditioned conjugate gradients at each iteration. The algorithm was implemented using Matlabs fminunc function with a largescale optimization. The algorithm successfully determined the three translations and three rotations that the pressure film points would have to undergo in the foot reference frame necessary to lineup the pressure regi ons with the positions of the markers fastened to the floor (Figure 31). Foot Model To model the foot, it was divided into two segments. The first segment is defined as the heel and its motion is tracked from the three markers placed on the medial, later al, and posterior sides of the heel and a cluster of three markers on the midfoot. The second segment is the toes segment which is tracked by using the markers on the hallux nail, index toe nail, and the marker at the base of the index toe. The kinematics of each segment is determined by singular value decomposition of a matrix derived from the positions of the land marks [59] To reduce the complexity of the model eight viscoelastic elements were fixed in the foot reference frame and two elements were fixed in the toes reference frame. The locations of the elements were determined using the pressure sensitive film, the three point calibration routine, and kmeans clustering Kmeans clustering was used to reduce the number of elements in each segment to a manageable amount. The number of elements in the midfoot and heel sections was reduc e d to four and a single representative element was to replace the elements in the index toe and a single representative element replaced the elements in the hallux (Figure 33 ). Full B ody M odel The model was created in Mathworks ( Natick, MA USA ) Matlab using the SimMechanics toolbox. The axis definition has the y axis of the laboratory perpendicular PAGE 45 45 to the treadmill surface. The z axis is orthogonal to the y axis and points in the opposite direction of the treadmill belts motion (i.e. the z axis points forward). The x axis is the cross product of the y and z axes. The model and segmental coordinate s ystems are presented in Figure 34. Visualization of the model is done utilizing equivalent ellipsoids to represent each segment. The ellipsoid is determined from the segme nts mass and mass distribution. The foot is visualized as a convex hull of the contact elements used to model foot ground contact. The model consists of thirteen rigid segments (toes, feet, legs, thighs, pelvis, trunk, head, upper arms, an d lower arms) all interconnected with ball and socket joints. The joint locations were optimized. The optimization placed a fixed point in the proximal segment; it also placed another point fixed in the distal segment. The coincidence of these two points trajectory was the cost function. When the optimizer placed two points fixed in the proximal and distal segment reference frames that had coincident trajectories through space due to the motion of the respective segments, the average position of the two points was considered to be the joint center (Figure 35 ). The optimization converged using a gradient descent approach (Matlabs fminsearch). After the joint center optimization the pelvis was constrained to move according to the experimental data. The thigh was then connected to the pelvis at the optimized joint center. This optimized joint center contains small residuals (stddev = 3.2 mm) and which results in a translational motion of the thigh that varies from the original experimental data within the r esidual of the joint center optimization. This joint center residual superimposes its effect after the knee and ankle joint centers. Prior to joint constraint, the maximum residual from least squares m arker tracking on the foot was 2.8 mm. The PAGE 46 46 radius of th e markers used to track the system was 7.5 mm. Post joint constraint the origin o f the foot had a mean error of 4.3 mm from the nonjoint constraint system. The maximum deviation of the origin of the foot in the joint constraint system was 8.5 mm from the origin of the foot in the nonjoint constrained system and this residual occurred at heel strike. Visco Elastic Contact Elements As mentioned in the introduction of this chapter, the typical approach to modeling foot ground contact has been with viscoel astic elements. The typical form of such an element is given by an equation that represents a nonlinear damping element. = ( 1 + ) (3 1 ) In this equation is the penetration distance, is the penetration velocity, and is and exponent that can be included as a design variable since it is dependent on the hypothetical geometry of the contact element. The characteristic use of this equation allows the simulation of a hysteresis in the simulation of the vertical ground reaction force. Two more design variables are c and b which are the spring constant and damping constant respectively. Since the vertical force is known at 1200 locations on the plantar surface of the foot after utilizing the measurement capability of the pressure film, the viscoelastic elements capability as an appropriate element can be explored. In order to do this a n optimization procedure is completed in which the height of the element in the segments reference frame, the spring constant, the damping coefficient, and the geometry exponent are all design variables. An element was placed at every point as dictated by the sensor positions in the pressure sensitive film. An optimization was performed on each individual element. The PAGE 47 47 optimization was to determine the parameters of the viscoelastic element that cause the element to successfully reproduce the ground react ion force as determined from its respective sensor measurement of the ground reaction force in the pressure film. The cost function was selected as the least squares error between the vertical force from the simulation and vertical force from experimental data. The vertical force from the simulation is the superposition of all the elements vertical reaction force. A comparison of the two data sets is made in Figure 36 Early in the step, the viscoelastic elements had trouble trying to model the character istics of the heel elements. This could be due to numerous reasons. The first reason is perhaps that the discrete natural of the elemental model does not contain a high enough resolution to accurately model the continuous surface of the heel. The second reason is that the compression of the heel pads m a y reach a maximum, beyond