Optimal Impedance Control for Task Achievementin the Presence of Signal-dependent Noise

Rieko Osu1, Naoki Kamimura2, Hiroshi Iwasaki2, Eri Nakano3,
Chris M. Harris4, Yasuhiro Wada2, and Mitsuo Kawato1

1. ATR Computational Neuroscience Laboratories, Kyoto, 619-0288, Japan
2. Nagaoka University of Technology, Niigata, 940-2188, Japan
3. ATR Human Information Processing Research Laboratories, Kyoto, 619-0288, Japan
4. Institute of Neuroscience, University of Plymouth, Plymouth PL4 8AA, U.K.
Abbreviated title (38/60): Optimal Impedance Control with TOPS-α
Corresponding Author
Mitsuo Kawato
ATR Computational Neuroscience Laboratories
2-2-2 Hikaridai, Seika-cho, Soraku-gun
Kyoto, 619-0288 Japanemail: kawato@atr.jp
TEL: +81-774-95-1058
FAX: +81-774-95-2647

Key Words
Impedance control, stiffness, signal-dependent noise, end-point accuracy, single-joint
movement, variability, TOPS-α model.

Abstract (243) There is an infinity of impedance parameter values, and thus different
co-contraction levels, that can produce similar movement kinematics, from which thecentral nervous system must select one. Although signal-dependent noise (SDN) predicts
larger motor-command variability during higher co-contraction, the relationship between
impedance and task performance is not theoretically obvious and thus examined here.
Subjects made goal-directed, single-joint elbow movements to either move naturally todifferent target sizes or voluntarily co-contract at different levels. Stiffness was estimatedas the weighted summation of rectified EMG signals through the index of muscle
co-contraction around the joint (IMCJ) proposed by Osu et al. (2002). When subjects
made movements to targets of different sizes, IMCJ increased with the accuracy
requirements, leading to reduced end-point deviations. Therefore, without the need for
great accuracy, subjects accepted worse performance with lower co-contraction. Whensubjects were asked to increase co-contraction, the variability of electromyography(EMG) and torque both increased, suggesting that noise in the neuromotor commandincreased with muscle activation. In contrast, the final positional error was smallest for
the highest IMCJ level. Although co-contraction increases the motor-command noise, theeffect of this noise on the task performance is reduced. Subjects were able to regulate
their impedance and control end-point variance as the task requirements changed, andthey did not voluntarily select the high impedance that generated the minimum end-point
error. These data contradict predictions of the SDN-based theory, which postulates
minimization of only end-point variance, and thus require its revision.

2

INTRODUCTION
The central nervous system (CNS) can control the mechanical impedance (inertia,
viscosity, and stiffness) of its motor apparatus (Hogan, 1985; McIntyre et al., 1996). For
example, viscosity and stiffness can be adjusted through mechanisms such as the
co-contraction of muscles. Several studies have examined impedance changes in
response to external perturbations. When trying to keep a hand position constant(Mussa-Ivaldi et al., 1984) or moving (Milner and Cloutier, 1993; Burdet et al., 2001)
against external perturbing forces, limb stiffness increases. Precise timed increases in
stiffness also occur in anticipation of predictable external events, such as catching a ball
(Lacquaniti et al., 1993). Impedance changes can also parallel the learning of novel tasks.
Milner and Cloutier (1993), Osu et al. (2002, 2003) and Franklin et al. (2003) reported
that a co-activation of flexors and extensors is observed in the early stages of dynamic
and kinematic learning, with co-activation decreasing with learning. Laursen et al. (1998)
and Gribble et al. (2003) found that muscle co-contraction estimated aselectromyographic (EMG) activity increased with the higher accuracy requirements ofmulti-joint arm movements. However, these previous studies did not examine thevariability in EMG activity, torque or position dependence on task conditions such as
stiffness or accuracy. Furthermore they did not address the effects of volitional
impedance changes on attained movement variability or accuracy. Consequently, it isunclear why a particular set of impedance parameters is chosen for a given task from theinfinite set of possible impedances that could produce the same endpoint kinematics andforces.

3

A similar problem has been investigated for the kinematics of movement. Given areaching task, an infinite number of hand paths and velocities can reach the goal. One
framework for selecting a trajectory is optimal control, in which the solution with thesmallest cost is selected. Several cost functions (Flash and Hogan, 1985; Uno et al., 1989;
Nakano et al., 1999; Wada et al., 2001) have tried to account for the data, but these models
are incapable of predicting a trajectory that takes into account the variance that changes
according to the changes in motor commands and/or impedance. Since the minimum-jerk
model

is

kinematic,

there

is

no

place

for

impedance.

Similarly,

theminimum-torque-change model considers joint torque but not muscle tension, so itcannotincorporate

impedance

withinits

framework.

The

minimum-motor-command-change model can actually deal with impedance because it
models the motor command for each muscle but always predicts the minimum motor
command, thus minimum impedance, which is at odds with the idea of different
co-activation levels dependent on different accuracy requirements (Laursen et al., 1998;
Gribble et al., 2003). These models neither have variance as a penalty term nor implement
noise in motor commands but only deal with a mean or desired trajectory. A recent
computational model of movement planning considering the presence of signal
dependent noise (Task Optimization in the Presence of Signal-dependent noise: TOPS)
may provide a unifying framework that could potentially account for both the kinematics
and impedance of a movement. In this framework, the motor commands are assumed tobe corrupted by noise, whose standard deviation increases with the level of the motor
command (Harris and Wolpert, 1998; Jones et al., 2002). Such signal-dependent noise

4

(SDN) plays through the muscle dynamics and limb impedance, leading to variability inmovement. Different motor commands should thus lead to different movement statistics.
The TOPS model proposes that the cost depends on these statistics (Miyamoto et al.,
2002; Hamilton and Wolpert, 2002). The motor command is selected to minimize the cost,thus minimizing the deleterious effects of SDN on task performance.
As the SDN plays through the impedance of the limb, the impedance can be changed
to modify the consequences of the noise. Similarly, the change in impedance alters thenoise in the system because the co-contraction of muscles leads to larger muscle activities,
resulting in a larger SDN. Assuming the presence of SDN, impedance should havecomplicated effects of both increasing and decreasing the movement accuracy. Given thatchanging either the motor command or impedance can affect the movement statistics, the
TOPS model has a potential capability to coherently explain both trajectory planning and
impedance control by including impedance parameters within its framework. Therefore,
the first aim of this study is to experimentally examine the effects of increased impedanceon movement accuracy.
The principle of the TOPS model is that the objective of motor planning is to optimize
task statistics in order to increase accuracy (cf. minimizing end-point variance in Harrisand Wolpert, 1998). If this principle holds, we expect the optimal impedance forminimizing movement error to be selected in voluntarily executed movements. Then, if
subjects are required to specifically increase impedance during movements, a decrease inaccuracy is predicted because non-optimal impedance should generate poorer task
statistics. On the contrary, if higher movement accuracy is required, subjects should

5utilize the same impedance as in ordinary conditions, where impedance is already
optimized to maximize accuracy. The second aim of this paper is to examine whether
these predictions can be experimentally confirmed.
To investigate the relationship between impedance and movement variability, we
examined two tasks. In the first task (target size), accuracy constraints were varied andmovement variability and impedance were estimated. In the second task (voluntary
co-contraction), impedance was voluntarily controlled and movement variability andaccuracy were examined. From the results of these two experiments, we found that
impedance does have the above mentioned complicated effects of both increasing SDN
and decreasing end-point accuracy and that the data contradicted the predictions made by
the TOPS model.

