letters to nature
between the birds was their experience in the earlier observational learning study14 Ð
observer, pilferer or observer + pilferer. These categories are directly related to the speci®cexperiences in the previous experiment, not the developmental histories of the birds.
Although we cannot state that birds in the pilferer group had never observed other birds
cache they did not do so in the previous study or in the current one. Importantly, theobserver birds have never pilfered another bird's caches.
The observer + pilferer (n = 7) group was ®rst tested at the University of California,
Davis (19±26 July 2000), with a replicate study performed at the University of Cambridge(15 January to 16 February 2001). Birds in the observer (n = 7) and pilferer (n = 7) groups
were tested at Cambridge (15 January to 16 February 2001). The interleaved trials study(observer + pilferer group only) was performed at Cambridge (27±28 February 2001). Allof the birds were housed individually in cages (45 ´ 76 ´ 76 cm). In Davis, birds werehoused in an outdoor aviary. In addition to natural lighting, we provided ¯uorescent strip
lighting. In Cambridge, the birds were housed in the same cages as Davis, but these wereplaced in a quarantined indoor room in which only ¯uorescent lighting was provided.
Birds were fed a mixture of Iams mini-chunk dog food biscuits and peanuts, both of whichwere provided in powdered form to ensure that the birds could not cache outside of theexperiment. Water was provided ad libitum.

Apparatus
The fronts of the individual cages were made of aluminium wire and the sides were madeof solid aluminium. The storer was placed in a cage located adjacent to a second cage
containing the observer, with a 20-cm-wide gap between the two cages. On the back ofeach cage was a 25 cm2 perspex panel, which allowed the storer and observer to see one
another. Caching trays10±13 were constructed from sand-®lled plastic ice-cube trays, eachcontaining a 2 ´ 8 array of moulds that were attached to a wooden board. Each caching tray
was made unique by attaching different con®gurations of Lego (Net®eld) bricks onto theboard behind the ice-cube tray.

Procedure
Birds were deprived of food overnight. Caching trials started at 10:00 (Davis) or 11:00(Cambridge) the following day. During every 15-min caching trial (n = 3 per cachingtreatment, observer + pilferer group in Davis and observer, pilferer and observer + pilferer
groups in Cambridge; n = 2 pairs, observer + pilferer group interleaved trials in
Cambridge) each subject received one sand-®lled caching tray and a bowl containing 50
wax worms. During the unobserved caching treatment (in private), a towel covered theback of the cage so that the view of the observer bird was completely obscured. During theobserved caching treatment, the observer had a clear view of the storer. At the end of eachcaching trial, the tray and food bowl were removed from each cage. The experimenters
recorded the number and location of wax worms cached in the tray. Any extraneous foodthat the birds did not cache was removed. The number of worms cached therefore refers to
the number of worms that remained in the tray at the end of the caching trial. Observers
were given 15 min in which they were allowed to eat their maintenance diet at the end of
the caching trial, and each bird was given three wax worms. In Davis and Cambridge, twocaching trays were placed inside the subject's cage during recovery trials. One of the trays
was unfamiliar to the bird. The other tray contained the previously cached worms and wasplaced in its original location. Three caching trays were placed inside the subject's cageduring recovery trials for the interleaved trials study. One tray was unfamiliar to the bird,whereas the other two trays contained the previously cached worms, which were placed in
the same locations as they were placed during the observed tray and in private tray cachingtreatments.
The subjects were allowed to recover the cached food items for 10 min. The number andlocation of the searches was recorded by direct observation, as well as the number of caches
recovered and whether these were eaten or re-cached. We also noted the location of the recached food items, and whether they were made in new sites compared to those places thebird had cached in during the previous caching trial (old site). We calculated the numberof searches to ®nd the ®rst cache (recovery accuracy), the proportion of worms recovered
and the proportion of those recoveries that were re-cached. All 21 birds had previous
experience with the presence of towels on their cages for at least three trials each. They wererestricted from caching during these trials, but were given powdered food and three wax
worms.

