Modeling the Role of the Foot, Toes, and Vestibular System in

Dissertation

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By

Laura R. Humphrey, B.S., M.S.

Graduate Program in Electrical and Computer Engineering

The Ohio State University

2009

Dissertation Committee: Dr. Hooshang Hemami, Adviser Dr. Ashok Krishnamurthy Dr. Andr`eaSerrani c Copyright by

Laura R. Humphrey

2009 ABSTRACT

The study of human balance and gait is a complex area that involves a large number of biological systems, including the musculoskeletal, somatosensory, vestibu- lar, visual, and central nervous systems. In contrast, computational models used for simulating motion of the human body tend to be relatively simple, especially with respect to the feet. Clinical research, however, has begun to more closely examine the mechanical and sensory contributions of the feet in balance and gait, leading to a disparity between the state of clinical research and models used for simulation. A model with a more complex foot would aid in the clinical diagnosis and treatment of motor control disorders, improvement of prostheses, and development of functional electrical stimulation for recovery of lost motor function.

This dissertation presents a computational model of a human with a more complex foot, which uses four rigid and connected segments to represent the heel, forefoot, and toes. Derivation of physical parameters, equations of motion, actuation based on human musculature, and control based on , i.e. body segment positions and velocities, will be discussed. Computation of ground reaction forces under the heel, forefoot, and toes will also be addressed. Simulations focusing on the role of the toes and toe muscles in static balance, forward leaning, and tip-toe stading will be presented. Contributions by the vestibular system will also be considered.

ii This dissertation is dedicated to all the great teachers in my life.

iii ACKNOWLEDGMENTS

I would first and foremost like to thank my advisor, Professor Hooshang Hemami, for his guidance, support, and knowledge, both in my research and my life.

I would also like to thank Professor Barin for his work and insight on our pending publications. Additionally, I would like to thank Professors Serrani and Krishna- murthy for serving on my Graduate Studies Committee. And, I would like to thank several people at the Ohio Supercomputer Center – Alan Chalker, Judy Gardiner,

Brian Guilfoos, Ben Smith, and Vijay Gadepally – for providing additional research opportunities, computing resources, and a fantastic work environment.

Finally, I would like to thank Jason Parker for being a very bad hobbit.

iv VITA

June 2004 ...... B.S. Electrical and Computer Engineering, The Ohio State University March 2006 ...... M.S. Electrical and Computer Engineering, The Ohio State University March 2006-present ...... Graduate Research Assistant, Ohio Supercomputer Center

PUBLICATIONS

L. Humphrey et al. Evaluating Parallel Extensions to High Level Languages using the HPC Challenge Benchmarks. Proceedings of the 2009 DoD HPCMP Users Group Conference, 2009.

B. Guilfoos, L. Humphrey, and J. Unpingco. Improvements to MPSCP. Proceedings of the 2008 DoD HPCMP Users Group Conference, pp. 376-378, 2008.

J. Unpingco, J. Gardiner, L. Humphrey, and S. Ahalt. Computationally Intensive SIP Algorithms on HPC. Proceedings of the 2007 DoD HPCMP Users Group Conference, pp. 271-276, 2007.

FIELDS OF STUDY

Major Field: Electrical and Computer Engineering

v TABLE OF CONTENTS

Page

Abstract ...... ii

Dedication ...... iii

Acknowledgments ...... iv

Vita...... v

List of Tables ...... ix

List of Figures ...... xi

Chapters:

1. Introduction ...... 1

2. Background ...... 3

2.1 Motor Control Theories ...... 3 2.2 The ...... 5 2.2.1 The and ...... 5 2.2.2 The Diencephalon and ...... 6 2.2.3 The ...... 7 2.2.4 The Primary, Secondary, Tertiary, and Association Areas ...... 7 2.2.5 The Basal Ganglia ...... 8 2.3 The Peripheral Nervous System ...... 9 2.3.1 The ...... 9 2.3.2 The Vestibular System ...... 10 2.4 The Musculoskeletal System ...... 12

vi 2.4.1 Bones, Joints, and Ligaments ...... 12 2.4.2 Overview of Skeletal Muscles ...... 13

3. Review of Literature ...... 16

3.1 Role of Foot Sole in Balance ...... 16 3.1.1 Effects of Aging ...... 18 3.1.2 Effects of Disease ...... 19 3.1.3 Effects of Inhibition ...... 22 3.1.4 Effects of Stimulation ...... 25 3.1.5 Balance on a Rotationally Disturbed Platform ...... 28 3.1.6 Summary ...... 34 3.2 Role of the Toes in Balance ...... 37 3.3 Role of the Vestibular System in Balance ...... 39

4. The Models ...... 41

4.1 Five-Link Sagittal Biped Model with a Foot ...... 41 4.1.1 Model Overview ...... 42 4.1.2 Model Parameters ...... 44 4.1.3 Muscles, Joints, and Ligaments ...... 47 4.2 Three-Link Sagittal Biped Model ...... 48 4.2.1 Model Overview ...... 49 4.2.2 Muscles ...... 50

5. Exploring the Role of the Foot Sole and Toes in Balance ...... 54

5.1 Control of the Five-Link Model Through the Muscles ...... 54 5.2 The Limits of Static Balance in a Healthy Subject ...... 58 5.3 The Limits of Static Balance in a Subject with Diminished Toe Mus- cle Strength ...... 64 5.4 Force Distribution and Foot Arch Angle during Forward Leaning . 68 5.5 Discussion ...... 71

6. Exploring the Role of the Vestibular System in Balance ...... 74

6.1 Recovery from Disturbance ...... 76 6.2 Fall from Disturbance ...... 79 6.3 Addition of a Vestibular Term ...... 79 6.4 Vestibular Modulation of Muscular Forces ...... 82 6.5 Simulation of Chorea-like Motion ...... 85 6.6 Discussion ...... 87

vii 7. Conclusions ...... 89

7.1 Summary ...... 89 7.2 Future Work ...... 91

Appendices:

A. Nomenclature ...... 95

B. Deriving Equations of Motion for the Five Link Model ...... 98

B.1 Basic Equations of Motion ...... 98 B.2 Terms due to Ground Reaction Forces ...... 102 B.3 Terms due to the Muscle Inputs ...... 103

C. Deriving Equations of Motion for the Three Link Model ...... 104

C.1 Basic Equations of Motion ...... 104 C.2 Terms due to the Muscle Inputs ...... 105 C.3 Initial Conditions due to a Sudden Horizontal Platform Disturbance ...... 107 C.4 Preventing Hyperextension ...... 108 C.5 Estimating Center of Pressure ...... 109

Bibliography ...... 111

viii LIST OF TABLES

Table Page

4.1 Lengths, masses, centers of mass, moments of inertia, and end points for major segments of the human body. Mass is given as a percent of total body weight. Center of mass is listed as a percent of the length, measured from the first listed endpoint. Abbreviations are: Cerv for cervicale, MDH for midhip, SJC for shoulder joint (center), EJC for elbow joint, WJC for wrist joint, DAC3 for tip of the 3rd digit (3rd dactylion), HJC for hip joint, KJC for knee joint, AJC for ankle joint, TTIP for tip of the longest toe...... 45

4.2 Lengths, masses, centers of mass, moments of inertia, and endpoints for larger segments of the human body derived using values from Table 4.1...... 45

4.3 Lengths, masses, centers of mass, moments of inertia, and endpoints for segments of the foot. Values were derived using information from [1]. Note that values are for the segments of one foot only...... 46

4.4 Parameters for the five-link biped model. I is moment of inertia in kg·m2 about the center of gravity, m is the mass in kg, l is the length in meters, and k is the distance to the center of mass from one joint of the segment in meters...... 46

4.5 Parameters for the three-link biped model. I is moment of inertia in kg·m2 about the center of gravity, m is the mass in kg, l is the length in meters, and k is the distance to the center of mass from one joint of the segment in meters...... 50

5.1 Gains and set points used to calculate muscle torques for the five-link biped model during normal standing...... 57

ix A.1 Symbols used for both models. “·” designates mixed units...... 95

A.2 Symbols used for the five-link model. “·” designates mixed units. . . . 96

A.3 Symbols used for the three-link model. “·” designates mixed units. . . 97

x LIST OF FIGURES

Figure Page

4.1 The five-link sagittal biped is composed of five rigid links. The term

ki, for i = 1 ... 5 is the distance from one end of link i to the center of mass of link i. This is shown for all links; however, it is labelled for

links 1 and 5 only. The term li is the length of link i. xa and ya are the coordinates of the ankle joint. Angles θi for each link are measured clockwise from the vertical. Fh is the horizontal ground reaction force at the heel, and Gh, Gm, and Gt are the vertical ground reaction forces at the heel, metatarsals, and toe, respectively...... 42

4.2 Medial (A) and lateral (B) views of a human foot skeleton. Features as numbered are 1-calcaneus; 2-sustentaculum tali; 3-talus; 4-head of talus; 5-navicular; 6-tuberosity of navicular; 7-medial cuneiform; 8-intermediate cuneiform; 9-base, 10-body, and 11-head of the first metatarsal; 12-base, 13-body, and 14-head of the proximal phalanx of the great toe; 15-base, 16-body, and 17-head of the distal phalanx of the great toe; 18-sesamoid bone; 19-lateral cuneiform; 20-cuboid; 21- base, 22-tuberosity, 23-body, and 24-head of the fifth metatarsal; and 25-tarsal sinus. Taken from [2]...... 43

4.3 Muscles used in the five-link sagittal biped model. Abbreviations are as follows: sl – soleus, ta – tibials anterior, fhl – flexor hallucis longus, ehl – extensor hallucis longus, fhb – flexor hallucis brevis, ehb – extensor hallucis brevis, aj – ankle joint, hj – hallux joint, and pa – plantar aponeurosis...... 47

4.4 The three link sagittal biped is composed of three rigid links and a

massless foot. The term ki for i = 1, 2, or 3 is the distance from the bottom of link i to the center of mass of link i. The term li is the length of link i. Angles are measured clockwise from the vertical with

θi ∈ (−π, π)...... 49

xi 4.5 The three-link sagittal biped with nine muscles: ga – gastronemius, sl – soleus, ta – tibialis anterior, sq and sh – short heads of the quadriceps and hamstrings, q and h – long heads of the quadriceps and hamstrings as two-joint muscles, p – paraspinal, and ab – abdominal. The gas- trocnemius, soleus, and tibialis anterior muscles connect to a massless foot, as shown in Figure 4.4 ...... 51

5.1 Minimum and maximum values of τfhb versus angle θ1 that allow the foot to remain on the ground are shown by the solid lines. The dashed

line represents the chosen equation for τfhb...... 59

5.2 Vertical forces under the heel, metatarsals, and toe for static poses with

θ1 varying from 0 rad to 0.45 rad. All inputs are as listed in Section 5.1. 60

5.3 Vertical forces under the heel, metatarsals, and toe for static poses

with θ1 varying from -0.15 rad to 0.0 rad. All inputs are as listed in Section 5.1...... 61

5.4 Static foot arch angle as θ1 varies from -0.15 to 0.45 rad. All inputs are as listed in Section 5.1...... 62

5.5 Tibialis anterior and soleus muscle torques as θ1 varies from -0.15 to 0.45 rad. All inputs are as listed in Section 5.1...... 63

5.6 Vertical forces under the heel, metatarsals, and toe for static poses

with θ1 varying from 0 rad to 0.45 rad and τfhb = 0. All other inputs are as listed in Section 5.1...... 64

5.7 Vertical forces under the heel, metatarsals, and toe for static poses with

θ1 varying from -0.15 rad to 0.0 rad and τfhb = 0. All other inputs are as listed in Section 5.1...... 65

5.8 Static foot arch angle as θ1 varies from -0.15 to 0.45 rad and τfhb = 0. All other inputs are as listed in Section 5.1...... 66

5.9 Tibialis anterior and soleus muscle torques as θ1 varies from -0.15 to 0.45 rad and τfhb = 0. All other inputs are as listed in Section 5.1. . . 67

xii 5.10 Angle of link 1, foot arch angle between links 2 and 3, and toe arch angle between links 4 and 5 during a movement of link 1 from 0 rad to 0.3 rad. All inputs are as listed in Section 5.1...... 69

5.11 Forces under the foot during a movement of link 1 from 0 rad to 0.3 rad. All inputs are as listed in Section 5.1...... 70

5.12 Position of the center of gravity (CoG) and center of pressure (CoP) during a movement of link 1 from 0 to 0.3 rad. All inputs are as listed in Section 5.1...... 71

6.1 CoG and CoP excursions of the biped versus time due to disturbance of the support surface. Muscular activities are nominal, and there is no vestibular input...... 77

6.2 The phase plane plots for the head, torso, hip, and center of gravity of the biped after the support surface undergoes a sudden backwards acceleration. Each point starts with a non-zero position due to the slightly stooped initial stance and a non-zero velocity due to the sup- port surface disturbance and then returns to equilibrium (0 m position, 0 m/s velocity)...... 78

6.3 CoG and CoP excursions of the biped versus time due to disturbance of the support surface. Muscular activity levels are set to half the nominal value, and there is no vestibular input...... 79

6.4 CoG and CoP excursions of the biped versus time due to disturbance of the support surface. Muscular activity levels are half the nominal value, and there is an additional vestibular term...... 81

6.5 CoG and CoP excursions of the biped versus time due to disturbance of the support surface. Muscular activity levels are 80% of the nominal value, and there is no modulation by the vestibular system...... 82

6.6 CoG and CoP excursions of the biped versus time due to disturbance of the support surface. Muscular activity levels are 80% of the nomi- nal value, and there is modulation by the vestibular system with the

modulation factor mf set to 2.5...... 83

xiii 6.7 The muscular activities in N·m for the quadriceps-hamstrings after a support surface disturbance when muscular activity levels are 80% of the nominal value, and there is modulation by the vestibular system

with the modulation factor mf set to 2.5...... 84

6.8 CoG and CoP excursions of the biped versus time due to disturbance of the support surface. Muscular activity levels are 80% of the nom- inal value, and there is modulation by the vestibular system through estimation of horizontal torso position and velocity. The modulation

factor mf is 2.5...... 85

6.9 CoG and CoP excursions of the biped versus time after the support surface experiences a sudden backward acceleration. Muscular activity levels are 80% of the nominal value, and there is modulation by the vestibular system using horizontal head position and velocity. The

modulation factor mf is 13...... 86

C.1 A free body diagram of the three-link model showing forces of connec- tion between links and ground reaction forces...... 109

xiv CHAPTER 1

INTRODUCTION

The study of human balance and gait involves a large number of systems within the human body, including the musculoskeletal, somatosensory, vestibular, visual, and central nervous systems [3]. Research in this area attempts to better understand the role of each of these systems in maintaining balance. In order to reduce the complexity of the problem, many studies focus on only one or two of these systems.

Here, we will first explore the role of the foot and toes in balance. A review of literature will focus on contributions of the foot and toes through both the somatosen- sory and musculoskeletal systems. Computational experiments will primarily focus on the musculoskeletal system, with some somatosensory proprioception, i.e. joint angles and velocities. The computational model used for these experiments will in- clude more complex feet than other models seen in the literature, which tend to use simple inverted pendulums for the body [4] and model the feet by either single points of contact, massless links, or triangular wedges.

We will also explore the role of the vestibular system in balance. The vestibular system is comprised of specialized organs located in the vestibulum of the inner . These organs the angular position, angular acceleration, and translational acceleration of the head [5]. A review of literature will discuss findings related to the

1 role of the vestibular system in balance. Computational experiments using a triple model will show how the vestibular system might aid in balance when the surface of support is disturbed.

With regard to both problems, an important requirement of balance – assuming that inertial forces are negligible – is that the center of gravity must lie within the base of the support [6], [7], [8]. The center of pressure lies within the base of support by definition. If disturbances move the center of gravity or center of pressure toward the edge of the support boundary, stepping strategies may be required to widen the base of support to prevent a fall [9], [10]. Thus, the center of gravity and center of pressure are commonly used to calculate measures such as body sway for evaluating balance [11]. Here, we will consider center of gravity and center of pressure for both problems.

The dissertation is organized as follows. Chapter 2 provides background on several relevant motor control theories and biological systems, including the centeral nervous system, the somatosensory system, the vestibular system, and the musculoskeletal system. Chapter 3 reviews literature related to the foot, toes, and vestibular system and their role in balance. Chapter 4 presents the two models we will use throughout the dissertation. Chapter 5 explores the role of the foot and toes in balance through several computational experiments. Chapter 6 explores the role of the vestibular system through several other computational experiments. Chapter 7 concludes the dissertation, providing an overview of findings presented throughout the disserta- tion. Appendices B and C derive the equations of motion for the models used in the computational experiments.

2 CHAPTER 2

BACKGROUND

In order to understand the problem of human balance, it is important to have an understanding of motor control and the biological systems involved. This chapter provides a basic background on motor control theories, the central nervous system, parts of the peripheral nervous system, and the muscles.

2.1 Motor Control Theories

Motor control is “the study of the nature and cause of movement” [12], including control of balance and movement of the body as well as the planning of movements.

The latter requires and , and thus, in order to understand motor control, one must understand these processes as well. Cognition allows us to establish goals and plan our actions based on our attention, motivation, and even emotional state. Perception [13] is important because it provides feedback on the state of the action or movement and the environment in which the action or movement is taking place.

Because much of the human body, especially the central nervous system (CNS), is still not fully understood, there is no single theory that can fully explain all aspects of motor control. Many theories have been developed, though each has its faults and

3 limitations. For instance, reflex theory [14] posits that complex movements are strings of reactions, or reflexes. While this is true for some movements, it does not explain voluntary movements, novel movements, the ability to override reflexes, etc. [13].

Another group of related theories are referred to as motor programming theories [12].

These theories focus on the idea of a “,” either a hardwired neural cir- cuit or a higher-level construct sometimes referred to as a “central pattern generator,” that is capable of generating a movement even in the absence of sensory input. This theory is supported by experiments such as those in [15], which demonstrated that locusts are able to generate rhythmic wing beating even when their sensory nerves are cut, and [16], which showed that spinal neural networks in cats can produce a locomotor rhythm with no sensory inputs or descending signals from the brain. How- ever, further experiments on cats demonstrated that sensory inputs can modulate the output of a central pattern generator [17]. Also, movements are undoubtedly affected by external variables such as gravity and friction, and so sensory input must be taken into account [18]. Thus, motor programming theories cannot explain all aspects of motor control.

Yet another approach to understanding motor control is the systems theory ap- proach [12], [18]. In systems theory approaches to motor control, the body is treated as a mechanical system that is influenced by inherent characteristics such as the masses and moments of inertia and sizes of its components, forces of connection, and initial states, as well as external forces such as those due to gravity or disturbances. System theory approaches consider properties of the system such as degrees of freedom (DOF) and how to reduce the DOF of the system in order to make it controllable. One of the founders of this approach, Nicolai Bernstein (1896-1966), hypothesized that in order

4 to solve the DOF problem, higher levels of the nervous system activate lower levels, which in turn active synergies, or groups of muscles constrained to act as a unit. For instance, Bernstein considered synergies for locomotion, maintenance of posture, and respiration. This theory is very broad, and it takes into account contributions from the nervous system as well as the musculoskeletal system. However, this theory of motor control does not currently consider interaction with the environment as much as some other theories.

There are many other motor control theories: hierarchical theory, dynamical ac- tion theory, parallel distributed processing theory, task-oriented theories, and ecologi- cal theory. See [12] for an overview and discussion of these theories. This dissertation will focus on the systems theory approach.

2.2 The Central Nervous System

The central nervous system (CNS) consists of the brain and spinal cord. This section gives an overview of the CNS, including the spinal cord and brainstem, as well as major parts of the brain that are involved with motor control.

2.2.1 The Spinal Cord and Brainstem

The spinal cord receives and processes sensory information from the skin, joints, and muscles of the limbs and trunk, and it contains motor neurons responsible for both voluntary and reflex movements [19]. Spinal nerves are formed by joining two types of neural roots, the dorsal and ventral roots. The dorsal roots carry sensory information into the spinal cord, and the ventral roots carry information away, innervating muscles through outgoing motor axons.