6

MATERIALS AND METHODS
Thirteen male subjects, aged 20-29 years, participated in at least one of the two
experiments, and 11 of them were naïve as to the purpose of the study. Nine subjects
participated in the target-size experiment and nine subjects participated in the voluntary
co-contraction experiment. For the five subjects who participated in both experiments,
the target-size experiment was first carried out and then the voluntary co-contraction
experiment was conducted on another day. Each experiment took several hours, since itconsisted of EMG electrode attachment, isometric force generation for parameter
calibration of IMCJ, preparatory trials to select movement times specific to individual
subjects, and main trials. Accordingly, a one- to two-hour lunch break and frequent short
rests were allowed. We did not try to estimate muscle viscosity in this study. However,from previous studies (Gomi and Osu, 1998; Osu and Gomi, 1999), we expected thatmuscle viscosity would co-vary with muscle torque, muscle stiffness, and EMG activity.
The Institutional Ethics Committee approved the experiments, and subjects gave
informed consent prior to participation.

Experimental apparatus
Subjects sat on a chair and rested their right elbow on a sponge fixed to a table (Fig.
1A). The table was adjusted to lift the subject’s arm to shoulder level so that movements
were made in a horizontal plane. The upper arm movement was constrained by fixing theshoulder position with a harness and the elbow position with the sponge. The subject’s

7

wrist was braced. To reduce friction between arm and table, the arm was attached to aboard that levitated above the table by an air sled (Fig. 1A). Subjects were asked to move
only their elbow joint. An OPTOTRAK 3020 was used to measure the position of amarker placed on the end of a 9-cm vertical bar, which was grasped by the subjects. The
marker position was sampled at 500 Hz and projected as a cross on a CRT screen, placedin front of the subject, representing current hand position (Figs. 1B and C). Subjects
performed the task while looking only at the CRT screen.

Electromyography
Surface EMG activity was recorded using pairs of silver-silver chloride surface
electrodes. The activities of two elbow muscles, the brachioradialis and the medial headof the triceps brachii, and two biarticular muscles, the biceps brachii and the long head of
the triceps, were recorded. The EMG signals were analog filtered at 25 Hz (high pass) and
1.0 kHz (low pass) with a Nihon Kohden amplifier (MME-3132) and then sampled at 2.0
kHz. All EMG comparisons between different conditions as well as the calibration of
IMCJ involved data that were recorded on the same day without removing the electrodes.
From a previous study using the same methods (Osu et al., 2002), we confirmed thestationarity of the EMG electrodes by comparing torque-EMG relationships before andafter the main experiment.

Stiffness estimation by IMCJ
Gomi and Kawato (1996) and Burdet et al. (2000, 2001) measured stiffness during8

multijoint arm movements by applying mechanical perturbations with a robotic
manipulandum. Previous studies suggested linear relationships among stiffness, torque,and surface EMG activity signals (Tsuji et al., 1995; Osu and Gomi, 1999). Based onthese previous studies, Osu et al. (2002) proposed an index of muscle co-contraction
around the joint (IMCJ) to evaluate joint viscoelasticity during movements. IMCJ was
defined as the summation of absolute values of antagonistic muscle torques around thejoint, and it was computed from the linear relation between surface EMG activity andjoint torque. Osu et al. (2002) confirmed that IMCJ during isometric contraction, as wellas during movement, correlated well with joint stiffness estimated by the conventional
method, i.e., applying mechanical perturbations, and proposed a formula to estimate
stiffness from EMG activity through the IMCJ. In this paper, based on Osu et al.’s (2002)
method, stiffness during movement was computed for the elbow joint in each trial.
Assuming that surface EMG activity is proportional to muscle tension, elbow torque τ
can be expressed as follows:τ = a1u1 − a2 u2 + a3 u3 − a4 u4 .

(1)Here, u1 and u2 denote surface EMG activity of the elbow flexor and extensor, and u3
and u4 denote surface EMG activity of the biarticular flexor and extensor. The
parameters ai include both the moment arm and conversion factor from the muscle
activity to muscle tension. Using the parameters ai , the IMCJ of the elbow is computed asfollows:

IMCJ = a1u1 + a2 u2 + a3 u3 + a4 u4 .

(2)

On each day of the experiment and before the main experiment, subjects were asked to9generate six different isometric force temporal patterns with three different levels of force
magnitude (maximum forces of 5, 10 and 15 N) with five repetitions (90 trials = 6 x 3 x 5).
The force temporal patterns were: i) extend and keep, ii) extend-flex-relax, and iii)
extend-relax-extend-relax-extend, with the same three patterns for flexion, all executed
within three seconds. The torques and EMG activity recorded in these calibration
experiments were used to estimate the parameters ai of the IMCJ by theleast-square-error method (Osu et al., 2002).
We note that the estimation of stiffness from IMCJ should not be taken as a rigorous
method. We well realize the existence of variable moment arms during movements,
nonlinear properties of muscle tension dependent on muscle length, shortening velocity,and motor commands, all of which work against simple linear models such as IMCJ toestimate the stiffness from EMG activity. However, quantitatively speaking, IMCJ was
found to be a good first approximation of stiffness even during movement, which wasthen independently measured by mechanical perturbations (Osu et al., 2002). At the very
least, IMCJ is a much superior and more systematic way of weighting EMG activity frommultiple muscles while estimating stiffness than any arbitrary method of weighting (cf.
Gribble et al., 2003). In order to demonstrate the robustness of the obtained results for theweighting of EMG activity from multiple muscles, we analyzed identical data using thefollowing two different weighting methods. One is to normalize each muscle’s EMG
activity by its maximum value, which is the conventional normalization technique but
does not lead to a stiffness estimate. The second method is to apply the smooth temporal
filter from EMG activity to muscle tension proposed by Koike and Kawato (1995) to data10

from both calibration and main experiments to represent the known electro-mechanicaldelay in computing IMCJ. All three methods of weighting EMG activity led to identicalstatistical results, and thus only the results obtained by IMCJ without filtering arereported here.
Furthermore, we note that hand paths were almost identical across trials and subjects
because only elbow rotation was included, and the joint torques that were generated werealso very severely controlled to be the same because the movement duration was
experimentally controlled. With these stringent experimental paradigms, we expect thatany change in EMG activity is mainly due to a change in co-contraction and stiffness, andnot to changes in movement trajectory, velocity, or torque profiles. This at least validates
the usage of IMCJ as a “relative” index for stiffness.
To confirm impedance changes in a more primitive way, we comparedroot-mean-square (RMS) EMG of each muscle. Assuming that EMG and stiffness have amonotonic relation, if the EMGs of all muscles are larger in a particular condition than inother conditions, we can conclude that stiffness was larger in that condition. The RMS
EMG was computed during the latter half of the movement, that is, 250 ms before the endof the movement to the end of the movement defined by an acceleration threshold (seebelow).

