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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