that, the calcaneous plays a significant role in the production of force. Compression of the calcaneous has a very high sensitivity to force production as is immeasurable with the eq uipment utilized in the current work. A third reason may simply be that the elemental penetration into the ground may not be measured accurately enough by the tracking markers. R esearchers have successfully utilized the elements for accurate force simulat ion beyond that seen in Figure 36 The approach taken by these researchers is to determine reaction force and moments at the ankle through standard inverse dynamics. The parameters of the viscoelastic elements are then determined in a forward dynamics se nse. That is, the kinematics of the foot are determined by driving the foot with the forces and moments at the ankle. During this process, the parameters of the PAGE 48 48 visco elastic elements are altered with an optimization until the motion of the foot best repre sents the experimental kinematics used in the initial inverse dynamic analysis, and also the corresponding ground reaction forces. Therefore, success has been due to the fact that the model is allowed to vary from the initial measured kinematics. Due to the high sensitivity of the ground reaction force production to kinematics, these slight variations from the measured kinematics can have profound effects on the ground reaction forces. To verify this hypothesis, a single element was selected which proved di fficult for the visco elastic element to model. To verify that slight variations in kinematics have profound effects on viscoelastic elements potential to model ground reaction forces, a single element is selected with an especially complex force penetr ation pattern (Figure 37 ). After the optimization, the spring constant value on this particular element was 2372 N/m. To determine the appropriate parameters such that the penetration and penetration velocities model ground reaction forces at an element, the spring constant, damping coefficie nt, and exponent in equation 3 1 are adjusted; however, the optimizations rarely produce a perfect fit. If one reverses the process and solv es the differential equation 31 for the penetration, x, by substituting in the true vertical forces from experimental data and the spring constant, damping coefficient, and exponent from the optimization, one will determine the penetrations that would produce the correct ground reaction force at that element. The differential equat ion which reveals the penetration trajectory that would satisfy the equation is given by: = ( 3 2 ) PAGE 49 49 To solve this equation, Matlabs ODE45 is utilized which is a numerical integrator that uses the RungaKutta modus operandi By perfor ming this check, it is noticed that variation in kinem atic measurement by less than 1mm sufficiently alters the penetrations to exactly match the experimental gr ound reaction forces (Figure 38 ). The reported accuracy of the motion capture system is on the order of 0.5 millimeters and the maximum residual on the segments tracking occurs on the toe segment and heel segment and is in the order of 0.75 millimeters. It can be concluded that the poor performance of the viscoelastic element to design a ground co ntact model is not due mathematical inconsistencies in the contact models development, but rather due to insufficient measurement capability of the motion capture system. It is also concluded that in order to successfully apply a model containing a reduced number of elements, it is vital to alter the kinematics of the model to obtain the correct viscoelastic properties of each element. Friction M odel Friction models are applied in various forms. In the current work a combination of viscous and Coulomb fri ction is utilized as well as the introduction of an elemental torsional friction associated with the rotational velocity of the element about the vertical axis. Coulomb F riction Coulomb friction is given as: =   (3 3 ) In equation 33 is the horizontal frictional force (a two dimensional vector) which is pointed in the opposite direction of the two dimensional horizontal velocity, PAGE 50 50 The Coulomb coefficient of friction is and is the magnitude of the vertical force (the force normal to the walking surface). This equation can be divided into it s two scalar equations for the xand ycomponents by: =   (3 4 ) for the x direction and: =   (3 5 ) For the z direction. Typically the frictional properties of a surface are independent of direction; however, in the case of treadmill walking this is not always true. On a treadmill it is possible that the belt slides in the medial/lateral direction (x direction) while its mo tion in the anterior/posterior (z direction) direction is dictated by the rollers of the treadmill. To verify this fact a 20kg mass was placed on the center of two different treadmills. A n approximate 100 N force was applied to the mass in the x direction and y direction separately. In the x direction on treadmill 1 (ADAL treadmill ; Techmachine, Andrezieux Boutheon, France) the weight and the belt moved 4mm in the direction of a 91.23 N force without the weight sliding on the belt (Figure 3 9) The belt sl id on the smooth surface under the belt. In the y direction the 95.1 5 N force did not cause any motion as the belt would not slide over the rollers (which were locked from rolling) and the weight would not slide over the belt (Figure 310) Similar results were found on treadmill 2 (Bertec treadmill; Columbus, Ohio, USA ). In the x direction the weight and belt moved but no motion was seen in the y direction. Again, flexibility in the belt allowed medial/lateral sliding in the x direction of the belt over the smooth surface under the belt but the high coefficient of friction between the rollers and the belt did not allow motion o f the belt in the fore/aft direction. PAGE 51 51 It can be concluded that in order to model horizontal friction on a treadmill, different coeffi cients of friction must be selected for the x and y directions. This is because there is generally a smooth surface under a treadmill belt and lateral belt flexibility allows for motion of the belt over this surface; however; the tensile flexibility of th e belt is very high in the walking direction and the coefficient friction between the bel t and the roller is very high; therefore, the belt will not slide in the y direction. Since motion of the subject is measured relative to the lab