BIOMECHANICS AND NEURAL CONTROL OF

LlMB POSITION

David W. Franklin

BSc. (Honours), Simon Fraser University, 1 995

THESIS SUBMITTED IN PARTIAL FLJLFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

MASTER OF SCENCE

in the School

o f

Kinesiology

O David W. Franklin 2000 SIMON FRASER UNMRSITY

April2000

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Multi-joint mechanical impedance of the a m is important in the control of

posture and movement. It determines how the arm responds to perturbations and

determines whether mechanical interactions with the environment will be stable.

Modification of muscle activation and sensory feedback gain allows adaptation of the

impedance of arm in response to the dynarnics of the task. In order to understand the

nature of control strategies employed by the neuro-muscular system it is necessary to

explore how mechanical impedance is modified for different tasks.

The multi-joint impedance of the human a m was measured by applying position

displacements to the hand and modeling the arm using a second order dynamic equation.

The subjects were asked to produce forces of up to 30% of their maximum voluntary

contraction (MVC) in four diflerent directions during a position control task. Endpoint

stiffness and joint stifmess were estimated and compared for a range of forces in the

different directions. The correlation beniveen joint stiffness and torque and joint viscosity

and torque for the shoulder and elbow were examined.

Endpoint stiffhess increased wi th force. Joint sti f i e s s increased linearl y wi th

both eIbow and shoulder torque. Previous work, using a force control task, had found that

shoulder stiffness increased with shoulder torque, double joint stiffness increased with

elbow torque and elbow stiffness increased with elbow torque. The results of this study,

using a position control task, also found that shoulder stiffness increased with elbow

torque and that double joint stiffhess and elbow stiffness increased with shoulder torque.

Simulations of the endpoint stifiess comparing these relations were performed. It

was found that, in the force directions studied in this expriment, a multivariate relation

between stiffness and joint torque produced more isotropic stiffness at the hand,

increasing the mechanical stability of the ami compared to a univariate relation between

stiffness and joint torque. This is consistent with the requirements for mechanical

stability in a position control task, which are greater than for a force control task. The

results of this study suggest that the central nervous system is able to adaptively regulate

the endpoint impedance of the arm according to the stability requirements of the task.

1 thank my supervisor Dr. Ted M ilner for al1 his support, enthusiasm and

knowledge, which has contributed so much to my understanding of this area of research. 1

would also like to thank Dr. Tony Hodgson and Dr. Shahram Payandeh for their

insightfbl comments and academic support. 1 am very tbanl6ul to al1 of the fellow

researchers in the Biomechanics Laboratory with whom 1 have spent much of the last few

years. 1 would especially Mce to thank Mike Grey, Dr. Etienne Burdet and Rebecca

Brumer. 1 also thank many people in the School of Kinesiology for their wonderfül

friendships and the times we have shared together.

1 owe a lot of thanks to my parents, Derek and York, for their support, fnendship,

encouragement and enthusiasm. 1 would ais0 like to thank al1 of my fnends and famil y

with special thanks to Sarah and Jen. The wonderful people who volunteered their time as

subjects deserve a huge thank-you for their contribution to this study. Finally, and most

importantly, 1 thank Robin for al1 of the wonderful support, eacouragement and love that

she has given me.

Table of Contents

Abstract

Acknowledgements

Table of Contents vii List of Tables

viii List of Figures

Background

Anatomy Mechanical impedance Stretch Reflex

Introduction

Materials and Methods

Apparatus Protocol Timing dunng Experiment

Mechanics Simulations

Results

Position Displacements Endpoint Stiffness Joint Stiffness Joint Stiffness Relation Inertia Joint Damping

Discussion

Endpoint Stiffness Joint Stiffhess Joint Stiffness Relation Inertia Joint Damping

Conclusion

References

List of Tables

Table 1. Single joint muscles of the elbow joint 4

Table 2. Double joint muscles of the shoulder and elbow joints 4

Table 3. Single joint muscles of the shouider joint 5

Table 4. The maximum voluntary contractions 39

Table 5. The intervals over which stiffness was estimated 44

Table 6. Slopes, intercepts and correlation coefficients fiom linear regression of joint stiffiiess tenns with joint torque 46

Table 7. Slopes, intercepts and correlation coefficients from linear regression of joint stiffness tenns with joint torque, when stiffness is allowed to Vary with both elbow and shoulder torque 50

Table 8. hertial parameters 62

Table 9. The intervals over which damping was estimated 63

Table 10. Slopes, intercepts and correlation coefficients calculated by performing linear regression on joint damping terms with joint torque 65

vii

List of Figures

Figure 1. The apparatus 23

Figure 2. Subject posture, coordinate h e , anthropometric parameters and experimental parameters 25

Figure 3. The eight directions in which the joystick displaced the subjects hand 27

Figure 4. The force perturbation used in the study 29

Figure 5. The stiffness ellipse 32

Figure 6. The mean peak displacements 40

Figure 7. The endpoint stiffness of Subject A 42

Figure 8. The four parameters characterizing hand stiffness 43

Figure 9. Joint stiffness plotteci against shoulder and elbow torque 45

Figure 10. Examination of relations seen due to constraint between shoulder and elbow torque 47

Figure I l , Elbow and double joint stiffhess are related to shoulder toque 48

Figure 12. Shoulder joint stiffness is related to elbow torque 49

Figure 13. The difference between the shape and orientation of the simulated stiffness ellipses and the measured stiffness ellipses 53

Figure 14. The difference between the size of the simulated stiffness ellipses and the measured stiffness ellipses 54

Figure 15. Cornparison of the stiffness ellipses calculated using the two different joint stiffness relations 56

Figure 16. Differences in the maximum eigenvalue between the univariate and the bivariate relations 58

Figure 17. Differences in the minimum eigenvalue between the univanate and the bivariate relations 59

Figure 18. Differcnces in the size of the ellipses between the univariate and the bivariate relations 60

Figure 19. Changes in the shape of the ellipses between the univanate and the bivariate reIations 6 1

Figure 20. The relations between joint damping terms and joint torque for al1 subjects 64

Figure 21. The relations between joint damping terms and joint torque for Subject B 66

Figure 22. The change in joint damping estimates when inertia is varied 68

Figure 23. The change in joint damping estimates when joint stiffness is varied 69

Figure 24. The relations between joint damping terms and joint torque for al1 subjects when joint damping was estimated with joint stifiess over 200 ms of the perturbation 71

Background

Hurnans can move quickly and interact with their environment while maintaining

postural stability. Stability can be defined as a system's ability to retum to the

equilibrium state aller small perturbations have been applied. Maintaining postural

stability while interacting with the environment is something which humans are able to

remarkably well. It has k e n suggested that inherent properties of the neuromuscular

system are largely responsible. Knowledge of the mechanisms underlying this ability to

interact with external objects in a stable manner is essential for understanding

neuromuscular control. This body of knowledge can then be used for improving the

control of robotic devices such as industrial robots, prosthetics, tele-robotics and haptic

interfaces. It can also be used for providing rehabilitation to patients with neurological

and muscular deficits.

Since the late 19 '~ century there has been a considerable research effort in both

characterizing the mechanical properties of the musculoskeletal system and

. understanding how the central nervous system controls it. The research has shown that

the mechanical impedance of muscle, or its resistance to movement, is of considerable

importance in the both in the control of posture and movement (Hogan, 198%).

However, few studies have examined multi-joint tasks involving the production of

isometric force. Isometric tasks are characterized by the application of force without

extemal motion. Such tasks include the maintenance of limb position while opposing an

external load. This study wiH investigate the mechanical impedance of the human arm

during such an isometric task.

Anatomy

The human arm is a complex mechanical system with at least 9 degrees-of-

freedom (independent axes of limb motion) not including the fingers. Five major joints in

the a m produce these movements.

The pectoral girdle consists of two pairs of bones, the scapulae and the clavicles.

The medial end of the clavicle, at the sternoclavicular joint is the only articulation of the

a m with respect to the axial skeleton. The scapula floats posteriorly and laterally to the

ribs and attaches to the clavicle at the acromioclavicular joint (Engin, 1980). The pectoral

girdle has three degrees-of-ffeedom, which are protraction/retraction,

eIevation/depression, and upward/downward rotation of the scapula.

The glenohumeral joint attaches the proximal head of the humerus to the scapula.

The head of the humerus is attached to the shallow glenoid fossa of the scapula by several

ligaments, although the stability of the joint is mainly due to the surrounding musculature

(Crouch, 1985). This permits a very large range of movement in three degrees-of

freedom: flexiodextension, abductiodadduction, and medialnateral rotation.

The attachent of the distal end of the humerus to the proximal ends of the radius

and ulna forms the eibow joint. There are two articulating surfaces that limit the

movement of this hinge joint to flexion and extension of the foream. The head of the

radius articulates with the capitulum of the humerus on the lateral side whereas medially

the trochlear notch of the ulna clasps the trochlea of the humerus. The olecranon of the

ulna fits into the olecranon fossa of the humerus when the foreatm is hl ly extended

preventing any hyperextension at the elbow.

The ulna and the radius are comected proximally at the head of the radius and the

radia! notch of the ulna, along their shafis by the interosseous membrane, and distally at

the ulnar notch of the radius and the head of the ulna. Together these three attachments

produce a one degree-of-freedom joint whereby pronatiodsupination of the hand occurs

by rotation of the radius around the ulna (Crouch, 1985).

The wrist joint is a two degree-of-freedom joint created by the articulation of the

distal ends of the radius and ulna with the convexity formed by the scaphoid, lunate and

triquetrium bones of the carpals (Crouch, 1985). The movements at this joint are

flexiodextension and abduction/adduction of the hand. The other carpal bones are the

pisiform, hamate, capitate, trapezoid and trapezium. Motion among al1 of the bones in the

carpals. especially between the proximal and distal rows, contributes to the movement

seen at the wrist.

The musculature of the arm contains muscles that span single joints and muscles

that span two joints. The former are referred to as uniarticular or single joint muscles and

the latter are called biarticular or double joint muscles. In the tables below each muscle is

3

listed as a single joint shoulder muscle, a single joint elbow muscle or as a double joint

muscle (one which crosses both the elbow and the shoulder joint). Some double joint

muscles cross both the pectoral girdle and the glenohurneral joint. However, these will be

listed as single joint muscles for the purpose of this study. The movement of the scapula

and the glenohurneral joint can be modeled as a single joint (Lenarcic and Umek, 1994).

Muscles with their primary action at the wrist joint which cross the elbow joint due to

origins on medial and lateral epicondyles of the humerus have not been listed in Table 1 .