Task
The task was a single-joint extension or flexion movement of the elbow, and it
followed the experimental paradigm of Gottlieb et al. (1989). In the target-size

11experiment, subjects were asked to move their hand from a start circle with a radius of 1.0
cm and enter a target circle with a different radius displayed on a CRT within a desired
movement duration (Fig. 1B). In the voluntary co-contraction experiment, subjects wereasked to move their hand from a start circle with a radius of 1.0 cm and come as close as
possible to a small target point (3-mm radius) displayed on the CRT within a desired
movement duration (Fig. 1C). The shoulder angle (θ1) was fixed to 94 degrees. The startand target positions were either an elbow angle (θ2) of 41 degrees or 97 degrees,
depending on whether the movement was extension or flexion (Fig. 1A). Extension
movements were first conducted in a batch, and then flexion movements were carried out.
The out-and-in movement duration was defined as the time (sec) between when the handexited the start circle and entered the target circle and was used throughout theexperimental task control. Following the isometric force generation, subjects conducted
flexion and extension movements of 20 trials each (preparatory experiment). Because thispreparatory experiment was conducted, we did not observe any apparent learning effects
in the main experiments. The desired out-and-in movement duration was predeterminedby its average over the last ten trials during the preparatory experiment to make themovement duration comfortable for each subject. The desired out-and-in movement
durations ranged between 0.232 and 0.325 (sec) for all subjects in the target-size
experiment and between 0.250 and 0.334 (sec) in the voluntary co-contraction
experiment. For each subject, only movements with an allowable error of ±10% of the
pre-specified out-and-in desired movement duration were recorded as successful
movements. A computer program checked whether the movement duration was within12the allowable range, and a warning of ‘too early,’ ‘too late,’ or ‘out of target’ was fed backto the subjects after each trial when the hand entered the target circle too soon, too late, or
did not end within the target circle, respectively. For the target-size experiment, average
success rates and standard deviations across subjects for large, medium, and small targets
were 70±14%, 62±14%, and 45±13% respectively. For the voluntary co-contraction
experiment, average success rates and standard deviations across subjects for normal,
medium, and high co-contraction were 58±20%, 66±11%, and 62±25% respectively.

Experiment I: Changed target size
The task was to move (flex or extend) the hand to different-sized target circles. The
radii of the target circles were 3.5 cm, 2.5 cm, and 1.5 cm for large, medium, and small
conditions, and they were presented in this order to subjects. Subjects were requested tobring their hand within a target at the end of the movement. We recorded 40 successful
trials in which the hand reached the target within the allowable time range for each target
size and for either flexion or extension; 240 successful trials (2 directions x 3 sizes x 40
trials) were recorded and used for analysis. The desired out-and-in movement duration
was corrected for the different distances that were traveled to the different target sizes. No
instructions were given on co-contraction. Although EMG signals were recorded, muscle
activity was not fed back to the subjects (Fig. 1B).

Experiment II: Changed co-contraction level
The task was to execute extension or flexion movement under three levels of muscle

13

co-contraction: a normal level with elbow muscles relaxed; a high level in which elbow
muscles were strongly co-contracted so that the arm became stiff; and a medium level ofco-contraction between normal and high. In order to avoid presentation of the target
region that explicitly instructs the permitted end-point variability, a small target point waspresented instead of a target circle. The subjects moved their hand from a start circle to as
close as possible to the target point. The out-and-in movement duration was measured as
the time interval between when the hand exited the start circle and entered a virtual
end-point-domain circle with a radius of 5 cm, whose range was shown by two tangent
lines on the CRT (Fig. 1C).
In the medium and high levels, subjects had to voluntarily co-activate their muscles
during movements. Because the strength of muscles varies between subjects, theactivation levels were set before the main experiment in preparatory trials. In this phase,
20 trials each for the normal and high conditions, in which the arm was relaxed or
stiffened, and for both flexion and extension, were preliminarily recorded when the handreached the virtual end-point-domain circle within the time limit. The elbow stiffness in
80 movement trials (2 co-contraction conditions x 2 directions x 20 trials) was estimatedby IMCJ. For each subject, the observed IMCJ range was divided into high, medium andlow ranges. For each trial in the main experiment, elbow IMCJ was computed todetermine whether it was within the predetermined range of the corresponding
co-contraction condition. The success or failure of achieving this required IMCJ level
was fed back to the subject at the end of each trial. To help subjects set an appropriate
co-contraction, the recorded EMG signals were shown on the CRT in real time to the14subjects from before the start of movement (Fig. 1C). We recorded 40 successful trials foreach level of co-contraction and for each direction of flexion and extension. Altogether,
240 successful trials were obtained (3 stiffness conditions x 2 directions x 40 trials). The
order of the three co-contraction conditions was counterbalanced across subjects.

Analysis
Position data were digitally filtered by a third-order Butterworth filter with an upper
cutoff frequency of 15 Hz. Derivatives of the position data were computed by applying athree-point local polynomial approximation. The start and end points of each movement
were determined using an angular acceleration with a 1.0 rad/s2 threshold. The movement
duration determined from these start and end points was called kinematic movement
duration, and it was nearly two times longer than the out-and-in movement duration. The
out-and-in movement duration was used only to control the movement duration during
the main experiment, and the kinematic movement duration was used for all data analyses.
The averages of kinematic movement durations for the three target sizes and forindividual subjects were within the range of 0.462–0.533 sec, and those for the voluntary
co-contraction experiment were within 0.520–0.616 sec. The actuated torque of theelbow τ was calculated as the product of the estimated inertial moment of the forearmfrom each subject’s body size and the angular acceleration of the elbow joint. We did notinclude a viscosity term in the calculation of the torque because most of the viscosity
measured around a joint arises from biochemical and mechanical processes within themuscle rather than from the properties of the joint (Akazawa, 1994). The effect of gravity

15was ignored because the arm was supported against gravity by the air sleds to reduce
friction.
We examined whether experimental conditions such as target sizes and the level ofco-contraction affect EMG activity, torque, position, and end points. Accordingly, we
first computed the ensemble-averaged temporal profiles over 40 trials of each signal.
Ensemble-averaged rectified EMG activity of the i-th muscle at time t was as follows:
EnsAveE i (t ) =1 40 j∑ ei (t )40 j=1(3)where ei (t ) denotes EMG activity of the i-th muscle at time t of the j-th trial.
jEnsemble-averaged torque at time t was
EnsAveT (t ) =1 40 j∑ τ (t )40 j=1(4)

where τ j (t ) denotes torque at time t of the j-th trial. Similarly, ensemble-averaged
x-position and y-position were computed as follows:
EnsAvePx (t ) =1 40 j∑ x (t )40 j=1(5)

EnsAvePy (t ) =1 40 j∑ y (t )40 j=1(6)where x j (t ) and y j (t ) denote x-position and y-position at time t of the j-th trial. Signals
are aligned so that the first 1.0 rad/s2 absolute angular acceleration threshold of ballistic
movements determines time zero. Movement duration for each signal was normalized
within the same experimental condition to mean movement duration of that condition,
without changing the amplitude of the signal (see below for detail). The
ensemble-averaged temporal waveforms of IMCJ and velocity profiles were computed in16the same way. These allow the deviation time course of each trial to be calculated as thedifference from the ensemble-averaged time courses. The ensemble-averaged deviation isdefined as the root mean square of this deviation time course over 40 trials. Therefore,
ensemble-averaged deviation of rectified EMG of the i-th muscle and ensemble-averaged
deviation of torque at time t are computed as follows: 1 402
EnsDevE i (t) =  ∑ EnsAveE i (t ) − eij (t )  40 j=1
(7)

 1 402
EnsDevT (t ) =  ∑ (EnsAveT (t ) − τ j (t ))  40 j=1


(8)

(

)

0.5

0.5

In other words, we studied the variability across trials at each point in time. As alreadymentioned in the description of stiffness estimation by IMCJ, we examined the effect offirst smoothing the EMG signal by the temporal filter of Koike et al. (1995), and the same
results were obtained regarding the relation between EMG amplitude and variance.
Finally, we computed the time-averaged magnitude and deviation on a per trial basis. The
time-averaged EMG of the j-th trial was computed as mean rectified EMG between thetime at the beginning and the time at the end of movement tf, summed over all muscles.
4

TimeAveE j = ∑tf

1eij (t )∑i=1 t f t=1(9)

The time averaged IMCJ magnitude of the j-th trial was computed in the same way:tf