Analysis
We used the Wilcoxon matched-pairs test on the mean values summed across trials foreach caching treatment. As the number of items recovered depends on the number of
items cached, and the number of items re-cached depends on the number of caches thatare recovered, we also analysed the proportion of caches that were recovered and the
proportion of those re-cached. For all analyses, alpha was set at 0.05.
Received 23 July; accepted 20 September 2001.
1. Zentall, T. R. & Galef, B. G. (eds) Social Learning. Psychological and Biological Perspectives (Lawrence
Erlbaum Associates, London, 1998).
2. Shettleworth, S. J. in Behavioral Brain Research in Naturalistic and Semi-naturalistic Settings (eds
Alleva, E., Fasolo, A., Lipp, H.-P. & Nadel, L.) 158±179 (Kluwer Academic, The Hague, 1995).
3. Vander Wall, S. B. Food Hoarding in Animals (Univ. Chicago Press, Chicago, 1990).
4. Clarkson, K., Eden, S. F., Sutherland, W. J. & Houston, A. I. Density dependence and magpie foodhoarding. J. Anim. Ecol. 55, 111±121 (1986).
5. Gibb, J. A. Populations of tits and goldcrests and their food supply in pine populations. Ibis 102, 163±
208 (1960).
6. Bednekoff, P. A. & Balda, R. P. Observational spatial memory in Clark's nutcrackers and Mexican jays.
Anim. Behav. 52, 833±839 (1996).

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7. Heinrich, B. & Pepper, J. W. In¯uence of competitors on caching behavior in the common raven,Corvus corax. Anim. Behav. 56, 1083±1090 (1998).
8. Bugnyar, T. & Kotrschal, K. Do ravens manipulate the others' attention in order to prevent or achievesocial learning opportunities? Adv. Ethol. 36. 106 (2001).
9. Heinrich, B. Mind of the Raven (Harper Collins, New York, 1999).
10. Clayton, N. S. & Dickinson, A. D. Episodic-like memory during cache recovery by scrub jays. Nature
395, 272±278 (1998).
11. Clayton, N. S. & Dickinson, A. D. Scrub jays (Aphelocoma coerulescens) remember the relative time ofcaching as well as the location and content of their caches. J. Comp. Psychol. 113, 403±416 (1999).
12. Clayton, N. S. & Dickinson, A. D. Memory for the contents of caches by Scrub Jays. J. Exp. Psychol.
Anim. Behav. Proc. 25, 82±91 (1999).
13. Clayton, N. S., Yu, K. & Dickinson, A. D. Scrub jays (Aphelocoma coerulescens) can form integrated
memory for multiple features of caching episodes. J. Exp. Psychol. Anim. Behav. Proc. 27, 17±29(2001).
14. Clayton, N. S., Grif®ths, D. P., Emery, N. J. & Dickinson, A. D. Episodic-like memory in animals. Phil.
Trans. R. Soc. Lond. B 356, 1483±1491 (2001).
15. Goodwin, D. Further observations on the behaviour of the jay. Ibis 98, 186±219 (1956).
16. Stevens, T. A. & Krebs, J. R. Retrieval of stored seeds by marsh tits (Parus palustris) in the ®eld. Ibis 128,513±515 (1984).
17. Hampton, R. R. & Sherry, D. F. The effects of cache loss on choice of cache sites in the black-cappedchickadee. Behav. Ecol. 5, 44±50 (1994).
18. Baker, M. C. & Anderson, P. Once-pilfered cache sites not avoided by black-capped chickadees. Anim.
Behav. 49, 1599±1602 (1995).
19. Hare, B., Call, J., Agnetta, B. & Tomasello, M. Chimpanzees know what conspeci®cs do and do not see.
Anim. Behav. 59, 771±785 (2000).
20. Hare, B., Call, J. & Tomasello, M. Do chimpanzees know what conspeci®cs know? Anim. Behav. 61,
139±151 (2001).
21. Whiten, A. & Byrne, R. W. Tactical deception in primates. Behav. Brain Sci. 11, 233±244 (1988).
22. Suddendorf, T. & Corbalis, M. C. Mental time travel and the evolution of the human mind. Genet. Soc.
Gen. Psychol. Monogr. 123, 133±167 (1997).
23. Heyes, C. M. Theory of mind in nonhuman primates. Behav. Brain Sci. 21, 101±148 (1998).
24. Clayton, N. S. & Dickinson, A. D. Motivational control of caching behaviour in the scrub jay,
Aphelocoma coerulescens. Anim. Behav. 57, 435±444 (1999).

