Exp Brain Res (2003) 151:145–157
DOI 10.1007/s00221-003-1443-3

RESEARCH ARTICLEDavid W. Franklin · Etienne Burdet · Rieko Osu ·Mitsuo Kawato · Theodore E. Milner

Functional significance of stiffness in adaptationof multijoint arm movements to stable and unstable dynamics
Received: 26 August 2002 / Accepted: 5 February 2003 / Published online: 29 May 2003
 Springer-Verlag 2003

Abstract This study compared the mechanisms of adaptation to stable and unstable dynamics from the perspective of changes in joint mechanics. Subjects were
instructed to make point to point movements in force
fields generated by a robotic manipulandum whichinteracted with the arm in either a stable or an unstablemanner. After subjects adjusted to the initial disturbing
effects of the force fields they were able to produce
normal straight movements to the target. In the case of thestable interaction, subjects modified the joint torques inorder to appropriately compensate for the force field. Nochange in joint torque or endpoint force was required orobserved in the case of the unstable interaction. After
adaptation, the endpoint stiffness of the arm wasmeasured by applying displacements to the hand in eight
different directions midway through the movements. Thiswas compared to the stiffness measured similarly during
movements in a null force field. After adaptation, theendpoint stiffness under both the stable and unstable
dynamics was modified relative to the null field. Adaptation to unstable dynamics was achieved by selective
modification of endpoint stiffness in the direction of theinstability. To investigate whether the change in endpoint
stiffness could be accounted for by change in joint torqueor endpoint force, we estimated the change in stiffness oneach trial based on the change in joint torque relative tothe null field. For stable dynamics the change in endpoint
D. W. Franklin ()) · R. Osu · M. Kawato
ATR Computational Neuroscience Laboratories,
2-2-2 Hikaridai, Seika-cho, Soraku-gun, 619-0288 Kyoto, Japane-mail: dfrank@atr.co.jp
Tel.: +81-774-951082
Fax: +81-774-952647
D. W. Franklin · T. E. Milner
School of Kinesiology, Simon Fraser University,
Burnaby, B.C., V5A 1S6, Canada
E. Burdet
Department of Mechanical Engineeringand Division of Bioengineering,
National University of Singapore, 119260 Singapore

stiffness was accurately predicted. However, for unstable
dynamics the change in endpoint stiffness could not be
reproduced. In fact, the predicted endpoint stiffness was
similar to that in the null force field. Thus, the change inendpoint stiffness seen after adaptation to stable dynamics
was directly related to changes in net joint torquenecessary to compensate for the dynamics in contrast to
adaptation to unstable dynamics, where a selective change
in endpoint stiffness occurred without any modification of
net joint torque.
Keywords Stability · Motor learning · Impedancecontrol · Endpoint stiffness · Inverse dynamics model

Introduction
We constantly interact with the world around us, moving
and manipulating objects in our environment. Tasks suchas opening a door, which involve a stable interaction withthe environment, are relatively simple to learn, because
similar motor commands will result in similar movements, so the dynamics can be easily identified andcompensated. For example, we may be surprised the firsttime we open a door with high friction, but will be able toopen the door smoothly on the second or third trial.
Unstable tasks are more difficult to learn, because they
are affected by different initial conditions, neuromotor
noise (Schmidt et al. 1979; Slifkin and Newell 1999), or
any small external perturbation, which can lead tounpredictable, inconsistent, and unsuccessful performance. For example, during sculpting, material irregularities can displace the chisel to the left or to the right of theintended path, and it requires extensive practice for a
sculptor to acquire the skill necessary to compensate forsuch instability.
Many tasks that humans perform, particularly those
involving tool use, are inherently unstable (Rancourt and
Hogan 2001). However, while the tasks may be unstable,
the mechanical impedance of the musculoskeletal system,
which is stabilizing in nature, counteracts the instability.

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Ultimately, it is the interaction between our limbs and theenvironment that determines whether or not a movement
will be stable. Coupled stability of the limb and theenvironment is a prerequisite for successful actions, as itprovides robustness to motor output variability andperturbations from the environment. Limb impedance
can be modified by changing the force at the hand sincemuscle stiffness inherently scales with activation level(Hunter and Kearney 1982). Modification of impedancegeometry, for example, shape and/or orientation of
endpoint stiffness, has been reported (Gomi and Osu
1998; Lacquaniti et al. 1993; McIntyre et al. 1996) but isgenerally associated with a change in applied force, or achange in the limb configuration. Such inherent modulation of impedance with applied force often cannot be used
to achieve stability, for example, when the task constrains
the direction or magnitude of the hand force. Similarly,
task demands may limit the variation in the endpoint
impedance that could be achieved by changing the limb
configuration. To deal with such situations, the CNS must
be able to modify the size, shape, or orientation of the
impedance independently of the force applied by thehand. This could be achieved by co-contracting specificgroups of muscles to select the geometry of the endpoint
stiffness (Hogan 1985). We have recently demonstrated
that the CNS can optimize the magnitude, shape, andorientation of the endpoint stiffness of the arm tocompensate for environmental instabilities (Burdet et al.
2001).
If the CNS is concerned about minimizing metabolic
cost then we would expect that the endpoint stiffness, intasks which are mechanically stable, would be directly
related to the minimum muscle torque necessary to movethe arm and compensate for any external dynamics.
Specifically, if metabolic energy is to be minimized, thenmuscles will be recruited with the minimal activation soas to produce the desired net joint torques. Accordingly,
we would expect a relatively linear relation between joint
torque and joint stiffness similar to that found understable isometric conditions (Gomi and Osu 1998; Perreault et al. 2001) as muscle and joint stiffness have beenfound to increase monotonically with muscle force andjoint torque, respectively (Gottlieb and Agarwal 1988;
Hunter and Kearney 1982; Kirsch et al. 1994; Milner etal. 1995). It should, therefore, be possible to predict thejoint stiffness from the net torque generated by muscles
during performance of the task. This is in contrast to tasks
which require greater stability than would be conferred by
the minimum muscle activation necessary to move thearm and compensate for any external dynamics. Under
such unstable conditions, co-contraction of opposing
muscles may be required above and beyond that ordinarily occurring with the movement. In such cases, the joint
stiffness should be higher than that predicted by the net
joint torque.
The present study was undertaken to analyze andcompare adaptation to environments producing stable orunstable interactions with the arm from the perspective ofchanges in limb mechanics. Previously, we measured theendpoint stiffness of the arm after adaptation to mechanical instability and found a selective increase in stiffness
in the direction of instability, which appeared to compensate precisely for the instability (Burdet et al. 2001).
We extend that work in the current paper by presenting
new data and analysis of stiffness after adaptation to
instability, as well as examining stiffness after adaptation
to stable dynamics. Adaptation to stable dynamics, which
has been examined previously by looking at kinematics
and EMG (Shadmehr and Mussa-Ivaldi 1994; Thoroughman and Shadmehr 1999), has never been characterized interms of the change in endpoint stiffness. In order toproperly compare and contrast the adaptation to stableand unstable environments we rigorously modeled thetrial to trial variation in the force response to displacement. We investigated whether the change in stiffness
observed after adaptation to the stable or unstableenvironment was controlled independently of the change
in joint torque needed to compensate for the new
dynamics. In an unstable environment, the change instiffness was independent of the change in joint torque.
However, in a stable environment, we found that changes
in endpoint stiffness were well correlated with changes injoint torque. Therefore, while the change in endpoint
stiffness in the unstable environment was directly controlled and could be attributed to adaptive co-contractionof antagonist muscle pairs, the change in endpoint
stiffness in the stable force field reflected only theadaptive change in joint torque.