These muscles could act as weak single joint elbow flexors and extensors when the wrist

joint is unable to move. Al1 tables are adapted from Anthony (unpublished), and Tortora

Muscle Origin Insertion Function Anconeus Lateni epicondyle of hurnerus Olecranon of ulna Extension of F o r m Medial H ~ r d of Triceps Middle posterior surface of Olecranon of ulna Extrnsion of Foream

humerus btcral Head of Triceps Upper latenl and posterior Olecranon of ulna Extension of Foream

surface of hurnsrus Brachiondialis Media1 and latrnl borders of Supcrior io styloid p r o c s of Fiexion. semipronation and - .

distal end of hurnerus ndius semisupinatic& of forrann Bnchialis Disml anterior surface of UInar tunerosity and coranoid Flexion of Formm

humerus process of ulna Supinator Latenl epicondyle of humems Latenl surface of proxinial Supination of F o r a m

and supinator crest of ulna 113 of ndius Pronator Tcm Mcdial epicondyle of humerus. Midlateral sudiice of radius Pronation and Fiexion of

connoid process of ulna forearm

Table 1. Single Joint Musclts of the Elbow Joint

Muscle Origin Insertion Function Long Head of Triceps Infnglenoid tubercle of scapula Olmranon of ulna Extension of fomnn. -

extasion of humcms Shon Htrd of Biceps Bnchii Supnglenoid tubcrcle of scapula Radial tuberosity and Flexion and supination of

bicipital aponeurosis foream. flexion. horizonral flexion of humerus

Long Head of Biceps Bnchii Concoid process of scapuia Radial tuberosity and Flexion and supination of bicipital aponeurosis foream. flexion. horizontal

flexion of humcrus

Table 2. Double Joint Muscles o f the Shoulder and Elbow Joints

Muscle Origin Insertion Function Subclavius First rib Clavicle Depresses clavicle Pectorrilis Minor Lateral surface of third to fifth Coracoid proccss of scapula Protnction. depression a n d

rib domward romtion of scapula Sern tus Antcrior Lateral surface of upper Costal surface o f medial Prouaction and upward

ei&t/ninc nbs border of scapula rotation of scapula T n p c ~ i u s Superior nuchal line. rxternal h e n l third o f claviclc, Retraction. elevation, upward

occipital protuberance. acrornion pmcess. uppcr rotation and depression o f l i ~ rnen tu rn nuchae and spinous bo rda of spine of scapula. uiapula proctss of C7-T 1 2 base of spine o f scapula

k v a t o r Scapulae Transverse process o f C 1 €4 Medial border o f scapula Elevation o f scrpula superior t o spine

Rhomboideus Major Spinous p r c m o f T2-TS Medial border o f scapula Retraction and downward inf&or to spine rotation

Rhomboidrus Minor L ipnen tum nuchae. spinous Mrdial end o f spine of Rctnction and downward p m e s s e s of CI-T I scaputa rotation o f scapula

Pr~tora l i s Major Clavicle. sternum. costal Grrater turbercle and Rexion. horizontal flexion. caniiagc of 2& IO 6'" ribs intenubercular sulcus o f and rncdial rotation o f

humcrus humerus h i s i r n u s Dorsi Spinous processes o f T7-Tl'. Intertubercular s u k u s o f Extension. horkzontal

thomcoiümbar fascia. cresr of humerus extension. adduction and ilium. I owa 4 ribs medial rotation of humerus

Teres Major Infcrior angle of scapula lntertubercular sulcus o f Extension. horizontal humerus extension. adduction and

medial rotation of hurnems Teres iMinor Inferior lataal border of scapuia Gfcater t u ~ r c l e of humerus Horizontal extension and

latenl rotation of humerus Infraspinatus Infiaspinous fossa o f scapula Greater t u rk r c l c of humrrus Horizontal extension and

latcnl rotation of humems Subscapularis Subscapular fossa o f scapula Lesser turbercle of humerus Mcdial rotation of humerus Supraspinatus Supraspinous fossa of scapula Greater turbcrcle of humerus Adduction of humcrus Anterior ûcltoid Laxml 1 /3 of claviclc Deltoid tuberosiry o f Flexion, horizontal flexion.

humcrus medial rotation and abduction of humerus

lMiddie Dcltoid Acrornion o f scopula Deltoid tukros i ty o f Abduction of hurnerus humerus

Posterior Deltoid Spine of scapula h l t o i d tuberosity o f Extension. horizontal humems extension, laterril rotation and

abduction of humerus Concobnchial is Coracoid ~ r o c e s s o f swr>ula Antero-medial surface o f Rexion. horizontal flexion

humerus and adduction o f humcrus Table 3. Single Joint Muscles of the Shouldcr Joint.

Mechanical Impedance

Mechanical impedance, or the resistance to movement, is composed of three

properties: stifiess, damping and inertia. Stifiess (K) is the resistance to a

displacement. It is defined by the change of force (3F) produced by a change in length

(at) divided by that change in length:

Muscle stiffness acts like a spring to resist changes in length. Darnping or Viscosity (B)

is the resistance to rnovement at uniform velocity. It can be defined by the change in

force (3F) produced by a change in velocity (av ) divided by that change in velocity

Muscle damping acts like a damper to resist movement of the muscle in proportion to

velocity. The term viscosity was used primarily to describe this parameter in most past

research, however it has been suggested more recently that the term damping is more

appropriate (Zatsiorsky, 1997). Inertia (T) is the resistance of a body to acceleration. It

can be defined as the change in force (F) produced by an acceleration (a) divided by that

acceleration:

The inertia is related to the mass of the object.

Joint impedance is the resistance to rotational motion about a joint. For stiffness

and viscosity, the linear terms are replaced by the rotational equivalents in the equations

of motion. Inertia (mass) is replaced by the moment of inertia, which is a weighted

measure of the distribution of the mass of an object about its center of rotation.

The impedance of several joints contributes to the resistance to movement of the

a m . I f we consider the mechanics of the endpoint of the am, we must take into account

that force and motion are vector quantities with direction and magnitude. If linearity of

the impedance is assumed, they can be represented in matrix form (Hogan, 1985b).

Stimiess in the horizontal plane (K) is represented by four terms:

where K, is the stiffhess along the X axis due to a displacement along the X axis, K x y is

the stiffness along the X axis due to a displacement along the Y axis, K, is the stiffness

along the Y axis due to a displacement alonp the X axis and K>? is the stiffness along the

Y avis due to a displacement along the Y axis. Damping (B) is represented in a similar

marner. The combination of stiffness and viscosity (damping) is generally termed

viscoelasticity.

Mechanical hpedance of Muscle

Muscle has viscoelastic properties (Gasser and Hill, 1924; Huxley and Simmons,

197 1 ). However, the viscoelasticity o f muscle is non-linear and depends on a multitude

of factors such as activation, velocity, length, and prior history (Keamey and Hunter,

1990).

Muscle fiber stiffness is made up of two types: passive and active. Passive

stiffness is produced by elastic tissue in the muscle fiber. Passive stiffness is quite high

at extremes of muscle fiber length but generally low in the normal physiological range of

lengths. Active stiflhess is produced by the cross-bridges. Cross-bridges are the active

force generating connections between the actin and myosin molecules. When a length 7

change is imposed upon a muscle fiber, the cross-bridge has been suggested to stretch out

producing a resistive force. Resisting force is produced by the deformation of the myosin

head (Dobbie et al., 1 998), detachment and reattachment of myosin heads to actin

(Lombardi, Piazzesi and Linari, 1992) and rapid force recovery From the working stroke

of the attached myosin head (Huxley and Simmons, 197 1). The active stiffness is

dependent on the number of attached cross bridges in the muscle.

Muscle also has viscous properties (Cecchi, Griffiths and Taylor, 1986). The

viscosity of a muscle fiber is the rate at which force changes with velocity. By definition,

therefore, it does not depend on a fixed number of cross-bridges. It depends on how the

force each cross-bridge produces, and the total number of attached cross-bridges, Vary

with velocity. Muscle viscosity is largest near zero velocity and decreases as velocity

increases.

Many muscles produce torque about a joint. Al1 the muscles that have actions

across that joint will contribute to the viscoelnsticity. Joint stiffness has been shown to

increase linearly with joint torque under isometric conditions (Cannon and Zahalak,

1982; Hunter and Kearney, 1982; Weiss, Hunter and Keamey, 1988). Joint viscosity also

increases linearly with joint torque under isometric conditions (Hunter and Kearney,

1982; Weiss, Hunter and Kearney, 1988). The moment of inertia about a single joint

remains constant with respect to joint torque (Hunter and Kearney, 1 982).

Joint elasticity is non-linear in response to displacements. Stiffness is highest

with small displacements and decreases exponentially as displacement size increases

(Kearney and Hunter, 1982; MacKay, Crammond, Kwan and Murphy, 1986). This

occurs because cross-bridge bonds are broken as a muscle is stretched (Huxley and

Simmons, 197 1).

At the joint level, muscles can have antagonistic functions. Two active muscles

producing torques in opposing directions (cocontraction) may produce no net torque.

However, the impedance of the joint is the sum of the impedance of al1 muscles (Hogan,

1984). Cocontraction allows the impedance of a joint to Vary independently of joint

torque (Hunter and Kearney, 1990; Milner, Cloutier, Leger and Franklin, 1995).

Muscle-Tendon Mechanical Impedunce

In intact muscles, the muscle fibers are in series with the tendon. When the

muscle is stretched, the amount of stretch of the muscle fibers will depend on the

respective stiffness of the muscle fibers and the tendon (Grifiths, 199 1 ). As the

activation of a muscle is increased, the stifkess of a muscle increases. At the same time

the muscle will shorten by gradually stretching the tendon, causing the tendon stiffness to

increase (Ito, Kawakani, Ichinose, Fukashiro and Fukunaga, 1998). Eventually the tendon

stiffiiess reaches a constant level and fùrther increases in activation only increase muscle

stiffness. This means that at low levels of muscle activation, much of the stretch applied

to the whole muscle will occur in the muscle fibers. However, as activation is increased,

prog-essively more of the stretch will occur in the tendon than in the muscle. At fidi

activation the stiffness of the muscle and tendon are approximately the same (Cook and

McDonagh, 1996) although particular muscles would Vary in this regard. Overall, the

stiffness of the muscle and tendon complex will increase as the activation of the muscles

is increased.

Muïti-Joint Mechanicd Impedance

The multi-joint impedance was first examined by Mussa-Ivaldi, Hogan and Bizzi

( 1985), who developed a method for determining the magnitude of the passive stifkess,

at the hand, in different postures in the workspace. They perturbed the hand of the subject

in eight directions and measured the force once the hand was at rest in the new posture.

The change in force in response to the displacement not only had a component opposite

to the direction of displacement but also had a component along the perpendicular

direction. Equation [4] was then used to calculate the endpoint stiffness of the hand,

which could be represented as an ellipse afier removing the effects of non-conservative

forces. Similar to single-joint stiffness, the endpoint stiffness decreases with increasing

perturbation displacement (S hadmehr, Mussa-Ivaldi and Bizzi, 1 993).

The postural behavior of the st i ffness has strong directional c haracter (anisotropy)

and varies in a regular way with workspace position (Mussa-Ivaldi et al., 1985). The

endpoint stifmess was highest along the line joining the hand and shoulder (major axis)

(Mussa-Ivaldi et al., 1985; Tsuji, Morasso, Goto and Ito, 1995). The stiffness was lowest

along the perpendicular axis (minor axis). As the hand was positioned farther fiom the

body, the stiffness became more anisotropic; the length of the major axis increased and

the length of the rninor axis decreased (Mussa-Ivaldi et al., 1985; Flash and Mussa-lvaldi,

1990). The direction of the major axis also rotated slightly (1 5O) in the clockwise

direction (for the right am). As the posture moved laterally from a position in front of the

body. the direction of the major axis rotated clockwise ( 6 5 O ) such that it continued to be

oriented towards the shoulder (Mussa-Ivaldi et al., 1985; Flash and Mussa-Ivaldi, 1990).

The endpoint viscosity and inertia also Vary with posnire (Dolan, Freidman and

Nagurka, 1993; Tsuji, et al., 1995). The viscosity was rotated counter-clockwise slightly

(5.) with respect to stifiess and changed similarly to stiffness with changes in posture.

The inertia was found to be aligned with the forearm in al1 postures (Tsuji et al., 1995).

Tsuji and colleagues (1 995) investigated the effect of grasping the handle of the

mechanical device used to perturb the subject as compared to being coupled passively to

the handle. When the subjects grasped the handle, the size of both the stifiess and

viscosity increased. This was likely due to the increased activation of wrist muscles, used

in grasping the handle, which also cross the elbow joint. The increased activation would

increase the viscoelastic impedance of the elbow joint resulting in increased endpoint

stiffness and viscosity.

Stretch Reflex

The neuromuscular response to displacements cornes from the intrinsic muscle

mechanics and reflex muscle activation. The stretch reflex produces a short latency

1 I

response to displacements of a joint. The stretch reflex operates through a rnonosynaptic

comection between muscle spindle receptors and motor neurons innervating

homonymous and synergistic muscles (Liddell and Shemngton, 1924). The stretch reflex

also produces longer latency responses via polysynaptic pathways involving the spinal

cord and the cerebral cortex (Mathews, 199 1 ).