1
TimeAveIMCJ j = ∑ IMCJ j (t )t f t=1(10)

The time-averaged deviation of the j-th trial was computed as the square root of the meansquared deviation time course between the time at the beginning and the time at the end of17

movement tf:
 4 1 tf2TimeDevE j = ∑ ∑ EnsAveE i (t ) − eij (t )  i=1 t f t=1


(11)

 1 tf
2
TimeDevTj =  ∑ (EnsAveT (t ) − τ j (t ))  t f t=1


(12)

(

)

0.5

0.5

 1 tf
2
2
TimeAvePj =  ∑ (EnsAvePx (t ) − x (t )) − (EnsAvePy (t ) − y (t ))  t f t=1


{

0.5

}

(13)

The amount of scatter at end-points, that is, end-point deviation, was indicated by thestandard deviation of end-point positions. On the other hand, the end-point error was
defined as the root mean square of the distance between the end-point positions and thecenter of the small target point. End-point deviation and end-point error are generally
different because the average end-point position deviates from the target center (Gribble
et al., 2003). Also, end-point error is not an appropriate terminology for an analysis of thetarget-size experiment because subjects were not required to reach for the centers oftarget circles, and movements ending within target circles were regarded as successful. To
examine the effect of SDN to various level of movement variance on a trial-by-trial basis,
we investigated the correlation between the time-averaged IMCJ of an individual trial(Eq. 10) and the time-averaged deviations of the torque (Eq. 12), position (Eq. 13), andend-point of the hand in that trial. In addition, we investigated the correlation between thetime-averaged EMG activity (Eq. 9) and the time-averaged EMG deviation (Eq. 11). The
correlation coefficients obtained from these trial-by-trial analyses are shown in Fig. 11.
To investigate SDN during movements, we computed correlation coefficients and slopes

18

for liner relationships between ensemble-averaged EMG (Eq. 3) and ensemble-averaged
EMG deviation (Eq. 7) at each time point for each muscle of each subject.
In the computations of ensemble-averaged temporal waveforms and time-averaged
deviations of position and torque, the time was normalized within the same experimental
conditions to mean movement time of that condition. That is, for each trial, each value(position, velocity and torque) was first computed, where the movement beginning andend was defined by the acceleration threshold, as explained above. Then each value wasscaled only in time domain to mean movement time of each condition by re-sampling
using spline interpolation. Therefore, the normalization did not change the amplitude ofeach signal. We also adopted two other methods of ensemble averaging without timenormalization. In both methods, the physical time was used in ensemble averaging over
40 trials, and all data in one condition were first aligned at the time of movement start. In
the second method, all data in one condition were cut at the time of the termination of the
movement whose duration was shortest in that condition. In the third method, all data in
one condition were cut at the time of the termination of the movement whose duration
was longest in that condition, and then the values between the termination of eachmovement and the time of data cutting were fixed to the value at the end of eachmovement to avoid distortion due to feedback correction or oscillation. Here, we show
the results using the first method because the results obtained from all three methods are
similar.
In order to examine the effect of the target-size or muscle co-contraction levels onmean performance, we carried out a 2-way repeated measures analysis of variance19

(repeated measure ANOVA, target-size or co-contraction levels x movement directions),
with subjects as a random factor, for time-averaged IMCJ, time-averaged deviations of
EMG activity, torque and position, and end-point as well as end-point errors and RMS
EMG using data for all subjects. F-ratios were computed by treating the target size or
co-contraction levels and movement directions as within-subject factors. This wasfollowed by Tukey’s HSD test.

20

RESULTS

Signal-dependent noise during movements
In order to examine signal dependent noise during movements, we computed signal
magnitude and its deviation at each time point of movements. Because EMG signals
during movements are not stationary, time courses of EMG magnitude were computed as
an ensemble average of rectified EMG time course over 40 trials (Eq. 3). Then, timecourses of EMG deviations were computed as the root mean square of difference between
each one and the ensemble-averaged EMG time course over 40 trials (Eq. 7). Then, EMG
deviation at each time point was plotted against EMG magnitude at that time point foreach muscle and each movement direction, for each condition of each experiment, and foreach subject. Figure 2 shows plots of a typical subject during a voluntary co-contraction
experiment. Blue, red, and green dots denote normal, medium, and high co-contraction
conditions, respectively. In all comparisons, EMG deviation linearly increased with EMG
magnitude. The mean and standard deviation of correlation coefficients across muscles,
subjects and conditions were 0.843±0.092 for extension and 0,821±0.099 for flexion in
the target-size experiment, and 0.814±0.087 for extension and 0.814±0.088 for flexion in
the voluntary co-contraction experiment. We also computed slopes between EMG mean
and deviation for each comparison. Figure 3 shows the histogram of the slopes for eachmovement direction. The slopes widely raged from around 0.5 to 3 with a mean of
1.192±0.422, which include the predicted slopes of uniform, Gaussian, and Laplacian21

distribution (0.58, 0.76, and 1.00, respectively) within the 2SD region, suggesting that theobserved linear relationship is not a trivial mathematical result (see Discussion for detail).
The slopes were significantly larger when muscles were working as antagonists than as
agonists for both target size (t-test, t(214) = 4.72; P < 0.0001) and voluntary
co-contraction experiment (t(214) = 3.60; P < 0.0005).

Experiment I: Changed target size
The averages and standard deviations of the kinematic movement durations acrosssubjects for the large, medium and small target sizes were 0.498±0.069, 0.497±0.069, and
0.491±0.062 sec, respectively, and were not significantly different. Figure 4 shows thehand paths, end-point distributions (A), angular velocity profiles (B), torque profiles (C),EMG time courses of the four muscles (D), and stiffness profiles (E) of a typical subject
recorded for the different target sizes for extension movements. These characteristics arein good agreement with previous studies (Gottlieb et al., 1989). The end-points of thehand were distributed along an arc within target circles (Fig. 4A). Therefore, theend-point deviation increased as the target size increased in accordance with ourexperimental design (see below and Fig. 7F). The angular velocity profiles werebell-shaped, and corrective movements were not observed. Although the hand paths (A),velocity (B), and torque profiles (C) were similar for the three different target sizes,
co-activation of bi-articulator muscles (D) and resulting IMCJ increase (E) were evident
in the latter half of movement for the more stringent accuracy requirements.
Figure 5 shows the ensemble-averaged time courses across all subjects and all trials

22

classified by target size for the summed EMG activity over the four muscles (A) (Eq. 3),
IMCJ (B), torque (C) (Eq. 4), and velocity (D), as well as the ensemble-averaged
deviations across all subjects for the summed EMG activity (E) (Eq. 7; summed over thefour muscles before taking square root), torque (F) (Eq. 8), and position (G). The datafrom extension (upper panel) and flexion (lower panel) are shown separately. Blue, red,and green indicate large, medium, and small target sizes, respectively. While average timecourses for torque (C) and velocity (D) were similar for different target sizes for both
extension and flexion, the summed EMG activity (A) and IMCJ (B) profiles of the small
target size were larger (green curves; see Figs. 7A and B for statistics), suggesting thatco-contraction increased as target size decreased. In good agreement with the assumptionof SDN, the EMG deviations were also generally larger for the small target size (Fig. 5E).
However, no general or marked differences between different target sizes were observed
for the torque and position deviations (Figs. 5F and G), except for the position deviation
around movement end as shown in the magnified inset (Fig. 5H), suggesting that noise
observed in the motor command was attenuated at torque and position levels. Further,around 0.4 sec after movement onset, three curves representing the positional deviations
for the three target sizes came very close in extension and crossed in flexion. Accordingly,
after 0.4 sec the positional deviation for the smaller target size was smaller in accordancewith the task requirements, despite the large EMG deviation.
To confirm that co-contraction increased later in the movement for higher accuracy
requirements, we plotted the late RMS EMG of each muscle for each subject andmovement direction against target sizes (Fig. 6). For all four muscles, late RMS EMG for23