Materials and methods
Six healthy individuals participated in the study (20–34 years ofage; two females and four males). The institutional ethics
committee approved the experiments and the subjects gave
informed consent prior to participation.
Apparatus
Subjects sat in a chair and moved the parallel-link direct drive airmagnet floating manipulandum (PFM) in a series of forward
reaching movements performed in the horizontal plane. Theirshoulders were held against the back of the chair by means of ashoulder harness. The right forearm was securely coupled to the
PFM using a rigid custom molded thermoplastic cuff. The cuff
immobilized the wrist joint, permitting movement of only theshoulder and elbow joints. The subjects’ right forearm rested on a
support beam projecting from the handle of the PFM. Motion was,therefore, limited to a single degree of freedom at the shoulder andat the elbow. The manipulandum and setup are described in detailelsewhere (Gomi and Kawato 1996, 1997). In brief, the subjects’
hand position was measured with joint-position sensors(409 600 pulse/rev) and the force exerted by the hand wasmeasured using a force sensor located between the handle and themanipulandum (resolution 0.059 N). Position and force were
sampled at 500 Hz.
Subjects performed reaching movements from a start circle
located 0.31 m in front of the shoulder to a target circle located0.56 m in front of the shoulder (total distance 0.25 m). The start andtarget circles along with the instantaneous hand position were
projected down onto an opaque horizontal surface located directly
overtop of the subjects arm. This surface also removed the subject’s
arm from view.

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Force fields
The experiment examined stiffness and EMG adaptation in two
force fields: a velocity-dependent force field (VF), producing astable interaction with the arm, and a position-dependent (divergent) force field (DF), producing an unstable interaction. Results
were compared to those in a null field (NF). The force (Fx, Fy) (inN) exerted on the hand by the robotic interface in the VF wascomputed as:
 

 
Fx
13 18 x_
ð1Þ
¼c
Fy
y_
18 13
where (ẋ, ẏ) is the hand velocity (m/s) and the scaling factor, c, wasadjusted to the subject’s strength (2/3c1). In particular, c was 2/
3 for women, 15/18 for most men, and 1 for the largest men. The
DF produced a negative elastic force perpendicular to the target
direction with a value of zero along the y-axis, i.e., no force wasexerted when trajectories followed the y-axis, but the hand was
pushed away whenever it deviated from the y-axis. The DF was
implemented as:   
Fx
bx
ð2Þ
¼
Fy
0
where the x-component of the hand position was measured relative
to the shoulder joint. b>0 (N/m) was adjusted for each subject sothat it was larger than the stiffness of the arm measured in NF
movements so as to produce an unstable interaction. Both force
fields were inactivated once the subject reached the target position.
Learning
All subjects practiced making movements in the NF on at least
1 day prior to the experiment. These training trials were used toaccustom the subjects to the equipment and to the movement speed
and accuracy requirements. Subjects were randomly assigned to
one of two groups. Group 1 initially performed the experimentswith the DF, and then proceeded to the VF, whereas group 2adapted to the fields in reverse order. Stiffness was normallymeasured the day after adaptation to a particular force field once
sufficient practice in the field had taken place (see impedance
estimation below).
Subjects first practiced in the NF until they had achieved
50 successful trials. Successful trials were those which ended inside
the 2.5-cm-diameter target window within the prescribed time
(0.6€0.1 s). All movements were recorded whether successful ornot. Movements were self-paced so subjects were able to rest
between movements if they wished. At the completion of 50 successful trials, the force field was activated. No information wasgiven to the subjects as to when the force field trials would begin.
Subjects then practiced in the force field until achieving 75 successful trials. They took a short break and then performed 100 more
movements, 20 of which were random trials in the NF. The NF
trials were called after-effects and were recorded to confirm that
subjects adapted to the force field.
Impedance estimation
We measured stiffness in VF and DF movements after extensive
learning, as well as in NF movements. The method is described indetail by Burdet et al. (2000). Prior to each stiffness measurement
session, the subjects were retrained in the force field to ensure thatthey had readapted to the field (NF: 40 trials; DF, VF: 80 trials).
For three subjects, 160 movements were then performed in eachforce field, of which 80 were randomly selected for stiffness
measurement. For one subject, in the case of all three force fields,
and for two other subjects, in the case of two of the force fields,
only 80 movements were performed. Half of these were randomly
selected for stiffness measurement (as in Burdet et al. 2001). On
these trials, displacements were introduced at the midpoint of the