5 The spinal cord consists of white matter – neural axons organized into tracts – which surrounds a butterfly-shaped column of gray matter – neural dendrites, neural cell bodies, and glial cells [20]. Dorsal columns are part of the white matter that runs up the dorsal or posterior side of the spinal cord, and they primarily carry afferent somatosensory information to the brainstem. The lateral columns, which compose the white matter in front of the dorsal columns, include axons that ascend to higher levels of the CNS as well as axons that project from nuclei in the brain stem and cortex down to motor neurons and interneurons in the gray matter of the spinal cord.

The spinal cord continues rostrally to become the brain stem, which consists of three parts: the medulla, , and [21]. The brain stem receives sensory information from areas near the head, and it provides motor control for muscles of the head. The medulla, along with the pons, regulates blood pressure and respiration.

The pons also relays information to the cerebellum, which wraps around the brain- stem, and it contains fibers that descend from the cerebral cortex to control muscles of the head, limbs, and trunk [20].

2.2.2 The Diencephalon and Thalamus

The diencephalon is composed of the hypothalamus and thalamus [19], the latter of which is very important for motor control. The thalamus relays sensory input to the primary sensory areas of the cerebral cortex and information about motor behavior to the motor areas of the cortex [20]. It also mediates motor functions by transmitting information from the cerebellum and basal ganglia to the motor regions of the frontal lobe, such as the primary motor cortex and higher-order motor areas.

6 The thalamus contains nuclei that can be split into two groups: relay nuclei and diffuse-projection nuclei. Relay nuclei process either a single sensory modality or input from a distinct part of the motor system, project to a specific region of the cerebral cortex, and receive recurrent input from this region of the cerebral cortex. The diffuse- projection nuclei have more widespread connections. They make connections to other areas of the thalamus as well as making connections in the cerebral cortex.

2.2.3 The Cerebellum

The cerebellum receives somatosensory input from the spinal cord, motor infor- mation from the cerebral cortex, and input about balance from the vestibular system

[19]. It contributes to planning, timing, and patterning of muscle contractions for movement and the maintenance of posture.

Inputs from the spinal cord provide feedback about movements, while inputs from the cerebral cortex provide information about planned movements [12]. Due to these connections, the cerebellum is thought to compare intended outputs with sensory signals and correct movements, i.e. perform feedback control. It also modulates the range and force of our movements and is involved in motor learning.

2.2.4 The Primary, Secondary, Tertiary, and Association Areas

The primary, secondary, and tertiary areas lie in the cerebral cortex, and each has components that are involved in motor control [19]. The primary motor cortex is part of the frontal lobe of the cerebral cortex and contains neurons that project into the spinal cord and onto motor neurons to allow for voluntary movements of the limbs and trunk. The primary sensory areas receive information from visual,

7 auditory, somatic, and gustatory areas. The secondary and tertiary areas process more complex aspects of motor control. They include a portion of the posterior called the posterior parietal cortex, which integrates somatic, visual, and movement information.

There are three large areas called association areas that lie in the cerebral cortex outside the primary, secondary, and tertiary areas that are involved in perception, movement, and motivation. The parietal-temporal-occipital association cortex re- ceives information from the parietal, temporal, and occipital lobes and integrates them to form complex . The prefrontal association cortex lies in the frontal lobe and participates in the planning of voluntary movements. The limbic association cortex is devoted mostly to motivation, emotion, and memory.

To summarize, the primary cortices receive and do initial processing of sensory information. This information is then sent to the secondary and tertiary cortices, which in turn connect to the association areas to provide a link between sensation and planned action. The secondary and tertiary cortices then send information back to the primary cortices, which has direct control over motor neurons.

2.2.5 The Basal Ganglia

The basal ganglia lie deep within the cerebral cortex and consist of the globus pallidus, the caudate nucleus, and putamen. The latter two structures are collec- tively known as the corpus striatum [20] and are involved in regulating the speed of movements. They also contribute to cognition [19]. The basal ganglia receive input from most areas of the cerebral cortex and send outputs to the motor cortex via the thalamus.

8 2.3 The Peripheral Nervous System

The peripheral nervous system includes parts of the nervous system other than the brain and spinal cord. These systems are important in providing feedback from areas such as the skin, eyes, and muscles to the CNS. This section covers two pe- ripheral systems that contribute to motor control: the somatosensory system and the vestibular system. The also contributes heavily to motor control, but will not be considered in this dissertation.

2.3.1 The Somatosensory System

The somatosensory system, or somatic sensory system, processes four types of sensations: discriminative touch, proprioception or the sense of position and move- ment of the limbs, or due to tissue damage, and temperature [22].

Sensory information travels through afferent nerve fibers bundled together to form peripheral nerves [21]. Individual peripheral nerves also contain efferent nerves for the same part of the body. As they approach the spinal cord, peripheral nerves join together to form spinal nerves. Near the spinal cord, the afferent fibers separate from the efferent fibers and enter the spinal cord through dorsal roots. Thus, the spinal cord is the first relay point for somatosensory information.

The spinal cord and brainstem contain a system called the Dorsal Column-Medial

Lemniscal system, which is the principal pathway for somatosensory information [20].

The first part of this system is the dorsal column of the spinal cord. Afferent fibers in this column synapse on cells in the dorsal column nuclei, located in the medulla.

Axons of postsynaptic neurons in these nuclei cross to the contralateral side of the brain in a fiber bundle called the medial lemniscus, which ascends through the pons.

9 The pons contains pontine nuclei whose axons run to the contralateral half of the cerebellum and contribute to cerebellar control of movement and posture. The medial lemniscus then terminates in the thalamus.

2.3.2 The Vestibular System

The vestibular system [12], [23] is housed within the of the . The vestibular portion of the membranous labyrinth consists of a pair of organs called the and the , as well as three semicircular ducts or canals. The otolith organs sense angular position and translational acceleration of the head; the sense rotational acceleration of the head.

Accelerations are registered through the bending of specialized vestibular hair cells. This bending occurs due to movement of fluids in the membranous labyrinth.

The membranous labyrinth is surrounded by a fluid called , and it is filled with a fluid called , which is denser than water and has inertial char- acteristics. When the head experiences a rotational acceleration, the inertia of the endolymph in the semicircular canals creates a force against a diaphragm-like mass called the cupula, which stretches between the roof of a structure called the ampulla and a raised structure on the ampulla called the ampullary crest. Vestibular hairs on the ampullary crest are bent when the endolymph exerts a force on the cupula, causing a change in the firing rates of associated receptors. The semicircular canals lie in mutually perpendicular planes, so that rotational acceleration in any direction can be measured. Receptors respond to accelerations as low as 0.1◦/s2 [23].

The utricle and saccule use similar processes to measure the effects of gravity and translational accelerations. The utricle has an area called the macula, where receptor

10 cells with projecting hair cells are located. These hair cells project into a gelatinous membrane embedded with crystals of calcium carbonate, called or otoconia.

The macula of the utricle lies in a horizontal plane of the head. If the head is tilted or accelerates horizontally, the otoliths deform the gelatinous membrane, causing the vestibular hair cells to bend. The saccule is similar, but its macula lies in a vertical plane, so that it responds to vertical accelerations.

To summarize the function of the vestibular system, the semicircular canals, utri- cle, and saccule all have a dynamic function. They sense either rotational or transla- tional acceleration. The utricle and saccule additionally have a static function. They sense the static angle of the head in space [5], and thus contribute to the control of posture.

As far as connections are concerned, neurons from the otolith and semicircular canals go through the 8th auditory nerve. Their cell bodies are in the . The axons of these neurons enter the brain in the pons. Most go to the

floor of the medulla, where the are located: the medial nucleus, the lateral nucleus, the superior nucleus, and the inferior or descending vestibular nucleus. A certain portion of the vestibular neurons go from the sensory receptors to the cerebellum, , thalamus, and the cerebral cortex.

The lateral vestibular nucleus, also known as Deiters’s nucleus, receives input from the utricle, semicircular canals, cerebellum, and spinal cord. Its output contributes to the vestibulo-ocular tracts and lateral , which activates antigrav- ity muscles in the neck, trunk, and limbs. It also contains neurons that respond to tilting of the head. Some have a response that increases with increasing angle; others

11 respond whenever the angle changes. The dorsocaudal part of the nucleus receives inhibitory input from Purkinje cells in the vermis of the cerebellum.

The medial and superior nuclei receive inputs from the semicircular canals. The medial nucleus sends outputs to the medial vestibulospinal tract (MVST) to help control the neck muscles and coordinate head and eye movements. Neurons from both nuclei ascend to motor nuclei of the eye muscles to aid in stabilizing gaze.

The inferior vestibular nucleus receives and integrates inputs from the semicircular canals, utricle, saccule, and . Outputs are part of the vestibulospinal tract and vestibuloreticular tracts. It affects centers at higher levels in the brain stem, perhaps the thalamus.

2.4 The Musculoskeletal System

The musculoskeletal system, innervated by the nervous system through motor neurons, allows the body to move. This section provides a brief overview of the bones and joints and a more thorough description of the muscles and other associated components. Unless otherwise noted, the information contained here can be found in

[24].

2.4.1 Bones, Joints, and Ligaments

Bones of the human body generally make contact through three types of joints:

fibrous joints, cartilaginous joints, and synovial joints. Fibrous joints, such as sutures of the skull, are relatively immobile. Cartilaginous joints, such as the intervertebral disks, are slightly moveable. Synovial joints, such as the hip and elbow, are much more mobile. Ligaments attach the bones at a synovial joint, and friction is reduced by lubricated articular cartilage that covers the bone surfaces that form the joint.

12 Synovial joints approximate pinned joints used in rigid body models and may have one to three degrees of rotational freedom with a limited range of rotational motion about each axis.

2.4.2 Overview of Skeletal Muscles

Movement of bones about joints is caused by the contraction of skeletal muscles, one type of muscle found in the human body. Skeletal muscles are relatively com- plicated organs that consist of many parts. At the highest level, muscles consist of muscle fibers that are linked together by three types of collagenous connective tissue.

These include endomysium, which surrounds individual muscle fibers; perimysium, which collects bundles of fibers into groups called fascicles; and epimysium, which surrounds the entire muscle.

The muscle fibers themselves can vary from 1 to 400 mm in length and from 10 to

60 µm in diameter. They are composed of even smaller myofilaments that are enclosed in cell membranes called the sarcolemma, which also holds in a fluid called sarcoplasm.

Within the sarcoplasm and connected to the sarcolemma lies a hollow, membranous system called the sarcoplasmic reticulum, which assists the muscles in conducting commands from the nervous system. The sarcolemma also has invaginations called transverse tubules that also assist in conducting nervous system commands.

The myofilaments that comprise a muscle fiber have a repeating banded structure made of sarcomeres. Sarcomeres are the basic contractile unit of muscle. They are composed of two types of partially overlapping filaments: thick filaments and thin

filaments. Thick filaments contain myosin and several myosin-binding proteins, while thin filaments are composed mostly of actin as well as tropomyosin and troponin,

13 which regulate the interaction between actin and myosin. The thin and thick fibers

are held in alignment by the cytoskeleton, which also aids in the transmission of force

generated by actin and myosin to intramuscular connective tissues and the skeleton.

So how these components of the muscle work together to generate force? When a

muscle receives a signal from an innervating motor neuron, the neural action potential

is converted to a sarcolemmal action potential through a process called neuromuscular

propagation, which involves the neurotransmitter ACh and the movement of Na+ and K+ across the sarcolemma. This sarcolemmal action potential is converted to muscle force through a series of processes known collectively as excitation-contraction coupling. First, the sarcolemmal action potential is propagated down the transverse tubule. Then, Ca+2 conductance of the sarcoplasmic reticulum is increased. This is also referred to as Ca+2 disinhibition. The Ca+2 ions bind to troponin of the thin

filaments, most likely uncovering myosin-binding sites on the actin, so that the thin and thick filaments can form what are known as cross-bridges. The Ca+2 is then removed by the sarcoplasmic reticulum, and the process can repeat.

The cross-bridge cycle is used to describe the interaction of actin and myosin that occurs due to Ca+2 disinhibition. During part of this cycle, the interaction between actin and myosin causes the thick and thin filaments to slide past one another. This is sometimes known as the power stroke portion of the cross-bridge cycle. Since actin and myosin are connected during this movement, a force is exerted on the cytoskeleton. Muscle force appears to be due to many concurrent cross-bridge cycles.

14 This type of activity in the muscle is known as a contraction. Note that a con- traction does not always correspond to the shortening of a muscle. Muscles may also perform isometric contractions, in which they provide a force but their length does not change, or a lengthening contraction, in which they provide a force while the length of the muscle grows.

15 CHAPTER 3

REVIEW OF LITERATURE

In this chapter, we provide a review of literature on topics related to the foot, toes, and vestibular system in balance. The first section discusses research regarding the somatosensory role of the foot soles in balance. The second section discusses research related to the role of the toes. And the third section discusses research regarding the effects of the vestibular system.

3.1 Role of Foot Sole Mechanoreceptors in Balance

The soles of the feet contain many types of mechanoreceptors, each capable of transmitting different types of cutaneous afferents. In [25], the authors mapped the distribution of foot sole mechanoreceptors by monitoring activity in the tibial nerve of thirteen healthy subjects when small indentations or vibrations were applied to different areas of the foot. By manually moving an electrode within the nerve, re- sponses for single units were isolated. Mechanoreceptors were classified as either slow adapting (SA) when they responded continuously to maintained indentation, or fast adapting (FA) when they responded only at the onset and removal of indentation or vibrations. Each receptor was further classified as type I or type II depending on its receptive field; type I receptors have small receptive fields with multiple hotspots, and

16 type II receptors have large obscure fields with a single hotspot and higher sensitivity.

In all, the authors identified 104 mechanoreceptors: 15 SA type I (median threshold of

36 mN), 16 SA type II (median threshold of 115 mN), 59 FA type I (median threshold of 12 mN), and 14 FA type II (median threshold of 5 mN). Variances in sensitivity thresholds were quite high, with some FA units responding to as little as 0.5 mN of force and some SA receptors responding to no less than 3000 mN of force. The mean activation threshold was 25 mN with a range of 0.5-150 mN for the toes, 80 mN with a range of 0.5-750 mN for the lateral foot, and 300 mN with a range of 0.7-3000 mN in the heel. The authors found that there was no activity in the absence of stimulation and suggest that “any activity from skin receptors in the foot sole may be important for signalling that the foot is in contact with the supporting surface” and that the

“wide dispersal of receptors throughout the foot sole would ensure that skin receptors would be able to code for contact pressures, and hence the position of the foot with the ground.”

Though it is clear that cutaneous afferents from mechanoreceptors located in the foot sole provide the CNS with information about the onset, removal, or continued application of forces or pressures under the feet, there is a great deal of debate about how this information is used in balance. The issue is complicated by the fact that vision, the vestibular system, and other aspects of the somatosensory system, such as proprioception, also make contributions to balance [3]. Separating these contributions is a difficult and ongoing area of research. A large number of experiments have been performed in order to provide insight into the role of cutaneous afferents in quiet and perturbed stance. The proceeding is a review of recent research in this area, divided into sections based on considered experimental factors: aging, disease, inhibition,

17 stimulation, and rotational platform disturbances. A concise summary of all research follows.

3.1.1 Effects of Aging

One class of experiments compares balance in young healthy subjects with balance in older healthy subjects. For instance, [26] compares quiet stance in thirteen young subjects (mean age 21.0 ± 1.6 years) and eight elderly subjects (mean age 71.4 ±

2.8 years). Tactile sensitivity of the big toe was measured in all subjects by running a contactor longitudinally over the bottom of the toe and recording the minimum distance, in increments of 1 µm, at which subjects reported movement. By this standard, sensitivity was much lower in the elderly group: 48.82 ± 11.02 µm versus 5.63 ± 4.87 µm in the right foot and 56.09 ± 12.98 µm versus 4.6 ± 3.98 µm in the left foot. The authors measured body sway index or SI (in this case, a standard- deviation-like measure based on center pressure) during quiet standing under four conditions: A) eyes open, B) eyes closed, C) standing on a 10-cm-thick foam pad with eyes open, and D) standing on the foam pad with eyes closed. Maximum great toe pressure was also measured in cases A and B.

In case A, sway values were not significantly different between the young and elderly groups. SI was significantly higher for the elderly group in the eyes closed versus eyes open cases, i.e. case A compared to case B and case C compared to case

D. Differences between these cases were found for the young group as well; however, they were not nearly so large. The foam padding also had an effect. SI was only slightly higher for the elderly group than the young group in condition B, and SI was much higher for the elderly group in case D compared to case B. In contrast,

18 differences between SI values for cases B and D for the young group were small.

In addition, maximum great toe pressure was higher in the elderly group for the

cases in which it was measured. The authors conclude that “reduced sensory input

(tactile and proprioceptive ) from the foot, deprivation of visual information,

and insufficiency of muscle control in the toes are all important factors associated

with postural instability.” In a paper describing the same experiment [27], the authors

suggest that, “It might be hypothesized that the older people used great toe pressure

in order to intensify sensory input from the great toe as well as to maintain balance.”

3.1.2 Effects of Disease

Another class of experiments compares balance between healthy subjects and pa-

tients with diseases that affect balance such as neuropathy, Charcot-Marie-Tooth

disease (CMT), or Parkinson’s disease (PD). Generally, these experiments find that

such conditions affect balance negatively.

For instance, in [28], the authors measured balance and sensibility in ten control

subjects (mean age 50 years) and thirty-five patients (mean age 62 years) with pe-

ripheral neuropathy due to either diabetes, hypothyroidism, chemotherapy-induced

neuropathy, or other unknown causes. Sensibility was measured by recording 1- and

2-point static touch thresholds on the pulp of the big toe, the medial heel, and dorsum

of the foot. Sway area was measured in all subjects during quiet standing in eyes open

and eyes closed conditions. The control group had lower pressure thresholds for 1-

and 2-point discrimination and low sway surface area: 27.3 ± 6 mm2 with eyes open

and 33.4 ± 7 mm2 with eyes closed. Those with moderate peripheral neuropathy had comparable 1-point discrimination pressure thresholds but much higher 2-point

19 discrimination thresholds, while those with more advanced peripheral neuropathy lost

2-point discrimination and had higher 1-point discrimination thresholds, and those with severe peripheral neuropathy had neither 1-point nor 2-point discrimination. For neuropathy patients, sway results were 52.2 ± 31 mm2 with eyes open and 158.5 ±

150 mm2 with eyes closed. Sway area was larger for those with more severe peripheral neuropathy.

In [29], the authors recruited three groups of subjects: eighteen young and healthy subjects (average age 27, range 18-42 years), eighteen elderly and healthy subjects

(average age 59, range 42-79 years), and eighteen idiopathic Parkinsonian disease

(PD) patients (average age 64, range 51-75 years). PD patients were examined during an “on phase” and when they were free from dyskinesia. Subjects were either asked to stand still with their arms at their sides using a normal posture, to lean forward or backward as far as possible, or to stand in an intermediate posture that was chosen spontaneously without visual feedback. Heels and toes remained on the ground, and hips and knees were bent as little as possible. Subjects stood with their eyes open (EO) or eyes closed (EC). Measurements included instantaneous center of foot pressure (CFP), its mean position (± standard deviation) along the anteroposterior axis, and sway area (SA). In two normal subjects and two patients, an LED was placed at approximately the level of the projection of the center of mass. In six normal subjects and six PDs, surface EMGs were recorded for the soleus (Sol), tibialis anterior (TA), flexor digitorum brevis (FDB), and extensor digitorum brevis (EDB).

CFP measurements showed that mean CFP was more posterior for PDs than for healthy young or elderly subjects. They also found that: sway area increased by the same relative amount in all groups in the EC case compared to the EO case, young

20 subjects had the largest angles during extreme postures and PDs had the smallest, and sway area was larger for extreme postures. In young subjects with EO, the maximum anteroposterior displacement of CFP was about 60% of foot length, and with EC it was about 50%; in healthy elderly subjects, these values were about 40%

(EO) and 30% (EC); in PDs, these values were about 30% (EO) and 20% (EC).