small target was significantly larger than that for medium and large targets (repeatedmeasures ANOVA; see figure legend and insets for statistics). This suggests thatco-contraction was utilized to control end-point accuracy.
Figure 7 shows observations similar to those shown in Fig. 5 and their statistics intime-averaged data. Figs. 7A and B again demonstrate an increased motor command witha decreased target size. Figure 7A shows the means and standard deviations across all
subjects for each target size of the summed EMG activity that was time-averaged over theentire movement duration. Figure 7B shows those for the time-averaged IMCJ. The graylines denote extension and the black lines denote flexion data. The horizontal lines in Fig.
7 indicate significant comparisons in post-hoc tests (p < 0.05). The main effect of thetarget size was significant for the amount of muscle activity in terms of the time-averaged
summed EMG activity (F(2,16) = 10.86; P < 0.005) (Fig. 7A) as well as thetime-averaged IMCJ (F(2,16) = 12.22; P < 0.001) (Fig. 7B). The muscle co-contraction,
examined either as the time-averaged summed EMG activity or the time-averaged elbow
IMCJ, was significantly higher in the small target than the medium and large targets forboth flexion and extension. Other figures demonstrate that target size has a different
effect on different deviations. Figs. 7C, D, and E show the means and standard deviations
across subjects for each target size of the deviations that were time-averaged over theentire movement duration for EMG activity, torque, and position, respectively (seeMethods for equations of time-averaged deviations). Figure 7F shows the mean andstandard deviation of the end-point deviation across subjects for each target size. The
main effect of the target size was significant for the time-averaged deviation of EMG

24

activity (F(2,16) = 12.27; P < 0.001), and the end-point deviation (F(2,16) = 157.80; P <0.0001). The EMG variability was significantly largest for the small targets both forextension and flexion, which is consistent with SDN.In contrast, the end-point deviation
was significantly larger in the large target than the medium target and significantly larger
in the medium target than the small target; consequently, the experimental manipulationof the end-point variability went well for the target-size experiment.
Similar trends to Figs. 5 and 7 were observed when EMG activity, torque, andposition ensemble averages and deviations were calculated using the shortest movement
duration of each condition (the second method; see MATERIALS AND METHODS) or
when using duration of each movement and extrapolate by the final value until longest
movement duration (the third method; see MATERIALS AND METHODS). The
summed EMG activity (F(2,16) = 8.35; P < 0.005 for shortest movement duration,
F(2,16) = 11.90; P < 0.001 for longest movement duration) and IMCJ (F(2,16) = 9.12; P< 0.005 for shortest movement duration, F(2,16) = 11.85; P < 0.001 for longest
movement duration) were significantly different and were largest for the small targets.

Experiment II: Changed muscle activation level
The means and standard deviations of the kinematic movement durations acrosssubjects for the normal, medium and high co-contraction conditions were 0.581±0.073,
0.558±0.071, and 0.537±0.062 sec, respectively, and decreased significantly for thehigher co-contraction level, although the predetermined desired out-and-in movement
duration was the same for the three conditions. Figure 8 shows the hand paths, end-point25

positions, angular velocity profiles, torque profiles, EMG time courses of the fourmuscles, and IMCJ profiles recorded for the three levels of co-contraction for the same
subject, in a similar format to Fig. 4. The end-points of the hand were distributed along an
arc around the target point more narrowly for the larger co-contraction level (Fig. 8A).
The angular velocity profiles were bell-shaped, and corrective movements were notobserved. Although the EMG activity of all four muscles (Fig. 8D) and IMCJ (Fig. 8E)
were much larger over the entire movement duration for the higher level of co-contraction,
the ensemble-averaged hand paths (Fig. 8A), velocity (Fig. 8B), and torque profiles (Fig.
8C) were similar among the three co-contraction conditions. However, we must note thatthe velocity profiles for high co-contraction were more symmetrical than those for low
co-contraction (Figs. 8B and 9D). The velocity peak in low co-contraction condition wasslightly, but significantly earlier than in medium and high co-contraction conditions(F(2,16) = 20.50; P < 0.0001). Mean velocity peak in low co-contraction condition was
significantly earlier than the mid point of movement duration (t-test, t(8) = 2.67; P < 0.05
for extension, t(8) = 3.41; P < 0.01 for flexion) while that in high co-contraction condition
was not significantly different (t(8) = 1.05; P = 0.32 for extension, t(8) = 1.83; P = 0.10
for flexion), suggesting that velocity profiles for high co-contraction condition weresignificantly more symmetrical.
Figure 9 shows the ensemble-averaged time courses of the summed EMG activity,
IMCJ, torque, and velocity as well as the ensemble-averaged deviation time courses ofthe summed EMG activity, torque, and position classified by co-contraction levels
combining data from all subjects and all trials, in a similar format to Fig. 5. Blue, red, and26

green indicate normal, medium, and high co-contraction conditions, respectively. While
the average time courses for the torque and velocity were similar for different
co-contraction levels (Figs. 9C and D), the summed EMG activity and IMCJ profiles forthe larger co-contraction conditions were larger (Figs. 9A and B), which shows that
subjects actually followed the instruction and could voluntary control co-contractionlevels. In good agreement with SDN assumption, the time courses of EMG activity,
torque, and position deviations were generally larger for the larger co-contraction
conditions (Figs. 9E, F and G), except for the position deviation around movement end as
shown in the magnified inset (Fig. 9H). Around 0.5 sec after the movement onset,
however, three curves representing the positional deviation for the three co-contractionlevels converged and/or crossed. That is, after 0.5 sec, the positional deviation for thedifferent co-contraction levels became nearly indistinguishable, suggesting attenuation of
SDN. Data from extension and flexion were qualitatively similar.
Figure 10 shows observations similar to those shown in Fig. 9 and their statistics intime-averaged data. Figs. 10A and B show the mean and standard deviation across all
subjects of the time-averaged summed EMG activity and time-averaged IMCJ,
respectively, for each co-activation requirement. The gray lines denote extension, and theblack lines denote flexion. The horizontal lines indicate significant comparisons in
post-hoc tests (P < 0.05). The main effect of the muscle co-activation requirement wassignificant for the amount of muscle activity in terms of time-averaged summed EMG
activity (F(2,16) = 29.09; P < 0.0001) as well as time-averaged IMCJ (F(2,16) = 17.16; P< 0.0005). Furthermore, the muscle activity, that is, arm IMCJ, was significantly smallest

27

in the normal level and significantly largest in the high level. Accordingly, the main task
design of the voluntary co-contraction experiment was fulfilled. Other figuresdemonstrate that the co-activation level has different effect on different deviations. Figs.
10C, D, and E show the mean and standard deviations across all subjects fortime-averaged deviations of EMG activity, torque, and position, respectively, for eachco-contraction requirement. Figs. 10F and G show the mean and standard deviations
across all subjects for the position deviation at the end point as well as the end-point error
for each co-contraction requirement. A larger variability in EMG activity, torque, andposition was observed for the higher muscle activation level, although there was no such
trend for the end-point deviation, and the relationship was reversed for the end-point error.
The main effects of the muscle activation level were significant for variabilities in EMG
activity (F(2,16) = 40.04; P < 0.0001), torque (F(2,16) = 66.53; P < 0.0001), position(F(2,16) = 11.74; P < 0.001), and end-point error (F(2,16) = 6.26; P < 0.01). The
variabilities in EMG activity and torque were significantly larger in the high level andsmaller in the normal level. The end-point error in the normal level was significantly
larger than in the medium and high levels for both extension and flexion.
The same trends shown in Figs. 9 and 10 were observed when the EMG activity,
torque and position deviations were calculated using the shortest movement duration ofeach condition (the second method; see MATERIALS AND METHODS) or when using
the duration of each movement and extrapolating by the final value until the longest
movement duration (the third method; see MATERIALS AND METHODS). That is,these variabilities were significantly different (EMG deviation; F(2,16) = 28.59; P <28