The muscle spindle is a receptor that is sensitive to both magnitude and velocity

of stretch of a muscle (Mathews, 1964). The sensitivity of the muscle spindle to these

inputs can be controlled by the central nervous system through gamma motor neuron

innervation of the intrafüsal fibers (Hunt and Kuffler, 195 1 ; Mathews, 1964). The

modulation of this sensitivity allows for control of the gain of the stretch reflex system.

The short latency stretch reflex circuit consists of a simple feedback pathway

from the muscle to the spinal cord and back to the muscles at the joint (Liddell and

Sherrington, 1924). Group Ia afferent fibers from the muscle spindles transmit action

potentials to the dorsal horn of the spinal cord. There they make monosynaptic excitatory

connections with motor neurons exciting the homonymous and synergistic muscles. The

Ia fiber also inhibits motor neurons of mtagonist muscles through a disynaptic pathway,

involving the Ia inhibitory interneuron.

The stretch reflex response varies with the initial force level, or muscle activation,

before dispIacement. The electromyographic (EMG) response of the stretch reflex

increases with background torque (Keamey and Hunter, 1983; Marsden, Merton and

Morton, 1976). Up to 50% maximum votuntary contraction (MVC), the force produced

by the stretch reflex increases linearly with background torque (Carter, Crago and Keith,

1990). In ankle flexors, this force then declines as background torque increases (Sinkjaer,

ToFt, Andreassen and Hornemann, 1988; Tofi, Sinkjaer, Andreassen and Larsen, 199 1 ).

However, in the biceps brachii this force increases linearly to torques near MVC (Stein,

Hunter, Lafontaine and Jones, 1995)

The stretch reflex also varies wi th the amplitude and veloci ty of a perturbation.

Increasing the amplitude of perturbation will increase the reflex EMG (Lee and Tatton,

1982; Smeets and Erkelens, 199 1 ; Stein and Kearney, 1 995) and reflex force (but not

stiffness) (Sinkjaer et al., 1988; Stein and Kearney, 1995). As the velocity of the

perturbation increases, the reflex EMG and reflex force increases (Cody and Plant, 1989;

Gielen and Houk, 1984; Gonlieb and Agarwal, 1979; Stein et al., 1995).

The multi-joint actions of the stretch reflex are more complicated. Double joint

muscles and dynamic interactions provide coupling between joints which need to be

controlled. Muscles can be activated by the stretch reflex even during perturbations that

move the joint in the direction of action. Lacquaniti and Soechting (1 986a) found that the

biceps muscle could be activated when the elbow was flexed as a result of torque applied

at the shoulder. Although the biceps is a double joint muscle, it appeared that the biceps

muscle length was shortened overall during the flexion of the elbow. Brachialis and

brachio-radialis, which are single joint elbow flexors, have been s h o w to have similar

reflex activation patterns (Lacquaniti and Soechting, 1986a,b). The activation of the

elbow flexors is not entirely dependent on motion of the elbow joint alone but also on the

motion of the shoulder (Lacquaniti and Soechting, 1 986b; Soechting and Lacquaniti,

1988). Reflexes, which are elicited at one joint and act at another, are called

heteronyrnous reflexes.

The stretch reflex latency consists of delays due to neural and muscular

conduction of action potentials, synaptic transmission and muscle excitation-contraction.

The EMG response in the shoulder and elbow muscles occurs at a delay of approximately

20 ms afier the onset of a perturbation (Smeets and Erkelens, 199 1 ; Stein et al., 1995).

The short latency reflex response occurs through a monosynaptic pathway and produces

EMG that occurs during the intervat 20 and 50 rns following the onset of the perturbation.

The force produced from the short latency reflex starts at 50 ms and peaks at

approximately 70 ms afier onset (Stein et al., 1995).

A long latency component of the stretch reflex acts through the cerebral cortex

and produces EMG during the interval 50-75 ms after the onset of stretch (Gielen,

Ramaekers and van Zuylen, 1988; Smeets and Erkelens, 199 1). The force produced by

this increased activation would start at approximately 80 ms, peaking by 100 ms,

following a stretch.

The short and long latency responses to multi-joint perturbations appear to have

different stimuli and effects. The short latency reflex responds to changes in the

kinematics of the joint although this is modulated by input from other joints (Lacquaniti

and Soecht ing, 1 986b; Soech:ing and Lacquaniti, 1 988). Heteronymous input from wrkt

flexors has been shown to affect short latency reflexes of biceps and triceps muscles

(Cavallari and Katz, 1989). Generally, flexor activity at one joint has an excitatory effect

for flexors at another joint and an inhibitory effect for extensors (Cavallari and Katz,

1989; Lacquaniti and Soechting, 1986b; Smeets and Erkelens, 1991). The short latency

reflex response increases linearly with pre-load activity in al1 muscles, even those not

shortened by the displacement (Smeets and Erkelens, 199 1 ). Because short latency

reflexes respond pnmarily to the kinematics of the joint that the muscle crosses, the

reflex response is generally excitatory in response to a stretch and inhibitory in response

to a shortening of the muscle. However, this simple response can be modified by

heteronyrnous reflexes fkom other joints. For example short latency reflexes can be

absent in muscles stretched by the perturbation (Lacquaniti and Soechting, 1986b) or

present in muscles not effected by the stretch (Smeets and Erkelens, 199 1 ). In the case of

a ball-catching task, Lacquaniti and Maioli ( 1987 and 1989) have even shown that the

short Iatency reflex can undergo reversal of sign. in rhis case, both the extensor and

flexor reflexes are coactivated to build up the resistance to the disturbing effect of the bal1

catching. However, the general findings suggest that the short latency reflex counteracts

the effect of the perturbation at the local joint.

The long latency reflex, on the other hand, appears to produce a coordinated

response to the perturbation such that the overall stability of the limb is maintained

(Gielen et al., 1 988). For example, the long latency reflex EMG has been shown to be

more highly correlated with the net torque change about the joint than with kinematic

variables (Lacquaniti and Soechting, 1 986b; Soechting and Lacquaniti, 1 988). Because o f

this, the sign of the long latency reflex can even be opposite that of the short latency

reflex (Soechting and Laquaniti, 1988). Like the short latency reflex, the Iong latency

reflex increases with the pre-load activity of the perturbed muscles (Smeets and Erkelens,

1991).

Viscoelastic properties of the human a m are essential for control of posture and

movement (Hogan, 1985b). They determine how the arm reacts to perturbations, interacts

with the environment, and stabilizes the end of movements. Viscoelastic properties of the

a m are dependent on the activation level of the muscles acting at the joints and the

reflexive gains of sensory receptors (Rack, 198 1). By changing the muscle activation and

feedback gain, humans can Vary the a m ' s viscoelasticity to adapt to a variety of

conditions. The arm can remain stable during many different tasks by changing its

viscoelasticity.

Understanding the variation of viscoelastic parameters of the a m dunng different

tasks is essential for many applications. The information can be used to further our

understanding of neuromuscular control and modeling of the a m . It is also necessary for

the design and implementation of prosthetic devices, telerobotics and haptic interfaces for

applications such as teleoperation. The combined viscoelasticity or impedance of both the

operator and a manipulated object determines the stability of the total system (Hogan,

198Sa). When one or both of these parts of the system can be actively modified,

knowledge of the functional dependence and variability of the viscoelastic parameters is

essential for system stability analysis.

Stability of the human a m can be defined as the resistance of the arm to

disturbances away from its original position. If a finite disturbance is applied to the hand

for a finite duration and the resulting force acts to restore the hand to its original position,

the a m has postural stability. The a m will have greater stability if the restoring force is

larger (e.g. the stiffness is larger). Therefore, the arm may not be equally stable in al1

directions. More formally, given the convention found in related neurophysiological

research of representing muscle stiffness as positive, the force field is considered stable if

al1 of the eigenvalues are greater than zero (Ogata, 1970).

The a m can be used to produce forces on extemal objects by controlling the torque at

each joint. To increase force, joint torque must be increased by increasing the activation

of the musculature- At the sarne time, the reflex feedback gain may increase. The

viscoelastic parameters change with activation of the muscles and with reflex feedback

gain. Both joint stiffiiess and joint viscosity have been shown to increase linearly with

joint torque (Akazawa, Milner & Stein, 1983; Cannon and Zahalak, 1982; De Serres &

Milner, 199 1 ; Hajian & Howe, 1994; Hunter and Keamey, 1982). The joint stiffness is

composed of two parts: intrinsic stiffness and reflexive stiffness. Intrinsic stiffness is the

elastic property of the muscle, without reflex feedback. Reflexive stiffness is due to the

increased activation of a muscle by the stretch reflex. Perturbations that ramp and hold

the position of the limb away from the original posture measure contributions from both

intrinsic and reflexive stiffness.

Previous studies have shown that single joint impedance shows task dependence

(Akazawa et al., 1983, Doemges and Rack, 1 WZa & b). The two tasks examined were

force control and position control. During force control tasks, the subject generates the

desired force in a particular direction and the environment is equally stiff in al1 directions.

An example of this is pushing on a fixed handle. On the other hand, during a position

control task the environment exerts a force against the subject who must maintain a target

position, while opposing this force- The subject must also stabilize the limb suFiciently

to maintain this posture. An example of this is holding a pole in a flowing river. The

stiffness has been shown to be higher in the position control tasks (Akazawa et al., 1983,

Doemges and Rack, 1992a & b). Doemges and Rack (1 992a & b) also showed that the

long latency reflexes are larger in a position control task. Consequently, higher reflex

stiffness couId have k e n expected than that occumng in a force-control task at similar

force levels.

Several studies have examined multi-joint viscoelasticity during force control

tasks in the a m (Gomi & Osu, 1998) and the f'nger (Milner & Franklin, 1998). Single

joint stiffness during multi-joint force control has been shown to increase lineariy with

joint torque for both the elbow and shoulder. The double joint stifiess increased linearly

with elbow torque. Single joint damping of both the shoulder and elbow joints increased

Iinearly with the respective joint torque. Double joint damping increased linearly with

elbow joint torque, although the correlation coefficients were small (Gomi & Osu, 1998).

Mclntyre et al. (1996) investigated joint stiffness during a position control task for

force levels up to 60 N produced along only one axis. Gomi and Osu ( 1998) detemined

both joint stiffness and joint viscosity during force regulation tasks up to 20 N. However,

the average human is capable of producing forces at the hand of up to 200 N. The limited

range of force directions or levels previously examined limits the ability of the

experimenters to accurately determine the trends of viscoelasticity with joint torque. in

order to fully characterize how joint viscoelasticity changes with joint torque, endpoint

forces of up to 30% of the subject's maximum voluntary contraction were used in the

present study.

Previous work of Gomi and Osu (1 998) found that joint stiffness increased

linearly with joint torque in a force control task. in the position control task, the stability

requirements are higher. In single joint studies, it has been shown that the subjects

cocontract their muscles in order to increase the stability while producing the sarne

amount of joint torque. This study was designed to investigate differences in multijoint

impedance during position control compared to force control. It was hypothesized that if

cocontraction is used during position control, changes would be expected in the relations

found between joint stiffiiess and joint torque that would be consistent with an increase in

stability. Examining how the central nervous system adapts impedance in a position

control task will provide insight into the arnount of control it has over the endpoint

stability of the a m .

While the original intention of this thesis was to separate the intrinsic and

reflexive contributions to multi-joint impedance. the manipulandum controller was not

capable of stable PD control at the high gains required for rapid responses. Rather than

quick displacements, force perturbations were used for estimating impedance. However,

because stiffness depends on displacement amplitude (Shadmehr, Mussa-Ivaidi and

Bizzi, 1993), use of constant amplitude force perturbations would lead to biased estimates

of stiffness in the muIti-joint system. A method of shaping the force perturbations to

avoid biased stiffhess measurements was developed. The multi-joint impedance of the

human a m during a position control task was measured using these force perturbations at

force Ievels up to 30% of the subject's maximum voluntary contraction.

Apparatus

A two degree-of-fieedom, computer-controlled, robot manipulandum (joystick)

was used to measure the endpoint impedance of the subject's m. The subject's hand was

fixed to the joystick handle while performing position control tasks. The joystick

perturbed the subject's a m in two-dimensional spherical coordinate space using shaped

force pulses. Displacements and forces produced at the hand in response to the

disturbances were recorded with a computer data acquisition system.