EMG measurements showed that during quiet standing, Sol was active in all sub- jects. Forward leaning increased Sol activity and induced tonic activity in FDB. The increases in FDB activity were larger than in Sol as inclination increased. Backward leaning resulted in tonic TA activity and bursts of activity in the EDB. The relation- ship between these EMG levels (toes versus ankle) is similar to that of SA versus CFP position. I.e., Sol activity was strongly correlated with CFP during quiet standing, and this correlation decreased as inclination increased during forward leaning; mean- while, correlation between CFP and FDB increased as inclination increased. In both cases, no significant difference was found between EO and EC cases. This seems to imply that the toes have a more pronounced role during quiet standing with extreme forward-leaning postures. During backward inclination, TA contraction acted as a brake mechanism, and cross-correlation with CFP was negative, and there was no significant correlation between CFP and EDB in this case. The authors state that this is unsurprising, because the EDB has no mechanical advantage in this situation.

Furthermore, the authors note that there seems to be an “economy zone” that lies in an area starting in front of the malleolus and ending 15-20% of total foot length past this point. In this zone, sway area does not increase much even though it corresponds to considerable body inclination. The authors note that “the moment of the centre of mass relative to the ankle is small, and the stiffness of the active

21 antigravity muscles can prevent or slow any inclination at its onset. Further, this is an ‘economy zone,’ since little static and dynamic active work is needed for small corrective actions; normal stance corresponds in fact to the least sum of leg muscle

EMG amplitudes.” Interestingly, during extreme postures, deviations from the mean

CFP do not result in straightforward corrections, i.e. movements seem to be some- what random. The authors theorize that “this is the effect of many small movements deliberately issued in order to seek balance and keep the requested inclination despite changes in gravity load (average inclination plus instant body sway). Therefore, sway area might depend more on central control of muscle synergies than on the automa- tisms and reflexes normally operating during quiet stance. In turn, this movement strategy would increase the cutaneous and proprioceptive sensory input to the CNS, thus allowing better organization of the postural command.”

3.1.3 Effects of Inhibition

Another class of studies consists of inhibiting cutaneous afferents originating from feet without inhibiting any other systems. The methods used to inhibit these afferent signals vary. Results show different effects that arise due to this type of inhibition.

In [30], anesthesia was applied to the soles of the forefeet of ten healthy subjects

(28.6 ± 8.5 years old). The anesthesia was said not to affect joint or muscle affer- ents, but was not applied to the toes due to risk of ischemia (restriction of blood supply). In the first experiment described in this paper, subjects completed twelve

35 second trials while performing one of three tasks: bipedal stance with eyes closed

(bipedal-EC), bipedal stance with eyes open (bipedal-EO), and unipedal stance with eyes open (unipedal-EO). The results were compared to a control condition in which

22 no anesthesia was applied. The effect of the anesthesia was to increase mean sensory threshold level (STL) from about 9 mN to about 350 mN. Under the effects of anesthe- sia, mean COP velocity increased in all directions by about 10%, and anteroposterior

COP median frequency increased by about 36% for the unipedal-EO condition. In the bipedal-EC task, mediolateral COP velocity and COP area increased by about

11%, mostly in the mediolateral direction. No significant effects were found during the bipedal-EO task.

In the second experiment, anesthesia was applied to the heels, lateral soles, and metatarsal heads of six healthy males (26 ± 10 years old) using 24 injection sites.

The subjects then stood with eyes closed for 35 seconds. The effect of the anesthesia in this case was to increase the mean STL from about 4 mN to about 4 N, with all sites remaining above the desired threshold level of 115 mN throughout experiment.

Results were again compared to a control condition. When anesthesia was used,

COP velocity increased by about 11% in the planar dimension (anteroposterior and mediolateral directions combined) and there was reduced persistence in the COP trajectory over short time intervals. The authors summarize by stating that, “Forefoot anesthesia produced mostly mediolateral postural effects, increasing the mediolateral

COP velocity during both bipedal-EC and unipedal-EO tasks. Whole-sole anesthesia, however, appeared to produce an anteroposterior effect, decreasing persistence over short time intervals as well as increasing COP velocity.”

In [31], body sway was induced in sixteen healthy young adults (mean age 24.7 years, range 22-27 years) by vibration of the calf muscles. During application of this stimulation, the subjects stood upright and relaxed on a force platform with their arms over their chests. Spontaneous sway was recorded 30 seconds before the

23 onset of calf muscle vibration, and the vibration was turned off and on according to a pseudo-random binary sequence schedule that lasted a total of 205 seconds.

Vibration periods were 0.8-6.4 seconds, yielding an effective bandwidth of

0.1-2.5 Hz. Test conditions included eyes open (EO) and eyes closed (EC), with or without hypothermic anesthesia of the feet. Hypothermic anesthesia was obtained by placing the feet of the subjects in ice water for 20 minutes. The depth of the water was only 2 cm, to minimize the effects on the Achilles tendon and calf muscles. This form of anesthesia decreased pain sensation, vibration detection, and touch sensation in the feet, though it did not significantly affect two point discrimination.

The authors found that initial anteroposterior and lateral torque responses to calf stimulation were larger with hypothermic feet than without, though these responses reached normal levels after about 50–100 seconds. Total, low frequency (< 0.1 Hz) and high-frequency (> 0.1 Hz) torque variances were dependent on test conditions and time of occurrence during periods of calf vibration; only high-frequency torque variances were dependent on test conditions during quiet stance. Also, vibrations applied to the calf induced more torque with eyes closed, regardless of thermal con- ditions of the feet. They also found that test order had an effect; for instance, the effects of hypothermia were more pronounced if they occurred during the first sets of testing conditions, i.e. previous experience with the calf stimulation when the feet were not cooled seemed to improve postural control in this case when the feet were later cooled.

24 3.1.4 Effects of Stimulation

There are several studies in which the authors attempt to stimulate mechanore- ceptors in the soles of the feet without stimulating any other systems. The methods used to stimulate the foot soles vary; however, these studies generally show that stim- ulation results in sway, implying a role for cutaneous afferents in standing balance.

For instance, in [32] the authors applied low-amplitude (0.2 mm to 0.5 mm), high-frequency (100 Hz) vibrations to areas under the heels and metatarsal heads of then healthy subjects (22-55 years) who stood upright with their eyes closed. The frequency of the vibrations was said to be high enough to activate the cutaneous receptors without activating the muscle spindles. The vibrators were activated in various combinations throughout ten trials. The authors found that vibration applied to one or both soles produced involuntary body sway in all subjects, and the direction of this sway depended on the point of application. To summarize, sway was in the direction opposite the stimulation, e.g. stimulation of the left heel resulted in sway that was forward and to the right. When two corresponding areas of the feet were stimulated simultaneously, e.g. both of the heels, the resulting sway was roughly the addition of the sways induced when each area was stimulated individually. When vibration was applied to all areas at once, postural instability increased, but no clear direction for the sway was evident.

In a similar study [33], the same researchers applied vibrations of different fre- quencies (20, 60, 100 Hz) to the heels and metatarsal heads of nine healthy volunteers

(22-55 years) who stood upright with their eyes closed. The first nine test conditions consisted of co-stimulating either the two anterior zones, the two posterior zones, or all zones of the soles. In the last three test conditions, the zones being stimulated and

25 the frequency of the stimulation varied throughout the test. In two ‘static contrast’ test conditions, a 1 second stimulation at 100 Hz was applied to both the forefeet and the heels, then one or the other was terminated for 2.5 seconds. In a third

’dynamic contrast’ test condition, the frequency of the vibrators was varied between

50-100 Hz sinusoidally, with stimulation frequencies at the heels and forefeet being

90◦ out of phase. As with the last study, stimulation to the heels produced sway in the anterior direction, and stimulation to the forefeet produced sway in the poste- rior direction. Stimulation applied to both areas simultaneously produced instability without a definitive direction. The highest postural responses occurred for the 100

Hz vibration, the lowest for the 20 Hz vibration, and the 60 Hz vibration produced intermediate results. The mean latency of the responses was 0.9 ± 0.4 seconds. Re- sults for the static conditions were similar to those obtained during the first nine test conditions; sway was opposite the direction of the remaining stimulation site and had approximately the same magnitude and speed. The sinusoidal stimulation produced a sinusoidal sway with a mean phase of 52 ± 30o.

The authors state that rapidly and slowly adapting cutaneous receptors respond proportionally to vibrations between 1 and 200-300 Hz. Thus, since slowly adapting receptors are known to encode static pressure and changes in pressure, it would appear that forces felt at the feet produce proportionate responses in sway.

In yet another study [34], the same researchers attempt to measure the effects of proprioception and cutaneous afferents by providing direct stimulation. Toward this end, nine healthy adults (age range, 24-52 years) were directed to stand barefoot on a foot rest with their hands at their sides and their eyes closed. Two mechanical vibra- tors were placed on the ankles to provide proprioceptive stimulation. Electromagnetic

26 vibrators were placed under the metatarsal heads to induce tactile stimulation of the soles. The subjects were told not to resist any vibration induced body tilts. The amplitude of the postural responses was assessed by the position of the anteropos- terior projection of the COP after 2.5 seconds of vibration. The authors found that stimulation of the tibialis anterior muscles always resulted in forward sway, and stim- ulation of the forefeet produced backward sway. When vibration was applied to the forefoot, the soleus EMG activity first increased while the EMG activity of the tib- ialis anterior muscle remained at its base level. As body tilted backwards, the EMG activity increased in the tibialis anterior and decreased in the soleus muscles. When vibration was applied to the tibialis anterior, the authors observed “(1) an increase in the tibialis anterior activity whilst the soleus activity was stable, (2) a decrease in the soleus activity whilst the tibialis anterior remained silent, (3) a concomitant increase in the tibialis anterior and decrease in the soleus muscle activities...Then, as soon as the body tilted forwards, a strong increase in the soleus activity occurred together with a decrease in the tibialis anterior activity to counteract the body tilt.”

EMG responses, COP shifts, and ankle angle deviations had longer latencies for pro- prioceptive stimulation compared to tactile stimulation: 119 ± 28 vs 166 ± 86 ms,

251 ± 111 vs 286 ± 215 ms, and 434 ± 170 vs 612 ± 291 ms, respectively.

Both tactile and proprioceptive responses were proportional to the frequency of vibration. At low vibrations, tactile stimulation induced greater postural effects, and at higher vibrations, proprioceptive stimulations induced greater postural responses.

When both types of vibrations were applied simultaneously at the same frequency, the result was a summation of the previous results, with 20 Hz and 40 Hz producing a backwards sway, and 60 Hz and 80 Hz producing a forwards sway. When the applied

27 frequencies were different, the sensory channel stimulated at the higher frequency dominated. The same differences at lower frequencies produced more backwards sway when the tactile stimulation was at the higher frequency (e.g. tactile stimulation at 40

Hz and proprioceptive stimulation at 20 Hz); the same differences at higher frequen- cies produced more forwards sway when proprioceptive stimulation was at the higher frequency (e.g. proprioceptive stimulation at 80 Hz and tactile stimulation at 60 Hz).

To summarize, “The postural responses corresponded to the sum of the effects ob- tained upon stimulating separately the two sensory modalities.” Furthermore, “Taken together, these findings suggest that tactile and proprioceptive afferents from soles and ankle muscles could subserve complementary functions for postural purposes: the regulation of small amplitude body sways would be predominantly assigned to tactile inputs, whereas ankle muscle proprioception would be mainly involved in the regulation of larger body sways.”

3.1.5 Balance on a Rotationally Disturbed Platform

Several studies test use a platform that rotates in the sagittal plane in order to test the effects that rotational disturbances have on balance and to observe the effects of cutaneous afferents in such cases. These types of disturbances can either be a sudden

“toes-up” or “toes-down” disturbance, or a sinusoidal disturbance that lasts several cycles.

In [35], the authors use a sudden “toes-up” platform rotation to investigate the role of joint receptors, muscle receptors, and plantar cutaneous mechanoreceptors in modulating postural reflexes. They classify these reflexes as short latency (SL, 30-69 ms after stimulus), medium latency (ML, 70-119 ms after stimulus onset), and long

28 latency (LL, 120-200 ms after stimulus onset) depending on the time between stimulus onset and the resulting muscle reaction. The authors hypothesized that “(1) the SL reflexes are initiated only by the stretch stimulation in the posterior muscles; (2) the

ML reflexes are correlated to the joint movement and plantar pressures that stimulate the joint receptors and the plantar cutaneous mechanoreceptors, respectively; (3) the

LL reflexes are mediated mainly by the plantar pressures under the foot; and (4) it is the time derivative of the pressure change between the forefoot and the rear foot regions that plays a major role in regulating these postural reflexes (such as

ML and LL).” In order to test their hypotheses, the authors measured EMG signals from the gastrocnemius (GAS) and tibialis-anterior (TA) muscles of subjects who stood on various support surfaces with their eyes closed and their ankle joints either free to move or immobilized. The support surfaces were then rotated “toes-up” in the sagittal plane with an amplitude of 8◦ and at a speed of up to 60◦/s. In some cases, the surface was hard; in others, either a 1- or 2-inch foam pad was used. The subjects included fifteen healthy males (age 2.478 ± 4.07 years). Means and standard deviations of the ankle rotation at 50 ms and 300 ms were recorded. Pressures under the forefoot and heel of the right foot were measured as well, and their time derivatives were computed.

The authors found that when the ankles were immobilized, ankle rotation was only about 1◦ maximum, as opposed to 1.2◦ at 50 ms and up to 8◦ at 300 ms when the ankles were free to move. These values were independent of surface conditions.

Only small differences in pressure were seen between ankle conditions, within 10 kPa in the forefoot and less than 20 kPa in the rear foot, both occurring toward the end of platform movement. The magnitude of the pressure derivative for both cases

29 decreased as the surface became more compliant, and in all cases, peak pressures correlated well to ground reaction forces. With respect to muscle latencies, SLs were not significantly different among surface conditions or ankle conditions; however,

MLs and LLs on foam surfaces were longer than those on a hard surface. Also, when the ankle joints were immobilized, the MLs were significantly delayed on all three surfaces by about 2.6 ms on average. For the LLs, the only significant delay was on the hard surface. Smaller plantar pressures in some conditions also led to longer delays. ML responses were most correlated to the maximum positive slope of the differential pressure (forefoot pressure - heel pressure), followed by static pressure

(pressure before platform rotation). They were least correlated to the pressure slope for either region. The LL responses were most correlated to the maximum negative slope of the differential pressure and least to the pressure slopes in the forefoot.

The authors conclude that for SL responses, the muscle spindles play the most important role, since even in the immobilized case, some movement was allowed, and this may have been enough to trigger them. The cutaneous receptors and joint receptors appeared to make no significant contribution with respect to SL responses.

ML responses appeared to be mediated by pressures sensed by the mechanoreceptors, since responses were delayed on softer surfaces. Since ML responses for the GAS were longer when the ankle joints were immobilized, joint receptors may contribute in this case as well. Since LL responses were longer only on a hard surface when the ankles were immobilized, it may be that these responses are related to the ankle joint, but modulated by pressures sensed by the cutaneous mechanoreceptors.

30 The authors note that this study does not include the effects of the vestibular

system, and that “the results in this study cannot exclude a possible effect of mus-

cle receptors on ML and LL responses. It might be possible that both ML and LL

responses are initiated by muscle receptors and the initiation is modulated by joint

receptors and mechanoreceptors.” To summarize, “(1) the short latency was not af-

fected by the changes in both plantar pressure and ankle joint movement; (2) the

medium latency was regulated by the plantar pressures under the foot as sensed by

the cutaneous mechanoreceptors in the sole of the foot and by the ankle joint move-

ment as perceived by the joint receptors in the ankle joint; (3) the long latency was

also related to the ankle joint movement, but this relation seems to be modulated

by the plantar pressures under the foot; and (4) the ML and LL were well correlated

with the time derivative of the differential pressures between the forefoot and the rear

foot regions.”

A sinusoidally disturbed platform was used in [36] to measure the effects of cuta-

neous afferents and vestibular afferents. Toward this end, six healthy subjects (36 ±

9 years old) and four vestibular loss patients (35 ± 3 years old) were asked to bal- ance on platform made to tilt in the sagittal plane. Goggles depicting a homogenous blue surface were worn to eliminate visual input. A pin matrix capable of producing small indentations in the skin at frequencies of 0.05, 0.1, 0.2, and 0.4 Hz was placed under the subjects’ forefeet. First, skin indentation was applied during quiet stand- ing. In normal subjects, this resulted in small COP excursions of about 0.5 cm or less that were essentially counter-phase at all frequencies of application. Responses for vestibular loss patients were similar, though with slightly larger phases. When the support platform was rotated ± 2◦ at speeds of 0.05 Hz, 0.1 Hz, 0.2 Hz, and

31 0.4Hz, indentations in the forefeet were produced either in phase with the platform

(indentation occurring when the platform was tilted forward), out of phase, or were constant. The authors found no substantial differences in responses when compared to those from [37], which analyzed the effects of vestibular loss. Given this result, the authors suggest that somatosensation in the feet is not used for postural control, but to judge the quality of surfaces and protect the feet from injury, and that propri- oception is the main mechanism for equilibrium control. They also suggest that the

Golgi tendon organs contribute to “graviception” [37], especially at low frequencies, and that fusion with vestibular information helps healthy subjects maintain balance at higher frequencies. It is interesting to note, however, that the indentation method used in these experiments produced COP excursions of less than 0.5 cm, which is relatively small compared to the excursions of 2-4 cm produced in [32].

It is also interesting to examine results obtained in [38], which seem to contradict the importance of proprioception for balance on a sinusoidally disturbed surface. In this study, the authors compared the eyes open (EO) and eyes closed (EC) responses of twenty normal subjects (ages 29-77) and twenty-seven neuropathic patients (43-

77 years) during static balance and dynamic balance. Static balance consisted of quiet standing, and dynamic balance consisted of balance while standing on a sinu- soidally rotating platform. In the latter case, the platform was set to rotate at 0.2 Hz with a peak-to-peak displacement of 60 mm in the anteroposterior direction. Of the twenty-seven neuropathic patients, fourteen suffered from diabetic polyneuropathy and thirteen from Charcot-Marie-Tooth (CMT) disease. Of the CMT cases, five were

CMT1A (demyelinating) and eight were CMT2 (axonal).

32 The results showed that during quiet standing, sway area (SA) increased during

EC compared to EO in both normal subjects and patients, and CMT2 and diabetic patients had larger sway than healthy subjects. However, there were not significant differences in SA between normal subjects and CMT1A patients under either of the visual conditions. Interestingly, no significant relationships were found between SA values and overall NDS (neurological disability) scores, nor were any relationships found between SA and the individual components of NDS (strength of Sol and TA muscles, Achilles’ and quadriceps tendon reflexes, touch-pressure, vibration, joint position, and pricking pain sense). No relationship was found between NS scores, which are derived from electrophysiological assessment of large motor and sensory nerve fibers of lower limb muscles. The authors did find that a decrease in conduction velocity (CV) of group II fibers was associated with increased SA in the EC case.

However, there appeared to be a minimal relationship between CV of motor fibers and SA. The authors also found that in the EC case, the head generally oscillated more than other segments, whereas with EO the head was stationary in space. In normal subjects, the head anticipated in the EO case and lagged behind platform movement in the EC case. In patients, the head always lagged platform movement, but the lag in the EC case was similar to that of normal subjects in the EC case.

The authors summarize by noting that diabetics are generally more unstable than other groups during quiet stance, and those with CMT1A disease are generally not, especially those whose group II fibers are spared. The CMT2 patients were unsta- ble, and they had a decreased CV of group II fibers, as is the case with diabetics.