and the medial lateral displacement of the belt is unknown, it is justifiable and necessary to use different coefficients of friction for the x and y directions in equations 34 and 35 respectively Viscous Friction Viscous friction is given by: = (3 6 ) Th is friction is also separable into two components and different coefficients of friction are required for treadmill walking for the x and y directions as shown for Coulomb friction. Both Coulomb and viscous friction can be superimposed to determine linear friction in the model. Torsional Friction If a single element is to represent a group of elements then rotation of that group of elements about the representative element has the ability to produce linear frictional forces that oppose the rotation The fr ictional linear forces produce a pure moment about the representative element The cumulative frictional moment about the representative point can then be modeled by a form analogous to linear Coulomb and viscous friction : PAGE 52 52 = (3 9) In this equation is the moment about the element that opposes rotation. is the rotational coefficient of friction and is the angular velocity of the segment. Typically a set of forces acting on a rigid body can be replaced with the resultant force acting on the body and a pure moment. If the single element is to replace an infinite amount of individual elements on a continuous surface, then a means for producing the resulting pure moment must be made available. Equation 39 completes the frictional model of the representative element. Kinematic Optimization Since it has been shown that kinematic variation within experimental measurement tolerances affect the ability of the model to accurately represent the ground reaction forces and moments, a joint angle optimization was performed on the whole body model with the foot model attached. The optimization allowed for small variations in hip, knee, and ankle angles (less than 2 degrees) using particle swarm optimization but assumed that the hip joint center moved with the pelvis in a fixed fashion. The optimization utilized bsplines with fifteen control points for each joint angle trajectory. Within the joint angle optimization, two secondary optimizations are performed to determine the vis co elastic element parameters and the frictional parameters that achieve the measured ground reaction forces and moments. The optimizations were performed with and without the torsional compensation at each element and with and without kinematic adjustment The elemental forces in the y direction (vertical) are known experimentally from the pressure sensitive film; therefore the viscoelastic element parameters to determine PAGE 53 53 vertical force are first performed using optimization to solve for n in equation 33 and nonnegative regression to find c and b Since the vertical force (normal force) is now modeled at each element, the results can be used in an optimization to find the frictional parameters to match the remaining ground reaction forces and moments. Results and Discussion A kinematic adjustment was made to the joints of the full body model. Three degrees of freedom were allowed to vary at the hip, one (flexion/extension) at the knee, three at the ankle joint, and three at the toes joint. No angle was allowed to vary more than two degrees in any direction. The optimization was performed with two optimizers in parallel. A gradient descent method (Mathworks fmincon) and a particle swarm optimizer. Periodically (every hour on a Pentium 4 processor with 1GB DDR RAM) the best result from each optimizer was compared and the best result applied to the optimizer with the worse result. Initially the particle swarm algorithm provided the best solutions and as the algorithms began to converge the gradient descent m ethod provided better results. The optimization continued for 6 hours and 43 before the minimum was reached. The resulting joint angles are presented in Figure 311 In Figure 312 the change in the force penetration curve for an element at the heel is presented as an example. Prior to the kinematic optimization, the force penetration curve exhibits a nonphysical damping characteristic. The penetration reaches a certain point and the force increases drastically with no change in the element penetration. This is not a functi on that can be modeled with a visco elastic element. After slight variations in the model kinematics, a more physically viable force penetration curve is noticed which lends itself to being PAGE 54 54 modeled with a viscoelastic element. The fin al results of the ground reaction forces and moments of the model are presented in Figure 313. The gains ranged from 5.43x103 to 3.29x106 N/m for the visco elastic element. After the kinematic optimization the mean squared error of the knee joint from it s original position was 7.3 mm. The position of the ankle joint center had a mean squared error of 22.4mm from its original measured trajectory. Conclusion A foot contact model for simulation has been developed herein. Viscoelastic elements were utilized to create the dynamic relationship between kinematics and the vertical ground reaction forces. With the utilization of a pressure sensitive film, it has been shown that the success of a foot ground contact model has a high dependency on measurement accurac y and that small kinematic variations within measurement tolerances improve the ability of a viscoelastic element to model the reaction forces. It has been shown that a foot contact model that is developed from data collected on a treadmill will most like ly have to utilize directionally dependant coefficients of friction since high quality, high performance instrumented treadmills still contain medial/lateral belt movement that is not present in the walking direction of the treadmill. A torsional frictiona l model has been proposed to account for the pure moment that is produced about a representative element from continuous surface area in the immediate vicinity of the representative element. The model was deemed necessary due to a dimensionality reduction using kmeans clustering and a breadth first search algorithm. The clustering relied on a three point calibration that aligned the elements of the pressure sensitive film accurately within the foot reference frame. PAGE 55 55 In combing all of the above aspects, a mod el is developed that reflects all six components of ground reaction force during gait. PAGE 56 56 Figure 