movement in one of eight directions chosen randomly from the set{0, 45, 90, 135, 180, 225, 270, 315}. The PFM briefly
displaced the hand by a constant distance from a prediction of theundisturbed trajectory (Burdet et al. 2000). This displacement had
an amplitude of 8 mm and lasted 300 ms. This was composed of a
100-ms ramp away from the current trajectory, a 100-ms holdportion, and a 100-ms ramp back toward the predicted trajectory.
During the hold phase of the perturbation, the hand was displacedwith the predicted velocity of the unperturbed movement. Assuming that the prediction is perfect there would be no difference invelocity between the perturbed and unperturbed trajectories,
eliminating any contribution of damping to the change in measured
endpoint force. Although the prediction is not perfect, our results
indicate that the errors are small and that the average prediction
over several trials is very close to the average of the actual
trajectory (Burdet et al. 2000). Therefore, we can be quite confident
that forces due to damping did not introduce error in the stiffness
measurements. Using the average force and displacement during a
60-ms interval toward the end of the hold portion of theperturbation window, an estimate of the 22 endpoint stiffness
matrix (K) was obtained by linear regression of the mean change in
hand force and the mean change in position, as represented by theequation:


 
DFx
Dx
ð3Þ
¼K
DFy
Dy
The stiffness in different directions was represented in terms of anellipse by plotting the elastic force produced by a unit displacement(Mussa-Ivaldi et al. 1985).
In the perturbation trials in the VF, the force field was activatedprior to and after the perturbation to ensure that the same trajectoryand motor commands were used in the perturbation and nonperturbation trials. In the trials where the hand was displaced in theDF, the force field was not activated prior to the displacement inorder to avoid amplification of trajectory deviations that would
contribute to error in the stiffness estimates. Since subjects could
not detect the absence of the force field on these trials we were able
to rule out the possibility that changes in stiffness were reactive tothe environment.
The joint stiffness (R) was calculated from the endpoint
stiffness (K) using the relation:
@JT
Rss Rse
R¼
F
ð4Þ
¼ JT KJ þ
Res Ree@qwhere J is the Jacobian, a matrix which represents the geometrictransformation of small changes in joint angles to small changes in
endpoint position, and F is the endpoint force (McIntyre et al.
1996).
Stiffness dependence on joint torquePreviously, it had been shown that joint stiffness linearly increases
with joint torque during postural tasks (Gomi and Osu 1998).
Recently however, we demonstrated that the endpoint stiffness after
adaptation to an unstable force field (DF) was selectively increased
along the x-axis without any corresponding mean change in net
joint torque (Burdet et al. 2001). However, even if the mean change
in joint torque is zero, this does not mean that the stiffness could not
be produced by variations in the joint torque from trial to trial
which sum to zero over the entire experiment. Previously, we
looked for a simple correlation between the actual endpoint force inthe x-direction and the expected force based on the measured
stiffness and the size of the displacement. A more rigorous
approach has been taken in the current work, which considers theforces and stiffness in both the x- and y-directions. We wish to
examine whether trial to trial variation in net muscle torque iscorrelated with the variation in measured stiffness. In the previous
approach we calculated muscle torque from the endpoint force
recorded during the perturbation. This value includes not only theforce which would be produced by muscles during a normal

148
movement but also the elastic force resisting the perturbation.
Consequently, part of the measured force would have beencorrelated a priori with the expected change in force calculated
from the measured stiffness. To avoid this confounding effect wehave estimated the force necessary to compensate for the force fieldusing our trajectory prediction algorithm (Burdet et al. 2000). For
each trial, we use the trajectory in the first portion of the movement
along with information from previous trials to predict what the
trajectory would have been without the perturbation. From thistrajectory we could determine the force that the force field would
have applied to the subject to obtain a more accurate estimate of the
net muscle torque at each joint. In this way, it is possible toinvestigate whether or not the changes in stiffness in the VF and DF
are dependent or independent of any change in joint torque while
avoiding the confounding effects of the perturbations.
The following formulas use three different symbols to denote
differences in variables: d, d, and D. For example, when referring totorque, d refers to the change in joint torque, dt, produced bydisplacing the hand, i.e., the torque resisting the displacement, drefers to the difference between the resisting torque, dt, in one force
field and dt in a different force field, and D refers to the difference
between the joint torque required to move the arm in one force field
and the joint torque required for the same movement in a different
force field.
When the arm is constrained to move in the horizontal plane atshoulder height, as in this experiment, it can be modeled as a twodegree-of-freedom mechanical system. The associated 22 joint
stiffness matrix R is defined by:

 


dts
Rss Rse dqs
¼
ð5Þ
dte
Res Ree dqe
where the change in joint torque dt=[dts dte]T is the elastic
resistance to a differential displacement dq=[dqs dqe]T. The stiffness
R can be estimated during arm movements by measuring therestoring force in response to small perturbations relative to a
prediction of the undisturbed trajectory (Burdet et al. 2000). As wewish to examine the change in stiffness that occurs during theadaptation to the force fields, the change in torque associated withjoint stiffness in the NF
 NF   NF


dqs
dts
Rss RNF
se
¼
NF
NF
NF
dqe
dte
Res Reeis subtracted from the change in torque associated with joint
stiffness in the force field FF   FFdqs
Rss RFF
dts
se
FF ¼
FF
FF
dqe
Res Reedte
resulting in

 
DRss
dts
¼
dte
DRes

DRse
DRee



dqs
dqe



where,
  FF
DRss DRse
Rss  RNF
ss¼
NF
DRes DRee
RFF
es  Res  FFdts
dts  dtNF
s
¼
NF
dte
dtFF
e  dte