Acknowledgements
This work was supported by the National Institutes of Health, National Institute on Aging
and Whitehall foundation grants to N.S.C. N.J.E was funded by a Medical ResearchCouncil programme grant to E. B. Keverne. We thank S. Hettige, D. Jennings and V. R.
Metcalf for help in running the experiments. We also thank the University of California,
Davis, for allowing us to conduct the ®rst trials there and for providing the necessary
facilities. We thank S. Baron-Cohen, E. B. Keverne, K. N. Laland, D. I. Perrett and C. J.
Saldanha for comments on the manuscript. We are especially grateful to T. J. Bussey,
A. Dickinson and N. J. Mackintosh for discussion and comments on the manuscript.
Correspondence and requests for materials should be addressed to N.S.C.
(e-mail: nsc22@cam.ac.uk).

.................................................................

The central nervous system stabilizes
unstable dynamics by learning
optimal impedance
Etienne Burdet*²§k, Rieko Osu²k, David W. Franklin§¶,
Theodore E. Milner¶ & Mitsuo Kawato²§* Department of Mechanical Engineering, National University of Singapore,
119260, Singapore² Kawato Dynamic Brain Project, ERATO, JST, Hikaridai, Seika-cho,
Soraku-gun, Kyoto, 619 0288, Japan§ ATR Human Information Science Laboratories, Hikaridai, Seika-cho,
Soraku-gun, Kyoto, 619 0288, Japan¶ School of Kinesiology, Simon Fraser University, Burnaby,
British Columbia V5A 1S6, Canadak These authors contributed equally to the work..............................................................................................................................................

To manipulate objects or to use tools we must compensate for any
forces arising from interaction with the physical environment.
Recent studies indicate that this compensation is achieved bylearning an internal model of the dynamics1±6, that is, a neuralrepresentation of the relation between motor command and© 2001 Macmillan Magazines Ltd

NATURE | VOL 414 | 22 NOVEMBER 2001 | www.nature.com

letters to naturebarobot produced no force when trajectories followed the y axis, but
the hand was pushed away whenever it deviated from the y axis (redarrows in Fig. 1a). The force (F x ; F y ) exerted on the hand was" #  
Fx
bx
1

Fy
0
where b . 0 (N m-1) was chosen to be larger than the stiffness of thearm measured in NF movements so as to produce instability.
The NF movements (green in Fig. 1b) had trajectories that wereapproximately straight. Almost all trials ended on target and the
distribution of endpoints was bell-shaped. However, the initial
movement direction varied slightly from trial to trial owing tomotor output variability. The DF ampli®es such variations bypushing the hand with a force proportional to the deviation fromthe y axis (red arrows in Fig. 1a). Consequently, the initial trials in
the DF exhibited unstable behaviour, diverging widely either to theright or to the left. Most trials ended either on the left or right safetybarrier (black vertical lines in the left plot of Fig. 1c) with a roughly
symmetrical distribution of positions. With practice, however,subjects gradually became pro®cient at producing straighter trajectories, similar to those in the NF (compare Fig. 1b and the middle
plot of Fig. 1c). Most trials then ended on target with a bell-shaped
endpoint distribution. The hand-path error (see Methods, equation(2)) decreased signi®cantly between the ®rst ®ve trials and the last®ve trials (P  0:002; Fig. 2a), indicating that the subjects learned toadapt to the dynamics of the DF. At the end of the learning trials, nosigni®cant differences in the hand-path error were found betweenaLearning in DF