0.0001 for shortest duration, F(2,16) = 30.45; P < 0.0001 for longest duration, torquedeviation; F(2,16) = 32.19; P < 0.0001 for shortest duration, F(2,16) = 30.78; P < 0.0001
for longest duration, position deviation; F(2,16) = 10.10; P < 0.005 for shortest duration,
F(2,16) = 8.84; P < 0.005 for longest duration) and larger at the high level but smaller at
the normal level.
The observed reversed relationship between motor command magnitude andend-point error suggests that SDN is attenuated as the space shifts from intrinsic motor
command space to extrinsic task space. Noise was highly positively correlated withmagnitude of motor command at EMG or torque level but was negatively correlated attarget achievement level. To further demonstrate the degradation of correlation between
motor command magnitude and deviations, we computed the correlation coefficients
between the co-contraction magnitude and EMG deviation, torque deviation, position
deviation, end-point deviation, and end-point error on a trial-by-trial basis (Figure 11).
For each movement trial of an individual subject, such as those shown in Fig. 8, thefollowing seven quantities were first computed. The first two were the summed rectified
EMG activity and its deviation, both of which were time-averaged over movement
duration. The equations are given in Materials and Methods (Eqs. 9 and 11). The third to
fifth were IMCJ and the deviations of torque and position, all of which weretime-averaged over movement duration. The equations are given in Materials andMethods (Eqs. 10, 12, and 13). The last two quantities were end-point deviation and theend-point error. Then, for each subject and for each movement direction, a correlation
coefficient was computed from 120 movement trials (40 trials x 3 stiffness conditions)29between two quantities for each of the following five pairs. The first is between thetime-averaged summed EMG activity and its time-averaged deviation. The second is
between the time-averaged IMCJ and the time-averaged torque deviation. The third is
between the time-averaged IMCJ and the time-averaged position deviation. The fourth is
between the time-averaged IMCJ and the end-point deviation. The fifth is between thetime-averaged IMCJ and the end-point error. Figure 11A plots these 18 correlation
coefficients (2 movement directions x 9 subjects) for the voluntary co-contraction
experiment while connecting the same subject by the thin line and representing the same
movement direction by the same symbol. The light and dark shaded regions show theinsignificance ranges of correlation coefficients, outside of which the coefficient isstatistically different from zero for an individual data item and the data from all subjects,
respectively. For the relationship between EMG activity and EMG variability, thecorrelation coefficients of all subjects were positive and very close to 1 (Fig. 11A
leftmost). The correlation coefficients of IMCJ and torque variability were positive andrelatively high and significant for all but one subject (Fig. 11A second from left). The
correlation coefficients of IMCJ and position variability were positive for 15 out of 18
cases and relatively small and insignificant for most subjects (Fig. 11A middle). The
correlation between IMCJ and end-point deviation was positive or negative depending on
the subject and movement direction, and most coefficients were small in magnitude andinsignificant (Fig. 11A second from right). The correlation coefficients between IMCJ
and end-point error were negative in 13 out of 18 cases and some were significant (Fig.
11A rightmost). Figure 11A also shows, by thick black lines and asterisks, the correlation30

coefficients for the five relationships computed from all 2,160 trials of all subjects. The
correlation coefficient was close to 1 and significant for EMG deviation. It was positiveand significant for torque deviation. It was slightly positive but significant for position
deviation. It was slightly negative but significant for end-point deviation, and negativeand significant for end-point error. Figure 11B plots the same correlation coefficients forthe relationships from the target-size experiment and shows similar trends.

DISCUSSION
We showed that when subjects made single-joint elbow movements to targets of different
diameters, IMCJ and EMG variability increased with the required accuracy of the task
and reduced end-point variability. When subjects were asked to increase co-contraction
during movements, the temporal profiles of EMG activity and torque both increased invariability. However, final positional error was largest for the lowest and normal
co-contraction levels. The qualitative trends of EMG activity, torque, and position
variabilities and end-point deviations and error with respect to increased IMCJ weresimilar between these two experiments, although the range of co-contraction in response
to different target sizes was over a much smaller range than could be voluntarily elicited(an 11% versus 196% increase). SDN suggests high correlation between motor command
magnitude and movement deviation, but this is true only for intrinsic parameters such as
EMG and torque. For the parameters in the task space (Cartesian space), the correlations
are low and at the final task level (endpoint error); furthermore, correlation was negative,
meaning that SDN no longer holds. This demonstrates that SDN is not sufficient to31

explain movement variance, but actually SDN is attenuated by impedance at the task
level. Such reversed relationship between EMG variability and positional variability has
also been observed when movement speed was increased (Darling and Cooke, 1987).

Signal-dependent noise during movement
When subjects were asked to move with larger co-contraction levels, they
increased the motor command signals of the flexor and extensor simultaneously. It has
been suggested that the standard deviation of the neuromotor commands increases withits mean based on both motoneuronal firing studies (Clamman, 1969; Matthews, 1996)
and surface rectified EMG signals (St-Amant et al., 1998; Clancy and Hogan, 1999)
during static force tasks. Psychophysical studies of isometric force production haveshown a strong relationship between mean force level and force variability as measured
by standard deviation (Schmidt et al., 1979; Sherwood and Schmidt, 1980; Sherwood etal., 1988; Jones et al., 2002). Sherwood et al. (1988) also reported a strong relationshipbetween mean force level and force variability in the elbow-joint movement task whendifferent loads were added to a hand-held bar.
These previous behavioral studies investigated the change in force variability inisometric tasks (Schmidt et al., 1979; Sherwood and Schmidt, 1980; Jones et al., 2002) ormoving conditions under normal co-contraction (Sherwood et al., 1988). The current
study is the first to examine the relationships between stiffness, EMG activity, torque, andposition variabilities and end-point deviation and error during movements. Analysis of
EMG variability demonstrated a strong linear relationship with the rectified EMG level

32

(Fig. 2). This is not a trivial mathematical result of rectifying an AC signal. For astochastic variable x with zero mean and even probability distribution, there is generally
no linear relationship between the standard deviation and mean of the rectified signal. If
the distribution changes its shape with different motor command levels, not only thelinear relationship but also the monotonically increasing relationship of the deviation as afunction of mean may not hold. For an evenly distributed zero-mean random variable x,
the variance of the rectified signal is given as follows: V(|x|) = V(x) – E(|x|)2. From this,the observed linear relationship holds if and only if V(x) is proportional to E(|x|)2, which is
not a generic property of the zero-mean even distribution. This condition is satisfied forwell-known uniform, Gaussian and Laplacian distributions, where the predicted slopes
are approximately 0.58, 0.76 and 1.00, respectively. That is, if either uniform, Gausssian,or Laplace distribution is maintained for different levels of motor commands, the linear
relationship between the mean and the standard deviation is mathematically derived. For
our data, the slope between EMG deviation and magnitude widely raged with a mean
larger than that of Laplace distribution (Fig. 3), suggesting that the EMG signal iscorrupted by larger noise when the mean level of the signal becomes large. Our results
also suggest that movement EMG distributions ranged wider than those in aconstant-force, constant-angle condition, which fell between Gaussian and Laplacian
distributions (Clancy and Hogan, 1999). Furthermore, the deviation is measured as a root
mean squared distance from the ensemble-averaged rectified trajectory, showing truevariability in the non-stationary signal. This allows EMG variability to be measured
during movement for the first time. Consequently, signal-dependent noise was confirmed

33during single-joint movements.