ð6Þ

NF
RFF
se  RseFF
Ree  RNF
ee

and the change in joint angle dq is assumed to be the same in allforce fields.
Joint stiffness has been shown to be well correlated with joint
torque for single joints such as the ankle (Hunter and Kearney1982), elbow (Cannon and Zahalak 1982), wrist (Milner et al.
1995), and interphalangeal joints of the thumb (Akazawa et al.
1983) and index finger (Carter et al. 1990) during maintenance of
posture, and for the elbow during movement (Bennett 1993).
Similar correlations between joint stiffness and joint torque havebeen found during maintenance of multijoint posture of the arm(Gomi and Osu 1998; McIntyre et al. 1996). Gomi and Osu (1998)

showed that: (1) the shoulder stiffness Rss was well correlated withshoulder torque |ts| and weakly correlated with the elbow torque |te|,(2) the cross-joint stiffness elements Rse and Res are of similar
magnitude, well correlated with |te| but poorly correlated with |ts|,and (3) the elbow stiffness is well correlated with |te| and weakly
with |ts|. However, as the biarticular muscles should contribute to
all four elements of the joint stiffness matrix, and as biarticular
muscle activation may be much higher during movement ascompared to isometric tasks (van Groeningen and Erkelens 1994;
Karst and Hasan 1991; Tax et al. 1989, 1990a, b), the shoulder
stiffness term may vary as a function of both |ts| and |te|. The
presence of the elbow torque term in the expression for shoulder
stiffness reflects the contribution of the biarticular muscles whichhave been shown to be activated primarily to produce elbow torque(Buchanan et al. 1986; Flanders and Soechting 1990; Gomi and Osu
1998; Osu and Gomi 1999; Wadman et al. 1980). In particular, we
assumed that the dependence of joint stiffness on joint torque could
be represented as:    NF  a3 tNF  
a1 ts þ a2 tNFe
 e  þRp ;a1 :::a4 constants
RNF ¼
ð7Þ
a4 tNF 
a3 tNF 
e

and
RFF ¼



e

 
 
 
 þ a2 tFF  a3 tFF  
a1 tFFs FF  e e  þRp ;a1 :::a4 constantsa3 te a4 tFFeð8Þ

where Rp is the passive stiffness. Subtracting Eq. 7 from Eq. 8yields:
a1 Djts j þ a2 Djte j a3 Djte j
DR ¼;a1 :::a4 constants
ð9Þ
a3 Djte j
a4 Djte jwhere   FF   NF  t  t
Djts j¼  sFF   sNF 
Djte jte  teEquation 6 was transformed from joint co-ordinates to endpoint coordinates to obtain the relation:


 T
dqs
ð10Þ
DF ¼ J1 DR
dqe
As the trajectories of movements in the VF and DF after learning
were similar to movements in the NF and as J(q) changed little in
the region of stiffness measurement, we assumed that J(q) wasconstant. To examine the correlation between the change instiffness relative to the NF and the corresponding change in joint
torque using individual trials, Eq. 9 can be substituted into Eq. 10
and rearranged as:
2 3

 a1
6 a2 7
 T dqs Djts j dqs Djte j dqe Djte j0
6 7
DF ¼ J100
dqs Djte j dqe Djte j 4 a3 5
a4
ð11Þ
where J(qo) is calculated from the joint angles of each subject at the
midpoint of the movements qo. The predicted trajectory was used in
combination with the force field equation to calculate the endpoint
force applied to the hand by the force field as described above.
From the endpoint force and kinematics, the joint torques were thencomputed using inverse dynamics as explained in the next section.
In this way, the difference in joint torque (D|t|) for force fieldmovements, relative to NF movements could be estimated. On theother side of the equation, the actual difference in force (DF)produced by the perturbation in the force field relative to the NF
was calculated directly from the difference in measured restoring
force (dF) in the force field and the NF:
  FF
dFx  dFxNF
DFx
DF ¼
¼
ð12Þ
dFyFF  dFyNF
DFy

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To determine whether the stiffness was being controlled
independently of the joint torque necessary to cancel the force
exerted by either force field, the parameters ai in Eq. 11 were
determined by linear regression for each subject using the trials inwhich stiffness was measured. An R2 value close to 1 would
indicate that endpoint stiffness could be explained by the joint
torque required to perform the task. An R2 value close to zerowould indicate that overall, from trial to trial, endpoint stiffness
was controlled independently of joint torque.
Torque estimation
Time varying muscle torque at the shoulder and elbow wascomputed using the equations of motion for a two-link planar arm(c.f. Hollerbach and Flash 1982). The joint torque was calculatedas:ts ¼€qs ð2X cos qe þ Y þ Z Þ þ €qe ðX cos qe þ Y Þ  q_ 2e X sin qe2q_ s q_ e X sin qe  ðl1 sin qs þ l2 sin ðqs þ qe ÞÞFxþðl1 cos qs þ l2 cos ðqs þ qe ÞÞFyte ¼€qe Y þ €qs ðX cos qe þ Y Þ þ q_ 2s X sin qe
l2 sin ðqs þ qe ÞFx þ l2 cos ðqs þ qe ÞFy

ð13Þ

EMG was smoothed using a 125-point moving average andaveraged over 20 trials for the purposes of illustration.