31. Alignment of the pressure film in the foot reference frame relative to the motion tracking markers (using a three point calibration) and also the resul ts from the breadthfirst algorithm search which found only the relevant force elements and clustered the elements into anatomically relevant groups PAGE 57 57 Figure 32. Calibration of the pressure film. Top: Original data normalized by body weight. Middle: Time synchronized data using a cross correlation to determine lag. Bottom: Final result from calibration. PAGE 58 58 Figure 33. Dimensionality reduction using kmeans clustering. All the elements in the segment are replaced with the points reflected by the large solid dots. Figure 34. Anthroprometric model with (left) and without (right) segmental coordinate system definititions. The large number of coordinate systems at the feet are associated with the contact elements in the foot and toes reference frames. PAGE 59 59 Figure 35. The convergence of an optimization to find the left shoulder joint center from a poor initial guess. PAGE 60 60 Figure 36 Comparison of the viscoelastic models reproduction of the experimental data that was used to train the model PAGE 61 61 Figure 37 Comparison of an optimized viscoelastic element to experimentally collected data. PAGE 62 62 Figure 38 The comparison of an example experimentally measured element penetration and a hypothetical penetration that would cause a viscoelastic element to exactly match its corresponding ground reaction force. The difference in the measurement falls well within the experimental error. PAGE 63 63 Figure 39. On an ADAL treadmill, the weight with no force acting on it is at rest at 50cm ( left). With a 9.3kg force al ong the x axis (normal to the walking direction of the treadmill) the belt slides 4mm to the right (right). The weight does not slide on the belt. PAGE 64 64 Figure 310a. The weight is at rest on the treadmill belt with no force being applied. PAGE 65 65 Figure 3 10b. With a 9.7 kg force being applied along the y axis (along the walking direction on the treadmill) no sliding motion of the belt or the weight on the belt occurs. PAGE 66 66 Figure 311. Kinematic alterations in joint angles produce foot motion that accommodates foot/ground contact modeling. PAGE 67 67 Figure 312. The effect of kinematic variations removes the nonphysical negative damping characteristic seen in the original experimental data. PAGE 68 68 Figure 3 13. Resulting force and moment production of the foot contact model post kinematic optimization. A torsional friction model is utilized at each element and differing linear coefficients of friction are used in the x and z directions due to medial/lateral treadmill belt flexibility. PAGE 69 69 CHAPTER 4 ANGULAR MO MENTUM CONTROL DURING GAIT Introduction The impact of stroke on walking is significant [1], with only 37% of stroke survivors able to walk after the first week. Even among those who achieve independent ambulation, si gnificant residual deficits persist in balance [2, 3] and gait speed with a 73% incidence of falls among individuals with mild to moderate impairment 6 months post stroke [4]. A primary disability associated wit h post stroke hemiparesis is the failure to make rapid graded adjustment of muscle force and to sufficiently activate muscles [5]; consequently, a reduced ability to rapidly respond to perturbations that influence balance is inferred. Various compensatory walking patterns have been suggested to increase balance in hemiparetic gait (e.g., increased double support time); however, it is difficult to state whether kinematic or kinetic variations during paretic gait are adaptations done by choice to increase stability or the result of an impaired neural controller and decreased power production. Regardless, persons with hemiparesis do not have robust control over dynamic balance during gait and little is understood about how to quantify their ability to retain balance. There is a substantial need to assess dynamic balance during walking in order to ascertain who is at increased risk for falls and whether a targeted intervention has improved dynamic balance. While little has been done to quantify balance during walking in the general population, much less has been done to quantify balance during gait in persons with neurological deficits such as stroke. Approaches for determining who is at risk for falls hav e usually relied on clinical measurement scales as opposed to quantitative evaluation of either stability of the walking pattern or balance mechanisms PAGE 70 70 active during walking. Numerous clinically inspired tests have been used to predict falls and classify su bjects into fallers and nonfallers [60] For example, the Berg Balance Scale was designed to provide an indication of a subjects static balance [61, 62] However, interpretations of these measures are based on statistical evidence and do not quantify a subjects balance performance during walking. Furthermore, these as sessments are not based on balance control mechanisms present in healthy subjects, which would likely be a more helpful approach to improve the efficiency of balance rehabilitation. To explain and ultimately improve dynamic balance during walking, it is necessary to interrogate balancespecific neural muscular systems during gait. In order to understand the complexities of balance during gait, we believe that it will prove necessary to quantify destabilization of gait following perturbation and determine the biomechanical mechanisms by which the consequent stabilization is achieved. To maintain upright walking, the whole body AM should maintain oscillation about zero, since a constant nonzero AM implies continuous rotation of all segments about an axis (other than the vertical axis) which would most likely represent falling [14] Previously it has been noted that inter segmental AM cancellations are mostly responsible for a close to zero whole body AM and various successful control schemes have been developed based on this zero moment assumption in the robotics literature [15] In highly sophisticated bipedal robots, control systems that maintain a constant zero AM, despite perturbation, result in stable and robust multi segmental bipedal gait. This has been described as the only reliable stability index of dynamic bipedal locomotion in robotics [16] Despite the success of such control systems in robotics, it is noted from experimental results that humans do not