50

Hand-path error (cm2)movement5,7. In these studies interaction with the physical environment was stable, but many common tasks are intrinsicallyunstable8,9. For example, keeping a screwdriver in the slot of ascrew is unstable because excessive force parallel to the slot cancause the screwdriver to slip and because misdirected force cancause loss of contact between the screwdriver and the screw.
Stability may be dependent on the control of mechanical impedance in the human arm because mechanical impedance cangenerate forces which resist destabilizing motion. Here we examined arm movements in an unstable dynamic environment created by a robotic interface. Our results show that humans learn tostabilize unstable dynamics using the skilful and energy-ef®cient
strategy of selective control of impedance geometry.
How do we succeed in performing mechanically unstable tasks?
One way would be to stabilize the arm by means of feedback control.
However, neural feedback, whether operating through involuntary(re¯ex) or voluntary commands to our muscles, is delayed by thesensory feedback pathways, which would tend to increase rather
than reduce instability10,11. Alternatively, stability might be achievedby controlling mechanical impedance (resistance to imposed
motion); speci®cally, the spring-like property of muscles12,13.
Demonstrating impedance control is not trivial, because it isnecessary to show that the shape or orientation of the impedance
can be voluntarily modi®ed independently of the force applied by
the hand. Modi®cations in impedance that depend on changes inapplied force do not constitute proof of impedance control becausemuscle stiffness inherently scales with activation level14,15. To
demonstrate impedance control, we compared multi-joint arm
movements in a null ®eld (NF) and in a divergent force ®eld(DF), which produced an unstable interaction with the arm, but
required no change in applied force relative to the NF.
Subjects made horizontal point-to-point movements away fromthe body, along the y axis of our coordinate system (Fig. 1a). The
hand was linked by means of a stiff brace to a handle at the end of arobotic manipulandum (PFM) that exerted computer-controlled
forces during movement (see Methods). The DF produced anegative elastic force perpendicular to the target direction. The

0

0

20

c

Null field, NF

Divergent field, DF

60
40
Number of trials

80

c 15

Endpoint force

Endpoint force

b 15

100

Fx (N)

PFM
55
Targety (cm)

DF
Start
y

0

0

–10

–10

15

15

0

0

x
30
–3 0 3x (cm)

InitialtrialsAfterlearning

Aftereffects

Figure 1 Experimental set-up and learning in a divergent force ®eld (DF). a, The PFM
exerts forces on the hand during horizontal point-to-point arm movements away from thebody. The red arrows graphically depict the DF. b, Five consecutive trajectories of one
subject are shown for movements in a null force ®eld (NF, green). At the top of trajectoryplot, a histogram shows the distribution of endpoint positions (based on curvature greaterthan 0.03 m-1) for the corresponding ®ve movements of all subjects. c, Learning in the DF(red). Five trajectories of the same subject are shown, which include the ®rst ®ve trials(left), the last ®ve trials (middle), and ®ve consecutive after-effect trials (right). For safety
reasons the DF was turned off when the trajectory deviated more than 3 cm from they axis. The black lines indicate this safety zone.
NATURE | VOL 414 | 22 NOVEMBER 2001 | www.nature.com

Fy (N)

[0,0]

–10

0

0.7

–10

S1

S2 S3

S4

S5

Time (s)Figure 2 Error evolution during learning the DF, endpoint force±time pro®les after
learning for the same subject as in Fig. 1, and forces at the movement midpoint for all
subjects. The hand-path error is de®ned in the Methods. a, Hand-path error as a functionof the number of learning trials for the DF. The solid curve is a least-squares ®t of an
exponential function to the data. b, Mean endpoint force pro®les with standard deviationsin the x and y directions after learning, for NF (green) and DF (red). The force is similar in
the DF and NF. c, The mean x and y forces and standard deviations at the midpoint of the
trajectory in the NF (green) and DF (red) for all ®ve subjects.