Results
The two force fields investigated in this study produced
distinctive perturbations of the trajectories prior toadaptation. Initial trials in the stable VF perturbed thesubjects to the left (Fig. 1A), but they quickly learned tocompensate for the force field and soon began to makestraighter movements. By the fifth trial, subjects usually
succeeded in moving to the final target in a single
movement. After learning, the trajectories were relatively
straight and consistently reached the final target position.
Movements in the unstable DF were initially perturbed
either to the right or the left (Fig. 1B), depending on theinitial deviation in the path. However, again subjects wereable to adapt to the force field, successfully completingthe task on most trials and exhibiting straight trajectories
to the final target. After learning, trajectories in both thewhere:
X ¼ m2 l1 cm2 þ mc lc cmc
Y ¼ I2 þ m2 c2m2 þ Ic þ mc c2md
Z ¼ I1 þ m1 c2m1 þ ðm2 þ mc Þl21
t is joint torque, q is joint angle (defined according to theconvention of Mussa-Ivaldi et al. 1985), I is moment of inertia
about the center of mass (cm) of the segment, l is segment length,
and m is segment mass. The subscript s refers to the shoulder joint(or the upper arm), e to the elbow (or the lower arm), and c to thewrist cuff. The mass and inertia of the subject’s arm segments were
estimated from the weight and segment lengths of each subject
based on anthropometrical scaling relations (Winter 1990).
EMG measurement of DF adaptation
To examine the muscle activity after learning in the DF, surface
EMG was recorded in the NF, after learning in the DF, and during
after-effect trials in the DF for four of the six subjects. All EMG
was recorded on the same day with the same electrode placement.
Activity was recorded from six muscles producing torque at theshoulder and elbow joints. The muscles included two monoarticularshoulder muscles, the pectoralis major and the posterior deltoid,
two biarticular muscles, the biceps brachii and the long head of thetriceps, and two monoarticular elbow muscles, the brachioradialis
and the lateral head of the triceps. The EMG was recorded by using
pairs of disposable silver–silver chloride surface electrodes in abipolar configuration with a separation distance of approximately
2 cm. The skin was thoroughly cleaned with alcohol and preparedby rubbing in electrode paste. Excess paste was wiped from the skinprior to attaching the electrodes. The resistance of each electrodepair was tested to ensure that it was less than 10 kW. EMG signals
were filtered at 25 Hz (high pass) and 1 kHz (low pass) andsampled at 2 kHz.
EMG during the DF after-effect trials was compared to EMG
during NF movements and EMG after complete adaptation to the
DF. The rectified EMG was integrated over the entire movement,from 100 ms prior to movement onset until 800 ms after movement
onset. Twenty trials were used from each condition and the data forall subjects was used in an ANOVA with subjects as a randomvariable. A post hoc test was then performed on the three conditions
using Scheffe’s test with a significance level of 0.05. The rectified

Fig. 1A, B Movements in the force fields, A in the velocitydependent force field (VF) and B in the position-dependent(divergent) force field (DF). Movements are shown for the initial
movements in the force field (trials 1–6) and the late portion oflearning (trials 65–70). The black lines on either side of trials 1–6
for the DF indicate the safety zone, outside of which the field wasturned off for safety reasons

150

VF and DF were similar to NF trajectories. The recorded
after-effects in the VF were similar to those reported inprevious studies using similar force fields (Shadmehr andMussa-Ivaldi 1994), confirming that subjects adapted tothe endpoint forces imposed by the force field. This has
been previously suggested to indicate the formation of aninternal model of the environmental dynamics (Shadmehr
and Mussa-Ivaldi 1994). In the DF, after-effect trajectories deviated less from the straight-line trajectory than did
NF movements (Burdet et al. 2001).
Endpoint force after learning
The similarity of trajectories in the VF and the DF to
those in the NF indicates that the subjects learned tocompensate for the force fields. To compare howcompensation for the dynamics was achieved in the twocases, we measured the endpoint force after learning.
Figure 2 shows the endpoint force of subject 1 in the NF,
VF, and DF averaged over ten movements with standard
deviations. Subjects adapted to the VF by counteracting
the force produced by the field. The force was morepositive in the x-direction and more negative in the ydirection than in the NF. During the early stages oflearning in the DF, subjects experienced large forces
which could be either positive or negative depending on
the initial movement direction. However, once subjects
had adapted to this field, similar endpoint forces wererecorded in the DF as had been recorded in the NF.
Figure 3 shows mean endpoint force and standard
deviations at the midpoint of the trajectory (time ofstiffness measurement) in the three force fields for six
subjects. The endpoint force after learning in the VF was
significantly larger (positive) in the x-direction andsignificantly smaller (more negative) in the y-direction
compared to the endpoint force in the NF. For no subject
was the x- or y-force at the midpoint of the movement
trajectory in the DF significantly different from that in theNF (Fig. 3). In contrast to the VF, adaptation in the DF
did not require a change in the endpoint force.
In order to further examine the difference in adaptationthat was required in these two force fields we computed
the joint torques using inverse dynamics (Fig. 4). In
comparison to the NF, the adaptation to the VF required alarger extensor torque at the shoulder throughout theentire movement and a small extensor torque at the elbow
late in the movement to compensate for the force field.
Since no change in endpoint force occurred in the DF, themean joint torque after adaptation was not different fromthe NF. There was, however, a larger variation in the joint
torque in the DF than the NF because force was applied tothe hand whenever the trajectory deviated from the y-axis.
Because this force acted to increase the deviation,
variation in the trajectory was also greater in the DF
than the NF.

Fig. 2A, B Force profiles after adaptation to the force fields. Mean
endpoint force profiles for the x- (A) and y-directions (B) during the
movement after learning in the null field (NF; gray), VF (dotted),and DF (solid black). Standard deviations are shown for movements
based on the last 20 successful trials. Data are shown for one
subject

Fig. 3A, B Mean endpoint force after adaptation to the force fields,
A in the x-direction and B in the y-direction. The mean x- and yforces and standard deviations at the midpoint of the movements in
the NF (squares), VF (diamonds), and DF (stars) for all six
subjects. The time of measurement corresponds to the time at which
stiffness was estimated in the movements

Fig. 4A, B Joint torque profiles after adaptation to the force fields.
A Shoulder torque. B Elbow torque. The shoulder and elbow joint
torque profiles are shown for the NF (gray), VF (dotted), and DF(solid black). Adaptation to the VF required an extensor torque atthe shoulder whereas no changes in joint torque were observed in
the DF. Data are shown for the same subject as Fig. 2

Endpoint stiffness changes with adaptation to force fields
The endpoint stiffness of the arm, measured after
adaptation to the NF, VF, and DF is shown in Fig. 5.
The ellipses represent stiffness at the midpoint of the
movement. In both the VF and DF, the endpoint stiffness
was modified in shape and orientation compared tostiffness in the NF. In the VF, the endpoint stiffness