Thus, there is generally an inverse relationship between SA and the CV of group II

fibers across all patients. In CMT2 and diabetics patients, increased sway occurred

33 even with EO, suggesting group II inputs may be required for full use of vision in balance. Surprisingly, in the dynamic task, performance was only slightly worse in

CMT1A, CMT2, and diabetics. The authors suggest possible reasons for this: “1) the sinusoidal perturbation provides larger sensory input than when standing qui- etly, and perhaps supplementing the missing lower limb sensitivity; (2) during the dynamic platform task, compared to quiet stance, vestibular or cutaneous inputs are engaged in controlling balance and their effects add to the residual proprioception;

(3) since after the first few cycles the platform perturbation becomes predictable, pa- tients learn to exploit anticipatory postural strategies.” These results “challenge the notion that leg muscle Ia input is the main feedback for stance control,” but because the head lagged the platform movement in the EO case for patients, proprioception may contribute to the production of anticipatory actions.

3.1.6 Summary

The soles of the feet contain slow adapting and fast adapting type I and type

II mechanoreceptors, and the “wide dispersal of receptors throughout the foot sole would ensure that skin receptors would be able to code for contact pressures, and hence the position of the foot with the ground” [25].

Several researchers have found a relationship between decreased sensitivity of the foot sole and increases in instability under certain conditions. Links were found between decreased toe sensitivity and increased body sway during quiet standing, especially with eyes closed and standing on compliant surfaces [26], leading to the idea that those with less toe sensitivity might have “used great toe pressure in order to intensify sensory input from the great toe as well as to maintain balance” [27].

34 Similarly, higher 1- and 2-point static touch thresholds on various areas of the foot sole in neuropathic patients were correlated to increased sway area when patients stood with their eyes closed [28]. [29] found that sway area increased when the eyes were closed and during more extreme forward- or backward-leaning postures, but that there seems to be an “economy zone” around the normal standing position in which inclination increases but sway area does not increase significantly. At more extreme forward inclinations, the flexor digitorum brevis seems to have a larger effect on center of foot pressure than the soleus, implying a significant mechanical and/or sensory role for the toes in this case.

Forced decreases in sensitivity through use of anesthesia have also resulted in changes in balance and stability. For instance, anesthesia applied to the forefoot was found to increase mean COP velocity and anteroposterior COP frequency during unipedal stance with eyes open, and it increased mediolateral COP velocity and

COP area during bipedal stance with eyes closed. And anesthesia applied across the sole of the foot increased COP velocity in general and COP trajectories became less persistent over short time intervals [30]. Hypothermic anesthesia applied to the feet caused initial anteroposterior and lateral torque responses to calf stimulation to become larger, though responses reached normal levels after about 50–100 seconds, and previous experience with the calf stimulation seemed to improve postural control even under anesthesia [31].

Other researchers have found a relationship between induced sensation in the foot soles and increased body sway. For instance, low-amplitude, high-frequency vibra- tions under the foot designed to stimulate cutaneous receptors induced involuntary

35 body sway during quiet standing that was opposite the area of stimulation, e.g. stim- ulation of the left heel resulted in sway that was forward and to the right. Also, stimulation appeared to be additive, e.g. stimulation to both the heels resulted in larger forward sway [32]. A similar study found that vibrations applied at lower fre- quencies produced less sway, and those at higher frequencies produced more. Also, removing such stimulation from one area of the foot led to body sway corresponding to stimulation in the remaining areas of the foot. Stimulation applied to the forefeet and heels that varied sinusoidally and out of phase between the two regions produced sinusoidal sway with a mean phase of 52 ± 30◦ [33]. Another study that additionally stimulated proprioceptors in the ankle found that the modality that was stimulated at the higher frequency dominated, and that proprioception was more effective at higher frequencies of stimulation and cutaneous afferents at lower frequencies. The result of these two simultaneous stimulations seemed to be a summation of the individual ef- fects, with regulation of small amplitude body sways predominantly by tactile inputs and regulation of large body sways by ankle muscle proprioception [34].

Results from platform rotations have been somewhat conflicting. In [35], subjects experienced a “toes-up” platform rotation while standing with their eyes closed and with ankles either immobilized or free to move. Results showed that short latency postural responses were mostly likely triggered by the muscle spindles and were not affected by joint or cutaneous mechanoreceptors; medium latency responses appeared to be mediated by pressures sensed by the mechanoreceptors with a possible contri- bution from the joint receptors; and long latency responses might be primarily due

36 to joint receptors, with modulation from pressures sensed by cutaneous mechanore- ceptors. Modulation from sensed plantar pressures seemed to correlate best with the maximum positive slope of the differential pressure (forefoot pressure - heel pressure).

A sinusoidal platform disturbance was applied to subjects and vestibular loss pa- tients in an eyes closed case in [36]. A pin matrix was used to indent the skin under the forefeet at different frequencies. During quiet standing, these indentations resulted in small COP excursions of about 0.5 cm or less that were essentially counter-phase to the indentation applications, with larger phases for vestibular loss patients. When these indentations were applied during platform rotations, responses were not signif- icantly different from those observed when no indentations were applied. However, the sway induced by this method of stimulation is small (0.5 cm) compared to that induced in [32] (2-4 cm). In addition, results obtained from diabetic and CMT patients in [38] found that, for sinusoidal perturbations of the support sur- , only the conduction velocity of group II fibers in an eyes closed case seemed to negatively affect sway area. The authors conclude that proprioceptive input did not have as large an effect as anticipated, and that the relatively good performance of patients may be due to anticipatory strategies, the platform disturbance producing larger sensory input, or higher levels of vestibular and tactile involvement. This seems to contradict the importance of proprioception expressed in [36].

3.2 Role of the Toes in Balance

Though many studies have examine the role of the feet during balance, fewer have isolated the role of the toes. There is some evidence that the toes contribute through

37 cutaneous afferent signals. From a neurological perspective, foot sole mechanorecep- tors are widely spread throughout the foot, including the toes [25]. From a functional perspective, elderly subjects with lower toe sensitivity have higher sway during quiet standing, especially with eyes closed and on compliant surfaces [26]; they also have higher maximum great toe pressure on non-compliant surfaces, leading to the possi- bility that the elderly use “great toe pressure in order to intensify sensory input from the great toe as well as to maintain balance.” And during extreme forward-leaning postures, there is correlation between average center of foot pressure and flexor dig- itorum brevis activity, which could imply a sensory and/or mechanical role for the toes [29].

It is also likely that the toes play a mechanical role in balance. In addition to the previous sensory/mechanical evidence, it has been shown [39] that patients with diabetes and transmetatarsal amputations (TMAs) wearing standard pairs of shoes with toe fillers are not able to reach as far during Functional Reach Tests as age- matched control subjects: an additional 19.1 ± 8.6 cm versus 31.5 ± 9.1 cm “beyond an arm’s length.” This may be due to shortened foot length as well as loss of foot and toe muscles and proprioceptors. Furthermore, a study of fifty-seven young women

[40] found that toe flexor/abductor power mildly correlated with sway locus length per environmental area (representing fine postural control) during one-leg standing with the eyes open and closed as well as environmental area (representing degree of equilibrium impairment) during one-leg standing with the eyes open.

38 3.3 Role of the Vestibular System in Balance

An overview of the vestibular system was given in Section 2.3.2. In terms of the its role in balance, it is well known that the vestibular system is involved in control of head position, eye position, and overall body movement [41]. When the neck muscles are voluntarily made very stiff so that the head does not move relative to the torso, the head, neck, and torso act as a single rigid body. In this situation, it is simple for the CNS to estimate the position of the torso based on the position of the head.

Thus, the torso can be controlled using the same strategy used to control the head.

If the hip and knees are also locked, the ensuing movement is referred to as the ankle strategy. In this case, the head control strategy may similarly be applied to the whole body, which acts as an inverted pendulum [42].

The vestibular system has been shown to directly affect postural responses in several ways, such as through a tonic effect on the muscles. During platform dis- turbances, EMG recordings show that short latency (∼40-120 ms) reflex responses of the quadriceps; medium latency (∼120-220 ms) balance-correcting responses of the quadriceps, tibialis anterior, paraspinal, and trapezius; and long latency (∼240-

500 ms) stabilization responses of the tibialis anterior, paraspinal, and trapezius are altered by vestibular loss [43].

39 Furthermore, galvanic vestibular stimulation (GVS), which is used to artificially stimulate the vestibular apparatus, is capable of inducing whole body sway [5]. The induced sway is greater in cases in which proprioceptive and visual information are not available, leading to the hypothesis that vestibular, visual, and somatosensory inputs are weighted with by adjustable gains [44]. GVS may be useful for resolving cue conflicts, such as those experienced during or visually enhanced control of remote automata [45] or virtual reality simulations.

40 CHAPTER 4

THE MODELS

This chapter describes the two computational humanoid biped models that will be used in this dissertation. The first is a five-link, rigid body model that is restricted to lie in the sagittal plane, i.e. it can move forward and backward but not side to side. It uses one link to represent the upper body and legs and four links to represent the foot. This model is particularly useful for the studying the role of the foot and toes in balance. The second model is a three-link rigid body inverted pendulum, also restricted to lie in the sagittal plane. This model is used to study the effects of the vestibular system in balance. It is assumed to stand on a support surface that is subject to a sudden horizontal translational disturbance. The following will provide an overview of the models and their parameters as well as their musculature.

Nomenclature is listed in Appendix A.

4.1 Five-Link Sagittal Biped Model with a Foot

In this section, we describe the five-link sagittal biped model. We begin by giving an overview of its structure. Then, we show how model parameters are derived based on anthropometric data. Finally, we discuss muscle models used to actuate the biped model.

41 Figure 4.1: The five-link sagittal biped is composed of five rigid links. The term ki, for i = 1 ... 5 is the distance from one end of link i to the center of mass of link i. This is shown for all links; however, it is labelled for links 1 and 5 only. The term li is the length of link i. xa and ya are the coordinates of the ankle joint. Angles θi for each link are measured clockwise from the vertical. Fh is the horizontal ground reaction force at the heel, and Gh, Gm, and Gt are the vertical ground reaction forces at the heel, metatarsals, and toe, respectively.

4.1.1 Model Overview

The structure of the five-link sagittal biped model is shown in Figure 4.1. This model can be understood by picturing the profile of a person standing upright, head 42 fixed relative to the torso and arms held at the sides parallel to the torso. Bilateral symmetry is assumed, with link 1 representing the head, arms, and torso (or HAT) and legs; link 2 the calcaneus of both feet; link 3 the metatarsals, navicular, cuneiforms, and cuboid of both feet; link 4 the proximal and intermediate phalanges of all the toes; and link 5 the distal phalanges of all the toes.

Figure 4.2: Medial (A) and lateral (B) views of a human foot skeleton. Features as numbered are 1-calcaneus; 2-sustentaculum tali; 3-talus; 4-head of talus; 5-navicular; 6-tuberosity of navicular; 7-medial cuneiform; 8-intermediate cuneiform; 9-base, 10- body, and 11-head of the first metatarsal; 12-base, 13-body, and 14-head of the prox- imal phalanx of the great toe; 15-base, 16-body, and 17-head of the distal phalanx of the great toe; 18-sesamoid bone; 19-lateral cuneiform; 20-cuboid; 21-base, 22- tuberosity, 23-body, and 24-head of the fifth metatarsal; and 25-tarsal sinus. Taken from [2].

43 The point at which links 1, 2, and 3 connect can be considered roughly the talus or ankle joint. The foot contacts the ground at the bottom of the calcaneus or heel, the head of the metatarsals or forefoot, and the head of the distal phalanges of the toes. The heel is chosen as the point at which horizontal friction forces affect the model, as this is the major weight bearing point when θ1 = 0 rad [46]. Figure 4.2 shows the bones of a human foot for comparison.

4.1.2 Model Parameters

Parameters for this model are derived from relatively recent anthropometric stud- ies. These studies use imaging procedures such as computed tomography and mag- netic resonance imaging to accurately measure the lengths, moments of inertia, and centers of mass of major body segments. Table 4.1 shows average results for 100 men as reported in one recent study [47], which makes adjustments to values derived by

Zatsiorsky, Seluyanov, and Chugunova (see [48], [49], and [50]). The average mass of the subjects is 73 kg.

These measurements can be used to derive similar measures for the arms (upper arm, forearm, and hand), then the upper body (torso with the head held upright and arms held parallel to the torso), and then the HAT and legs together. Resulting estimates are listed in Table 4.2. Also used in these derivations is the approximate distance from the vertex of the head to the shoulder joint, about 31.43 cm (see distance from vertex to suprasternale in [47]).

Measurements for the foot can be found in [1]. Masses, lengths, and moments of inertia of the heel and forefoot are given explicitly. Approximations for the length and mass of each segment of the toe can be obtained by halving the values given

44 2 Segment Length (cm) Mass (%) CM (%) Iz (kg·m ) End Points Head 20.33 6.94 59.76 0.0296 vertex to cerv Trunk 60.33 43.46 51.38 1.0812 Cerv to MDH Upper arm 28.17 2.71 57.72 0.0114 SJC to EJC Forearm 26.89 1.62 45.74 0.0060 EJC to WJC Hand 18.79 0.61 36.24 0.0009 WJC to DAC3 Thigh 42.22 14.16 40.95 0.1994 HJC to KJC Calf 44.03 4.33 43.95 0.0371 KJC to AJC Foot 25.81 1.37 44.15 0.0040 heel to TTIP

Table 4.1: Lengths, masses, centers of mass, moments of inertia, and end points for major segments of the human body. Mass is given as a percent of total body weight. Center of mass is listed as a percent of the length, measured from the first listed endpoint. Abbreviations are: Cerv for cervicale, MDH for midhip, SJC for shoulder joint (center), EJC for elbow joint, WJC for wrist joint, DAC3 for tip of the 3rd digit (3rd dactylion), HJC for hip joint, KJC for knee joint, AJC for ankle joint, TTIP for tip of the longest toe.

2 Segment Length (cm) Mass (%) CM (%) Iz (kg·m ) End points Arm 73.85 4.94 40.39 0.1138 SJC to DAC3 Upper Body 80.66 60.28 60.06 2.1506 vertex to MDH HAT and legs 166.91 97.26 71.21 9.5711 vertex to AJC

Table 4.2: Lengths, masses, centers of mass, moments of inertia, and endpoints for larger segments of the human body derived using values from Table 4.1.

45 2 Segment Length (cm) Mass (%) CM (%) Iz (kg·m ) Distal toe 3.15 0.0012 50 0.0000078 Proximal toe 3.15 0.0012 50 0.0000078 Metatarsals, etc. 15.5 0.008 50 0.00082 Calcaneus 9.60 0.003 50 0.00015

Table 4.3: Lengths, masses, centers of mass, moments of inertia, and endpoints for segments of the foot. Values were derived using information from [1]. Note that values are for the segments of one foot only.

Link I (kg·m2) m (kg) l (m) k (m) 1 9.5711 71.0 1.669 0.4805 2 0.1 0.44 0.096 0.048 3 0.5 1.16 0.155 0.0775 4 0.05 0.17 0.0315 0.0158 5 0.05 0.17 0.0315 0.0158

Table 4.4: Parameters for the five-link biped model. I is moment of inertia in kg·m2 about the center of gravity, m is the mass in kg, l is the length in meters, and k is the distance to the center of mass from one joint of the segment in meters.

for the entire toe. Centers of mass are assumed to lie at the center of each segment.

Moments of inertia for each toe segment can be estimated by assuming each segment makes an equal contribution to the total moment of inertia of the toe. Final estimates for foot segment masses, lengths, centers of mass, and moments of inertia are listed in Table 4.3.

Finally, values from Tables 4.1, 4.2, and 4.3 can be used to select parameters for the model. These parameters are listed in Table 4.4. For each link, five parameters are specified. Parameter I refers to the moment of inertia in kg·m2 about the center of gravity of the link. Parameter m refers to the mass of the link in kg. Parameter l

46 refers to the length of the link in m. Parameter k refers to the distance to the center of mass from one end of the link in m, designated by a line in Figure 4.1. Note that for segments of the foot, moments of inertia were chosen to be larger than those given in Table 4.3, as the very small values given in [1] resulted in some ill-conditioned matrices during simulation.

4.1.3 Muscles, Joints, and Ligaments

Figure 4.3: Muscles used in the five-link sagittal biped model. Abbreviations are as follows: sl – soleus, ta – tibials anterior, fhl – flexor hallucis longus, ehl – extensor hallucis longus, fhb – flexor hallucis brevis, ehb – extensor hallucis brevis, aj – ankle joint, hj – hallux joint, and pa – plantar aponeurosis.

47 Actuation for the model is provided by pairs of muscles. Passive effects of joints and ligaments should also be considered. An overview of the muscles, joints, and ligaments of the foot can be found in [2]. Here, we will consider three pairs of muscles, two joints, and one ligament, as shown in Figure 4.3. The muscle pairs include the soleus and tibialis anterior, whose primary purpose is to control movement of the legs and upper body; the flexor hallucis longus and the extensor hallucis longus, which allow flexion and extension of the distal segment of the big toe; and the flexor hallucis brevis and extensor hallucis brevis, which allow flexion and extension of the proximal segment of the big toe. The joints considered here are the ankle joint or talus and the hallux joint or toe joint. The ligament used in this model is the plantar aponeurosis or plantar fascia, which runs under the arch of the foot. It should be noted that each muscle and ligament can only provide torque in one direction with respect to each link, i.e. each muscle or ligament can only pull, not push. The two joints, however, can provide torque to a single link in either direction. Equations for torques due to the muscles, joints, and ligament will be discussed in the next chapter and also in

Appendix B.3.

4.2 Three-Link Sagittal Biped Model

In this section, we describe the three-link sagittal biped model used to study the effects of the vestibular system on balance. This biped is assumed to stand on a support surface that undergoes a sudden horizontal translational disturbance. We begin by giving an overview of the structure of the model and list model parameters.

Then, we discuss its musculature.

48 Figure 4.4: The three link sagittal biped is composed of three rigid links and a massless foot. The term ki for i = 1, 2, or 3 is the distance from the bottom of link i to the center of mass of link i. The term li is the length of link i. Angles are measured clockwise from the vertical with θi ∈ (−π, π).

4.2.1 Model Overview

This model, which is similar to those used in [51] and [52], is composed of three rigid links that are restricted to lie in the sagittal plane, as shown in Figure 4.4.

Bilateral symmetry is assumed, with link 1 representing both of the the calves, link

2 both of the thighs, and link 3 the head, both of the arms, and the torso. The vestibular system is assumed to be located at the top of link 3.

49 Link I (kg·m2) m (kg) l (m) k (m) 1 0.17 5.05 0.7 0.21 2 0.5 18.67 0.412 0.26 3 2.1 41.87 0.619 0.24

Table 4.5: Parameters for the three-link biped model. I is moment of inertia in kg·m2 about the center of gravity, m is the mass in kg, l is the length in meters, and k is the distance to the center of mass from one joint of the segment in meters.

The model uses a massless link, about 20 cm long [53], to represent the feet.

This link is used to estimate horizontal excursions of the center of pressure. The model is developed such that the support surface can experience a sudden horizontal translational disturbance. Also, the model prohibits hyperextension at the knees and hips, as discussed in Appendix C.4.

For each link, four parameters based roughly on those of an adult human are specified. These parameters are listed in Table 4.5. Parameter I refers to the moment of inertia about the center of gravity of the link, m refers its mass, l refers to its length, and k refers to the distance of its center of mass measured from the lower joint. The gravity constant g is 9.81 m/s2.

4.2.2 Muscles

The three-link biped model is equipped with a minimal set of nine muscles, as shown in Figure 4.5. Note that the soleus, tibialis anterior, and gastrocnemius muscles connect to the feet. The level of muscle activation is represented in a vector of

50 Figure 4.5: The three-link sagittal biped with nine muscles: ga – gastronemius, sl – soleus, ta – tibialis anterior, sq and sh – short heads of the quadriceps and hamstrings, q and h – long heads of the quadriceps and hamstrings as two-joint muscles, p – paraspinal, and ab – abdominal. The gastrocnemius, soleus, and tibialis anterior muscles connect to a massless foot, as shown in Figure 4.4

proportional gains α and derivative gains β. Here,

α = [500, 300, 300, 300, 300] N · m,

β = 0.25α N · m · s.