The joystick was attached to a gimbal mechanism that allowed two torque motors

to apply pianar forces to a handle (Figure 1). Two axial air gap DC servomotors

(MAVILOR MOTORS MT 2000) were mounted at right angles to one another. Each

rnotor shaft was extended to the gimbal joint that comected the motor axis independently

to the handle shaft (Adelstein, 1989). The distance from the center of the gimbal joint to

the center of the handle was 26.5 cm.

The instrumented joystick operated under control from a computer. A six-axis

force-torque sensor (AT1 FT 3 175, precision O. 1 N) was located between the handle and

the handle shaft. The ends of the motor shafis opposite the gimbal joint were

instnimented with bnishless resolvers (MICRON Part No. 1 l), which measured any la r

4 monitor

resolver

torque m ofor

Figure 1. Thc apparatus. The subject is anached to the joystick through a splint that is bolted to the top of handlc dircctly abovc the force transducer. The end of the handle is capable of movcment on a spherical

surfacc sirnilar to thc horizontal plane of this figure for small movcrnents. Forccs arc applicd to the handle by thc torquc motors.

position. The output from the resolvers was input to a resolver to digital convertor (CS1

168 4800 16 bit) that outputs digital position (resolution 0.005O) and analog velocity

signals (resolution 0.07°/s) for both axes. The velocity signals were filtered before digital

conversion using a one-pole analog lowpass RC filter (cutoff frequency of 100 Hz). The

23

digital angular position signals for each mis, and the endpoint force and torque signals

were transmitted to the computer over parallel interfaces. The analog velocity signals

were acquired by a 16-bit A/D converter (National Instruments AT-MIO- 16X). Al1 of the

signals were sampled at 1 .O kHz by an iBM 486 computer. An analog control signal

specimng the torque for each motor was sent from the computer at a 1.0 kHz update rate

to a PWN curent amplifier that powered the torque motor.

Protocol

The hand endpoint impedance of six healthy right-handed subjects was rneasured

using shaped force perturbations. Three male and three female subjects (age range: 2 1 -

42) were recruited from colleagues at Simon Fraser University. The experimental

protocol confonned to the guidelines of the Helsinki Convention and was approved by

the Simon Fraser University Ethics Review Commi ttee- Subjects gave infomed consent

to the procedures. Subjects were present for three days of testing. The first day was a set-

up day that was used both to accustorn the subject to the experimental apparatus and to

determine the parameters that would be used for the force perturbations. The next two

days consisted o f the actual experiments with identical protocols.

The subject was seated in an adjustable chair with his or her right hand firmly

attached to the end of the joystick. The trunk of the subject was restrained in the chair

with straps, limiting movement of the subject's nght a m to the shoutder (glenohumeral

joint and shoulder girdle) and elbow joints. The subject's right hand was splinted with a

24

ThermoplastTM cast that was bolted to the joystick handle. The wrist joint was also

splinted to prevent any movernent at that joint. The subject's a m was supported in the

horizontal plane, level with their Rght shoulder, using a sling (suspended from above)

Iocated proximal to the elbow joint.

The mechanical impedance and reflex activity of the a m were examined at a

single posture within the reachable workspace. In this posture, the shoulder joint was at

45 degrees and the elbow joint was at 90 degrees as illustrated in Figure 2. The actual

posture can be seen in Figure 1.

force , r directions

Figure 2. Subjject posture, coordinate fi-ame, anthropomctric parametcrs and experimental paramcters are ilIustntcd. The force directions used in this expcrïment are shown with the light gray arrows originating at

the hand.

The subject produced five isometric force levels in each of four different

directions. The four directions of force were in the 45, 135,225, and 3 15 degree

directions in the horizontal coordinate h e shown in Figure 2. On the first day of the

experiment, maximum voluntary contractions (MVC's) were measured for each subject

in each of these four directions by clamping the handle of the manipulandum in a fixed

position. The force levels used in subsequent experiments were percentages (0%

(passive), 7.5%, 15%, 22.5% and 30%) of the MVC in a given direction.

The subject produced a given force dunng the experiment while controlling joystick

position. The joystick generated a force, which had to be matched by the subject's equal

but opposite force. This force was gradually ramped to the desired level over 4 seconds.

The motion of the joystick was not constrained during this time. The subject had to

maintain zero net force in the direction perpendicutar to the joystick force in order to hold

the correct position.

At the begiming of the trial the subject positioned his or her hand and produced the

desired force: the joystick then perturbed the subject's hand in one of the eight directions

shown in Figure 3. The subjects were instructed not to respond voluntarily to the

displacement. Joystick and target positions were displayed on a computer screen. Once

the subject had positioned the joystick at the desired position, the cursor, representing

joystick position, changed colour. The torque motors then gradually ramped up to the

target force while the subject resisted. The subject then stabiiized the joystick at the

target position. The target window was 4mm square. A stabilization period between one

and three seconds was randomly assigned for each trial. Once the subject's hand position

had remained within the target window for the required time, the screen display was

frozen and the joystick produced an open loop force perturbation in one of the eight

directions. The resulting displacements, velocity, resisting force of the hand and EMG

were recorded 325 ms prior to and 15 15 ms subsequent to the onset of the displacement

to monitor for any voluntary reaction to the displacement.

Legend

Force Perturbation

Resul tant Position Displacement

center the

Initial Hand Position

Figure 3. The eight directions in which the joystick displaced the subjects hmd. The star in the indicates the hand position. The lighter outside arrows rcpresent the force pcrturbations produced by joystick. The dark middlc arrows indicate the position displacements that resultcd from the forcc

perturbations. Note that the resulting displacement is not always colinear with the force.

To keep displacement amplitude constant, the magnitude of the force

perturbations were adjusted for force direction, force level and perturbation direction to

compensate for differences in hand stifiness. Equal sized force perturbations applied to

the hand in various directions will nonnally perturb the arm different distances in each

direction. This in turn will lead to biased estimates of stiffness because stiffness has been

27

shown to Vary with displacement amplitude (Shadmehr et al, 1993). Values were chosen

that would approximate a ramp displacement of approxirnately IOmm in al1 conditions.

The force perturbations were adjusted for different conditions so as to achieve

fairly constant amplitude displacements. In panicular, the perturbation had two phases, an

initial force pulse and a constant force offset. These can be seen in Figure 4. The initial

force pulse was used to initiate the movement and generally acted against the inertia of

the a m , which dominated the mechanical impedance dunng the eady portion of the

perturbation. As the inertia of the a m depends on the direction of movement, the size of

the force pulse was adjusted according to the perturbation direction. The second part of

the force perturbation is the force offset, designed to hold the subject's hand at a constant

distance frorn the initial position. This offset acted to oppose the stiffness of the subject's

arm so it had to be adjusted to compensate for factors that affected the magnitude of

stiffness. This offset was, therefore, a function of the perturbation direction, force

direction and force level. Determination of how the magnitudes of the force puise and

offset should be adjusted for each of these factors was performed for each subject

separately on the day of preliminary testing.

-20 -10 O 10 20 30 40 50 60 70 80 Time (rns)

Figure 4. The force perturbation used in the study.

Timing during Experiment

Al1 trials during an experiment were performed in a random order. The subjects

performed three trials of each combination of force direction, force level and perturbation

direction on each of the two days. Trials were also recorded using a bi-directiocal force

perturbation not included in this experiment. This amounted to 8 16 trials per day or 1632

trials in total. The subjects were allowed to rest berween any trials, and generally took at

least one 30 minute break during the experiment at some point during each day. Each

recording session generally lasted for 4-5 hours.

Analysis

Mechanics

All of the data analysis was performcd off-line using MATLAB" 5. The

rnanipulandum angular position and velocity were converted to linear position and

velocity of the endpoint. This was done using the following equations fiom Adelstein

(1 989). The variables a and f3 are the angles of the motor shafi measured by the

resolvers, and & is the length of the handle shafi.

cosa - sin p -Tm = Ro

Ji -sin2 a -sin2 /?

sina scosp y,,, = -Ro

JI - sin2 a - sin' P

sina -sin/? -cos2 j? X, = -R ,a ] + R o b [ r o s a - E O S P (I -sin2 a .sin2 p F (I -sin2 a sin2 p r

] V I

cosa -cos p sina - sin p . cos' p ym = - R o a

(I -sin2 a -sin2 py - s in2a -sin2 p y4 ] 181

Endpoint Stiffness Model

The data was modeled using a static mode1 to calculate the endpoint

stiffness in Cartesian space. The stiffness of the ann at the hand was calculated using the

methods described by Mussa-Ivaldi, Hogan and Bizzi ( 1 985). The static mean force and

position vectors were calcuiated over a 320 ms interval before the onset of the

perturbation. The mean position and force vertors were calculated again over a 25 ms

30

interval staning at 300 ms following the onset of the displacement. The displacement and

force vectors, A r and A F were calculated from the difference between the initial and final

values of position and force. These vectors were then used to compute the coefficients of

a 2 x 2 stiffhess matrix, K, fkom the vector equation AF = U r . This is expressed in

matrix notation in equation [4].

The coefficients of the stiffness matrix were calculated for each subject and

condition using forty-eight pairs of difference vectors, from the eight displacement

directions and six trials per displacement direction. The data recorded from the two

experimental sessions were combined for the analysis. Using [4] the stifiess matrix was

de termined using a standard linear least squares method.

The characteristics of the stiffness ellipse were then determined by calculating the

singular value decomposition of the stiffness matrix (Gomi and Osu, 1998):

K = u - s - T ~ (91

where:

The singular value decomposition of the stiffness matrix does not require the calculation

of the symmetric matrix. In the cases of symmetric stifTness matrices, this method

produces the same results as the method of Mussa-Ivaldi et al. (1985). In order to

determine the variation of endpoint stiffness with force level and force direction, four

parameters describing the ellipse were calculated. These parameters were size, maximum

eigenvalue, shape and orientation. They were calculated from results of the singular value

decomposition and are represented in Figure 5. The larger of the ?wo eigenvalues is

shown as the major axis (a, ). The minimum eigenvalue is represented by the minor

axis (a ,,, ). Size was calculated from the area of the ellipse or:

Shape was calculated by dividing the minimum eigenvalue by the maximum eigenvalue.

T h e closer to 1 the shape, the more circular, or isotropie, the ellipse. The closer to O the

shape, the more elongated, directional or anisotropic the stiffness ellipse. The orientation

of the ellipse is the direction of the major axis or:

The stiffness matnx can be visually represented as an ellipse with major axis of a, ,

minor axis of a,, and orientation cp, .

Figure 5. The stiffhess ellipse. The major and minor axes and the orientation of the ellipse are shown.

Joint Mechanics Model

In order to examine the relation between joint stifkess and damping and joint

rorque and compare with previous shidies, the joint mechanics must be calculated.

Initially the endpoint kinematics and kinetics must be converted to joint kinematics and

kinetics using the jacobian transformation matrix (J). Joint position can be calculated

from the equation:

q = J-'r

where q = [::], r = [;] and

The endpoint forces can be converted to joint torque from the equation:

where T = [::] and F = [:] Joint velocity (a) and acceleration (ij) were calculated

from the joint position using a dynarnic optimization method with a smoothing factor of

5.0 x 1 O-''' (Busby and Trujillo, 1985).

The two-link human arm dynamics were modeled using a similar method to that

of Gomi and Kawato (1997). The dynamics were modeled for motion in the horizontal

plane using the following second-order nonlinear differential equation:

[(di + ~( i l .9 ) =t , (q.q.u)++ ,

where r,,, denotes the external torque applied to the joints and si, denotes the torque

generated by the muscles, dependent on position, velocity, and activation (u). 1 denotes

the inertia matrix:

and H denotes the Coriolis-centrifuga1 force vectcrr. In order to estimate the joint

viscosity and stiffness, by applying small displacements, the following equation. which

assumes thet u is constant, was used:

I f we represent joint viscosity (D) and stiffness (R) matrices such that:

where the subscripts 'ss' represent single joint shoulder effect, 'ee' represents single joint

elbow effects and 'se' and 'es' represent double joint effects, then we can rewrite (1 7) as:

aH a1q aH H and 1 (and therefore - and - + - ) can be written in terrns of structural

ail aq as

panmeters (Zi, Zz, and Z3) which are independent of posture:

I I and I2 denote the inertia for each link and Igi and Ig denote the distance from each joint

to the center of gravity for each link.