151
Fig. 5 Stiffness geometry after
adaptation to different force
fields. Stiffness ellipses areshown for six subjects in the NF(light gray filled ellipse), VF(dotted ellipse), and DF (darksolid ellipse). For the ellipses ofsubject 6, a gray arrow illustrates movement direction anddotted and solid arrows show
the direction of the external
force in the VF and direction of
instability in the DF, respectively

increased along the direction of force compensation(Fig. 3), although this occurs only for specific force
directions (cf. Gomi and Osu 1998). In the DF thestiffness increased dramatically in the direction of theinstability (indicated by small black arrows for subject
S6), but there was relatively little change in the movement direction. The increase in stiffness in the direction of
instability was achieved without a change in the endpoint
force (Fig. 3). This illustrates the ability to control the
endpoint stiffness of the arm independently of endpoint
force as previously described by Burdet et al. (2001).
The endpoint stiffness can be decomposed into two
components: a symmetric component (conservative) andan antisymmetric component (rotational) (Mussa-Ivaldi etal. 1985). Neither during estimation nor visualization
procedures, did we constrain the endpoint stiffness to besymmetric. For comparison to previous work, the curlcomponents of the endpoint stiffness were estimated as inMussa-Ivaldi et al. (1985). In particular, we estimatedZmean which is the square root of the ratio of thedeterminant of the antisymmetric stiffness to the determinant of the symmetric stiffness. Expressed as apercentage, we found the following Zmean values(mean € SD of six subjects) for stiffness under the three
conditions, NF: 23.0€10.8; VF: 15.5€11.1; DF:
24.6€12.9. In general, these are larger than those foundunder isometric conditions at rest (Mussa-Ivaldi et al.
1985). Differences in Zmean across force fields was tested
using an ANOVA with subjects as a random effect. There
was no significant difference in the contribution of theantisymmetric component to the endpoint stiffness after

adaptation to the DF compared to the NF (P=0.83) or after
adaptation to the VF compared to the NF (P=0.28).
Stiffness in VF can be predicted only from change
in joint torqueEndpoint stiffness increased in specific directions both formovements in the VF and the DF. However, the reasonsfor the increase in stiffness were likely different. To
address this issue we performed an analysis to determine
whether or not stiffness was being controlled independently of the joint torque necessary to compensate for theforces applied by the force field. We used the linear
relation represented by Eq. 11 to predict the stiffness andso determine whether the increased stiffness could be
explained by increased joint torque relative to NF
movements. The change in endpoint force produced by
the perturbation used to measure stiffness was estimatedfor each trial based on the prediction of the unperturbed
trajectory and the field strength. Figure 6 compares themeasured and estimated changes in endpoint force
relative to endpoint force in the NF (DFx and DFy). The
parameters ai were estimated from the data obtained inthe VF (left) and DF (right) for each subject using trials
where stiffness was measured. From the results it isevident that the stiffness in the VF was relatively wellcorrelated with the applied force, whereas in the DF thecorrelation was generally poor. The difference in slope for
the VF and DF relations demonstrates clearly thatstiffness changed for different reasons. For the VF, theslope was close to one indicating that the difference in152

joint torque accounts well for the difference in resistive
force due to the difference in stiffness relative to the NF.
In contrast, for the DF the slope was close to zero for all
subjects indicating that the difference in resistive force
was independent of the difference in joint torque.

Fig. 7A, B The stiffness geometry predicted by modeling joint
stiffness as a linear function of joint torque (Eq. 10; broken ellipse)
with the stiffness measured in the NF (gray) and force field (solid)
superimposed. A VF. B DF. Stiffness was computed for individual
trials and averaged, using the force data for each subject. The
ellipses represent the stiffness averaged across all six subjects

To further investigate how much of the difference inendpoint stiffness could be explained by the difference injoint torque we used Eq. 9 to compute the difference injoint stiffness in the force field relative to the NF, which
was then added to the NF stiffness. The parameters ai
were estimated for each subject using the respective data
for the VF and DF. The stiffness was computed using
force data from individual trials and averaged. The
resulting stiffness ellipses in Fig. 7 illustrate the stiffness
geometry that would have resulted if the stiffness scaled
directly with the calculated joint torques. The modeled
stiffness for movements in the VF shows an increase instiffness along the same direction as the observed
stiffness, of almost identical size and shape (Fig. 7A).
In the DF, the modeled stiffness is essentially the same as
the NF stiffness so the model fails to reproduce theobserved increase in stiffness along the direction of
instability (Fig. 7B). This indicates that the increased
stiffness in the DF was not due to any increase in net joint
torque accompanying adaptation to the DF and must,
therefore, have been independently modulated due to cocontraction of antagonist muscles or contribution fromreflexes.
Joint stiffness changes under VF and DF
The joint stiffness was calculated from the endpoint
stiffness matrix (McIntyre et al. 1996). The elements of
the joint stiffness matrix in the NF are shown in Fig. 8A.

Fig. 6 Relations between reconstructed and measured changes inforce for displacements in the VF (left) and DF (right). The
measured change in force is calculated from the recorded force data
during the experiments (Eq. 13). The reconstructed force is basedon estimates of the joint torque, the displacement size, and theestimates of a (Eq. 12). Changes of endpoint force in the x- and ydirection relative to force in the NF (DFx and DFy) are plotted foreach subject. R2 values for the relation, calculated using data fromboth the x- and y-directions, are placed in the top left corner of eachpanel. The slope of each relation is shown with the solid line. For
reference purposes, lines of slope one (VF) and slope zero (DF) areoverlaid on each plot with dotted lines

153

Fig. 8A–C Changes in joint stiffness associated with the change inendpoint stiffness. A Joint stiffness in the NF. The bar graph shows
the mean stiffness for each term in the matrix, whereas the joinedlines show the individual values for five subjects. B Mean joint
stiffness after adaptation to the VF (black bars) relative to the joint
stiffness in the NF (gray bars) (same as in A). C Mean joint
stiffness after adaptation to the DF (black bars) relative to the joint
stiffness in the NF (gray bars)