We refer to this set of numbers as the nominal activity level. This activity level represents the amount of input provided to the muscles by proprioception, learned by experience. α1 and β1 correspond to the soleus/tibialis anterior, α2 and β2 to the

51 short heads of the quadriceps/hamstrings, α3 and β3 to the the abdominal/paraspinal pair, α4 and β4 to the long heads quadriceps/hamstrings, and α5 and β5 to the gastrocnemius, which is assumed to have an antagonist muscle group.

The values for α were chosen based on findings by Peterka in [54] – that muscle activity levels are roughly one-third larger than those necessary to overcome gravity.

For a single link model with angle ω clockwise from the vertical, moment of inertia I, an approximate weight of 65 kg, a center of mass about 1 m from the ground, gravity estimated at 10 m/s2, and with a feedback term −αωˆ − βˆω˙ , the equation of motion

is that of an inverted pendulum,

Iω¨ = 650 sin(ω) − αωˆ − βˆω,˙

or with sin(ω) ≈ ω for ω ≈ 0 rad,

Iω¨ = 650ω − αωˆ − βˆω.˙

Thenα ˆ = 4/3(650) = 800 is 1/3 larger than is needed to counteract gravity, and with βˆ arbitrarily chosen as βˆ = 0.25ˆα, solutions for ω(t) tend towards 0 rad for

initial conditions not too far from ω(0) = 0 rad.α ˆ corresponds to muscle torque

at the ankle, which is α1 + α5. We choose α1 = 500 and α5 = 300 for a stronger

soleus/tibialis anterior pair, and choose α2 = α3 = α4 = 300 for comparable muscle

strength.

The muscles provide input to the system in the form

τ = −Aq − Bq,˙

52 where   α1 + α2 + α4 −α2, −α4 A =  −α2 α2 + α3 + α5 −α3  , −α4 −α3 α3 + α4

T B has a similar form, and q = [θ1, θ2, θ3] . In this case, since β = 0.25α, B = 0.25A.

This input is derived in Appendix C.2 and [51] and stabilizes the model in the region of q = [0, 0, 0]T ,q ˙ = [0, 0, 0]T .

53 CHAPTER 5

EXPLORING THE ROLE OF THE FOOT SOLE AND TOES IN BALANCE

In this chapter, we will explore how movement of the body affects the foot and vice versa. This will be done using the five-link model described in Section 4.1 with nomenclature listed in Appendix A. First, we will discuss control of the system through the muscles. Then, we will explore the limits of static balance in a healthy subject with the toe muscles at full strength. That is, we will determine the range of angles at which the subject can hold the bulk of the body without removing the foot from the ground. Next, we will explore the limits of static balance when muscle toe strength is greatly diminished. Finally, we will examine how forces are redistributed and the foot arch falls or lengthens as forward leaning occurs.

5.1 Control of the Five-Link Model Through the Muscles

In this section, we discuss control of the five-link model shown in Figure 4.1 of

Section 4.1. Stability of the foot and tracking of the body arise though a combination of inputs provided by the muscles, joints, and plantar aponeurosis as shown in Figure

4.3. The muscles actively control movement of the main body and the angle of the toes. They also affect the distribution of ground reaction forces. The ankle joint, toe

54 joint, and plantar aponeurosis help to passively stabilize the foot. The equations of

motion for the model are derived in Appendix B.1 and take the form

∂C(q)T J(q)¨q = −B(q, q˙)q ˙ − G(q) + τ + F , (5.1) m ∂q gr

T where q = [xa, ya, θ1, θ2, θ3, θ4, θ5] , J(q) is a 7x7 positive definite inertia matrix,

B(q, q˙) is a 7x7 matrix of centripetal and Coriolis terms, G(q) is a 7x1 matrix of

gravity terms, τm are inputs from the muscles, joints, and plantar aponeurosis, and

∂C(q)T T ∂q Fgr where Fgr = [Fh,Gh,Gm,Gt] is due to ground reaction forces.

The term τm takes the form   τsl   0 0 0 0 0 0 0 0 0  τta     0 0 0 0 0 0 0 0 0   τaj       −1 1 0 0 0 0 0 1 −1   τpa      τm =  1 0 −1 −1 0 0 0 0 0   τehb  , (5.2)      0 −1 1 1 1 −1 0 0 0   τfhb       0 0 0 0 −1 1 −1 0 0   τhj    0 0 0 0 0 0 1 −1 1  τehl  τfhl as derived in Appendix B.3. The first two rows are 0 since movement of the ankle joint coordinates xa and ya are not directly affected by these torques, and each column has a 1 and -1 since each muscle, joint, or ligament connects two links and provides equal but opposite torques to each. τsl is due to the soleus, τta to the tibialis anterior,

τaj to the ankle joint or talus, τpa to the plantar aponeurosis, τehb to the extensor

hallucis brevis, τfhb to the flexor hallucis brevis, τhj to the hallux or toe joint, τehl to

the extensor hallucis longus, and τfhl to the flexor hallucis longus.

As noted in Section 4.1.3, the muscles and plantar aponeurosis can only provide

torque in one direction with respect to each link, i.e. they can only pull. The joints,

however, can provide torque in both directions. In order to ensure the muscles only

55 provide torque in one direction, let us define the function

 x if x > 0 U[x] = 0 if x ≤ 0.

Now we can define the muscle inputs used for standing when the foot remains on the ground. Using circular springs to model all but the plantar aponeurosis, which is modeled by a linear spring, the torques provided by each muscle, joint, and ligament will be chosen as

τsl = U[−τ1 − τfhl]

τta = U[τ1 + τfhl]

¯ ˙ ˙ τaj = αaj(θ2 − θ3 − θaj) + βaj(θ2 − θ3)

τpa = U[αpal2 cos(θ2)[l3 sin(θ3) − l2 sin(θ2)] +

˙ 2 2 ˙ βpa(l2l3 cos(θ2) cos(θ3)θ3 − l2 cos (θ2)θ2)]

τehb = 0

¯ τfhb = min(U[αfhb(θ1 − θ1t)], 20)

¯ ˙ ˙ τhj = αhj(π + θ4 − θ5 − θhj) + βhj(θ4 − θ5)

τehl = 0

¯ ˙ τfhl = U[−αfhl(θ5 − θ5) − βfhlθ5] (5.3) where ¯ ˙ ¯˙ ˆ ¯¨ τ1 = −gk1m1 sin(θ1) − α1(θ1 − θ1) − β1(θ1 − θ1) + I1θ1. (5.4)

¯ The term τ1 is a function of θ1 and a desired tracking trajectory θ1. This term is ˆ based on the standard linearization of an inverted pendulum, where I1 is the moment of inertia about the bottom of link 1, −gk1m1 sin(θ1) cancels the effects of gravity,

56 Parameter Value αfhl 1000 N·m βfhl 100 N·m/s ¯ θ5 1.97 rad αhj 50 N·m βhj 1 N·m/s ¯ θhj 1.79 rad αfhb 57.17 N·m ¯ θ1t 0.081 rad αpa 1000 N·m βpa 100 N·m/s αaj 500 N·m βaj 10 N·m/s ¯ θaj 2.34 rad α1 358.62 N·m β1 194.73 N·m/s ˆ 2 I1 25.96 kg·m

Table 5.1: Gains and set points used to calculate muscle torques for the five-link biped model during normal standing.

¯ ˙ ¯˙ and the term −α1(θ1 −θ1)−β1(θ1 −θ1) provides feedback. τ1 enters into the equations for τsl and τta, as these are the two muscles connected to link 1. The terms τehb and

τehl are chosen to be 0, as their use does not provide any significant benefit during ¯ ¯ normal quiet standing [29]. θaj and θhj are desired equilibria or set points for the angles under the ankle joint and hallux joint, i.e. the angle between links 2 and 3 and the angle between links 4 and 5. The term τfhb is a function of θ1, and this muscle ¯ becomes active when θ1 > θ1t. This corresponds to observations in [29], which found that the extensor brevis hallucis is more active during extreme forward leaning. It will be shown in subsequent sections that this effectively lengthens the base of support of the biped model, a requirement for balance when θ1 becomes large.

57 Parameters values are listed in Table 5.1. αaj, βaj, αpa, and βpa, are roughly based on parameters from [46]. αhj and βhj were chosen to be smaller but similar in proportion to αaj and βaj. α1 and β1 were chosen to provide tracking according to ˆ ¯ ¯ findings in [54], and I1 follows from the model parameters listed in Table 4.4. θ5, θaj, ¯ and θhj are chosen to reflect realistic angles for the bones of the foot [1], [2]. αfhl and

βfhl were chosen to be the same as αpa and βpa. ¯ θ1t and αfhb were chosen through experimentation. In order to choose these values, we varied θ1 between 0 rad to 0.4 rad and τfhb between 0 and 20 N · m. Simulations were run under these conditions until the system approximately reached equilibrium.

For each angle, the maximum and minimum values of αfhb that resulted in the foot remaining on the ground, i.e. Gh > 0, Gm > 0, and Gt > 0, were recorded.

Minimum and maximum values for τfhb are shown using solid lines in Figure 5.1.

The final formula for τfhb, shown in part by the dashed line, is equivalent to a line that passes through the average x-intercept and average y-intercept of the minimum and maximum values, with a maximum imposed value of 20 N · m. Thus,

¯ τfhb = min(U[αfhb(θ1 − θ1t)], 20)

as given in (5.4). The role of τfhb will be exploring in the proceeding sections.

5.2 The Limits of Static Balance in a Healthy Subject

In this section, we explore the role of the toes in static balance in a healthy subject. Muscle activation levels and inputs from the joints and plantar aponeurosis are as given in the previous section.

Figure 5.2 shows vertical forces under the heel, metatarsals, and toe when the ¯ model is at equilibrium with θ1 = θ1 varying from 0 to 0.45 rad. Figure 5.3 shows

58 20

18

16

14

12 m) ⋅ 10 (N fhb τ 8

6

4

2

0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Angle of link 1 (rad)

Figure 5.1: Minimum and maximum values of τfhb versus angle θ1 that allow the foot to remain on the ground are shown by the solid lines. The dashed line represents the chosen equation for τfhb.

the same forces for θ1 varing from -0.15 to 0 rad. What can be observed from these two plots is that as the biped model leans forward, i.e. as θ1 increases, the force on the heel decreases. The force under the toe remains very small until the model leans ¯ significantly forward, i.e. θ1 > θ1t = 0.081 rad, at which point the force under the toe increases. The force under the metatarsals increases at first, until the flexor hallucis brevis becomes active around the same point. From these plots, we see that θ1 can range from about -0.142 to 0.4160 rad without the foot lifting off the ground, i.e. without Gh, Gm, or Gt becoming negative.

59 Figure 5.4 shows the angle of the foot arch, i.e. the angle between links 2 and 3, as θ1 varies between -0.15 and 0.45 rad. The foot arch angle is smallest, i.e the arch is highest, when θ1 is at 0 rad. The foot arch angle grows, i.e. begins to fall, as θ1 either increases or decreases. This is due to muscle torques from the tibialis anterior and soleus, as shown in Figure 5.5. These torques stabilize link 1 during forward and backward leaning, but also put strain on the foot arch, causing it to flatten. This is similar to observations discussed in [55], which reports that the arch lengthens or falls as the calf muscles become active before the push-off phase of gait.

700 G h G 600 m G t 500

400

300

200 Forces under the foot (N) 100

0

−100 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Angle of link 1 (rad)

Figure 5.2: Vertical forces under the heel, metatarsals, and toe for static poses with θ1 varying from 0 rad to 0.45 rad. All inputs are as listed in Section 5.1.

60 800 G h G 700 m G t 600

500

400

300

200 Forces under the foot (N)

100

0

−100 −0.16 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0 Angle of link 1 (rad)

Figure 5.3: Vertical forces under the heel, metatarsals, and toe for static poses with θ1 varying from -0.15 rad to 0.0 rad. All inputs are as listed in Section 5.1.

61 2.02

2

1.98

1.96

1.94

1.92

1.9 Foot arch angle (rad) 1.88

1.86

1.84

1.82 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 Angle of link 1 (rad)

Figure 5.4: Static foot arch angle as θ1 varies from -0.15 to 0.45 rad. All inputs are as listed in Section 5.1.

62 140 τ sl τ 120 ta

100 m) ⋅ 80

60 Muscle torque (N 40

20

0 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 Angle of link 1 (rad)

Figure 5.5: Tibialis anterior and soleus muscle torques as θ1 varies from -0.15 to 0.45 rad. All inputs are as listed in Section 5.1.

63 5.3 The Limits of Static Balance in a Subject with Dimin- ished Toe Muscle Strength

In this section, we explore what happens to the limits of static balance when

muscle toe strength is lessened. In this case, we will set τfhb = 0. All other muscle

activation levels and inputs from the joints and plantar aponeurosis are as given

Section 5.1.

1000 G h G 800 m G t

600

400

200

Forces under the foot (N) 0

−200

−400 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Angle of link 1 (rad)

Figure 5.6: Vertical forces under the heel, metatarsals, and toe for static poses with θ1 varying from 0 rad to 0.45 rad and τfhb = 0. All other inputs are as listed in Section 5.1.

64 800 G h G 700 m G t 600

500

400

300

200 Forces under the foot (N)

100

0

−100 −0.16 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0 Angle of link 1 (rad)

Figure 5.7: Vertical forces under the heel, metatarsals, and toe for static poses with θ1 varying from -0.15 rad to 0.0 rad and τfhb = 0. All other inputs are as listed in Section 5.1.

65 2.05

2

1.95

1.9 Foot arch angle (rad)

1.85

1.8 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 Angle of link 1 (rad)

Figure 5.8: Static foot arch angle as θ1 varies from -0.15 to 0.45 rad and τfhb = 0. All other inputs are as listed in Section 5.1.

66 150 τ sl τ ta

100 m) ⋅

Muscle torque (N 50

0 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 Angle of link 1 (rad)

Figure 5.9: Tibialis anterior and soleus muscle torques as θ1 varies from -0.15 to 0.45 rad and τfhb = 0. All other inputs are as listed in Section 5.1.

67 Figure 5.6 shows vertical forces under the heel, metatarsals, and toe when the

model is in equilibrium with θ1 varying from 0 rad to 0.45 rad. Figure 5.7 shows the same data for θ1 varing from -0.15 rad to 0 rad. As in the previous section, we see that when the biped model leans forward, i.e. as θ1 increases, the force on the heel decreases. However, with τfhb = 0, the force under the toe remains very small throughout the entire range of θ1, and the force under the metatarsals strictly increases as θ1 increases. From the plots, we see that θ1 can only range from -0.142 to 0.301 rad without lifting the foot off the ground, i.e. without Gh, Gm, or Gt becoming negative. There has been very little change in the range of θ1 = -0.15 to 0 rad; however, with τfhb = 0, there is no mechanism to redistribute the force between the metatarsals and toes, and the heel begins to lift off the ground at 0.301 rad instead of 0.416 rad when τfhb is active.

Figure 5.8 shows the angle of the foot arch, i.e. the angle between links 2 and 3, as θ1 varies between -0.15 and 0.45 rad. The plot is similar to the one from before, but with a slightly higher arch angle. Figure 5.9 shows muscle torques due to the tibialis anterior and soleus muscles. Again, the plot is similar to the one from before, but with slightly larger torques.

5.4 Force Distribution and Foot Arch Angle during Forward Leaning

In addition to studying static poses, the five-link model can also be used to study the shape of the foot and distribution of forces under the foot during movement.

Here, we will use a forward leaning movement, in which link 1 starts at 0 rad and moves to 0.3 rad. Figure 5.10 shows the actual trajectory of link 1, θ1, resulting from ¯ the desired trajectory θ1, as well as the foot arch angle between links 2 and 3 and the

68 angle between toe links 4 and 5. This figure shows that the model successfully tracks ¯ θ1; the foot arch lowers as the model leans forward, i.e. the angle between links 2 and

3 grows; and the same happens to the angle between links 4 and 5 as the toes clench.

0.4

0.2 θ¯1 (rad) 1

θ 0 θ1 −0.2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 2

1.9

Foot arch (rad) 1.8 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

2.37

2.36

2.35 Toe arch (rad) 2.34 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (s)

Figure 5.10: Angle of link 1, foot arch angle between links 2 and 3, and toe arch angle between links 4 and 5 during a movement of link 1 from 0 rad to 0.3 rad. All inputs are as listed in Section 5.1.

Figure 5.11 shows the forces under the foot during this movement. The horizontal force under the heel, Fh, remains relatively small throughout the movement. The

2 vertical force under the heel, Gh, starts at about 500 N or about 3 the total weight of the model as reported in [46], and decreases as link 1 rotates forward. The vertical force under the metatarsals, Gm, stays around 200 N throughout the movement. And

69 the vertical force under the toe, Gt, starts at about 0 N and increases once θ1 > 0.081 rad.

50

(N) 0 h F −50 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 1000

(N) 500 h G 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 400

(N) 200 m G 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 400 (N)

t 200 G 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (s)

Figure 5.11: Forces under the foot during a movement of link 1 from 0 rad to 0.3 rad. All inputs are as listed in Section 5.1.

Figure 5.12 plots the estimated center of gravity (CoG) and center of pressure

(CoP) throughout the movement. The CoG is estimated as the position of the center

of mass of link 1 along the ground, as this link comprises more than 97% of the total

model mass. The CoP is the average location of pressure under the foot, based on a

weighted average of the vertical forces under the foot and their points of application.

This figure shows that the CoP drops behind the CoG as link 1 starts to move forward,

reflecting the fact there is a positive torque on link 1 at the start of the movement

70 that causes link 1 to accelerate. The dynamics of this are explained in [3]. Likewise, the CoP moves ahead of the CoG as link 1 slows at the end of the moment, reflecting the fact that there is a negative torque on link 1 at this time.

CoG CoP 0.2

0.15

0.1

0.05

Center of gravity (CoG) and pressure (CoP) (m) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (s)

Figure 5.12: Position of the center of gravity (CoG) and center of pressure (CoP) during a movement of link 1 from 0 to 0.3 rad. All inputs are as listed in Section 5.1.

5.5 Discussion

In this chapter, we have developed a basic control for the five-link model that allows it to stand upright, i.e. with link 1 near 0 rad. Muscles and the plantar aponeurosis provide torque in only one direction with respect to each link, i.e. they pull but do not push, and the joints provide torque in both directions. Parameters for muscle, ligament, and joint stiffnesses and elasticities were based on findings from

[46] and also through experimentation. Torques provided by the soleus and tibialis

71 anterior were designed to provide tracking control for link 1 that matches the transfer

function experimentally derived in [54].

2 When θ1 = 0 rad, Gh is equal to about 3 of total body weight as reported in [46],

1 Gm is about 3 body weight, and Gt is very small, as shown in Figure 5.2 and at the beginning of the simulation in Figure 5.2. With τfhb providing torque during static forward leaning postures according to the function shown in Figure 5.1, the base of support is large enough to allow the angle of link 1 to range from about -0.142 to

0.416 rad without the foot leaving the ground, i.e. without Gh, Gm, or Gt becoming negative. When τfhb is inactive, the base of support only allows the angle of link

1 to range from about -0.142 to 0.301 rad. The reduced range of forward leaning in the latter case is consistent with findings in [39], which found that patients with transmetatarsal amputations, i.e. no toes, were not able to reach as far forward as healthy subjects. It could also offer an alternative explanation for larger maximum toe pressures in the elderly during quiet standing as reported in [26]. In this paper, the authors suggest that elderly patients may be compensating for loss of sensitivity in the toe by increasing pressure under the toe. However, increasing toe pressure could be also be a preventative measure, to increase the base of support in case of a sudden disturbance that causes the body to rotate forward. Note that τfhb was not designed to be active during backward leaning, as it has no mechanical advantage in this case [29].