This allows equation (1 9) to be linearized with respect to the unknown parameters (N):

where p is the parameter vector:

and

el = Ml, e, = Aq2 -

The inertia and stiffness can be estimated independently, using this equation, by

taking advantage of the relative timing of acceleration, velocity and displacement peaks.

The joint stiffness R can similarly be estimated at the time of peak displacement. At this

time the contributions to the joint torque from inertia and darnping are close to zero. The

stiffness was estimated fiom the equation:

AT = iùîq E221

using linear regression over a 26 ms interval around the time of peak displacement for

each subject. The joint s t i f iess matrix was estimated using data from the forty-eight

triais for each condition.

The inertial matrix I was estimated over the first 20 ms afier the onset of the

perturbation. This is at a time when the acceleration is highest and the velocity and

position are still low which means that the relative contributions to the torque from

damping and stiffness are negligible. The inertia was estimated from the equation:

Ar = iq 1231

by performing linear regression over the fint 20 ms of al1 perturbations for each subject.

Joint damping can then be estimated by removing the torque due to inertia and

stiffness. The darnping was estimated over 40 ms as:

AT = D ~ + - A T , +AT,

using linear regression where:

Ar, = W q .

In order to determine the approximate time interval for estimating the joint

damping parameters, the following criteria were used. The ideal time to estimate damping

is when velocity is high and acceleration and elastic force are low. Because inertial forces

are generally large during the perturbation it is critical to choose a time when the

acceleration is low. In particular we should estimate damping when W T for the damping

is maximal and F P T for the acceleration is minimal where T are the dynarnics used to

assess the parameters (Slotine and Li, 199 1). This minimizes the chance that errors in our

estimate of the inertial parameters affect the damping estimates. For inertia:

For damping:

Intervals were, therefore, chosen where the maximum singular value of F P T for

acceleration was low and the minimum singular value of W T for damping was high.

The relation of joint stifkess to joint torque during these position control tasks

was determined by linear regression. Slopes, intercepts and correlation coefficients were

calcuIated.

Simulations

The relation of joint stiffness terms with torque was then examined. By speci@ng

the endpoint forces, this relation was used to calculate joint stiffness, which was

converted into endpoint stifiess. The endpoint or hand stiffness was calculated from the

joint stiffness using the formula: e

where:

The characteris tics of the stifiess ellipses were then calculated as descri bed

previously using the singular value decomposition of the stifiess matrix. Cornparisons of

the stiffness characteristics produced by various stifiess torque relations were

perfomed. The effects of various terms in the measured joint stiffness relation were then

exarnined.

The MVC's in the four force directions used in this experiment were recorded for

each of the six subjects. The values are listed in Table 4.

Force Direction l MVC for Subiects (N)

31 5 1 212 / 179 1 51 1 160 1 268 1 135 Table 4. The maximum voluntary contractions for al1 subjects are listed for the four force directions chat

(deg rees) 45 135 225

were used in this experiment.

Position Displacements

Force perturbations that produced displacements in eight directions were used.

The peak displacements occurred at a mean of 2 12 ms afier the onset of the force

E 141 284 162

A 1 B

perturbation. The peak displacements averaged between 8 and 13 mm for each person.

C 50 39 69

F 93 172 142

136 259 1 54

The displacement data was compared with respect to force direction, force level and

. , D 86 189 1 27

98 204 1 52

displacement direction (Fig. 6) . There were differences with respect to these factors.

However, the means generaIly deviate by less than 1 millimeter from the 10.2 mm

average.

u A B C D E F

Subi-

V

O 7.5 $5 22.5 30 Force Level (%MVC)

Figure 6: The mean peak dispIacements together with the respective standard deviations are show for the factors: subjects, force direction, force Ievcl and pemrbation direction. Each factor is s h o w separately.

averaged over atl other factors.

The displacements produced in this experiment should allow for unbiased

estimates of the mechanics of the human am. Equal sized force perturbations could cause

biased estimates of stiffness because stifiess has been s h o w to Vary with displacement

direction, displacement amplitude (Shadmehr et al, 1993) and background force (Gomi

and Osu, 1998). However, by tuning the size of the force perturbation based on

displacement direction, force direction, and force level, the displacements seem to exhibit

little or no trend with respect to those factors. While the total scatter of displacement

amplitudes was high because there was no explicit control of the endpoint position, the

mean displacement for any given condition varied less than a millimeter from the overall

mean. Therefore, the force perturbations in this expriment should not cause any bias in

the measured mechanical properties.

Endpoint Stiffness

The endpoint stifiess of al1 six subjects was calculated for each condition using

forty-eight separate trials. This stiffness can be plotted as an ellipse for visual

representation. The stiffness ellipses of one representative subject (subject A) are shown

in Figure 7. The stiffness ellipses were charactensed by calculating four parameters: size,

maximum eigenvalue, shape and orientation. The means of afl subjects were detennined

along with the standard deviations (Fig. 8). Size increases with force level in al1

directions, as does maximum eigenvalue. The shape of the stiffness ellipse, which is a

ratio of the maximum eigenvalue over minimum eigenvalue, was quite variable. The

ellipses were most isotropic in the 3 15" force direction. Orientation can be described in

reference to the stiffness ellipse when the a m was relaxed. Ellipses for force directions

of 45" and 225" tended to the same orientation, If the ellipse was rotated it was generally

in the clockwise direction. In contrast, the ellipses tended to be rotated in the anti-

clockwise direction by about 40" in the force directions of 135" and 3 15"

lncreasing Force in 135 deg Direction

lncreasing Force in 225 deg Direction

Subject A

lncreasing Force in 45 deg Direction

Increasing Force in 31 5 deg Direction

Figure 7. The endpoint stiffiiess of Subject A is shown for al1 seventeen conditions. The central ellipse reprcsents the stimicss with the a m relaxcd. The other cllipscs arc organized outwards in the direction that

the force was applied in order o f increasing % MVC. Al1 ellipses are drawn to the sanie scale, wliich is shown at the bonom center.

B : Maximum Eigenvalue A : Size

C : Shape D : Orientation

Figure 8. The four parameters characterizing hand stifiess. Values are prescnted in the same way as F ig rc 7. Bars represent the mcan value for that condition over al1 six subjccts. The error bars indicatc

standard dcviation. Panel C: al1 shape values rnust be betwecn O (linc) and 1 (circle). Pancl D: the orientation is ploned as the difference from a baseline of 80" (rnean valuc of test condition).

Joint Stiffness

Joint s t i f iess was estirnated for each subject over a 26ms interval around the

peak displacement (Table 5). At this time both velocity and acceleration were very close

to zero, so damping and inertia would have minimal effect on estimation ofjoint

stiffness. A plot of the variation in the terms of the joint stiffness matnx with joint torque

for al1 six subjects is shown in Figure 9.

Table 5. The interva~s over which stiffness was estimated- Times are given with respect to the onset of the perturbation.

The relations with the highest correlation (Fig 9) were shoulder stifThess (R,)

with shoulder torque, double-joint stifiess (R,: and &) with elbow torque and elbow

sti ffness (L.) with elbow torque. The dope, intercepts and correlation coefficients were

determined for these four relations for each subject and al1 subjects using linear

regression (Table 6). Note that there was a large variation in stifiess when the elbow

torque was zero. This is investigated later, but for the purposes of caIculating the slope

and intercept these values were ignored.

C

2 13

239

Subject

Stan (ms)

End (ms)

F

239

265

D

229

255

A

244

270

E

199

225

B r

278

304

l LU, 1

Figure 9. Joint stiffness plotted against shoulder and elbow torque. Each elemcnt o f the joint s t i f i e ss rnatrix is ploned against both shoulder torque and elbow torquc. The correlation coefficient (8) o f each

relation is s h o w in the top lefi-hand corner o f each plot.

Subject

a) Rss vs Ts A 8 C D E F

Al l

b) Rse vs Te A

---- c) Res vs Te

A B C D E F

Al l

---- d) Ree vs Te

A B C D E F

All

Table 6 . Slopcs

Slope + 9% Confidence fa lntercept + 95% Confidence Int

1 intercepts and correlation coefficients from lincar rcgression of joint sti

joint torque.

Correlation Coefficient

0.81 O -89 0.91 0.77 0.95 0.87 0.88

0.90 0.82 0.88 0.74 0.81 0.88 0.79

ness t c m with

In Figure 9 it can be seen that the plots that have very low correlations, for

example shoulder joint st i f iess versus elbow torque, still appear to have well defined

relations. This result stems from the limi ted number o f conditions investigated (one joint

configuration, and four force directions), which produced a constraint between shoulder

torque and elbow torque. A simulation was performed in order to examine how the

constraint would affect the relation between stiffness and torque (Fig 10).

Ts [Nm] Te [Nrn]

D 80 O

Te [Nm] Figure 1 O. Examination of relations seen due to the constraint between shoulder and elbow torquc. A) Plot of theoretical relation between joint stifiess and shoulder torque for al1 six subjects uoder the conditions used in this experirnent. B) Stiffness plotted against elbow torque given the constraint benvcen shoulder torque and elbow torque C) Theoretical relation between stifiess and elbow torque. D) Stiff3ess plotted

against shoulder torque, givcn the constraint bewecn shoulder torque and clbow torquc.

Companng the simulations of Fig. 1 O to the plots Fig. 9 it c m be seen that most of the

features of the relation between R, and Tc, and between R,, R, or %, and Ts can be

explained simply by the constraint between torque at the two joints. However, this does

not explain the relatively large elbow and double joint stiffhess seen when the elbow

torque was zero (Fig. 9). These stiffhess terms were plotted against shoulder torque for

zero elbow torque (Fig. 11). It is apparent in this figure that the elbow (L) and double

joint stiffness (R, and &) are correlated with shoulder torque when elbow torque is

zero. Similarly a plot exarnining shoulder stifiess shows that shoulder joint stiffness

may also depend on elbow torque (Fig. 12). Fig. 12 compares the slopes of R, versus

shoulder torque in conditions where the elbow torque is zero or non-zero. The shoulder

joint stiffness is higher for a given shoulder torque when T, is non-zero than when it is

zero. This indicates that Te affects the values of R,.

01 I 1 I 1 1 1 I I l 1 1 1

-25 -20 -1 5 -1 0 -5 O 5 10 15 20 25 Ts [Nm]

Figure 11. Elbow and double joint stiffhess are related to shouldcr torque when elbow torque is zero. Only stiffness terms estimated in conditions o f zcro clbow torquc are plotted. Correlation coefficients for each

rclation arc shown in the upper lefi corner o f each plot.

-25 -20 -15 -1 O -5 O 5 10 15 20 25 Ts [Nm]

Figure 12. Shoulder joint stifiess is related to elbow torque. The shoulder stimiess is ploned separately for zero eIbow torque (circles) and non-zero eibow torque (diarnonds). The solid line represents the linear regrcssion for zero T, and the dotted line for non-zero T,. The shoulder stiffness is higher in cases whsn

elbow torque is non-zero. This indicates a dependence of shoulder stiffness on elbow torque.

The results suggest that in a position controt task al1 joint stiffness tenns depend

on both shoulder and elbow torque. This was hrther examined by performing multiple

linear regession on the data using the equation:

Values for the dopes of stiffness versus joint torque (m 1 and m2), the intercept (b) and

the correlation coefficient were determined for each subject and al1 subjects (Table 7).