The relative values for all subjects were similar in the NF.
After adaptation to the VF, the shoulder joint stiffness(Rss) and cross-joint stiffness (Rse and Res) increased by
36.7, 11.7, and 13.8 Nm/rad, respectively, while theelbow joint stiffness (Ree) increased by only 2.9 Nm/rad(Fig. 8B). The greatest changes were in the shoulder joint
and cross-joint stiffness terms. This indicates that the
major contribution came from the shoulder extensor
muscles, followed by the biarticular extensor muscle.
The changes in joint stiffness terms, seen after
adaptation to the DF, present a different pattern(Fig. 8C). In this case, the increases in the shoulder joint
stiffness term (82.1 Nm/rad) and cross-joint stiffness
terms (49.5 and 60.8 Nm/rad) were more comparable,with a somewhat smaller increase in single joint elbow

stiffness (39.3 Nm/rad). As the biarticular muscles must
contribute to the single joint elbow and shoulder stiffness,
as well as to the cross-joint stiffness terms it seems thatthe single joint muscles likely contribute little to theincreased joint stiffness.
One noteworthy point is that in the NF the cross-joint
stiffness term, Rse, is larger than the elbow stiffness, Ree,for all five subjects. The other cross-joint term, Res, isalmost the same as Ree. After adaptation either to the VFor the DF, the cross-joint stiffness terms increased bymore than the single joint elbow stiffness term. Thisfinding could be explained by biarticular muscles contributing more to cross-joint stiffness than to single joint
elbow stiffness in this condition. The actual contribution
to single joint stiffness relative to shoulder stiffness or
cross-joint stiffness will be dependent on the ratios of the
moment arms of the biarticular muscles at the elbow andshoulder joints.
Stiffness measurements in the DF were performed withthe force field turned off prior to the applied perturbation.
In order to determine if this could affect the value of thestiffness measurement and to confirm that the stiffness
adaptation is preprogrammed, the EMG after adaptation
was compared to that during after-effects. Figure 9
compares the EMG during after-effect trials of the DF
with that during NF trials. The EMG was much higher
during DF trials compared to NF trials (P<0.001 for all
six muscles), illustrating the large increase in co-contraction in antagonist muscle pairs, and corresponding to theadapted impedance. Similarly, the after-effect EMG wasmuch higher than NF trials (P<0.001 for all six muscles)
although these movements were also performed in the
NF. Furthermore, the EMG during after-effect trials wasvery similar in magnitude and shape to DF trials. For
three muscles there was no significant difference in themagnitude between after-effect and DF trials, posteriordeltoid (P=0.17), pectoralis major (P=0.25), and brachioradialis (P=0.96), whereas for the other three muscles the
DF EMG was slightly higher than the EMG during aftereffects, triceps long head (P=0.017), biceps brachii(P=0.005), and triceps lateral head (P=0.022). Although
the DF was off during the after-effect trials, the subjects
were not aware on which trials the DF was on or on which
it was off, and assumed that it was always on. Thus, theEMG during the after-effect trials reveals a preprogrammed motor command to compensate for the DF. The
impedance was predictively and not reactively controlled.
Similarly, this indicates that having the force field offprior to the stiffness measurement should not haveaffected the stiffness measurement. The small differencesin EMG between DF and after-effect trials, occurring in
the later half of the movement, likely represent the neural
feedback response to perturbations caused by the DF.
Such perturbations would not have occurred during trials
in which stiffness was being measured, regardless ofwhether the DF had been activated or inactivated prior to
the interval of stiffness measurement. Because thedifferences in EMG were small compared to the overallmagnitude of the EMG and because their onset tended to154Fig. 9 Rectified, averaged, andsmoothed surface EMG during
the DF after-effect trials for
subject 1. Black dashed curves
denote EMG in the DF-on trialsof the after-effect experiment(DF). Black solid curves denote
EMG in the NF trials of theafter-effect experiment (AE).
Gray curves denote EMG in NF
trials, where the DF was neverpresented (NF). In each plot themean integrated EMG activity
over the entire movement for all
subjects is shown with bar
graphs. The asterisks indicate
significant differences between
the three conditions using
Scheffe’s post hoc comparison:* P<0.05, ** P<0.01,*** P<0.001

be later than the onset of stiffness measurement, we canbe quite certain that the muscles were in the samemechanical state on normal DF trials and on DF trials
when stiffness was being measured.

Discussion
This study compared changes in endpoint impedance after
adaptation to stable and unstable environments in order tofurther understand the processes of learning and adaptation. The endpoint stiffness of the arm was modified
during adaptation to both environments, but for different
reasons. In the stable environment (VF), the change instiffness was due to changes in the net joint torquesneeded to counteract the large environmental force. In
contrast, the change in stiffness after adaptation to theunstable environment (DF) far exceeded what would havebeen predicted from the change in joint torques necessary
to cancel the environmental force. Furthermore, thechange in stiffness was uncorrelated with the change injoint torque. Instead, there was a directionally selective
increase in stiffness which was likely brought about by
biased co-contraction of biarticular muscles.

Stiffness in stable dynamics is a side effect of change injoint torquePrevious studies which investigated the adaptation of arm
movements to novel dynamics did not measure how
endpoint stiffness changed after learning, althoughchanges in the EMG of selected muscles were reported(Thoroughman and Shadmehr 1999). While these changes
in muscle activity patterns likely corresponded to changes
in endpoint stiffness the latter has not been quantified.
This is the first quantitative investigation of the change inendpoint stiffness following learning of an internal model.
After adaptation to the VF, the stiffness ellipses
became elongated along the direction of compensation
for the force field. Such a change in endpoint stiffness
with force direction has been reported previously forstatic postures (Gomi and Osu 1998; McIntyre et al. 1996;
Perreault et al. 2001). The observed change in stiffness
orientation is similar to that reported by (Gomi and Osu
1998) for the corresponding force direction. We foundthat the difference in the elastic force, resisting displacement of the hand, relative to the NF was well correlatedwith the estimated difference in force based on anassumed linear relation between the difference in joint
stiffness and the difference in joint torque relative to the
NF. Furthermore, the same linear relation generated an
excellent prediction of the stiffness measured in the VF.