© 2001 Macmillan Magazines Ltd

447

letters to nature

S2

S1

S5

DF
NF

200 N m–1
S3

Movement
direction
S4

Figure 3 Adaptation of stiffness geometry to the DF. Stiffness ellipsesÐthe graphicaldepiction of the elastic restoring force corresponding to a unit displacement of thehandÐare shown for ®ve subjects in the NF and DF. The long and short axes of the ellipse
represent the directions of maximal and minimal stiffness, respectively. The stiffness inthe DF (red) changed relative to the NF (green). We note that in the DF the stiffness
increased signi®cantly in the direction of the instability (shown by small red arrows for
subject S4), but there was relatively little change along the movement direction (see also
Fig. 4).
448

ab

x component of stiffness
1,000

N m–1

movements in the NF and DF (P  0:906), indicating that movements had become stable and were similar under the two conditions. This is also indicated by the force applied to the PFM by thearm in the DF, which was not signi®cantly different from that in the
NF, after learning (Fig. 2b, c).
To examine the after-effects of learning the DF, the force ®eld was
unexpectedly removed on selected trials after learning. These aftereffect movements were characterized by trajectories that hardly
deviated from the y axis with endpoints concentrated in the middleof the target (Fig. 1c, right). The hand-path error revealed that they
were even straighter than NF trajectories (P  0:001).
How did subjects overcome the instability of their movements?
Because of the unpredictability of the direction of the disturbance aforward or inverse dynamics model5,7 could not have been used. In
contrast, an increase in mechanical impedance would provide
increased resistance to a disturbance regardless of its direction.
Perhaps subjects used generalized muscle coactivation to achieve an
isotropic increase in stiffness at the hand. This has been demonstrated previously during posture16. An analogous strategy is commonly employed in single-joint tasks when the environment isdestabilizing17,18. On the other hand, the stiffness might have been
selectively controlled in the direction of the instability. Such controlof limb impedance was ®rst proposed 16 years ago12, but has yet to
be demonstrated.
These possibilities were investigated by examining the stiffness at
the hand after learning (see Methods). We measured stiffness
around the midpoint of movement paths, using a technique19 thatprovides an unbiased and accurate estimate of stiffness and requires
fewer trials than previous methods20,21. On random trials, the PFM
brie¯y displaced the hand by a constant distance from a predictionof the undisturbed trajectory19. Stiffness was estimated from therestoring force, produced in response to the displacement, divided
by the amplitude of the displacement.
Stiffness at the hand in the DF was modi®ed in shape andorientation relative to stiffness in the NF (Fig. 3, compare red andgreen data). Although the x component of stiffness was signi®cantly
larger in the DF than in the NF (Fig. 4a, compare red and greendata), the y component of stiffness in the DF was not signi®cantly
different from that in the NF (Fig. 4b, compare red and green data).

y component of stiffness

1,000

DF

500

500
DF

Net
NF
0

S1 S2 S3 S4 S5

NF
0

S1 S2 S3 S4 S5Figure 4 Stiffness and stability. a, The x components of hand stiffness in the NF (green),
hand stiffness in the DF (red) and net stiffness of the hand and DF environment (Net, black)after learning. Vertical bars show 90% con®dence intervals. b, The y components of hand
stiffness in the NF (green) and in the DF (red).