Arch angles in the static poses and during the forward leaning movement shown in Figures 5.4, 5.8, and 5.10 show that arch angle is smallest, i.e. the arch is highest, when link 1 is at 0 rad, and arch angle increases, i.e. the arch falls, as link 1 moves away from 0 rad in either direction. Though it is difficult to find information on arch

72 angle during quiet standing while the body leans forward or backward, information on the foot arch during gait is available. For instance, [55] reports that the arch lengthens or falls as the calf muscles become active before the push-off phase of gait.

Arch falling or lengthening here is also due to the muscles. During forward leaning, the soleus muscle must activate to counteract gravity, and during backward leaning, the tibialis anterior must activate for the same purpose [29]. In the former case, there will be a torque on link 2 that causes the arch to flatten or widen, and in the latter case, there will be a torque on link 3 that causes the arch to flatten or widen.

It should be noted that the overall range of leaning for this model may be unrealis- tic when compared to that of human subjects. Our model is capable of leaning about

8 degrees backward and 17 degrees forward, whereas human subjects are observed to lean about 4 degrees backward and 8 degrees forward [56]. It could be that some of our model parameters are incorrect, particularly those used for stiffness and viscosity of the ankle joint and plantar aponeurosis. It could also be that, unlike the model used here, human subjects do not feel comfortable using their maximum theoretical range of motion. For instance, [29] reports that in healthy human subjects, maximum center of pressure displacement in the sagittal plane is on average less than 60% of total foot length. If this is the case, then the model’s range of leaning may not be completely unrealistic.

73 CHAPTER 6

EXPLORING THE ROLE OF THE VESTIBULAR SYSTEM IN BALANCE

In this chapter, we will show how signals from the vestibular system can be used to help the biped maintain balance during a sudden backwards disturbance of the support surface when other mechanisms are diminished or unavailable. The following uses the three-link humanoid biped model described in Section 4.2. Nomenclature is listed in Appendix A. The equations of motion for this system take the standard form

J(q)¨q = −B(q, q˙)q ˙ − G(q) + τ, (6.1)

T where q = [θ1, θ2, θ3] and τ is due to inputs from the muscles. This equation is derived in Appendix C.1. Inputs due to the muscles are

τ = −Aq − Bq,˙ (6.2) with   α1 + α2 + α4 −α2, −α4 A =  −α2 α2 + α3 + α5 −α3  , −α4 −α3 α3 + α4 and

α = [500, 300, 300, 300, 300] N · m,

β = 0.25α N · m · s (6.3)

74 as discussed in Section 4.2.2.

The goal will be to bring the biped towards equilibrium after the disturbance, i.e.

q = [0, 0, 0]T rad and

q˙ = [0, 0, 0]T rad/s.

We assume the main control scheme used by the CNS is based on proprioception, i.e. joint angles and velocities are known; however, no visual feedback is available, i.e. the biped stands with eyes closed.

In each of the simulations, the initial state of the biped at time t = 0 is assumed to be a slightly stooped stance, with

q(0) = [0.1, −0.1, 0.1]T rad, which corresponds to a slight bending at the knees and hip. Due to a sudden accel- eration of the support surface of the form

2 ap δ(t) m/s ,

2 where δ(t) is the Delta dirac function and ap = −0.5 m/s is the magnitude of the acceleration (the negative sign indicates acceleration in the negative x-direction), the biped has an initial velocity at time t = 0 of

q˙(0) = [1.279, −0.0327, 0.0013]T rad/s

as derived in Appendix C.3. In our simulations, we refer to this particular disturbance

simply as “disturbance of the support surface.” We also assume that the neck muscles

are very stiff, i.e. the head does not rotate relative to the torso, so that the position

75 of the vestibular system can be estimated from the torso position and vice versa.

Furthermore, we assume the location of the vestibular system and the head are the same, and both are located at the top of link 1.

For each simulation, position of the center of gravity (CoG) and the center of pressure (CoP) along the x-axis are plotted versus time. At equilibrium, these should both be 0 m; thus, these values are excursions from the equilibrium, with positive values representing movement forward. Calculation of the CoP is derived in Appendix

C.5 and assumes the model has a massless link representing the feet as shown in Figure

4.4. The feet are connected to the soleus/tibialis and gastrocnemius/antagonist group muscle pairs and are assumed to be approximately 20 cm in length.

6.1 Recovery from Disturbance

In the first experiment, we assume the muscles are at their nominal activity level, i.e. α and β are as given in (6.3), and there is no vestibular input or modulation.

Excursions of the CoG and CoP due to disturbance of the support surface are plotted versus time in Figure 6.1. In this case, there is a forward excursion of the CoG and the CoP, with a maximum CoP excursion of 19.43 cm. The center of gravity moves forward to a maximum of 10.67 cm and back towards equilibrium.

Figure 6.2 shows phase plane plots of position versus velocity for four vital points

– the head, torso, hip, and center of gravity of the biped. These phase planes show that the motions of the four points are similar, somewhat scaled versions of each other. This implies that the CNS could be capable, through learning and adapta- tion, of estimating other parameters of interest for balance based on signals from the vestibular system. This observation also supports the hypothesis of McCollum,

76 0.2 CoG 0.15 CoP

0.1

0.05

CoG/CoP position (m) 0 0 0.5 1 1.5 2 2.5 3 3.5 Time (s)

Figure 6.1: CoG and CoP excursions of the biped versus time due to disturbance of the support surface. Muscular activities are nominal, and there is no vestibular input.

Horak, and Nashner [57] about parsimony or simplicity in neural calculations for postural movements.

77 0.6 0.6

0.4 0.4

0.2 0.2

0 0 Head CoG vel. (m/sec) Torso CoG vel. (m/sec) −0.2 −0.2 0 0.1 0.2 0 0.1 0.2 Torso CoG (m) Head CoG (m) 0.6 0.6

0.4 0.4

0.2 0.2

0 0 Hip CoG vel (m/sec)

Biped CoG vel. (m/sec) −0.2 −0.2 0 0.1 0.2 −0.05 0 0.05 0.1 Biped CoG (m) Hip CoG (m)

Figure 6.2: The phase plane plots for the head, torso, hip, and center of gravity of the biped after the support surface undergoes a sudden backwards acceleration. Each point starts with a non-zero position due to the slightly stooped initial stance and a non-zero velocity due to the support surface disturbance and then returns to equilibrium (0 m position, 0 m/s velocity).

78 6.2 Fall from Disturbance

In this experiment, muscular activity levels are reduced to 50% of the values of the previous experiment, i.e. α and β are reduced to half the nominal values given in

(6.3). The total input to the system (6.1) is thus

τ = −0.5Aq − 0.5Bq.˙

Figure 6.3 shows that the excursions of the CoG and CoP due to disturbance of the support surface become very large as the biped leans forward and is unable to return to equilibrium.

1 CoG 0.8 CoP 0.6

0.4

0.2 CoG/CoP position (m) 0 0 0.5 1 1.5 2 2.5 3 3.5 Time (s)

Figure 6.3: CoG and CoP excursions of the biped versus time due to disturbance of the support surface. Muscular activity levels are set to half the nominal value, and there is no vestibular input.

6.3 Addition of a Vestibular Term

In the third experiment, signals derived from horizontal motion of the head as provided by the vestibular system are added to the weakened muscle signals from the last experiment. As horizontal motion of the head indicates that the biped may

79 be experiencing forwards or backwards movement, these signals are useful in limiting

excursions of the CoG. If we designate horizontal position of the head as xh, one

control strategy is to create an additional term τf that mimics the original control

input by replacing θ1, θ2, and θ3 and their derivatives with xh andx ˙ h and multiplying

by a constant, e.g.

T T τf = −0.25A[xh, xh, xh] − 0.25B[x ˙ h, x˙ h, x˙ h] .

After multiplying terms, this reduces to

T T τf = −0.25[α1xh, α5xh, 0] − 0.25[β1x˙ h, β5x˙ h, 0] , which is equivalent to increasing muscular activities of the soleus/tibialis and gas- trocnemius/antagonist group pairs, i.e. total ankle torque. In order to simulate transmission delays of vestibular signals, τf (t) is delayed. Thus, the total input to the system (6.1) in this case is

τ(t) = −0.5Aq(t) − 0.5Bq˙(t) − τf (t − µ), where µ = 0.002 s has been chosen for a medium latency 200 ms delay.

Results due to disturbance of the support surface are shown in Figure 6.4. The biped now moves toward equilibrium, with a maximum CoG excursion of 16.97 cm and a maximum CoP excursion of 22.2 cm.

80 0.3 CoG CoP 0.2

0.1

0 CoG/CoP position (m) −0.1 0 1 2 3 4 5 Time (sec)

Figure 6.4: CoG and CoP excursions of the biped versus time due to disturbance of the support surface. Muscular activity levels are half the nominal value, and there is an additional vestibular term.

81 6.4 Vestibular Modulation of Muscular Forces

In the following experiments, neural inputs to the muscular system are amplified or

modulated by signals from the vestibular system. First, let us assume that muscular

activity levels are diminished but closer to their nominal values, in this case, 80%. In

other words, input to the system (6.1) is

τ(t) = −0.8Aq − 0.8Bq.˙

Excursions of the CoG and CoP due to disturbance of the support surface is shown in Figure 6.5. The maximum CoG excursion is 15.11 cm and the maximum CoP excursion is 16.83 cm.

0.2 CoG 0.15 CoP

0.1

0.05 CoG/CoP position (m) 0 0 1 2 3 4 5 6 7 Time (s)

Figure 6.5: CoG and CoP excursions of the biped versus time due to disturbance of the support surface. Muscular activity levels are 80% of the nominal value, and there is no modulation by the vestibular system.

Now, let the muscular activities be modulated by a term based on horizontal head

position and velocity, as supplied by the vestibular system. Again, let horizontal

position of the head be xh. Let us define a total head disturbance term d as

d = |xh + 0.05x ˙ h|,

82 where | · | indicates magnitude. Let mf be a modulation factor or gain, and assume

that due to transmission delays, d(t) is delayed by µ seconds. Let total amplification of muscular activities be mf d(t − µ). When muscular activity levels are at 80% of their nominal values but are also modulated the vestibular system, total input to the system (6.1) is

τ(t) = (1 + mf d(t − µ))(−0.8Aq(t) − 0.8Bq˙(t)), (6.4)

where µ = 0.002 s for a 200 ms for a medium latency delay.

Let mf = 2.5. The biped is simulated under the same disturbance of the support

surface but with raised muscular activities as in (6.4). Excursions of the CoG and

CoP are shown in Figure 6.6. The biped moves towards equilibrium, with a maximum

CoG excursion of 11.66 cm and a maximum CoP excursion of 20.32 cm. Note that

the maximum excursion of the CoG is lower than before – 15.11 cm – at the cost of

a higher CoP excursion. The biped also approaches equilibrium much more quickly

than before.

0.25 CoG 0.2 CoP

0.15

0.1

0.05 CoG/CoP position (m) 0 0 1 2 3 4 5 6 7 Time (s)

Figure 6.6: CoG and CoP excursions of the biped versus time due to disturbance of the support surface. Muscular activity levels are 80% of the nominal value, and there is modulation by the vestibular system with the modulation factor mf set to 2.5.

83 The resulting muscular activity level of the quadriceps-hamstring pair is shown in

Figure 6.7. This can be compared with the experimental results of Horak and Nashner

[58]. However, the result here is more diffuse in time and possibly less intense.

100

0 m) ⋅ −100

−200

Quad−ham (N −300

−400 0 1 2 3 4 5 6 7 Time (s)

Figure 6.7: The muscular activities in N·m for the quadriceps-hamstrings after a support surface disturbance when muscular activity levels are 80% of the nominal value, and there is modulation by the vestibular system with the modulation factor mf set to 2.5.

If the head is held stationary relative to the torso, torso position could be esti-

mated using information from the vestibular system. Thus, let us repeat the same

experiment, but using horizontal torso position xt and velocityx ˙ t instead of head

position and velocity. Thus, let

dt = |xt + 0.05x ˙ t|,

with mf = 2.5 and µ = 0.002 s. Then the total input torque is

τ(t) = (1 + mf dt(t − µ))(−0.8Aq(t) − 0.8Bq˙(t)).

Excursions of the CoG and CoP due to disturbance of the support surface are shown in Figure 6.8. The result is similar to modulation based on head position and

84 0.2 CoG 0.15 CoP

0.1

0.05 CoG/CoP position (m) 0 0 1 2 3 4 5 6 7 Time (s)

Figure 6.8: CoG and CoP excursions of the biped versus time due to disturbance of the support surface. Muscular activity levels are 80% of the nominal value, and there is modulation by the vestibular system through estimation of horizontal torso position and velocity. The modulation factor mf is 2.5.

velocity. The biped tends towards equilibrium more quickly than with no modulation.

There is a slightly higher maximum CoG excursion of 12.26 cm and a slightly lower maximum CoP excursion of 19.11 cm.

6.5 Simulation of Chorea-like Motion

In order to test the feasibility of the muscular activities and the modulation factor mf , the modulation factor was increased significantly to mf = 13. Excursions of the

CoG and CoP due to disturbance of the support surface are shown in Figure 6.9. The biped approaches equilibrium with a maximum CoG excursion of 9.84 cm; however, the CoP is erratic and well outside the bounds of the feet, with a maximum of 129 cm. This type of motion may approximate Huntington’s Chorea.

85 1.5 CoG 1 CoP

0.5

0

−0.5 CoG/CoP position (m) −1 0 0.5 1 1.5 2 2.5 3 3.5 Time (s)

Figure 6.9: CoG and CoP excursions of the biped versus time after the support surface experiences a sudden backward acceleration. Muscular activity levels are 80% of the nominal value, and there is modulation by the vestibular system using horizontal head position and velocity. The modulation factor mf is 13.

86 6.6 Discussion

Experiments from the previous section have shown that inputs to the muscular system can be augmented by information from the vestibular system in order to maintain balance under a sudden backwards support surface disturbance. In some cases, this augmentation may be required when other mechanisms are diminished.

For instance, nominal muscular activity levels based on proprioception are sufficient for maintaining balance under such a disturbance. However, when proprioceptive input is halved, the biped loses balance under the same disturbance. In this case, a vestibular term with a 200 ms delay was used to increase torque at the ankle and bring the biped towards equilibrium again. This type of input could also be used to increase muscular activity of other muscles, such as the quadriceps or paraspinal muscles [59].

In other cases, this type of augmentation may not be required, but can improve balance. For instance, proprioceptive inputs at 80% of their nominal values are enough to stabilize the biped. However, modulating these inputs by a vestibular term and a modulation factor can decrease maximum excursions of the CoG and the amount of time needed for the biped to move towards equilibrium by increasing the maximum

CoP excursion. A substantial increase in this vestibular modulation gain causes erratic excursions of the CoP in our model. Biologically, the origin of such an increase in gain may not lie in the proposed modulator scheme, but rather in the failure of inhibitory mechanisms that are located further upstream, such as in the basal ganglia.

The failure may involve the caudate nucleus and the putamen.

The experiments involving vestibular modulation represent how the vestibular system, via the lateral vestibular tract and the Deiters’s nucleus, facilitates both the

87 gamma and alpha signals of muscles. In the human system this facilitation is more selective. The tonic excitation from cerebellar origins might selectively be applied to certain extensors and flexors. We have applied it uniformly to all the muscles. This may be why muscle torque profiles, such as provided by the quadriceps/hamstring pair as shown in Figure 6.7, are similar in character to those found by Horak and

Nashner [58] but more diffuse in time and less intense. It may also result in a better control strategy that further limits excursions of the CoG and CoP.

Either vestibular contribution scheme, i.e. an additional term or a modulation term based on head position and velocity, is comparable to theories in which dif- ferent sensory systems, e.g. the visual, vestibular, and somatosensory systems, are adjustably weighted depending on the situation [60]. In this case, when propriocep- tive inputs from the somatosensory system are weakened, an additional term can be added or gains on vestibular modulation can be increased to improve balance. Such gain increases could also be triggered by other factors such as becoming apprehensive, alert, startled, or tense.

88 CHAPTER 7

CONCLUSIONS

In this dissertation, we have explored the role of the foot, toes, and vestibular system in balance using two computational models: a five-link sagittal biped model with a detailed foot and a three-link sagittal biped model with a vestibular-like ap- paratus. Computational experiments have shown how the foot, toes, and vestibular system can aid in the maintenance of balance. In what follows, we summarize the work done in this dissertation and then discuss how this work could be expanded in the future.

7.1 Summary

A significant contribution of this dissertation is the development of a five-link sagittal biped model with a detailed foot. This model uses one link for the main body, which includes the head, arms, torso, and legs, and it uses four links for the foot. The four foot links represent the heel or calcaneus; the forefoot, which is composed several bones including the navicular, cuboid, cuneiforms, and metatarsals; the proximal and intermediate phalanges of the toes; and the distal phalanges of the toes. Actuation of the model is accomplished through several major muscles, including the soleus, tibialis anterior, flexor hallucis longus, flexor hallucis brevis, extensor hallucis longus,

89 and extensor hallucis brevis. Passive contributions of the joints and the plantar aponeurosis are also considered. This model is unique in that the foot is not treated a single link, wedge, or other simple shape. Thus, the model allows for exploration of the effects of the muscles on the bones of the foot, estimation of tension in the plantar aponeurosis, easy estimation of ground reaction forces under different portions of the foot and thus easy estimation of center of pressure, and exploration of the role of the toes in balance.

For instance, this model was used to show that the toes can expand the base of support during forward leaning through action of the toe muscles. Here, this is achieved by activating the flexor hallucis brevis, one of two muscles that cause dorsiflexion in the large toe, when the main portion of the body leans forward. This muscle helps redistribute ground reaction forces from the metatarsals to the toes in this case. When the flexor hallucis brevis is disabled, the amount of forward leaning achievable without the heel lifting off the ground is lessened. Backward leaning is not affected.

It was also shown that foot arch angle is affected by the amount of leaning. Larger amounts of leaning cause the arch to flatten more. This can be attributed to torques on the heel or forefoot caused by action of the the soleus or tibialis anterior, which stabilize the main portion of the body by counteracting gravity during leaning.

Finally, this model was used to estimate center of pressure and center of gravity during movement. The center of pressure falls behind the center of gravity when the main body accelerates forward, and the center of pressured moves ahead of the center of gravity when the main body accelerates backward, as expected.

90 Another contribution of this dissertation is an exploration of how inputs from the vestibular system could aid in balance. This was done using the three-link sagittal biped model. For instance, it was shown experimentally that the three-link model can maintain balance under a sudden backwards platform disturbance when muscu- lar activity based on proprioception is nominal. However, the biped loses balance when these muscular activity levels are halved. Balance can restored by including an additional term based on horizontal head position and velocity as provided by the vestibular system.

It was also shown that balance can be restored by modulating muscular activity levels by a vestibular term. When proprioceptive muscular activity levels were dimin- ished to 80% of their nominal values, the biped maintained balance during disturbance of the support surface. Moderate modulation by the vestibular system reduced excur- sions of the center of gravity. A similar scheme based on horizontal torso position and velocity, which could be estimated from head position and velocity when head is held stationary relative to the torso, had a similar effect. However, when the modulation factor was greatly increased, the center of pressure became erratic, and maximum excursion of the center of pressure was far outside the base of support. This could correspond to a choreaic-like condition.

7.2 Future Work

The five-link model could be expanded in many ways. First, the simple muscle and ligament models used here could be replaced with more complicated models, such as the Hill model used in [46], though this would make control of the model more difficult. Also, constraints at the heel can be broken when the vertical ground reaction

91 force becomes negative, allowing the heel to lift off the ground for tip-toe standing.

Preliminary simulations show that control of the model is somewhat difficult in this case due to the nature of the muscle inputs. These preliminary attempts used a combination of muscle, joint, and ligament inputs to successfully control the position and velocity of the main body, heel, and forefoot. However, the forces under the toes were unrealistic for biologically feasible poses. It could be the case that the control needs to be modified, or it could also be that the model should be modified to include additional muscles, ligaments, joints, or the effects of tissue in the toes. A realistic model for tip-toe standing would allow for exploration of short-term and long-term stability during tip-toe standing, as explored in [61].