Term S l o ~ e + 95% Int

A B C

Rss D E F

All

A B C

Rse D E F

Al l

c Table 7. Slopcs, intercepts and correlation coefi

Stiff ness Subject Slope + 950A Int

Ts Te Correlation

ients fiom linear regrc: - ssia

lntercept Value I 950A Int Coefficient

0.96 0.90 0.91 0.81 0.98 0.90 0.92

0.98 O -86 0.89 0.84 0.95 O -96 0.88

0.94 O -86 0.93 0.80 0.89 0 -94 0.88

0.97 0.85 O .go O -83 O -92 0.93 0.89

with - - -

joint toque, bhen stiffness is allowcd to Vary with both elbow and shouldcr torque.

Joint Stiffness Relation

The modified relation, with joint stiffness terms k i n g related to both elbow and

shoulder torque (bivariate relation) was compared to the relation used by Gomi and Osu

(1 998) where each joint stifiess term is related to either elbow or shoulder torque only

(univanate relation). These relations were compared by simulating the corresponding

endpoint sti fkess ellipses. The characteristics of stiffiness ellipses produced by each

relation were compared with respect to those measured in the experiment to examine

their similarity to the measured endpoint stifiess. Comparisons were also made between

characteristics of the endpoint stiffness ellipses calculated frorn each reiation.

Cornparison with respect to Measured Endpoint StifJiness

The characteristics of the endpoint stiffness ellipses produced by both relations

were compared as to how well they matched the measured endpoint stiffness. Ellipses

were simulated for each subject using both the univariate relation and the bivariate

relation for al1 conditions examined in the study. The difference in size, shape and

orientation between each simulated ellipse and the corresponding value for the measured

stiffness was calculated. The differences for al1 six subjects were then averaged for each

condition. Figure 13 shows the differences between the simulated and measured shape

and orientation. Figure 14 examines the differences in size between the simulated

endpoint s tiffness and the measured values.

The stiffness ellipses produced by the bivariate relation were generally closer in

orientation and shape in the 135O and 3 15" force directions than those produced by the

univariate relation. The univariate relation tended to produce much more anisotropic

ellipses than were measured in the subjects. In the 45" and 225" force directions, both

relations produced correctly oriented ellipses but the univariate relation predicted the

shape more accurately. In this direction both relations tended to produce ellipses that

were more anisotropic than the measured ellipses.

The error in the estimation of the size of the ellipses increased with force level.

Both relations overestimated the size of the ellipses at rest and in the 135" force direction

and underestimated the size in al1 other conditions. The only major difference in errors

between the two relations occurred in the 3 1 5" force direction where the sizes of the

bivariate stiffness ellipses were closer to that of the measured stimiess.

&variare Relationship

0

-80 -80 -60 -40 -20 O 20 40 60 80

X Force (N)

Univariate Relationship

-80 ' -80 -60 -40 -20 O 20 40 60 80

X Force (N)

Figure 13. The difference between the shape and orientation of the simulated stiffncss ellipses and the measured stimiess ellipses. Top: differences bctween the ellipses produced from the bivariate relation and the measurcd stiffhess ellipses. Bottom: differences from the univariate relation. Differences in shape are

rcpresented as the length of the arrow. Arrows directed to the right are more isotropie whcreas arrows directed to the left are more anisotropic. Differences in orientation are represented by the change in

direction from the horizontal.

BivaMte Relationship

80 0

X Force (Fi)

Univariate Rehtionship

-80 ( 1 1 1 -80 -60 -40 -20 O 20 40 60 80

X Force (N) Figure 14. The difference between the size of the simulated stifiess ellipses and the measured stiffiess

ellipses. Top: differences between the ellipses calculated using the bivariate relation cornpared to the measured stiffness ellipses. Bottom: differences when using the univariate relation. Differences in size are

reprcsented as the area of the circlc. Dark circlcs rcpresent smaller size whercas light circles represent larger site of simulated compared to measurcd ellipses.

Comparison of Univariate and Bivariate Stiffness Reiations

The bivariate relation was compared to the univariate relation in t e m s of the

endpoint ellipses and ellipse characteristics predicted by each model. The estimated

stiffness ellipses for the two different relations are shown in Figure 15 for a large range of

endpoint forces.

The ellipses produced by the bivariate relation tend to be more isotropie than

those produced by the univariate relation, especially in the 135" and 3 15" force

directions. Orientation of the ellipses is similar in the directions studied in this

experiment but differs by as much as 90 degrees in other directions. in order to examine

this variation in more detail, the differences between the characteristics of the ellipses

were determined over a force space of 120 N in the X and Y directions. Differences in

characteristics were expressed as the bivariate characteristics minus the univariate

characteristics. The differences in the maximum eigenvalue are shown in Figure 16.

Differences in the minimum eigenvalue are s h o w in Figure 17. Differences in size are

shown in Figure 18 and differences in shape are shown in Figure 19.

Force X (N) Figure 15. Cornparison o f the stiffhess ellipses calculated using the two different joint stiffness torque

relations. The univanate relation is plotted in the thui dark ellipses and the bivariate relation is plotted with the thick light ellipses.

The force directions that were examined in this study are the diagonals through

the center of each figure. The maximum eigenvalue for the bivariate relation is smaller in

the 135" and 3 15" directions but larger in the 45" and 225" directions (Fig 16). The

minimum eigenvalue is much larger in the 135" and 3 15" directions for the bivariate

relation. It is only slightly smaller in the 45" and 225" directions (Fig 16). The size of the

elIipse for the bivariate relation tends to be larger than the ellipse produced by the

univanate in the 135" and 3 15" directions. Similarly the shape tends to be more isotropic 5 6

in these two directions for the bivariate relation. in the other two directions the difference

in shape and size between the two relations is very small but the bivariate relation does

produce slightly smaller more anisotropic ellipses. Overall, the ellipses produced by the

bivariate relation tend to be larger and more isotropie in the directions used in this

experiment than do the ellipses produced by the univariate relation in the 1 3S0 and 3 1 5"

force directions. In the other two directions, little diflerence is seen in the stiffness ellipse

characteristics between the two relations.

Force Y (N) Force X (N)

-60 -40 -20 O 20 40 60 Force X (NI

Figure 16. Differences in the maximum eigen"&e betweeo the univariate and the bivariate relations of joint stiffness and torque. Values are expressed as a tünction o f the endpoint forccs in Cartesian space. The

top and bottom panels show the same relation fiom two differcnt angles for clarity.

Force Y (N) Force X (N)

Force X (N) Figure 17. Differences in the minimum eigenvalue between the univariate and the bivariate relations of joint stifiess and torque. Values arc expresscd as a function of the endpoint forces in Cartesian space.

Force Y (N) -60 -60 Force X (N)

r in4

Force X (N) Figure 18. Differences in the size of the ellipses bctween the univariate and the bivariate relations ofjoint

stificss and torquc. Values are expressed as a function of the endpoint forces in Cartesian space.

Force X (N) Figure 19. Differences in the shape of the ellipses between the univariate and the bivariate relations ofjoint

st i f icss and torque. Valües are expressed as a fiuiction o f the cndpoint forces in Cartesian space.

The joint-based inertial matrix components were estimated fiom the experimental

data along with the 95% confidence intervals (Table 8). The vafues calculated from an

anthropometric model based on body mass and arm and forearm lengths (Winter, 1990)

are also shown in brackets. The experimental values are al1 somewhat higher than the

anthropornetric values. However this is expected because the experimental values also

include the handle of the manipulandum and the thennoplastic cast. The estimated inenia

of the handle and cast is about 0.05 kgm2 about the elbow joint and 0.08 km2 about the

shoulder joint. These values would explain most of the differences between the estimated

inertial t ems and the anthropometric inertial terms. This indicates that the estimated

inertial parameters are accurate.

Subject I (0.3682)

0.3099 + -0022 (0.2749)

0.2159 + .O019 (O. 1966)

0.28 142 .O024 (0.2294)

0.4963 1 -0040 (0.44 19)

0.3226 f .O024

(0.1013) 0.1044 + .O015

(0.0669) 0.0962 -t -00 14

(0.0520) O. ll79I .O016 (0.0623)

0.1801 k .O022 (O. 1 128)

0.1 116-+.0015

(0.1013) O. 1024 + -0022

(0.0669) 0.095 1 i -00 19

(0.0520) 0.1216 k.0024

(0.0623) O. 1788 1 -0040

(O. 1 128) 0.11 18 + -0024

(O. 10 13) 0.1033 + .O015

(0.0669) 0.0903 + -00 14

(0.0520) 0-1 106 i -0016

(0.0623) O. 1594 + -0022

(O. 1 128) 0.1 104 1 .O015

1 (0.3 158) (0.0793) (0.0793) (0.0793) Table 8: Inertial parameters and their 95% confidence intervals deterxnined for each subject from the

experimentai data. Estimates of the ineniai matrices using an anthropomemc model (Winter, 1990) are shown below in bnckets.

Joint Damping

Joint damping was determined for each subject at each condition during a 40 rns

penod when the velocity was hi& and the acceleration was low. Intervals over which

damping was estimated for each subject are shown in Table 9. To examine the data for

relations with shoulder and elbow torque, linear regression was perforrned. The scatter

plots for ail subjects are shown in Figure 20 and the results of the linear regression are

listed in Table 10. In Figure 20, no correlation between the damping and joint torque is

apparent so the data for one of the two subjects where some correlation was found

(Subject Et) are shown in Figure 2 1.

perturbation.

Stan (ms)

End (ms)

The damping parameters are not strongly correlated with joint torque and are

generally quite small. Shoulder joint darnping (D,) showed the strongest correlation with

shoulder torque over al1 six subjects but even it was relatively weak. Only for subjects B

and C were damping terms highly correlated with joint torque. Overail, the highest

correlation was found between shoulder darnping and shoulder torque. It was also the

onIy relation that had a positive dope with joint torque in al1 six subjects. The mean

values for the damping estimates over al1 torque values were: ([D,, D,, D,, D,] = [0.4 1,

0.63,0.53, 1-68]} Nm- s/rad.

Table 9. The intervals over which dampuig was estimated. Times are given with respect to the start of the

75

115

7 3

113

65

105

80

120

75

115

65

105

C

-1 O O 1 O Te (Nm)

Figure 20. The relations between joint darnping t c m and joint torque for al1 subjects. Correlation coefficients are shown in the top lefi-hand corner o f each subplot.

- Subjec

- Dss A B C D E F

All

- - Dse

A B C D E F

All

Des A B C D E F

AI1

-

Dee A B C D E F

Al l

- Table 1 , Slopes, intercepts and correlation coefficients c

ShouMer Torque - - - -- - -

Corr Slope + 95Oh lntercept t 95Oh Caen

culated by performing linear regression on joint

Elbow Torque - Con dope t 95% lntercept + 95% Coeff

darnping tenns with joint torque.

-

L

-20 -10 O I O 20 Ts (Nm) Te (Nm)

Figure 21. The relations between joint damping terms and joint torque for Subject B. Correlation coefficients are shown in the top lefi-hand corner o f each subplot. For this subject damping t e m were

highly corrclated with joint torque.

An important question is to what degree the estimates of inertial and elastic tems

influence the damping estimates. To address this question, the inertia and stiffness were

varied independently in 10% steps from 50% to 150% of the onginal estimates and the

damping was re-estimated. The original darnping estimates were subtracted from the new

damping values and plotted against the change in the inertial or elastic parameters (Fig.

22 & 23). it is apparent from Figure 22 that changing the inertial parameters has

relatively little effect on the estimated damping values. This is expected since the method

for estimating the darnping parameters was chosen so that errors in the inertial parameter

would have as little effect on the damping estimate as possible. ï h e largest effect was on

D, where an increase in inertia produced a small increase in the estimated darnping term.

Changing the stiffness had a larger effect on the estimated darnping values but this effect

was still small. Generally an increase in the joint stiffness caused the estimates of al1

damping parameters to decrease whereas a decrease in the joint stifTness caused them to

increase.

- la c

"-51 I l I I 1 1 1 1 I 1

50 60 70 80 90 100 110 120 130 140 150 lnertia (% of Measured)

Figure 22. The change in joint damping estimates when inertia is varied. The inertia was changed fiom 50% to 150 % of the estimated value in 10% steps. At each point the damping values were re-estimated.