There was a selective increase in stiffness in the direction of
instability. After learning, the stiffness of the arm increased suf®ciently that the combined stiffness of the arm and the environment
in the x direction (Fig. 4a, Net, black data) was positive and similar
in magnitude to that measured during NF movements (Fig. 4a, NF,
green data). Trial-to-trial variations in x force did not correlate withchanges in stiffness in the x direction for any subject (Table 1),
providing even stronger support for impedance control. Furthermore, we can conclude that the adaptation was preprogrammed andnot reactive to the unstable environment because the DF was notapplied on those trials used for stiffness measurement. (This,
however, was not detected by the subjects.)
Previous impedance adaptation studies did not address thecapacity to modify stiffness geometry independently of force.
Adaptive increases in stiffness observed for single-joint tasks inunstable environments17,18,22 provide a limited view of the adaptive
capabilities of the central nervous system (CNS) because the only
parameter that can be varied is stiffness magnitude. Changes in
stiffness geometry, observed in ball catching20, occurred principally
while joint torque was changing to compensate for the momentumof the ball. The anticipatory increase in stiffness just before ball
impact involved generalized muscle coactivation, which serves
mainly to increase stiffness magnitude. The use of biarticularshoulder muscles to perform a postural task, requiring no net
shoulder torque9, might be considered a form of impedance control.
However, because joint stiffness clearly scaled with joint torque inthis task it is not clear whether the involvement of biarticular
muscles represents genuine impedance control or `hard-wired'muscle synergy. Similarly, the statistically signi®cant covariationof change in stiffness with change in external force found formovements in a velocity-dependent force ®eld23 could be attributed
largely to the linear relationship between joint torque andstiffness14,15. The modi®cations of stiffness observed during movements constrained by a mechanical channel24 were also probably
due to changes in force. In contrast, adaptation to the DF required
no change in force or joint torque relative to NF movements, yet
subjects skilfully adapted their stiffness to the instability of theenvironment so that stiffness increased only in the required direction and by the required amount. Using the DF, we have been able toshow for the ®rst time that the CNS can voluntarily control themagnitude, shape and orientation of the endpoint stiffness in a
predictive way that is independent of the force needed to compensate for the imposed dynamics.

Table 1 Correlation between stiffness and force
Subject

S1

S2

S3

S4

S5

R2

0.24

0.53

0.08

0.29

0.34

.............................................................................................................................................................................
R2 is the correlation between the change in stiffness and the change in force in the DF relative to the
NF. DF, divergent force ®eld; NF, null ®eld.

© 2001 Macmillan Magazines Ltd

NATURE | VOL 414 | 22 NOVEMBER 2001 | www.nature.com

letters to nature
Finally, why does the CNS use the dif®cult and computationallycostly strategy of controlling the full impedance geometry rather
than the simple strategy of co-contracting all muscles? The particular shape of endpoint stiffness learned in the DF, large in the
direction of instability and small in the direction of movement,suggests a trade-off between stability and high muscle activation.
Increasing impedance enhances the robustness to external perturbations, but co-contraction of muscles increases metabolic cost12.
Moreover, motor output variability increases with muscle
activation25,26. Therefore, reducing the impedance will decrease
both the metabolic cost and movement error. For these reasons, it
is desirable to optimize impedance. The control problem facedby the CNS is to optimize the magnitude, shape and orientationof impedance to achieve stability while minimizing metabolic
cost.
M

Methods

Nine subjects performed the experiment (24±34 years of age; two females and sevenmales). The learning analysis used only data from the ®ve naive subjects. The institutional
ethics committee approved the experiments and the subjects gave informed consent before
participation. The robot used is a parallel-link direct-drive air and magnet ¯oatingmanipulandum (PFM), which moves in the horizontal plane. Details of its design andoperation have been described previously19,21. The PFM was set up by T. Yoshioka and
H. Gomi. Subjects sat in a chair with a harness to constrain the trunk so that the elbow
and shoulder joints could only move in the horizontal plane (two degrees of freedom).
The forearm and wrist were held in a thermoplastic splint rigidly attached to the
manipulandum. Therefore, subjects could not vary the kinematics of the arm to change its
impedance. Subjects began by performing 100 movements in the NF, followed by 100±300
learning trials in the DF. Subjects then made 100 more movements in the DF, 20 of which
had the force ®eld removed unexpectedly (called after-effects). All movements wererecorded during these sessions, including those not reaching the target. The start position,
target (diameter 2.5 cm) and instantaneous hand position were displayed on a horizontal
screen slightly above the arm. The prescribed movement time of 600 (6100) ms wasindicated by acoustic signals.
Adaptation to the DF was quanti®ed by calculating the error relative to a straight line
joining the start position and target centre. The hand-path error E represents the area
between the actual movement path and the straight line, and was calculated from the start
time 0 (75 ms before crossing a hand-velocity threshold of 0.05 m s-1), to the termination
time T (when curvature exceeded 0.07 m-1):
E