follow this strategy PAGE 71 71 completely. Humans tend to allow oscillation in AM about zero during gait [12, 17] ; however, it has been proposed that ang ular momentum is controlled rigorously by the central nervous system and can be assumed to be zero to make predictions on COP location [12, 17] As opposed to the zero moment assumption, which assumes that the subjects strive to maintain a constant zero AM throughout the gait cycle, we hypothesize that inter segmental AM cancellations are not the primary mechanisms of AM stabilization. The hypothesis states that healthy and efficient walking requires fluctuation of the AM from zero, and it requires a mechanism whereby the AM can be directed to zero when necessary in normal gait or perturbed walking. We propose that this regulation occurs during normal and perturbed gait because AM is often manipulated purposefully during certain tasks [17] Phases of normal walking have been described as a controlled fall, which can be thought of as a nonzero fluctuation in AM. To control this fall well timed joint power production is necessary and specific foot placement and it is the goal of this work to investigate both characteristics. Methods Subjects Twenty o ne healthy subjects were used for to demonstrate theoretical step limits and a single healthy subject was used to in the perturbation study for demonstration purposes A single subject six months post stroke or longer also participated in the perturbation study. The subject s walked at self selected speed on an instrumented treadmill ( Techmachine, Andrezieux Boutheon, France) for three t rials of thirty seconds each. The instrumented treadmill is equipped to measure 3 orthogonal components of ground reaction forces at the subjects feet and three orthogonal components of ground reaction moments. These six measurements allow for the calculation of the center of PAGE 72 72 pressure (COP) under the foot. This can be done under both feet since the treadmill is a split belt treadmill. Data Collection Kinematic data were collected using a 12 camera VICON system and sampled at 200Hz to capture spatial positi ons of markers placed on the subject Rigid clusters of markers were placed on 13 of segments assumed to be rigid bodies. Clusters where placed on the feet, legs, thighs, pelvis, trunk, head, upper arms, and lower arms. A linear algebra method based on the singular value decomposition of a matrix constructed from the positions of markers in the clusters is used to track segment motion in a least squares sense [59] Subjects joint power productions were determined from standard inverse dynamics procedures. Model A description of the model utilized on the subject for the perturbation study in this chapter is provided in detail in Chapter 2. Perturbation Force perturbations have rarely been applied during gait because their application requires the design of significantly complex custom hardware. The accurate application of a force is required in which direction, magnitude and point of application are all known. In order to apply a quantified force to a subject walking on a treadmill, a sophisticated and rapidly responding feedback control system is needed in conjunction with a powerful and speedy actuator. Our research facility has recently had built (Aretec, LLC, Ashburn, VA) a custom piece of hardware that solved the problems associated with impulse response tests on a walking subject. The Force Pod is based on a master/ slave configuration that maintains tension in a cable between the device and the subject PAGE 73 73 without applying a net force until commanded to do so. The configuration winds up cable slack and releases cable tension in an appropriate fashion to not affect the subject. At a c ommanded time, the system winds up the cable and measures the force being applied to the subject. The system then alters the length of the cable to ensure that the movement of the subject does not influence the magnitude or direction of the force the cable is inducing on the subject. In conjunction with an advanced safety harness system, state of the art motion capture hardware, and a split belt instrumented treadmill already in place, the Force Pod can determine the impulse response of a human subject to perturbation in a controlled and safe fashion. The F orce Pod was utilized to apply a perturbation to the subject just after toe off. The force as applied at should height pulling the subject forward. At this point it should be noted that the subject was walking on the treadmill and a forward pulling force could bring the subject to the front of the treadmill. Trying to avoid coming off the front of the treadmill may have played a part in the subject s response, but this didnt seem to be the case here. Whole Body Angular Momentum During the Gait Cycle The gait cycle can be divided into 6 regions: 1) initial double support, 2&3) first and second half of single limb stance respectively, 4) second double support and 5&6) first and second half of ipsilateral swing respectively. Sagittal Plane: In healthy subjects, each of the regions shows characteristic fluctuations in sagittal plane angular momentum. Just prior to each heel strike, there is a rapid change in angular momentum, which reflects a forward free fall [63] d uring the second half of swing phase (either region 3 or 6). This forward freefall is an intentional fluctuation in angular PAGE 74 74 momentum. Subjects tend to land on the swinging leg which is aptly placed such that the ground reaction forces acting on the foot cr eate a moment about the COM which results in the angular momentum returning to a value above or close to zero at the end of double support (regions 1 and 4). If the foot is not placed in the correct position, the moment will not be sufficiently large enoug h and the whole body angular momentum will not be corrected. In the middle of swing (end of regions 2 and 5) the angular momentum reaches a maximum and begins its descent throughout the second half of swing (regions 3 and 6) to complete the limit cycle for each step. Thus, there are two angular momentum cycles in the sagittal plane for every gait cycle and the transition from one rotational state to the next occurs during double support. Angular Momentum in the Frontal plane In order to regulate angular mom entum in the frontal plane (lateral balance) during swing, the stance leg is responsible for generating ground reaction forces that will redirect the