The line represents the best-fit line to al1 dampïng values. Values for al1 subjccts are included.

P A m 1

C .- 0 rn C <O c t I I O

I I

O - 5 - l I I 1 1

50 60 70 80 90 100 110 120 130 140 150 Stiff ness (Oh of Measured)

Figure 23. The change in joint damping estimates when joint stimicss is vaned. The st i f iess was changed from 50°h to 150 O h of the estimated value in I W O steps. At each point the damping values wcre re-

estimated. The Iine rcpresents the best-fit line to al1 damping values. Values for al1 subjects are included.

For cornparison with the current method of estimating damping, the joint damping

was also estimated together with stiffhess over 200 ms of the displacement. Again, this

was perforrned at times afier the initial high acceleration period to avoid biases from

possible errors in estimated inertia. Darnping was estimated over a 200 ms interval

following the start times indicated in Table 9. The damping results are plotted in Figure

24 against elbow and shoulder torque with the correlation coefficients. There are no

correlations. The mean values of darnping were ([D,, D,, D,, D,] = [-1.68,4.85,

-0.50,4.25]) Nm. s/rad.

. - .

-20 -10 O 10 20 Ts (Nm)

Figure 24. The relations between joint damping te- and joint toque for al1 subjects when joint damping was estimated with joint stiffness over 200 ms of the perturbation. Correlation coefficients arc s h o w in thc

top lefi-hand corner of each subplot.

Discussion

Endpoint Stifhess

The endpoint stiffness of the arm exhibits several general trends with respect to

the initial force vector. As the force level produced by the subject is increased, stiffness

increases. This is in terms of both the total area (Figure 8A) and the magnitude (Figure

8B) of the major eigenvalue of the stiffness matrix. This is similar to what has been seen

previously (Gomi and Osu, 1998; Mcintyre et al., 1996)-

Shape varies quite widely among the subjects especially in the 45' and 135" force

directions (error bars Fig. 8c). The stifikess was most isotropic when force was applied in

the 3 15" direction. These results were similar to those of Gomi and Osu (1 998) although

there is no evidence of the extreme anisotropic effect that they saw while force was

exerted in the 135" direction. It is possible that this difference stems from the use of a

position control task in this study rather than the force control task utilized by Gomi and

Osu (1998). In a force control task, the subject's actual position is controlled by the

manipulandum such that the subject does not need to be stiff in the direction

perpendicular to the force. However, in a position control task, the subject must both

produce the correct force and stabilize the position. In this case the subject would likely

increase the stiffness in the direction perpendicular to the force direction. This would

explain the more isotropic ellipses seen in the 1 3 5" and 3 1 5" force directions.

Orientation of the endpoint stifiess ellipse also follows a pattern for the force

directions used in this study. The orientation of the ellipse generally rotates towards the

direction of applied force. The results of both Gomi and Osu (1998) and the simulations

performed in this expriment suggest this is a feature particular to the directions where

either the shoulder or elbow torque is zero. Changes in orientation fiom one force

direction to another can be as large as 50". This rotation also occurred to a limited degree

in Gomi and Osu's ( 1998) study although the arnount of rotation was on the order of 10"

for the same force directions. The reason that larger rotations were seen in this study may

be because higher force levels were used. It is apparent fkom simulations using the

univariate relation between stiffness and torque (Fig 1 S), that rotation of the stiffness

ellipse could have been just as prominent had they used higher force levels. This occurs

because at low force levels the characteristics of the ellipse are detennined most

prominently by the passive stiffhess properties of the arm. Only as the force levels are

increased will the stimiess of the activated muscles dominate the shape and orientation of

the stiffness. The stiffness ellipses fiom McIntyre et al. (1996) are also oriented in the

direction of applied extemal force at high force levels. The arnount of rotation in

McIntyre's data is at inost 20" over a 60 N force range in the 90" and 270" force

directions examined, which is less than that found in the simulations using either the

univariate or bivariate relations.

Joint Stiffness

The estirnated joint stiffhess values are similar to previous results. This study

found mean relaxed joint stiffhess values (intercept of regression) of [R,, kjj, L] =

[7.36,6.45, 13-32] Ndrad where R, is the cross joint stiffness defined as (R, + &)/2.

In cornparison with previous studies, Gomi and Osu (1998) found relaxed joint stiffhess

values of [lO.8,2.7, 8.71 Nm/rad, Tsuji et al. (1995) found values of [8.3 3.1, 7-51 N d r a d

and Mussa-Ivaldi et al. (1985) found values of [îW, 10.3,28.9] Nmhd. Estimating

from their figures, McIntyre et al. (1996) found values of approximately [IO, 5, 201

Nm/rad. It can be seen that the estimated stiffness in the present study is similar to that

found in other studies. The study of Mussa-lvaldi et al. (1 985) required that subjects grip

the handle of the manipulandum. Tsuji et al. (1995) found that handle gripping forces

caused the stifiess of the a m to increase. Tsuji et al. (1995) and Gomi and Osu (1998),

like this study, used an experimental set-up that did not require the subjects to grip the

manipulandum. Stiffness in this study is still slightly higher than that of Tsuji et al.

(1 995) and Gomi and Osu (1 998). One possible reason for the difference could be

differences in moment anns of elbow muscles due to the orientation of the foream. In

previous studies, the forearm was supinated. whereas in the present study the fore-

was pronated. This would likely increase the moment m s of elbow flexors, which

would in tum increase joint stiffness. Differences between subject groups with regards to

factors such as muscle mass may also have contributed. Another factor is the task itself.

While the no force condition should result in similar relaxation in both a position and

force control task, it is possible that the subjects performing position control in this study

had a higher level of CO-contraction because the apparatus was different. The joystick

used in this study was in unstable equilibrium whereas the horizontal manipulanda of the

other studies would have been in neutral equilibnum.

Relation with Joint Torque

Previous studies, in both single (Cannon and Zahalak, 1982; Hunter and Keamey,

1 982) and mu1 ti-joint studies (Gomi and Osu, 1 W8), have shown that joint stifkess is

rdated to joint torque. Similady, Mcintyre et al. (1996) found linear increases in joint

stiffness as endpoint force increased. Gomi and Osu found that in a multi-joint isometric

force control task shoulder stiffness increased linearly with shoulder joint torque, whereas

cross joint and elbow joint stimiess increased linearly with elbow torque. In this study,

using an isometnc position control task, ail joint stiffhess terms were found to increase

linearly with both shoulder and elbow joint torque. The mean slopes of the linear

regression between the stifiess terms and joint torque found by Gomi and Osu (1998)

were [R, vs Ts, R, vs Te, R, vs Te, R, vs Tc] = [2.86,2.51,2.69, 6-82] rad-'. This is

similar to the slopes found in this study [3.83, 2.94, 3.3 1.4.351 rad-'.

Correlations between shoulder stiffness and elbow torque, between cross joint

stiffness and shoulder torque and between elbow stifhess and shoulder torque were also

found in the present study. The slopes of these relations are [1.09, 1.35, 1.24, 1.271 rad-'.

These relations are most likely task dependent. It is possible that Gomi and Osu (1998)

did not find these correlations because of the lower force levels used. However, a more

likely explanation is that such correlations only occur in position control tasks. In the

position control task, the subject is required to stabilize the hand not only in the direction

in which the external force is applied but also in the perpendicular direction. As the force

leveI increases, small errors in the direction of the applied force would produce motion in

the perpendicular direction. This might induce subjects to increase CO-contraction of

muscles to increase stiffhess perpendicular to the direction of the external force. Tenns

such as shoulder joint stiffness might then increase linearly with eibow torque as

observed in this study. These relations are less likely to occur in a force-control task

because the hand is stabiIized both in the direction of and perpendicular to the extemal

force.

This study extends the work of Gomi and Osu (1 998) by investigating force

production tasks up to 85 N compared to 20 N. Over this extended range of forces. which

correspond to 30 % of the isornetric MVC's, the relations of joint stiffness to joint torque

were still found to be linear.

This shidy used six subjects with a wide range of anthropometric characteristics

and strengths. When the data fiom al1 subjects were pooled together, to perforrn linear

regression against joint torque, clear relations with hi& correlation coefficients were

found. In contrast, Gomi and Osu (1998) simply averaged the slopes and intercepts from

each subject and no overall correlation coefficients were given.

Joint Stiffness Relation

The clear linear relation between joint stiffness and joint torque may explain the

changes in endpoint stifiess ellipses with force. To examine the implications of this joint

stiffness -joint torque relation in detail, simulations were performed to compare the

endpoint stiffhess characteristics produced by the univanate and bivariate relations.

The univariate relation was reported by Gomi and Osu (1 998) in a force control

task. In that task, the subject produced a desired force level but no control of their hand

location in space was required because the hand was stabilized. The stifiess ellipses

tended to be quite anisotropic or directional. Gomi and Osu found that the shoulder

stiffness was correlated with the shoulder torque and the cross joint stiffness terms and

elbow stiffness were correlated with elbow torque. In cornparison, using a position

control task, a bivariate relation between joint stiffness and torque was found. In the

position control task, the subjects were required to stabilize their hand position. Under

these conditions, it was found that the ellipses tended to be more isotropie than in the

force control task. The shoulder stiffness was still correlated with shoulder torque, but

was also increased with elbow torque. Similarly, cross joint and elbow stiffness were

correlated with elbow torque but also with shoulder torque. These secondary relations of

joint stifTness with joint torque had a mean dope of 1.24 or about 35% of that of the

major relations.

Coniparison with Respecî to Measured Endpoint Stiffness

From cornpansons of the characteristics of the simulated stiffness ellipses to the

characteristics of the measured stiffness ellipses, it is apparent that the bivariate joint

stiffness relation more closely predicts the measured endpoint stiffness. This is true,

particularly for the shape and orientation characteristics of the endpoint stiffness. In the

45" and 225O force directions both the bivariate and univariate relations tend to

underestimate the isotropy of the endpoint stiffness. In fact, the univariate relation

underestimates the isotropy in almost every condition. It is apparent that the endpoint

stiffness of the a m during position control tasks is more isotropie than that resulting from

the relation found by Gomi and Osu (1998) in force control tasks.

Comparison of Univariate and Biwariate Stiffness Relations

The univariate and bivariate relations of joint stifiess with joint torque were

compared in terms of the charactenstics of simulated endpoint stiffness ellipses. To what

degree do these relations differ in their effects on the endpoint ellipses that govem

stability of the arm? Figure 13 gives an overall picture of how the hand stifiess matrix

changed with different endpoint forces for the two relations and the foIlowing figures

examine the differences in the ellipse characteristics between the two relations. What is

apparent is that using the univariate relation or bivariate relation has little effect in the

45" and 225" force directions, but increases stability greatly in the 135" and 3 15" force

directions. The stiffness ellipses measured by Gomi and Osu (1998) were most

anisotropic in these directions. As well, ellipses for these directions were oriented most

closeiy to the directions of the external force (Gomi and Osu, 1998). In their study,

therefore, ellipses in these directions had the least stabil ity perpendicular to the direction

of applied force. Ln a force control task, such as theirs, this would not affect successful

performance of the task. However, if the same control strategy had been employed in the

position control task, it is unlikely that the subjects would have been able to stabilize the

hand in the target position. It is Iikely that a control method was utilized by the central

nervous system that produced the bivariate relation between stiffness and torque because

it increased the stability in the 135" and 3 15' force directions. This theory is also

supported by a comparison of the studies of McIntyre et al- (1996) and Gomi and Osu

( 1 998). The stiffness ellipses in the 90" force direction appear to be more isotropic in a

position control task (Mclntyre et al., 1996) than in a force control task (Gomi and Osu,

1998).

Using the univariate or bivariate relation of joint stiflness had little effect on the

effective stability provided by the stifiess ellipses in the 45" and 225" force directions.