Computationally, balance of the model could be studied under a variety of distur- bances, such as translational, rotational, or random disturbances of the support sur- face. Small random disturbances could be used to study how greater muscle strength in the legs and toes allows for finer postural control, as reported in [40]. Vertical ground reaction forces under the foot could also be used to supplement control based on proprioception, perhaps using a scheme similar to that presented in [62], though this control applies to a robotic foot with no toes. This control scheme treats the cen- ter of pressure under a wedge-shaped foot as a variable to be controlled. The control itself is a function of the differential forces under the heel and toe of the model. This may correspond to findings in [35], which found that postural reflexes were correlated to the time derivative of pressure changes between the heel and forefoot. With such a control, one could also study the effects of inhibition as in [30] and [31]; artificially induced sensation as in [32], [33], and [34]; and disturbances of the support surface, as in [35], to name a few.

92 With regards to both models, the same issues and arguments could also be applied to three-dimensional versions. For the five-link model, this may require more complex modeling of the foot to account for mediolateral effects due to the forefoot segments, as discussed in [63]. For the three-link model, the control would need to be modified to utilize rotational motion of the head, the whole body, or the torso for enhancing balance in transverse plane.

Both models would require further clinical experimentation for validation and refinement. For instance, for the five-link model, one could record forces under a subject’s feet using a force plate for a range of forward and backward leaning postures.

The results could be compared to those of the model, with model parameters adjusted to match those of the subject. Also, though it is difficult to measure muscle torques in a live subject, one could estimate muscle activities using EMG recordings for the same leaning postures and compare the results to muscle torques in the model. Forces and muscle activity levels could also be measured and compared during forward and backward movements of the body. Finally, tension in the plantar aponeurosis of the model during normal standing, i.e. when the body is not leaning, could be compared to tension measured in the foot of a cadaver when a force equivalent to the model weight is applied to the talus or ankle joint.

For the three-link model, the simulated experiments performed here differ from previous experiments by modeling individual body segments and muscle pairs. Past experiments have treated the body as a single inverted segment or pendulum. Often in these types of cases, the support surface is disturbed, and a transfer function between the disturbance and center of gravity or center of pressure is examined. For instance, in [54] and [37], sinusoidal rotations of the platform are applied, and transfer functions

93 are compared between normal subjects and those with conditions such as vestibular loss. However, this type of formulation only considers the relationship between the input stimulus, i.e. support surface disturbance, and the output, i.e. center of pressure or center of gravity. It does not easily allow for analysis of vestibular effects on major body segments or individual muscles pairs.

Clinical experiments that include a sudden backwards disturbance of the support surface could be used to compare the responses of the three-link model to those of healthy subjects and vestibular loss patients. If the results are inconsistent, new hypotheses regarding vestibular contributions to balance could be generated, and the control scheme could be modified and retested.

94 APPENDIX A

NOMENCLATURE

Many symbols are used throughout the dissertation. They are defined here for reference, grouped according to the model(s) to which they apply.

Symbol Meaning Unit li Length of link i m ki Distance from one end of link i to its center of mass m θi Angle of link i relative to the vertical rad mi Mass of link i kg 2 Ii Moment of inertia of link i about its center of mass kg·m g Gravity constant m/s2 q State of the system · J(q) Inertia matrix · B(q, q˙) Matrix of Coriolis/centripetal terms · G(q) Matrix of gravity terms ·

Table A.1: Symbols used for both models. “·” designates mixed units.

95 Symbol Meaning Unit Fh Sheer ground reaction force under the heel N Gh Vertical ground reaction force under the heel N Gm Vertical ground reaction force under the metatarsals or forefoot N Gt Vertical ground reaction force under the tip of the toe N Fgr Vector of ground reaction forces N C(q) Matrix of constraints m xa Horizontal position of the ankle joint m ya Vertical position of the ankle joint m τsl Active torque provided by the soleus N·m τta Active torque provided by the tibialis anterior N·m τaj Passive torque due to the ankle joint or talus N·m τpa Passive torque due to the plantar aponeurosis N·m τehb Active torque provided by the extensor hallucis brevis N·m τfhb Active torque provided by the flexor hallucis brevis N·m τhj Passive torque due to the hallux or toe joint N·m τehl Active torque provided by the extensor hallucis longus N·m τfhl Active torque provided by the flexor hallucis longus N·m τm Vector of generalized torques due to muscles, joints, and ligaments N·m τgr Vector of generalized ground reaction torques N·m τq Vector of all generalized torques N·m U[x] Function such that U[x] = x if x > 0 and U[x] = 0 otherwise · τ1 Torque that provides tracking for link 1 N·m ¯ θ1 Desired trajectory for link 1 rad α1 Proportional gain for control of link 1 N·m β1 Derivative gain for control of link 1 N·m/s ˆ 2 I1 Moment of inertia of link 1 about the ankle joint kg·m ¯ θaj Desired ankle joint angle rad αaj Stiffness of the ankle joint N·m βaj Viscosity of the ankle joint N·m/s αpa Stiffness of the plantar aponeurosis N·m βpa Viscosity of the plantar aponeurosis N·m/s αhj Stiffness of the hallux joint N·m βhj Viscosity of the hallux joint N·m/s ¯ θ1t Threshold for flexor hallucis longus activity based on link 1 angle rad αfhb Proportional gain of the flexor hallucis brevis N·m ¯ θ5 Desired angle for link 5 rad αfhl Proportional gain for flexor hallucis longus activity N·m βfhl Derivative gain for flexor hallucis longus activity N·m/s

Table A.2: Symbols used for the five-link model. “·” designates mixed units.

96 Symbol Meaning Unit α1 Proportional gain for soleus/tibialis anterior activity N·m β1 Derivative gain for soleus/tibialis anterior activity N·m/s α2 Proportional gain for quadriceps/hamstring short heads activity N·m β2 Derivative gain for quadriceps/hamstring short heads activity N·m/s α3 Proportional gain for abdominal/paraspinal activity N·m β3 Derivative gain for abdominal/paraspinal activity N·m/s α4 Proportional gain for quadriceps/hamstring long heads activity N·m β4 Derivative gain for quadriceps/hamstring long heads activity N·m/s α5 Proportional gain for gastrocnemius activity N·m β5 Derivative gain for gastrocnemius activity N·m/s A Matrix of proportional muscle gains N·m B Matrix of derivative muscle gains N·m τ Vector of generalized muscle torques N·m 2 ap Horizontal support surface acceleration m/s xh Horizontal position of the head m xt Horizontal position of the torso m τf Feedback from the vestibular system N·m mf Modulation factor for vestibular feedback · d Head or torso “disturbance” m µ Signal delay s

Table A.3: Symbols used for the three-link model. “·” designates mixed units.

97 APPENDIX B

DERIVING EQUATIONS OF MOTION FOR THE FIVE LINK MODEL

This appendix derives all the equations needed to simulate the motion of the

five-link sagittal biped shown in Figure 4.1 of Chapter 4. First, we derive the basic equations of motion using the Euler-Lagrange method [64]. Next, we derive an ex- pression for the ground reaction forces using constraints based on movement relative to the ground. Finally, we derive expressions for inputs provided by the muscles.

B.1 Basic Equations of Motion

According to the Euler-Lagrange method, the equations of motion for this system take the form d ∂L ∂L − = τ (B.1) dt ∂q˙ ∂q q where

L = K − P (B.2)

and K is the kinetic energy of the system, P is the potential energy, τq is a vector of the

T generalized forces/torques, and q = [xa, ya, θ1, θ2, θ3, θ4, θ5] is a vector of generalized coordinates.

98 It is simple in this case to derive the kinetic and potential energies of the system.

The kinetic energy is the sum of the rotational and translational energies of the

system, and the potential energy is due to gravity. These quantities can be written

in terms of the angles θi and positions of the centers of mass (xi, yi) of each link i as

5 1 X h i K = I θ˙2 + m (x ˙ 2 +y ˙2) (B.3) 2 i i i i i i=1 and 5 X P = g miyi. (B.4) i=1 In order to put (B.3) and (B.4) in terms of the generalized coordinates, we may write each xi and yi in terms of q as

x1 = xa + k1 sin(θ1)

y1 = ya + k1 cos(θ1)

x2 = xa + k2 sin(θ2)

y2 = ya + k2 cos(θ2)

x3 = xa + k3 sin(θ3)

y3 = ya + k3 cos(θ3)

x4 = xa + l3 sin(θ3) + k4 sin(θ4)

y4 = ya + l3 cos(θ3) + k4 sin(θ4)

x5 = xa + l3 sin(θ3) + l4 sin(θ4) + k5 sin(θ5)

y5 = ya + l3 cos(θ3) + l4 sin(θ4) + k5 sin(θ5). (B.5) Thus, (B.5) may be used to write (B.3) and (B.4) in terms of the generalized coor- dinates, and (B.3) and (B.4) can then be substituted into (B.1) and (B.2) to derive the equations of motion for the system. The final equations take the form

J(q)¨q = −B(q, q˙)q ˙ − G(q) + τq, (B.6)

99 where  m1 + m2 + m3 + m4 + m5 0  0 m1 + m2 + m3 + m4 + m5   k1m1 cos(θ1) −k1m1 sin(θ1)   k2m2 cos(θ2) −k2m2 sin(θ2)   (k3m3 + l3(m4 + m5)) cos(θ3) −(k3m3 + l3(m4 + m5)) sin(θ3)   (k4m4 + l4m5) cos(θ4) −(k4m4 + l4m5) sin(θ4) k5m5 cos(θ5) −k5m5 sin(θ5)

k1m1 cos(θ1) k2m2 cos(θ2) −k1m1 sin(θ1) −k2m2 sin(θ2) 2 m1k1 + I1 0 2 0 m2k2 + I2 0 0 0 0 0 0 J(q) = (k3m3 + l3(m4 + m5)) cos(θ3) (k4m4 + l4m5) cos(θ4) −(k3m3 + l3(m4 + m5)) sin(θ3) −(k4m4 + l4m5) sin(θ4) 0 0 0 0 2 2 m3k3 + I3 + l3(m4 + m5) l3(k4m4 + l4m5) cos(θ3 − θ4) 2 2 l3(k4m4 + l4m5) cos(θ3 − θ4) m4k4 + I4 + l4m5 k5l3m5 cos(θ3 − θ5) k5l4m5 cos(θ4 − θ5)  k5m5 cos(θ5) −k5m5 sin(θ5)   0   0   k5l3m5 cos(θ3 − θ5)   k5l4m5 cos(θ4 − θ5)  2 m5k5 + I5

100  0 0  0 0   0 0   0 0   0 0   0 0 0 0

−k1m1 sin(θ1)θ˙1 −k2m2 sin(θ2)θ˙2 −k1m1 cos(θ1)θ˙1 −k2m2 cos(θ2)θ˙2 0 0 0 0 0 0 0 0 0 0

B(q, q˙) = −(k3m3 + l3(m4 + m5)) sin(θ3)θ˙3 −(k4m4 + l4m5) sin(θ4)θ˙4 −(k3m3 + l3(m4 + m5)) cos(θ3)θ˙3 −(k4m4 + l4m5) cos(θ4)θ˙4 0 0 0 0 0 l3(k4m4 + l4m5) sin(θ3 − θ4)θ˙4 −l3(k4m4 + l4m5) sin(θ3 − θ4)θ˙3 0 −k5l3m5 sin(θ3 − θ5)θ˙3 −k5l4m5 sin(θ4 − θ5)θ˙4

 −k5m5 sin(θ5)θ˙5 ˙ −k5m5 cos(θ5)θ5   0   0   k l m sin(θ − θ )θ˙  5 3 5 3 5 5  k5l4m5 sin(θ4 − θ5)θ˙5  0

and  0   m1 + m2 + m3 + m4 + m5     −k1m1 sin(θ1)    G(q) = g  −k2m2 sin(θ2)  .    −(k3m3 + l3(m4 + m5)) sin(θ3)     −(k4m4 + l4m5) sin(θ4)  −k5m5 sin(θ5)

The term τq comes from two sources: the ground reaction forces Fh, Gh, Gm, and Gt as well as actuation provided by the muscles and joints. Thus,

τq = τgr + τm, (B.7) where τgr is due to ground reaction forces and τm is due to muscle inputs. These will be derived in the next two sections.

101 B.2 Terms due to Ground Reaction Forces

To derive τgr in (B.7), we first write an expression C(q) for the points at which the forces are applied. This term takes the form   xa + l2 sin(θ2)  ya + l2 cos(θ2)  C(q) =   .  ya + l3 cos(θ3)  ya + l3 cos(θ3) + l4 cos(θ4) + l5 cos(θ5) Then, using standard results [64], the generalized forces and torques due to the ground reaction forces can be expressed as ∂C(q)T τ = F , gr ∂q gr where T Fgr = [Fh,Gh,Gm,Gt] . Then, the equations of motion (B.6) become ∂C(q)T J(q)¨q = −B(q, q˙)q ˙ − G(q) + τ + F . (B.8) m ∂q gr If we assume that the points of contact do not move, i.e. the foot remains on the ground and the heel does not slip, then C˙(q) = C¨(q) ≡ 0, and thus ∂C(q) ∂ ∂C(q)  C¨(q) = q¨ + q˙ q˙ = 0. (B.9) ∂q ∂q ∂q Rearranging (B.8),q ¨ can be written as  ∂C(q)T  q¨ = J(q)−1 −B(q, q˙)q ˙ − G(q) + τ + F , (B.10) m ∂q gr where J(q)−1 is invertible since it is a positive definite inertia matrix. Then, using (B.10) in (B.9), Fgr can be expressed as

T !−1 ∂C(q) −1 ∂C(q) Fgr = J(q) ∂q ∂q     ∂C(q) −1 ∂ ∂C(q) J(q) (B(q, q˙)q ˙ + G(q) − τm) − q˙ q˙ . ∂q ∂q ∂q Note that the term ! ∂C(q) ∂C(q) T J(q)−1 ∂q ∂q

102 ∂C(q) is invertible as long as ∂q is full rank, which is true for this model in all realistic cases, e.g. the toe does not lie parallel to the ground, the toe does not bend below ground, the ankle joint does not lie parallel to the ground, and the ankle joint does not bend below ground.

B.3 Terms due to the Muscle Inputs

The term τm in (B.5) is due to inputs from the muscles, joints, and ligaments in the foot, as shown in Figure 4.3 of Chapter 4. Individual muscle inputs are discussed in Section 5.1. By examining the layout of the muscles, joints, and plantar aponeurosis in Figure 4.3, it is easily seen that the final form of the muscle inputs in (B.7) is   τsl   0 0 0 0 0 0 0 0 0  τta     0 0 0 0 0 0 0 0 0   τaj       −1 1 0 0 0 0 0 1 −1   τpa      τm =  1 0 −1 −1 0 0 0 0 0   τehb  ,      0 −1 1 1 1 −1 0 0 0   τfhb       0 0 0 0 −1 1 −1 0 0   τhj    0 0 0 0 0 0 1 −1 1  τehl  τfhl where τsl is due to the soleus, τta to the tibialis anterior, τaj to the ankle joint or talus, τpa to the plantar aponeurosis, τehb to the extensor hallucis brevis, τfhb to the flexor hallucis brevis, τhj to the hallux or toe joint, τehl to the extensor hallucis longus, and τfhl to the flexor hallucis longus. Note that each column has a 1 and -1 entry, since each muscle, joint, or ligament connects two links, applying the same magnitude torque but in opposite directions. Also, the first two rows are 0 because the inputs do not directly affect the motion of coordinates xa and ya.

103 APPENDIX C

DERIVING EQUATIONS OF MOTION FOR THE THREE LINK MODEL

This appendix derives all the equations needed to simulate the motion of the three- link sagittal biped shown in Figure 4.4 of Chapter 4. As with the five-link model, we first derive the basic equations of motion using the Euler-Lagrange method. Next, we show how inputs from the muscles enter into the system. Then, we show how a sudden backwards translational disturbance of the platform affects initial conditions of the model. Next, we discuss how to prevent hyperextension, i.e. incorrect bending at the knees and hip. Finally, we discuss how to estimate center of pressure under the foot.

C.1 Basic Equations of Motion

As with the five-link model, the equations of motion for this system take the same general form as shown in (B.1) of Appendix B. However, the generalized coordinates T for this system are q = [θ1, θ2, θ3] and

3 1 X h i K = I θ˙2 + m (x ˙ 2 +y ˙2) (C.1) 2 i i i i i i=1 and 3 X P = g miyi. (C.2) i=1 Furthermore, equations for the center of mass positions of each link are

x1 = k1 sin(θ1)

y1 = k1 cos(θ1)

x2 = l1 sin(θ1) + k2 sin(θ2)

y2 = l1 cos(θ1) + k2 cos(θ2)

x3 = l1 sin(θ1) + l2 sin(θ2) + k3 sin(θ3)

y3 = l1 cos(θ1) + l2 cos(θ2) + k3 cos(θ3). (C.3)

104 Using the same process as in Appendix B with (C.1), (C.2), and (C.3) in (B.1), the equations of motion take the form

J(q)¨q = −B(q, q˙)q ˙ − G(q) + τm, (C.4) where

2 2 2 3 I1 + m1k1 + (m2 + m3)l1 (m2k2 + m3l2)l1 cos(θ2 − θ1) m3k3l1 cos(θ3 − θ1) 2 2 J(q) = 4 (m2k2 + m3l2)l1 cos(θ2 − θ1) I2 + m2k2 + m3l2 m3k3l2 cos(θ3 − θ2) 5 , 2 m3k3l1 cos(θ3 − θ1) m3k3l2 cos(θ3 − θ2) I3 + m3k3

2 3 0 (m2k2 + m3l2)l1θ˙2 sin(θ1 − θ2) m3k3l1θ˙3 sin(θ1 − θ3) ˙ ˙ B(q, q˙) = 4 (m2k2 + m3l2)l1θ1 sin(θ2 − θ1) 0 m3k3l2θ3 sin(θ2 − θ3) 5 , m3k3l1θ˙1 sin(θ3 − θ1) m3k3l2θ˙2 sin(θ3 − θ2) 0

2 3 [m1k1 + (m2 + m3)l1] sin(θ1) G(q) = g 4 (m2k2 + m3l2) sin(θ2) 5 , m3k3 sin(θ3) and the term τm is due to the muscles, which will be discussed in the next section.

C.2 Terms due to the Muscle Inputs

In this appendix, we describe the muscle model used to actuate the system by pro- viding τm in (C.4). There are four muscle pairs and the gastrocnemius, as shown in Figure 4.2.2. We assume that the gastrocnemius is also part of an agonist/antagonist pair, perhaps consisting of a combination of several muscles. Thus, there are effec- tively five muscle pairs. In our model, the force Fm for m = 1, 2, ..., 5 provided by each muscle pair is modeled by a circular spring, i.e. ˙ Fm = −αm`m − βm`m, (C.5) ˆ ˆ where `m = rmθm, rm is the radius of the spring, θm is the angle between corresponding body segments, αm is a proportional gain, and βm is a derivative gain. For simplicity, we will choose rm = 1 for each muscle pair. In this case, F1 corresponds to the soleus/tibialis anterior, F2 to the short heads of the quadriceps/hamstrings, F3 to the paraspinal/abdominal muscles, F4 to the quadriceps/hamstrings, and F5 to the gastrocnemius, as shown in Figure 4.2.2. Thus, ˆ `1 = θ1 = θ1 ˆ `2 = θ2 = θ2 − θ1 ˆ `3 = θ3 = θ3 − θ2 ˆ `4 = θ4 = θ1 − θ3 ˆ `5 = θ5 = θ2 (C.6)

105 with θ1, θ2, and θ3 as shown in Figure 4.4. Nominal gains for the muscle pairs are as given in (6.3), i.e. T T α = [α1, α2, α3, α4, α5] = [500, 300, 300, 300, 300] and T β = [β1, β2, β3, β4, β5] = 0.25α. We can now derive the form of the muscle input to the system by writing the generalized torques in the space q. Let T F = [F1,F2,F3,F4,F5] , T ` = [`1, `2, `3, `4, `5] , and T q = [θ1, θ2, θ3] . The generalized torques are then written as ∂` T τ = F, (C.7) ∂q T where τ = [τ1, τ2, τ3] . Using (C.6) ,  1 −1 0 1 0  ∂` T = 0 1 −1 0 1 . ∂q   0 0 1 −1 0 Substituting (C.5) and (C.6) into (C.7) yields ∂` T ∂` ∂` τ = (diag(α) q + diag(β) q˙), (C.8) ∂q ∂q ∂q where diag(·) indicates a diagonal matrix generated from the indicated vector. Note that this equation may also be written as ∂` T τ = u, (C.9) ∂q T where u = [u1, u2, u3, u4, u5] is a vector of torques provided by each muscle pair, with u1 corresponding to the soleus/tibialis anterior, u2 to the short heads of the quadriceps/hamstrings, u3 to the abdominal/paraspinal pair, u4 to the long heads quadriceps/hamstrings, and u5 to the gastrocnemius/antagonist group pair. Finally, (C.8) reduces to the simplified form of the muscle input as given in (6), τ = −Aq − Bq,˙ where   α1 + α2 + α4 −α2, −α4 A =  −α2 α2 + α3 + α5 −α3  −α4 −α3 α3 + α4 and B has the same form. In this case, since β = 0.25α, B = 0.25A as well.