angular momentum. An initial nonzero angular momentum (at the beginning of swing) is normally zero at the end of the first half of swing and changes sign in the second half of swing, and vice versa for an initial positive angular momentum. In either case, the angular momentum is required to go from an initial value at the beginning of swing to a zero value at a point close to midswing (since the sign of the angular momentum cannot change at midswing without the angular momentum first becoming zero). The initial angular momentum in swing is highly inversely correlated (r = 0.857; p<0.0001) with the change in angular momentum during the first half of swing; however, the initial angular momentum is not correlated (r = 0.096, p<0.0001) to the magnitude of the angular momentum at midpoint of swing (Figure 41). This shows that angular PAGE 75 75 momentum in the frontal plane is a controlled value and generally driven to a value close to zero during the first half of swing regardless of the initial conditions of swing. The Relation B etween Joint Power and Whole Body Angular Momentum Whole body angular momentum (AM) is the summation of segmental moment of inertia multiplied by angular velocity. Thus, it collectively quantifies the rotational velocity of the entire body about the center of mass (COM) in a single measure. It can be calculated from kinematic data where the AM of the entire body is expressed as the superposition of individual segmental AM. To find the whole body AM about any arbitrary point q, a summation is performed [12] = + p p ( v v ) ( 4 1 ) In the summation, is the whole body AM about point q. is the i th segments inertia tensor and is its angular velocity. p and v are the position and velocity of point q respectively while p and v are the position and velocity of the segments center of mass. is the segments mass. To calculate whole body AM from kinematics, an anthropometric model is required to determine m and I and to measure segmental kinematics. The change in whole body AM about point q can be related to the sum of the moments acting on the subject. = + ( ) ( 4 2 ) In equation 42, the summation on the left side represents the summations of all moments acting on the subject about point q. The acceleration of point q is and m is the mass of the subject. If point q is selected as the COM of the subjects the second term on the right hand side becomes zero since is the subjects COM. The moments PAGE 76 76 acting on the subject can be expressed as the ground reaction moment and the moment arm from the COM to the COP crossed with the ground reaction forces. GRM + ( COP COM ) GRF = ( 4 3 ) From the equation, it can be noticed that in order to change the whole body angular momentum about the COM the subject is required to alter the COP location (which is related to foot placement) and the GRF (which is related to joint power production). Thus, subjects with reduced power production and reduced coordination to accurately place the foot will have difficulty in correcting whole body AM to reestablish it to a regular limit cycle. When comparing the AM profile of a patient with hemiparesis to a control subject, it is noticed tha t the control subject needed fewer gait cycles to return the AM profile to within two standard deviations of the r egular AM trajectory (Figure 42 ). Although it can be experimentally verified that hemi paretic subjects require more time to reestablish AM t o a regular limit cycle, it is unknown to what extent power production inhibits the reestablishment of AM to its original trajectory. Experimental data does show the difference in power production in the response to a perturbation from paretic subjects and from controls (F igure 43 ). Step Length and Moment Impulses Relation to Angular Momentum Correction An external moment acting about a subjects COM will cause a linear change in angular momentum. The only way to counter act the change is to place a foot in the appropriate position and generate a ground reaction force that develops a near equal and opposite moment about the COM. In fact, the angular momentum is given by the impulse of the moment acting about the COM: dt = H (4 4) PAGE 77 77 i f the moments are summed about the subjects COM. Thus an angular momentum perturbation is quantifiable in units of Newton meter seconds. The only external moment available to the subject to counteract the change in angular momentum due to a perturbation is the ground reaction forces acting at the COP. The ground reaction moments at the COP are, by definition of COP, zero about the horizontal axes and rotation about the vertical axis does not imply falling It is only the linear ground reaction forces that ca n generate moments which oppose angular momentum perturbations which could result in falling Two aspects affect the production of a responsive moment about the subjects COM: the lever arm from the COP to the CO M and the force at the COP. A subject has to essenti ally produce a counter impulse to correct unwanted changes in angular momentum. An impulse of lower magnitude is to be applied over a longer period of time to have the same effect as a shorter impulse of high magnitude. In order to generate a sufficient impulse in a finite amount of time (i.e. before it s to o late to avoid catastrophic failure), the foot must be placed in a manner such that the angular momentum about the horizontal axis can be taken from a nonzero value to a close to zero value. Consider the sag ittal plane and consider falling forward to be an increase in negative angular momentum about the x axis. That is, the x axis points to the subjects right if a right hand coordinate system is used. Then assume that = Let = ( ) and = ( ) be the position of the at ipsilateral heel strike (HS) and contralateral toe off (TO) respectively. Assume there exists a function ( ) such that ( ) = ( ) + ( ) ( 4 5 ) Where ( ) = 0 = 1 = and 0 < ( ) < 1 < < PAGE 78 78 ( ) is a function that describes the weight shift from the trailing leg to the leading leg in double support which starts at and ends at To determine the foot plac ement in the medial direction it will be assumed that ( ) = ( ) < < where is the time at the end of the first half of swing or midswing (MS). The moment equation about the subjects COM can be written in general as: + ( ) = ( 4 6 ) where is the time rate of change of the angular momentum about the COM, is the ground reaction moment, m is the total mass of the