155This is strong evidence that the change in stiffness
observed in the VF was simply a byproduct of the change
in endpoint force required to adapt to the force field.
Stiffness in unstable dynamics is directly controlled
When subjects were faced with the unstable dynamics of
the DF, they responded by selectively adapting theirstiffness. This provided necessary stability to perform thetask successfully. This impedance control was performedby co-contracting antagonist muscle pairs in order toincrease stiffness without changing the net joint torque.
Such impedance control was first postulated by Hogan(1985). While EMG measurements during movements toa single point from various starting positions provided
some limited evidence that impedance may vary depending on past history (Gribble and Ostry 1998), no
conclusive evidence for the existence of impedancecontrol had been found until recently (Burdet et al. 2001).
While reflexes could have contributed to the measured
stiffness it is unlikely that they were responsible for theadaptation which permitted stable movement in the DF.
Reflexes would only act following displacement of thehand from the planned trajectory and would occur too lateto counteract the DF because its effect would be
increasing throughout the delay period. Since the trajectories in the DF after learning were similar to those in the
NF, with the hand moving smoothly to the final target,
there is no evidence that the hand was sufficiently
disturbed to evoke reflex responses.
Because measurements of stiffness are based on
multiple trials they contain no information about trialby-trial variations. Although unlikely, it is possible thatthe observed increase in stiffness was linked to the force
which the subject exerted to counteract the force appliedby the force field. This would vary from trial to trial due
to variations in the trajectory. If this had been the case,then the difference in stiffness, relative to the NF, shouldhave given a change in force which was correlated withthe difference in measured endpoint force. When this wastested we found that the two were essentially independent.
Similarly, stiffness ellipses generated from presumed
linear relations between joint torque and joint stiffness
failed to reproduce the measured endpoint stiffness in the
DF. From this, it is clear that the change in endpoint
stiffness in the DF with respect to the NF could not be
attributed to compensation for the small forces exerted by
the force field. Therefore, we can conclude that after
adaptation to the unstable DF, the stiffness was controlled
independently of the endpoint force and was selectively
tuned to the instability of the environment.
Change in stiffness is a byproduct of developmentof an inverse model
As the CNS learns to adapt to the environment, theactivity of the muscles is modified in order to counteract

the environmental dynamics. This is presumed to occur
during formation of an inverse model of the dynamics(Conditt et al. 1997; Flanagan and Wing 1997; Flanagan
et al. 2001; Krakauer et al. 1999; Shadmehr and MussaIvaldi 1994; Thoroughman and Shadmehr 1999). Becausemuscle stiffness increases linearly with muscle activationand force (Hunter and Kearney 1982; Kirsch et al. 1994),
any change in the activation produced by the developmentof an internal model should produce a corresponding
change in stiffness. We found that the change in limb
impedance was well explained by the change in the joint
torque that occurred during the learning in the VF. The
only other study which has attempted to address changes
in the impedance of the arm after adaptation to novel
force fields is the recent work of Wang et al. (2001). They
proposed that an inverse model would not have any effecton the impedance of a limb and specifically would notchange the intrinsic stiffness or the short latency reflex
response to a perturbation. The restoring force due tolarge perturbations during movements was attributed to
the long latency reflex and it was presumed that the gainof this reflex was changed due to a modification of thesensory-motor feedback control pathway (Wang et al.
2001). This conclusion is based upon the assumption thatthe use of an inverse model to adapt to the environment
would not affect the intrinsic or reflexive stiffness of amuscle. However, modifying the feedforward command
to muscles will automatically change both the intrinsic
and reflexive components of the muscle impedance. As
the descending drive to muscles (activation) is increased
the muscle’s force response to a disturbance (impedance)increases. This has been shown at both the muscle level(Kirsch et al. 1994) and joint level (Carter et al. 1990,
1993; Gottlieb and Agarwal 1988; Hunter and Kearney
1982; Milner et al. 1995; Weiss et al. 1988). Similarly,
previous work has also shown that both the short and longlatency reflexes are directly influenced by the level ofmuscle activation (Capaday et al. 1994; Dufresne et al.
1978; Marsden et al. 1976; Matthews 1986; Smeets andErkelens 1991; Stein et al. 1995). These results indicatethat any change in the activation level of a muscle due toa modification in the descending command, such as
would be produced by an inverse model would affect boththe intrinsic and reflexive stiffness of the muscles. This isalso seen in the multijoint arm where a change in the joint
torque has a direct effect on the joint stiffness and,
therefore, the endpoint stiffness (Gomi and Osu 1998;
Perreault et al. 2001). A key issue is the question ofwhether the change in impedance following adaptation tonovel dynamics is due to a change in intrinsic andreflexive stiffness which occurs as a direct effect of achange in motor command, or if it is due to themodification of the sensory-motor feedback pathway.
The results of our analysis on adaptation to the VF clearly
indicate that the change in impedance scales with thechange in joint torque, exactly as in isometric multijoint
tasks (Gomi and Osu 1998; Perreault et al. 2001) where
no change to the sensory-motor feedback pathways isrequired. This indicates that development of an inverse

156

model can directly affect the limb impedance. This must
be taken into account when using the force responses toperturbations to infer the properties of the motor control
system.
Acknowledgements The experiments were performed at ATR. We
thank T. Yoshioka for his technical assistance. The manipulandum
was setup by T. Yoshioka and H. Gomi. This research was
supported by the Telecommunications Advancement Organizationof Japan, the Natural Sciences and Engineering Research Councilof Canada, the Swiss National Science Foundation, and the Human
Frontier Science Program

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