Negative correlation of end-point error with stiffness
Analysis of torque variability in the voluntary co-contraction experiment (Figs. 10D
and 11A) showed that it increased with stiffness but with a weaker correlation than EMG
variability. The positional variability averaged over the movement duration also
increased significantly with IMCJ, but with an even weaker correlation (Figs. 10E and11A). In stark contrast, the final end-point error was significantly larger for the lowest
co-contraction level than for the medium or high level (Figs. 10G and 11A). For thehigher accuracy requirements of the target-size experiment, subjects chose a higher IMCJ
level and achieved a lower end-point deviation (Fig. 7). Therefore, subjects exploited therelationship between stiffness and accuracy as the task demands changed.
Although these findings seem counter-intuitive because higher stiffness involves
larger motor commands, resulting in larger signal-dependent noise, they can bereconciled within the extended TOPS framework as will be explained later.
Co-contraction has two effects within this model. First, it increases the noise produced by
the actuators; although some of the forces in co-contraction cancel each other out, thevariabilities from both agonist and antagonist are additive. Second, it changes the
impedance of the limb. The impedance determines how the noise plays through the motor
apparatus into the end-point error. Figure 11 most markedly demonstrates this effect of
decreased and reversed correlation with IMCJ as the signal is transformed from EMG
activity to torque, to position, to end-point variability, and finally to end-point error.

34

Decreased end-point error under higher co-contraction might be explained by twocontrol mechanisms. One is the impedance control that utilizes the muscle’s intrinsic
spring properties to realize a desired trajectory or end point in negative feedback control(Hogan, 1985; Burdet et al., 2001; Osu et al., 2002; Franklin et al., 2003). When muscle
stiffness is increased by feedforward and predictive co-contraction, the increased
feedback gains bring higher accuracy. The achievement of the desired trajectory andend-point is suggested because the end-point error decreased while the end-point
deviation showed no definite trend as IMCJ increased in the co-contraction control
experiment (Figs. 10F and G). Furthermore, in the target size experiment, the positional
deviation decreased with higher stiffness (small target size) only in the late stage ofmovement (Fig. 5G), corresponding to the larger EMG activity and IMCJ in only thelatter half of movement duration (Figs. 5A and B). A second possible control mechanismis the optimal feedback control proposed by Todorov and Jordan (2002), where time
variant feedback gains are optimally selected based on neural feedback computations
rather than muscle co-contraction. This computational theory seems to suggest larger
torque variability for higher gains of neural feedback loops, which seems at odds with our
results showing higher stiffness and accuracy with little change in torque variability (Figs.
7D and F). Therefore, we may conclude that co-contraction leads to a trade-off between
increased noise and reduced consequences due to the changed impedance. In our
experiment, the increased variability of position seen during movement can be interpreted
as the effects of increased SDN, whereas the reduced final error may be due to the35impedance change, which reduces variability.
In the co-contracted state, the limb is clearly more stable against external noise or
perturbations. However, because there was no explicit external perturbation in the current
experiments it is not appropriate to ascribe the observed reduction of final task error to therejection of external perturbation. One possible explanation is other uncontrolled internal
processes that can have an effect on final accuracy, such as for example an increase in theoverall level of attention, were triggered by the increase of co-contraction. That is,
accuracy and impedance are independently regulated and subject generated higher levelsof co-contraction while producing less variable feedforward commands. Another possible
explanation is that at some level of co-contraction the limb may also be more stableagainst internally generated noise. How the impedance could reduce the internallygenerated noise is an open question because the reference trajectory, or the equilibrium
position itself, could be perturbed by internally generated noise. Assuming that muscle
visco-elasticity changes according to muscle activation (Bizzi et al., 1984), theequilibrium position can be expressed as the difference of two antagonistic muscles’
activation divided by the summation of their activation (co-contraction), i.e. (uflex uext)/(uflex + uext) (Hogan, 1984). When noise is added to one or both of the twoantagonistic muscles, changes in the equilibrium position generated by the noise aresmaller when the denominator, that is, the amount of co-contraction, is larger. Therefore,
the effect of internally generated noise on the reference trajectory may possibly smallerwhen stiffness is larger. Thus, co-contraction could possibly be one of factors in reducing
the effect of internally generated noise on the final endpoint variability.

36

Shorter movement duration with higher stiffness
In the co-contraction control experiment, a significantly shorter kinematic movement
duration was observed with larger co-contraction, although the required out-and-in
movement duration was constant irrespective of the co-contraction level. In thetarget-size experiment, the same effect was observed, although it was much smaller andnot significant. A shorter movement duration for higher accuracy is apparently contrary toFitts’ law (Fitts, 1954; Hirayama et al., 1993) and to TOPS model, which predicts Fitts’law (Harris and Wolpert, 1998). A shorter movement duration with higher stiffness may
be interpreted as a consequence of the mechanical resonant frequency of the forearm link,
which increases in proportion to the square root of the elbow stiffness. The shorter
movement duration with the same movement amplitude leads to higher acceleration andlarger EMG activity, but this effect is negligible compared with large differences in EMG
levels. That is, movement duration change was only 1.48% and 7.52%, while stiffness
increase was 11% and 196% in the target-size and co-contraction control experiments,
respectively. Assuming that EMG linearly increases with muscle torque and stiffness, it isexpected that joint torque, and thus the reciprocal component of EMG, change with thesquare of duration change. Although a higher-order relationship between EMG andduration might exist because of the nonlinearity between EMG and stiffness, theorder-of-magnitude smaller change of movement duration compared with IMCJ increasesuggests that IMCJ increase was not merely due to reciprocal activation caused by joint
torque increase but mainly due to additional co-contraction. The movement trajectories

37

and torque profiles were very similar between different experimental conditions (see Figs.
5C, D; Figs. 9C, D), and their differences were too small to explain differences in EMG
activity and IMCJ changes. Because the kinematic and dynamic features of the
movements are almost invariant across different experimental conditions, IMCJ could be
regarded at least as a good relative measure for stiffness, as discussed in Materials andMethods. For similar movement trajectory, IMCJ has been demonstrated to linearly
increase with stiffness (Osu et al., 2002). Thus, we conclude that the stiffness changes
were caused by the different accuracy requirements and voluntary co-contraction.