T# jx tjjyÇ tjdt
02

To test whether learning occurred, we compared the hand-path error in the ®rst ®velearning trials and the last ®ve learning trials using an analysis of variance (ANOVA) withrandom factor subjects. An ANOVA was also used to compare the last ®ve trials in the NF
and the last ®ve trials of the learning phase. The presence of after-effects was tested by
performing an ANOVA comparing the hand-path error of the last 20 movements in the NF
to 20 after-effects trials.
Before each stiffness measurement session, the subjects were retrained in the force ®eld
to ensure that they had remembered the ®eld (NF, 40 trials; DF, 80 trials). Eighty
movements were then recorded in the force ®eld, of which 40, selected randomly,
incorporated displacements. These displacements, at the midpoint of each movement,
were randomly applied in one of eight directions (08, 458, 908, 1358, 1808, 2258, 2708 or
3158). The endpoint stiffness was estimated from the resulting mean change in hand force
and position over a 60-ms interval during the perturbation, starting from 120 ms after
onset of the perturbation19.
Linear regression was carried out to determine whether the change in stiffness for
movements in the DF relative to those in the NF could be explained by the corresponding
change in force. By de®nition, the difference between torque produced in response to a
perturbation dqs dqe T in the DF and NF is given by:




dqs
dts
 DKq
:
3
dte
dqe
where DKq is the difference between DF and NF joint stiffness, and dts and dte correspondto the difference between DF and NF shoulder and elbow torque owing to elastic resistancel, respectively.
It has been shown that the joint stiffness is linearly dependent on the joint torque inposture15. Making the same assumption for movement, we obtain the linear relation:
a1 Dts a3 Dte
DKq 4a3 Dte a4 Dte
where a1 ¼a3 are constants, and Dts and Dte are the difference between DF and NF
shoulder and elbow driving torque measured by the force sensor l. The difference inendpoint force can be obtained from equation (3) using the jacobian. Restricting theanalysis to the x direction, since endpoint stiffness in the DF was principally modi®ed inthat direction, we then have:
NATURE | VOL 414 | 22 NOVEMBER 2001 | www.nature.com

3
g1
6 7
Dte dqs 4 g2 5
g32

DF x  Dts dqs

Dte dqe5

where g1, g2 and g3 are constants that incorporate the ai and the coef®cients of thejacobian. R2 values were computed for each subject using a linear system of 40 equations(with g1, g2 and g3 as unknowns) representing the 40 trials where stiffness was measured.
An R2 value close to one would indicate that endpoint stiffness in the x direction wascompletely determined by the joint torques required to perform the task. An R2 value close
to zero would indicate that in every trial endpoint stiffness was controlled independentlyof the required joint torques. A more detailed derivation of these equations is available as
Supplementary Information.
Received 8 June; accepted 27 September 2001.
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Supplementary Information accompanies the paper on Nature's website(http://www.nature.com).

Acknowledgements
The experiments were performed at ATR. We thank A. Illiesch and T. Flash for discussions.
We thank A. Smith, S. Scott, D. Wolpert, Z. Ghahramani, and S. Schaal for giving valuablecomments on an earlier version of the manuscript. This research was supported by theSpecial Coordination Fund for promoting Science and Technology of the Science andTechnology agency of the Japanese Government; the Swiss National Science Foundation;
the Natural Sciences and Engineering Research Council of Canada; and the Human
Frontier Science Program.
Correspondence and requests for materials should be addressed to M.K.
(e-mail: kawato@atr.co.jp). Further information about this work can be found at
http://www.bioeng.nus.edu.sg/humanRobotics.htm.

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