However, it is possible that the stiffness is already high enough to guarantee stability in

this direction even in the more demanding task of position control. A cornparison can be

made between the ellipses predicted by the two relations in the 4 5 O and 225" force

directions to the measured stifiess ellipses. The stiffness measured by Gomi and Osu

(1 998) and that of this study is more isotropic than that predicted by the stiffness torque

relations.

The endpoint stifiess of the a m is actually composed of two terrns: the muscle

stiffness and the geometric stiffhess. The muscle stiffness is due to changes in force

produced by changes in muscle length. It c m be modified by varying the activation of the

muscles. Joint stiffness is only comprised of this term. The geometric stiffness is due to a

change in force produced simply by the change in endpoint position. It arises because the

endpoint force changes when the a m is pemirbed due to a change in joint angles. The

endpoint force produced by the same joint torque will now be different because the joint

angles have been changed. This effect scales with endpoint force but cannot be affected

by the muscle activation. If the geometric stiffhess were much larger than the muscle

stiffness, then the central nervous system would be unable to change the endpoint

stiffness of the arrn other than by varying the endpoint force or the posture. However, if

the muscle stiffness were comparable in size or larger than the geometric stiffiiess then

the central nervous system would be able to adapt the endpoint stiffhess of the a m . From

the simulations performed in this study, it can be seen that changes in the joint stiffhess

relation, Le. from univariate to bivariate, c m produce differences in the characteristics of

the endpoint stiffness. This means that the changes in the activation of muscles will

produce changes in the endpoint impedance of the m. Thus, the central nervous system

can modi@ the impedance of the arm not just through changing posture or external force

leveIs but also through the pattern of muscle activation.

The measured muscle stiffness is produced both by the intnnsic muscle properties

and by the reflex response to the perturbation. It has been shown previously that the

reflex response is increased when performing tasks that require increased stability. In

single joints, the amplitude of the long latency reflex is much larger in a position control

task due to an increased stretch reflex gain (Akazawa et al., 1983; Doemges and Rack,

1992 a&b). Using a negative stifiess task, Milner et al. (1995) found that the long

latency reflexes can produce excitatory responses even when shortened by the stretch.

Similarly, in a multi joint bali-catching task, Laquaniti and Maoli (1987 & 1989) found

that both extensor and flexor reflexes were excitatory at the time of bal1 impact. Such

studies have shown, therefore, that the reflexes are also modified to maintain stability of

the a m dunng tasks with large stability demands. It is iikely therefore that the increased

stiffness seen in this snidy during the position control task is due not only to increased

muscle activation but also to changes in the reflex gain.

The bivariate relation between joint stiffhess and joint torque produced equal or

greater stability in the directions shidied than that produced by the univariate relation.

However, it is apparent that in many other force directions, the bivariate relation could

actually cause the endpoint stability to decrease. It is unlikely that a control method

would be used by the centml nervous system that resulted in reduced stability in

situations where more stability was required. it would be more likely that the central

nervous system would utilize a control strategy that would increase stability in al1 force

directions. Such a control strategy may then not exhibit changes in stiflhess with joint

torque that are constant throughout al1 force directions. This may explain why both

relations tended to underestimate the isotropy of the stiffness in the 45O and 225" force

directions,

The endpoint stiffness is detemined by the joint stiffness terms, which are related

to activation of the muscles acting about each joint. in order to produce the endpoint

forces required in these multijoint isometric tasks, the muscles acting about a joint must

be activated to produce the correct joint torque. Activation of these muscles causes the

muscle stimiess and therefore joint stiffness to increase. Cocontraction may be needed in

certain directions for control of the endpoint force (Milner and Franklin, 1998). If the

leve1.s of cocontraction of these muscles do not Vary as the endpoint force is increased,

then the joint stifiess tenns will increase linearly with the joint torque. Such as result

was found by Gomi and Osu (1 998) where shoulder joint stiffness was found to increase

linearly with shoulder torque, and cross joint stiffness and elbow joint stiffhess were

found to increase with elbow torque.

However, in order to increase stability, it is aIso possible that the muscles acting

at a joint will be CO-contracted proportionally to the joint torque required at the other

joint. This was found in the present study. The level of contraction produced by the

elbow, double joint, and shoulder muscles may Vary with respect to each other as the

force direction changes to ensure stability in al1 directions. A further study measuring

stiffness for more force directions with various stability requirements needs to be

performed in order to examine this possibility. However, the results of this study do show

that the simple relation between joint stiffness and joint torque found by Gomi and Osu

( 1 998) does not exist under conditions that require more stability. Such conditions occur

frequently in the eveyday tasks performed by humans. For example, using a tool to

interact with the environment (chiselling, using a screwdriver, etc.) requires the a m to

generate the required stability to perform the task correctly. It appears that the central

nervous system can modulate the endpoint impedance of the a m in order to perforrn a

variety of tasks successfûlly.

Inertia

The inertial matrices estimated from the data were always larger than those

determined from anthropometric tables. However, this was expected as the estimated

inertial values also include the joystick handle and thennoplastic cast. When the values

were corrected for the inertia of the joystick handle and cast, the two different estimates

were similar. The inertial estimates should therefore be fairly close to the actual inertia of

the arm and joystick handle. Therefore, errors in the estimates of inertia should not have

unduly affected the estimates of damping.

Joint Damping

The joint damping estimates were small and sometimes negative. Negative values

for the darnping estimates likely occurred because the error of the damping estimates was

quite large and the values themselves were very close to zero. The intercepts from Table

1 1 give an indication of the relaxed damping values, which can then be compared to other

studies. Values found here were [D,, Dcj, D,] = [ - O 3 1, 0.56. 1.461 N m shd . Other

experiments found values of [O.63,0.18,0.76] Nm- s/rad (Gomi and Osu, 1998) and [0.7 1 ,

0.2 1,0.43] Nm- s h d (Tsuji et al., 1995). Generally, the values estimated in this

expenment are similar to those found previously with the exception of the negative

shoulder joint damping. Other investigators have also found negative damping estimates

(Gomi and Osu, 1996), which were likely due to small values of damping and relatively

large errors in the estimation procedure.

Relurion with Joint Toque

Previous studies have shown that joint damping terms exhibit a linear relation

with joint torque both in multi-joint (Gomi and Osu, 1998) and single joint cases (Hunter

and Keamey, 1982; Weiss et al., 1988). In this study, such relations, if found at all, were

weak. Using the initial method of estimating damping by itself, two subjects showed

strong correlations for the same relations found by Gomi and Osu (1 998) and overall

shoulder damping tended to increase with shoulder torque. However, using the second

method to estimate joint damping, either no relation was found, or joint damping

decreased with joint torque.

In a snidy examining the damping of the wrist and elbow dunng ball-catching

tasks with pseudorandom perturbations, Lacquaniti et ai. (1 993) found values of elbow

damping of 5-24 N m slrad. This value is much higher than those found in this shidy or in

that of Gomi and Osu (1998). The joint torque in the ball-catching task prior to and afier

catching the task was zero, so the difference in values must be due to large amounts of

cocontraction. The cocontraction in their task may be due to two separate causes:

preparation to stabilize the bal1 and stabilization against the effects of the pseudorandom

perturbations used for system identification. They did, however, find cross joint damping

terms of zero which are comparable to those found both here and Gomi and Osu (1 998).

Smceptibiiiiy to Changes in Inertia or St~xness Estimation

Damping estimates do seem robust to errors in the inertial or elastic element

estimates. Changing the inertial parameters by as much as 50% had very little effect on

the values of the damping estimates. As we can see in Figure 17, there was no net effect

on shoulder damping estimates, and small net effects on cross-joint and elbow damping

terms. Estimates of the inertia of the subject's a m should be quite accurate. A

cornparison of the estimates in this study to those from anthropometric data allows us to

have confidence in these values. This suggests that errors in damping estimates should

not be due to errors in the inertial estimates.

Variations in joint stiffness had a larger effect on the net joint damping values.

joint damping was estimated using the value ofjoint stiflkess at the peak displacement.

However, it has been shown that stifiess is lower for larger displacements (Kearney and

Hunter, 1982; Shadmehr et al, 1993). The estimated stiffness would likely be much

higher for the displacement where the damping was estimated than at the peak

displacement. However, increasing the joint stiffness caused the damping estimates to

become negative. Similarly, the second method of estimating joint damping and joint

stiflness together also produced negative damping values. While estimates of joint

stiffness appear to have a larger effect on the joint darnping estimates than does inertia,

neither appears to explain the negative estimates found and lack of relations with joint

torque.

It is possible that bad estimates for damping were found because the model of

damping is incorrect. If damping is non-linear, then the parametric mode1 used here

would be wrong and bad parameter estimates would be found. If damping depends on

velocity then the errors in the estimates would be greatest if made at high velocities or

during perturbations with high variability of velocities. The force perturbation used in

this study produced displacements with constant amplitude but widely variable peak

veloci ties. If damping depends on veloci ty, as has been suggested previousl y (Kirsch,

Boskov and Rymer, 1994), then these factors may explain the poor estimates of damping

obtained by using this linear rnodel of darnping.

Conclusion

This goal of this research was to examine how the mechanical impedance of the

human arm changes as forces are applied in different directions in a position control task.

The impedance of the a m was studied at forces up to 30% of the subjects' maximum

voluntary contraction. Endpoint stiff'hess eliipses generally increased in size as the

required external force was increased. Joint stiffness increased Iinearl y with joint torque

up to 30% of the subjects' MVC. Joint damping estimates were close to zero and

generally exhibited limited changes with joint torque.

Previous research using a force control task had found that each stiffness term

was dependent on either shoulder torque or elbow torque (Gomi and Osu, 1998).

Specifically, they found that shoulder stifiess increased with shoulder torque, double

joint stiffness increased with elbow torque and elbow stiffness increased with elbow

torque. However, in this study, using a position control task, which requires greater

stability, joint stiffness terms were found to be dependent on both shoulder and elbow

torque. Similar slopes were found for the same relations between joint stimiess terms and

joint torque found by Gomi and Osu (1998). That is, shoulder joint stiffness increasing

with shoulder torque and cross joint and elbow stiffness increasing with elbow torque.

However, relations with somewhat smaller slopes were found between shoulder stiffness

and elbow torque, double joint stiffness and shoulder torque and elbow joint stiffness and

shoulder torque.

Simulations of the stiffness ellipses created by the unvariate relation found by

Gomi and Osu (1 998) and the bivariate relation found in this study were performed.

Cornparisons of these two relations showed that the bivariate stiffness matrix had larger

minumum eigenvalues and therefore more isotropic ellipses specifically in the 135" and

3 15" force directions. in the two perpendicular directions, the characteristics of the

stiffness ellipses produced by the two relations were similar. Overall, in the four force

directions used in this study* the endpoint stability is increased by the found bivanate

relation compared to that produced by the simpler relation of Gorni and Osu (1 998). The

bivariate relation between joint stiffness and joint torque was likely found in this

experiment because the position control task places greater demands for the stability of

the lirnb than does a force control task. Such stability intensive tasks are found in

everyday activities performed by humans. Examples include most tasks involving tool

use such as writing, using screwdrivers, and eating with knife and fork or chopsticks.

The next step in this research will be a cornparison of position and force control

tasks in the multijoint a m for many force directions. It is expected that the stifiess will

be higher in the position control task due to increased activation to maintain stability. It is

expected that the dependence ofjoint stiffness terms on both joint torque terms rather

than ~ n l y one will be found only in the position control task. It is ais0 possible that the

slopes of the stiffness terms arising from cocontraction will Vary with force direction

such that the stability of the limb will be increased in al1 directions. Such a study would

also be able to examine the degree to which the central nervous system can regulate the

impedance of the a m , which would then help to explain how the variations found in this

study fit into the larger h e w o r k of motor control.

The results of this study do show that the univariate relation between joint

stiffness and joint toque does not occur in al1 conditions. To use such a relation to

descnbe the endpoint stifiess of the a m during different tasks at the sarne force levels

may grossly underestimate or overestimate the impedance of the arm during tasks that

have increased stability requirements. It also suggests that the central nervous system can

adapt the endpoint impedance of the arm not just by changing the posture but also by

changing the activation of the muscles.

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