subject, is the acceleration of the subjects COM, and is the gravitational constant. Expanding equation 2 and writing the equations that relate the angular momentum about the x and y axes to COM accelerations and the elements of it can be shown that: = ( ) ( 4 7 ) = ( ) (4 8 ) In equations 47 and 4 8 the subscripts denote the scalar elements of the corresponding vectors and = 9 81 /. Foot placement and sagittal plane angular momentum: The integral of over double support is: = ( ) ( 4 9 ) Substituting the equations and expanding the integrals, it can be shown that the change in angular momentum over double support can be expressed in terms of which is the position of the COP at toe off, a constant and an indicator of t he placement of the foot at heel strike. PAGE 79 79 = ( ) + ( ) ( ) ( ) ( ) (4 10) Note in equation 410 all terms are dependent on the trajectory of the COM. is the vertical distance between the COM and the ground. The criterion for taking the angular momentum from a negative to zero or a positive value through double support (to rectify a forward freefall by placing a foot suffi ciently far in front of the COM) is that ( ) 0 In order for this to be true, should have a minimum value as described in equation 411 ( ) ( ) ( ) ( ) ( ) ( ) (4 11) For f oot placement and frontal plane angular moment um, t he integral of over swin g is: = ( ) ( 4 12) Similarly to the sagittal plane angular mo mentum analysis, = ( )( ) + ( ) ( ) ( ) ( ) (4 13) In order to maintain stability in the frontal plane, the angular momentum in the frontal plane is required to go from a positive to a negative value and a negative to positive value during left and right leg swing respectively. Only the case for right leg swing will be presented as the left leg swing is analogous in symmetric gait with a sign PAGE 80 80 change. Going from a negative to positive angular momentum during a period of time requires that ( ) 0 Using this relationship and equation (413) a bound can be placed on to ensure the change in angular momentum that is needed for stable gait. Similarly to the derivation for the limits in the sagittal plane: ( ) ( ) ( ) ( ) ( ) ( ) (4 14) For any given COM trajectory, a minimum step length/width required to be able to rectify angular momentum during the first half of swing. A drawback of this approach is that a mixi ng function is assumed for a weight shift pattern which is fair for regular gait; however, this function may not apply if a subject is undergoing perturbation. Subjects adhere to these limits (Figure 44 ). In the frontal plane, subjects average limit was at 65.2611.52% of the average step width in relation to a step width that fluctuated by 4.82%. Subjects step width was approximately 35% wider than the limit. In the sagittal plane, subjects did not overextend their step length far beyond the limit. Thi s limit was 42.766.11% of the stride length from the COM of the contralateral leg at the previous mid swing of the ipsilateral leg and subjects placed their foot at 49.44.0% of the stride length, which reflects symmetric gait. No subject fell significant ly or consistently short of the determined limit. In simulation the biomechanical model was used to verify that a shortened step consequent to a perturbation places excess strain on the neural control system. Since biomechanical models tend to fall over anyway despite joint center optimization, COM/Inertial property optimization, and a highly advanced finely tuned foot contact model, a controller was applied to the rotational dynamics of the mathematical model to maintain stable gait without any perturbati on. A simple low gain PID controller was PAGE 81 81 applied to the COM of the subject. The controller applied a torque that opposed any change in angular momentum about the COM No linear controller for translational forces was needed maintain stable upright gait on the simulation due to the ground contact model only the angular momentum controller. The controller gains were set such that it the angular momentum of the walking model reflected that of the subject which the m odel was based on (Figure 45 ). The assumption was that the controller accommodates and model residuals or errors. Once the controller was in place to regulate angular momentum to a certain degree, the joint angles were defined such that the foot fell short of the above calculated limits by a foot length. The low gain controller could not control the angular momentum, the system became unstable, and the model fell forward (Figure 4 6 ) The angular momentum profile also increased without bound until the mod el struck the ground (Figure 4 7 ). In order to shorten the step length in the simulation, the joint angular velocity of the joint E uler angles was multiplied a 0.75. The result was then integrated to determine the new joint Euler angle and the first value in time of the initial Euler angle was used as an initial condition for the integration. Doing this for all four angles in the leg (hip, knee, ankle, and toes) fully defined the new step length. This reduction in joint angular velocity is synonymous with r educed joint power production. The Effects of Reduced Joint Power Produc tion on Resisting Perturbation D uring G ait The response of a 132lb subject undergoing a 10lb perturbation while walking on an instrumented treadm ill is illustrated in figure 4 8 The s ubject lowered the COM, leaned backward, and rapidly placed the lead foot relatively far in front of the COM. The perturbation force lasted for 0.5 seconds. By leaning back the subject generates a PAGE 82 82 negative moment about the x axis to counter act the perturb ation force. The PID controller gains were increased so that the controlled model could withstand the perturbation force. The model s joint angles were constrained to be the same as the experimental data initially Figure 49 shows the angular momentum pro file for the model and for the subject while figure 410 shows the perturbation force that was applied by the Force POD. To investigate the effect of reduced power production, the models joint angular velocity was reduced in the stepping leg (since joint power is torque multiplied by angular velocity) at varying increments to note the effect on angular momentum and on the stability of the entire system. All other joint angles were constrained to be the same as the experimentally measured angles. With a 15% reduction in joint angular velocity in the leg that was to be