106 C.3 Initial Conditions due to a Sudden Horizontal Platform Disturbance

In this section, we show that a sudden horizontal platform disturbance whose ac- celeration is modeled by a Dirac delta or impulse is equivalent to an initial angular velocity on an otherwise stationary model. First, let us assume that the horizon- tal displacement of the platform is xd. Then, assuming the foot does not slip, the equations for x1, x2, and x3 in (C.3) become

x3 = l1 sin(θ1) + l2 sin(θ2) + k3 sin(θ3) + xd

x2 = l1 sin(θ1) + k2 sin(θ2) + xd

x1 = k1 sin(θ1) + xd

and the equations for y1, y2, and y3 remain the same. Applying the new constraints to (B.1), (C.1), and (C.2) results in an equation similar to (C.4), except for an extra term. We will denote this term as H(q)x ¨d, where H(q) is the 3x1 matrix   [m1k1 + (m2 + m3)l1] cos(θ1) H(q) =  (m2k2 + m3l2) cos(θ2)  . m3k3 cos(θ3)

If the horizontal platform disturbance is such that the translational acceleration x¨d = ap δ(t), where δ(t) is the Dirac delta, then with the addition of this new term, (C.4) becomes J(q)¨q = −B(q, q˙)q ˙ − G(q) + ap δ(t)H(q) + τm. (C.10) Rearranging (C.10), we obtain

−1 q¨ = J (q)[−B(q, q˙)q ˙ − G(q) + ap δ(t)H(q) + τm], (C.11)

where J(q) is invertible because it is a positive definite inertia matrix. If we assume the biped is stationary at time t = 0− and integrate (C.11) over time t = 0− to t = 0+, we obtain + −1 q˙(0 ) = J (q(0))H(q(0))ap (C.12) where q(0) is the value of q at time t = 0. In Chapter 6, the model experiences a horizontal acceleration or disturbance of the support platform equal to −0.5 δ(t) m/s2. Thus, for an initial position

q(0) = [0.1, −0.1, 0.1]T rad,

by (C.12) and using the value ofq ˙ at time t = 0+ for time t = 0,

q˙(0) = [2.5580, −0.0654, 0.0027]T rad/s.

107 C.4 Preventing Hyperextension

The model used here incorporates protections against hyperextension at the knee and hip. This is done by applying a Dirac delta or impulse torque resulting from forces opposing the hyperextension at the corresponding joint. As a simplification, we define hyperextension at the knee as θ2 > θ1 in Figure 4.4 of Section 4.2 and hyperextension at the hip as θ2 > θ3. First, consider the case of hyperextension at the knee. Let θi(th) be the value of θi ˙ ˙ at time t = th. During simulation, if θ1(th)−θ2(th) < 0.001 and θ2(th)−θ1(th) > 0.001, hyperextension at the knee is about to occur. Thus, additional equal and opposite impulse torques of magnitudeτ ˆ δ(t − th) should be applied to links 1 and 2. Similar to the derivation of the equations of motion of Section C.3, the equations of motion for the system become

T J(q)¨q = −B(q, q˙)q ˙ − G(q) = [1, −1, 0] τˆ δ(t − th) + τm. (C.13) Rearranging (C.13), we obtain

−1 T q¨ = J (q)[−B(q, q˙)q ˙ − G(q) + [1, −1, 0] τˆ δ(t − th) + τm], (C.14) where J(q) is invertible because it is a positive definite inertia matrix. If we integrate − + (C.14) over time t = th to t = th , we obtain

+ − −1 T q˙(th ) − q˙(th ) = J (q(th))[1, −1, 0] τ.ˆ (C.15) ˙ + In order to prevent hyperextension, we can chooseτ ˆ so that θ(th ) is 0. Setting ˙ + θ(th ) = 0 and multiplying both sides of (C.15) by [1, −1, 0] results in

− − −1 T q˙2(th ) − q˙1(th ) = [1, −1, 0]J (q(th))[1, −1, 0] τˆ and thus

−1 T −1 − − τˆ = ([1, −1, 0]J (q(th))[1, −1, 0] ) [q ˙1(th ) − q˙2(th )].

+ Using values at time t = th for time t = th results in an a change in total torque at time th, withq ˙1(th) = 0 andq ˙2(th) = 0. ˙ ˙ When θ3(th) − θ2(th) < 0.001 and θ2(th) − θ3(th) > 0.001, hyperextension at the hip is about to occur. A similar derivation can be used to show that the counteracting torque in this case should be

−1 T −1 − − τˆ = ([0, −1, 1]J (q(th))[0, −1, 1] ) [q ˙2(th ) − q˙3(th )].

108 C.5 Estimating Center of Pressure

With a flat, massless link representing the feet connected to link 1 as shown in Figure 4.4, we can estimate the center of pressure during movement using a method similar to that described in [3]. This method equates ankle torque to the ground reaction force applied at some distance from the ankle joint. This distance is the center of pressure. Note that the only muscle pairs connected to the feet are the soleus/tibialis anterior pairs and the gastrocnemius/antagonist group pairs. Thus, torque at the ankle is equal to u1 + u5 in (C.9).

Figure C.1: A free body diagram of the three-link model showing forces of connection between links and ground reaction forces.

109 Figure C.1 shows the forces of connection between the three links of the model, and ground reaction forces at the foot. Assuming as an approximation that the foot has no height, the torque provided by Fg is 0 N · m. Thus, the final result is that the center of pressure xcop depends on the torque at the ankle and the vertical component of the ground reaction force Gg through the relation

xcop = −(u1 + u5)/Gg. (C.16)

Computation of xcop is performed after the main simulation is complete and re- T quires estimates ofy ¨1,y ¨2, andy ¨3. Since the model’s state is q = [θ1, θ2, θ3] , these values can be calculated using half the constraints listed in (C.3). Differentiating the relevant equations twice yields

¨ ˙2 ¨ ˙2 y¨3 = −l1θ1 sin(θ1) − l1θ1 cos(θ1) − l2θ2 sin(θ2) − l2θ2 cos(θ2) ¨ ˙2 −k3θ3 sin(θ3) − k3θ3 cos(θ3) ¨ ˙2 ¨ y¨2 = −l1θ1 sin(θ1) − l1θ1 cos(θ1) − k2θ2 sin(θ2) ˙2 −k2θ2 cos(θ2) ¨ ˙2 y¨1 = −k1θ1 sin(θ1) − k1θ1 cos(θ1). (C.17)

The net vertical forces on the centers of mass of each link can be calculated using the formula Gi = miy¨i and (C.17). Then, from Figure C.1, we see that we can calculate the interconnection forces and the vertical ground reaction force starting with link 3 using using the equations

G23 = G3 + g m3

G12 = G2 + G23 + g m2

Gg = G1 + G12 + g m1. (C.18) ˙ ¨ Assuming values for u1, u5, θ, θ, and θ are stored during the main simulation, (C.16), (C.17), and (C.18) can be used calculate xcop.

110 BIBLIOGRAPHY

[1] M. G¨unther and H. Ruder. Synthesis of two-dimensional human walking: a test of the λ model. Biological Cybernetics, 89:89–106, 2003.

[2] R. McMinn, R. Hutchings, and B. Logan. Color Atlas of Foot & Ankle Anatomy. Mosby-Wolfe, 1996.

[3] D. A. Winter. Human balance and posture control during standing and walking. Gait & Posture, 3(4):193–214, 1995.

[4] W. Gage, D. Winter, J. Frank, and A. Adkin. Kinematic and kinetic validity of the inverted pendulum model in quiet standing. Gait & Posture, 19:124–132, 2004.

[5] R. Fitzpatrick and B. Day. Probing the human vestibular system with galvanic stimulation. Journal of Applied Physiology, 96:2301–2316, 2004.

[6] P. Sardain and G. Bessonnet. Forces acting on a biped robot. center of pres- surezero moment point. IEEE Transactions on Systems, Man and Cybernetics, Part A, 34(5):630–637, 2004.

[7] Y.-C. Pai and J. Patton. Center of mass velocity-position predictions for balance control. Journal of Biomechanics, 30(4):347–354, 1997.

[8] Y. Jian, D. Winter, M Ishac, and L. Gilchrist. Trajectory of the body cog and cop during initiation and termination of gait. Gait Posture, 1:9–22, 1993.

[9] B. Maki, W. McIllroy, and S. Perry. Influence of lateral destabilization on com- pensatory stepping responses. Journal of Biomechanics, 29(3):343–353, 1996.

[10] D. Thelen, L. Wojcik, A. Schultz, A. Ashton-Miller, and N. Alexander. Age difference in using a rapid step to regain balance during a forward fall. Journals of Gerontology Series A: Biological Sciences and Medical Sciences, 2(1):M8–M13, 1997.

[11] J. Raymakers, M. Samson, and H. Verhaar. The assessment of body sway and the choice of the stability parameter(s). Gait & Posture, 21:48–58, 2005.

111 [12] A. Shumway-Cook and M. Woollacott. Motor Control: Theory and Practical Applications. Lippincott Williams & Wilkins, 1995.

[13] D. Rosenbaum. Human Motor Control. Academic Press, 1991.

[14] C. Sherrington. The Integrative Action of the Nervous System. Yale University Press, second edition, 1947.

[15] D. M. Wilson. The central nervous control of flight in a locust. Journal of Experimental Biology, 38:471–490, 1961.

[16] S. Grillner. Control of locomotion in bipeds, tetrapods and fish. In S.R. Geiger, editor, Handbook of Physiology, volume 2, pages 1179–1236. American Physio- logical Society, 1981.

[17] H. Forssberg, S. Grillner, and S. Rossignol. Phase dependent reflex reversal during walking in chronic spinal cats. Brain Research, 85:103–107, 1975.

[18] N. Bernstein. The Coordination and Regulation of Movement. Pergamon Press, 1967.

[19] J.P. Kelly and J. Dodd. Anatomical organization of the nervous system. In E.R. Kandel, J.H. Schwartz, and T.M. Jessell, editors, Principles of Neural Science, pages 273–282. Elsevier, third edition, 1991.

[20] J.P. Kelly. The neural basis of perception and movement. In E.R. Kandel, J.H. Schwartz, and T.M. Jessell, editors, Principles of Neural Science, pages 283–295. Elsevier, third edition, 1991.

[21] J.H. Martin and T.M. Jessell. Anatomy of the somatic sensory system. In E.R. Kandel, J.H. Schwartz, and T.M. Jessell, editors, Principles of Neural Science, pages 353–366. Elsevier, third edition, 1991.

[22] E.R. Kandel and T.M. Jessell. Touch. In E.R. Kandel, J.H. Schwartz, and T.M. Jessell, editors, Principles of Neural Science, pages 367–384. Elsevier, third edition, 1991.

[23] J.P. Kelly. The . In E.R. Kandel, J.H. Schwartz, and T.M. Jessell, editors, Principles of Neural Science, pages 500–511. Elsevier, third edition, 1991.

[24] R. M. Enoka. Neuromechanics of Human Movement. Human Kinetics, 2002.

[25] P. M. Kennedy and J. T. Inglis. Distribution and behavior of glabrous cutaneous receptors in the human foot sole. Journal of Physiology, 538:995–1002, 2002.

112 [26] T. Tanaka, S. Ino, and T. Ifukube. Tactile sense and pressure of toe contribution to standing in the healthy elderly. Journal of Physical Therapy Science, 8:19–24, 1996.

[27] T. Tanaka, S. Noriyasu, S. Ino, S. Ifukube, and M. Nakata. Objective method to determine the contribution of the great toe to standing balance and prelim- inary observations of age-related effects. IEEE Transactions on Rehabilitation Engineering, 4:84–90, 1996.

[28] I. Ducic, K. W. Short, and A. L. Dellon. Relationship between loss of pedal sensibility, balance, and falls in patients with peripheral neuropathy. Annals of Plastic Surgery, 52:535–540, 2004.

[29] M. Schieppati, M. Hugon, M. Grasso, and M. Nardone, A. Galante. The lim- its of equilibrium in young and elderly normal subjects and in Parkinsonians. Electroencephalography and clinical Neurophysiology, 93:286–298, 1994.

[30] P. Meyer, L. Oddsson, and C. De Luca. The role of plantar cutaneous sensation in unperturbed stance. Experimental Brain Research, 156:505–512, 2004.

[31] F. Stal, P. A. Fransson, M. Magnusson, and M. Karlberg. Effects of hypothermic anesthesia of the feet on vibration-induced body sway and adaptation. Journal of Vestibular Research, 13:39–52, 2003.

[32] A. Kavounoudias, R. Roll, and J.P. Roll. The plantar sole is a ‘dynamometric map’ for human balance control. NeuroReport, 9:3247–3252, 1998.

[33] A. Kavounoudias, R. Roll, and J.P. Roll. Specific whole-body shifts induced by frequency-modulated vibrations of human plantar soles. Neuroscience Letters, 266:181–184, 1999.

[34] A. Kavounoudias, R. Roll, and J.P. Roll. Foot sole and ankle muscle inputs contribute jointly to human erect posture regulation. Journal of Physiology, 532(3):869–878, 2001.

[35] G. Wu and J.-H. Chiang. The significance of somatosensory stimulations to the human foot in the control of postural reflexes. Experimental Brain Research, 114:163–169, 1997.

[36] C. Maurer, T. Mergner, B. Bolha, and F. Hlavacka. Human balance control during cutaneous stimulation of the plantar soles. Neuroscience Letters, 302:45– 48, 2001.

[37] C. Maurer, T. Mergner, B. Bolha, and F. Hlavacka. Vestibular, visual, and somatosensory contributions to human control of upright stance. Neuroscience Letters, 281:99–102, 2000.

113 [38] A. Nardone, M. Grasso, and M. Schieppati. Balance control in peripheral neu- ropathy: Are patients equally unstable under static and dynamic conditions? Gait & Posture, 23:364–373, 2006.

[39] M. J. Mueller, G. B. Salsich, and M. J. Strube. Functional limitation in patients with diabetes and transmetatarsal amputations. Physical Therapy, 9:937–943, 1997.

[40] Y. Katayama, M. Senda, M. Hamada, M. Kataoka, M. Shintani, and H. Inoue. Relationship between postural balance and knee and toe muscle power in young women. Acta Medica Okayama, 58(4):189–195, 2004.

[41] B. Peterson. Head movement: multidimensional modeling. In M. Arbib, editor, The Handbook of Brain Theory and Neural Netowrks, pages 450–454. The M.I.T. Press, 1995.

[42] A. Roy and K. Iqbal. Pid controller stabilization of a single-link biomechanical model with multiple delayed feedbacks. In IEEE International Conference on Systems, Man and Cybernetics, 2003.

[43] J. H. J. Allum and F. Honegger. Interactions between vestibular and propriocep- tive inputs triggering and modulating human balance-correcting responses differ across muscles. Experimental Brain Research, 121:478–494, 1998.

[44] B. Day and J. Cole. Vestibular-evoked postural responses in the absence of somatosensory information. Brain, 125:2081–2088, 2002.

[45] D. Woods, J. Tittle, M. Feil, and A. Roesler. Envisioning human-robot coordina- tion in future operations. IEEE Transactions on Systems, Man and Cybernetics, Part C, 34(2):210–218, 2004.

[46] W. Kim and A. Voloshin. Role of plantar fascia in load bearing capacity of the human foot. Journal of Biomechanics, 28(9):1025–1033, 1995.

[47] P. de Leva. Adjustments to Zatsiorsky-Seluyanov’s segment inertia parameters. Journal of Biomechanics, 29:1223–1230, 1996.

[48] V.M. Zatsiorsky, V.N. Seluyanov, and L. G. Chugunova. Methods of determining mass-inertial characteristics of human body segments. In G. G. Chernyi and S. A. Regirer, editors, Contemporary Problems of Biomechanics, pages 272–291. CRC Press, 1990.

[49] V. Zatsiorsky and V. Seluyanov. The mass and inertia characteristics of the main segments of the human body. In H. Matsui and K. Kobayashi, editors, Biomechanics VIII-B, pages 1152–1159. Human Kinetics, 1983.

114 [50] V.M. Zatsiorsky, V.N. Seluyanov, and L. G. Chugunova. In vivo body segment inertial parameters determination using a gamma-scanner method. In N. Berme and A. Cappozzo, editors, Biomechanics of Human Movement: Applications in Rehabilitation, pages 186–202. Bertec, 1990.

[51] H. Hemami, K. Barin, and Y.-C. Pai. Quantitative analysis of a sagittal biped under platform disturbance. IEEE Transactions on Neural Systems and Reha- bilitation Engineering, 4:470–480, 2006.

[52] K. Iqbal, A. Roy, and M. Imran. Passive and active contributors to postural sta- bilization. In IEEE International Conference on Systems, Man and Cybernetics, 2003.

[53] C. Saltzman, D. Nawoczenski, and K. Talbot. Measurement of the medial lon- gitudinal arch. Archives of Physical Medicine and Rehabilitation, 76(1):45–49, 1995.

[54] R. J. Peterka. Sensorimotor integration in human postural control. Journal of Neurophysiology, 88(3):1097–1118, 2002.

[55] J. Kayano. Dynamic function of medial foot arch. Journal of Japanese Or- thopaedic Association, 60:1147–1156, 1986.

[56] L. Nashner, F. Black, and C. Wall. Adaptation to altered support and visual conditions during stance: patients with vestibular deficits. The Journal of Neu- roscience, 2(5):536–544, 1982.

[57] G. McCollum, F. Horak, and L. Nashner. Parsimony in nerual calculations for postural movements. In Bloedel, editor, Cerebellar Functions, pages 52–66. Springer-Verlag, 1984.

[58] F. Horak and L. Nashner. Centeral programming of postural movements: adap- tation to altered support surface configurations. Journal of Neurophysiology, 55:1369–1381, 1986.

[59] J. Allum and F. Honegger. Interactions between vestibular and proprioceptive in- puts triggering and modulating human balance-correcting responses differ across muscles. Experimental Brain Research, 121:478–494, 1998.

[60] B. Day and J. Cole. Vestibular-evoked postural responses in the absense of somatosensory information. Brain, 125:2081–2088, 2002.

[61] L. Nolan and D. Kerrigan. Postural control: toe-standing versus heeltoe standing. Gait & Posture, 19:11–15, 2004.

115 [62] S. Ito, Y. Saka, and H. Kawasaki. A consideration on position of center of ground reaction force in upright posture. SICE 2002. Proceedings of the 41st SICE Annual Conference, 2:1225–1230, 2002.

[63] F. Buczek, M. Walker, M. Rainbow, K. Cooney, and J. Sanders. Impact of medio- lateral segmentation on a multi-segment foot model. Gait & Posture, 23:519–522, 2006.

[64] M. Spong, S. Hutchinson, and M. Vidyasagar. Robot Modeling and Control. Wiley, 2006.

116