placed the model fell over from the perturbation almost immediately. The PID no longer had sufficient gains to accommodate the 10lb perturbation force (Figure 411 ) Conclusion Persons with a r eduction in joint power are more likely to fall from a perturbation than those without joint power production. The complexities and redundancies of the human locomotors system make investigation into these principles difficult. An increasing application of biomechanical models is assisting in the understanding of the complex phenomena associated with human bipedal gait and the control thereof. Angular momentum is a gaining rapid recognition in the exploration of stable bipedal locomotion. In this work angular momentum successfully predicts foot placement preference in control subjects and should be developed further in the clinical setting to improve the versatility and robustness of gait in persons with stroke. PAGE 83 83 Although a simple PID controller cannot be co mpared to the complexity of the human brain (even one with neurological inconsistencies), the work herein does develop an understanding for the taxation on a control system from reduced power production. A simulation utilizing a model with foot/ground cont act interaction, which only required angular momentum control to achieve stable linear gait explored and validated the necessity for rapid foot placement and power production for the prevention of fall. PAGE 84 84 Figure 41 The angular momentum at the beginning of swing versus the angular momentum at mid swing (left). The angular momentum at the beginning of swing versus the change in angular momentum through swing (right). Data is from a right stance leg which is why the initial angular momentum is primarily positive. Figure 42 AM profiles for normal (blue) and perturbed (red) states. The shaded region represents three standard deviations from the mean trajectory. The perturbation occurs at the black line in the frames labeled Perturbed Step 1. PAGE 85 85 Figure 43 Joint powers from unperturbed (blue) and perturbed (red) states for a control and a hemiparetic subject. The perturbation occurs at the vertical black line in the window entitled Perturbation step. PAGE 86 86 Figure 4 4 : Calculated foot placement limits, COP trajectory, COM trajectory, and the foot placement density functions relative to the previous foot placement of the contra lateral leg for s ubjects walking at self selected speeds. The x distance is in the walking direction. Figure 45 Angular momentum of the model undergoing control via a simple low gain PID controller. The controller opposed changes in angular momentum enough to cause sta ble gait but did not hinder the model from achieving a natural motion. PAGE 87 87 Figure 46 The top row figures are from a simulation that reflects how a short step length for a given speed can result in a fall forward. The lower is the simulation in normal gait. A PID controller is applied in both cases. Figure 47 The angular momentum profile for 1 gaticycle of the model under PID control for the model falling over due to a shortened step length (see figure 45, top) and the model walking normally (see figure 45, bottom). PAGE 88 88 Figure 48 The kinematic response of the subject to an anterior perturbation force of 10 lbs applied just above the center of mass. Figure 4 9 The angular momentum profiles of the model undergoing perturbation with PID control gains tuned to give a similar response to experimental data and the experimental angular momentum profile of a subject undergoing the same perturbation. Figure 410. The perturbation force applied by the Force pod. The command was given to ap ply a 10lb ( 44.49N ) over a period of 0.5 seconds (100 Frames). PAGE 89 89 Figure 411. The response of the model to a 15% reduction in joint angular velocity in the left leg. The perturbation force of ten pounds caused a fall forward almost immediately. PAGE 90 90 CHAPTER 5 CONCLUSION AND DISCUSSION There is a 73% incidence of falls among individuals with mild to moderate stroke. 37% of patients that fall sustain injury that required medical treatment. 8% sustained fractures and the risk for hip fracture is ten times higher i n the stroke population. General risk of fracture is two to seven times greater following a stroke [4] Due to the impact falls have on the health and well being of persons with stroke, there is a substantial need to assess dynamic balance during walking in order to ascertain who is at increased risk for falls. While little has been done to quantify balance during walking in the general population, much less has been done to quantify the effect of neurological impairment on balance during g ait in persons with stroke. Approaches for determining who is at risk for falls have usually relied on clinical measurement scales as opposed to quantitative evaluation of joint power production ability However, interpretations of these measures are based on statistical evidence and do not quantify a subjects balance performance during walking To quantify and ultimately improve dynamic balance during neurologically impaired walking, it is necessary to quantify the biomechanical effects of altered or impaired joint power productions during paretic gait. These effects are relevant since a primary disability associated with post stroke hemiparesis is the failure to make rapid graded adjustment of muscle forces [5, 64] Reduction in, or ill coordinated, muscle forces result in inefficient and ineffective net joint power production. 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He started his academic career in the United States of America in 2001 as a student athlete at McNeese State University in L ouisiana. There he specialized in jet engine control systems as a mechanical engineering student. Consequently, Cameron was recruited to the University of Alabama as a track athlete where his research during his masters in aerospace e ngineering involved supersonic wind tunnel control systems, jet engine health monitoring systems, and laser Doppler velocometery with application of artificial intelligence. During this period, Cameron began to become interested in neural prosthesis and decided to pursue his doctoral degree at the Brain Rehabilitation Research Center in the Malcolm Randall VA Medical Center in Gainesville, Florida. There he obtained his PhD and went on to be an assistant research professor at the Medical University of South Carolina, South Carolina. 