Stiffness control dependence on tasks
The largest end-point errors were observed when subjects did not increase
co-contraction above the normal level. Similarly, subjects could increase their accuracy inresponse to a smaller target by increasing co-contraction. Consequently, without the needfor greater accuracy, subjects accepted worse performance but with lower stiffness. This
contradicts the principle of TOPS and suggests that minimizing the end-point deviation or
end-point error alone may not be the sole consideration for task optimization. There is an
extra dimension to stiffness control, which increases accuracy but also leads toundesirable factors. For example, the CNS may adopt criteria such as minimization of
fatigue (Dul et al., 1984), energy consumption (Alexander, 1997), commanded torquechange (Nakano et al., 1999), or motor command change (Kawato, 1992) in combination
with the maximization of task achievement. Miyamoto et al. (2002), and Nagata et al.
(2002) demonstrated that the TOPS-α model, where a cost term in the motor command

38

magnitude multiplied by α is added to the task achievement term, gives better predictionof trajectories than the TOPS model for a large number of point-to-point movements. The
TOPS-α model combines maximization of task achievement and minimization of motor
command magnitude. The relative contribution of these two terms to the overall cost isdetermined by the coefficient α, which is the weighting constant for motor command
magnitude. If α is zero, the TOPS-α model is the same as the TOPS model, which
maximizes task achievement. If α is large, the motor command magnitude is reduced at
the expense of performance. The TOPS-α model is consistent with the current results inthe sense that without the need for the greatest accuracy, subjects accept worse
performance but with lower stiffness, resulting in reducing motor command magnitude.
Slightly different movement duration and velocity profiles for different co-contractionlevels may also suggest that desired trajectories were computed taking into account themagnitude of stiffness.
The results of our experiments suggest that the central nervous system somehow
acquires the knowledge of the relationship between stiffness level and size of end-point
deviation and error, allowing it to generate the stiffness necessary for difficult tasks.
Furthermore, stiffness is reduced as long as the task demand is satisfied under skilled andnatural movements. Optimal impedance control may be accomplished in order togenerate the proper trajectory for the task requirements through a trade-off between the
stiffness level and task achievement.

39Generalization to multi-joint movements
The required movements were unnatural, and most subjects never rotated only theelbow without friction or gravity. There were three major reasons for this experimental
setting. First, by minimizing the gravity and frictional forces during the movements, we
could reliably estimate dynamic torques from surface EMG recordings. This wasimportant for stiffness estimation via IMCJ. Second, by avoiding the shoulder freedom,
we did not need to record from many shoulder muscles, thus shortening the experimental
duration and simplifying the IMCJ computation. Finally, and most importantly, the hand
path was forced to be identical between trials and subjects, and movement variability
could result only from changes in velocity profiles. This is very important because we
wanted to examine the effect of stiffness control on movement accuracy while
minimizing changes in trajectory shapes. For multi-joint movements, it is known that thehand paths themselves fluctuate between trials and change systematically between
subjects. Laursen et al. (1998) and Gribble et al. (2003) examined multi-joint movements
and measured muscle activation by EMG activity for different accuracy requirements.
Although, unlike us, they did not estimate joint stiffness and did not examine voluntary
co-contraction, they found qualitatively similar results for the effects of accuracy
requirements on co-contraction. Thus, we expect that at least a part of our results could be
extended to more natural movements.

ACKNOWLEDGEMENTS
This study was supported by the Telecommunications Advancement Organization of40

Japan and by grants to Mitsuo Kawato from the Human Frontier Science Program. We
thank David Franklin for his valuable comments on the manuscript.

41

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Figure Legends
Fig. 1 (A) Experimental setup and CRT display patterns for changed target-size
experiment (B) and voluntary co-contraction control experiment (C). The two arm
positions shown in (A) indicate either the start or end position depending on extension
or flexion movements. The CRT display patterns of (B) and (C) show extension
movement. In the EMG level display of (C), the two extensor-muscle activities areshown on the right and the two flexor-muscle activities are shown on the left by the xmarks.
Fig. 2 EMG deviation at each time point plotted against EMG magnitude at that time
point for a typical subject during voluntary co-contraction experiment. Green, red,and blue dots denote high, medium, and low co-contraction conditions, respectively.
Upper four panels are plots for extension movements and lower four are those forflexion movements. Middle four panels (cyan background) are plots when muscles
are working as agonists. The other four panels (magenta background) are plots whenmuscles are working as antagonists.
Fig. 3 Histograms of the slopes between EMG magnitude and EMG deviation estimatedfor each muscle of each subject. Upper two panels are histograms of target-size
experiments and lower two are those of co-contraction experiments. Vertical lines
denote slopes of uniform, Gaussian, and Laplace distributions.
Fig. 4 Results of a typical subject for the target-size experiment for extension movements.
The results for large, medium and small target sizes are shown in the left, middle, andright columns, respectively. (A) shows the hand paths and end-point distributions. (B),(C) and (E) show velocity profiles, torque profiles, and stiffness profiles, respectively,
as a function of time for 40 trials shown by thin gray curves. Ensemble-averaged timecourses are indicated by solid colored curves. (D) shows the average EMG timecourses of the four muscles. Ensemble averaging equations for C to E are given inMaterials and Methods.
Fig. 5 Ensemble-averaged time courses across all subjects and all trials classified by46

target size shown by three colors: blue for large, red for medium, and green for small
target size. (A), (B), (C), and (D) show the summed EMG activity over the fourmuscles, stiffness, torque, and velocity. (E), (F), and (G) show the ensemble-averaged
deviations across all subjects for the summed EMG activity, torque, and position,
respectively. (H) shows a magnification of (G) around the end of movement. The datafrom extension (upper panel) and flexion (lower panel) are shown separately.
Fig. 6 RMS EMG of each muscle for each subject and movement direction against target
size. Large, Medium, and Small indicate large, medium and small target sizes,
respectively. The open circles indicate flexion movements, and the crosses extension
movements. The p-values in each panel denote results of repeated measures ANOVA
among target sizes (F(2,16) = 6.98 for elbow flexor, F(2,16) = 10.04 for biarticular
flexor, F(2,16) = 12.52 for elbow extensor, F(2,16) = 6.74 for biarticular extensor).
The horizontal bars denote significant comparison in post-hoc test (Tukey’s HSD test;
P < 0.05).
Fig. 7 Means and standard deviations (vertical bars) of six indices across all subjects foreach target size. Large, Medium, and Small indicate large, medium and small target
sizes, respectively. The gray lines denote extension data, and the black lines denote
flexion data. The horizontal lines indicate significant comparisons. An interaction
between target size and movement direction was significant for all indices except
end-point deviation (P < 0.01). A and B show the means and standard deviations
across subjects for the summed EMG activity and stiffness, respectively, which weretime-averaged over the entire movement duration. C, D, and E show the means andstandard deviations of the deviations that were time-averaged over the entire
movement duration for EMG activity, torque, and position, respectively. F shows themean and standard deviation of the end-point deviation. See Materials and Methods
for the equations used to calculate the time-averaged deviations in C to E.
Fig. 8 Results of the same subject shown in Fig. 4 for the voluntary co-contraction control
experiment for extension movements. The results for the normal, medium and highlevels of co-contraction are shown in the left, middle, and right columns, respectively.
The format is similar to Fig. 4. (A) shows the hand paths and end-point distributions.
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(B), (C), and (E) show velocity profiles, torque profiles, and stiffness profiles,
respectively, as a function of time for 40 trials, and ensemble-averaged time courses
are indicated by solid colored curves. (D) shows the average EMG time courses of thefour muscles.
Fig. 9 Ensemble-averaged time courses across all subjects and all trials classified byco-contraction levels shown by three colors: blue for normal, red for medium, andgreen for high in a similar format to Fig. 5. (A), (B), (C), and (D) show the summed
EMG activity over the four muscles, stiffness, torque, and velocity. (E), (F), and (G)show the ensemble-averaged deviations across all subjects for the summed EMG
activity, torque, and position, respectively. (H) shows a magnification of (G) aroundthe end of movement. The data from extension (upper panel) and flexion (lower
panel) are shown separately.
Fig. 10 Means and standard deviations (vertical bars) of six indices across all subjects foreach co-contraction level in a similar format to Fig. 7. Norm, Med and High indicate
normal, medium and high co-contraction levels, respectively. The gray lines denote
extension, and the black lines denote flexion data. The horizontal lines indicate
significant comparisons. An interaction between co-contraction level and movement
direction was significant for summed EMG, EMG, torque and position deviation (P <0.01). A and B show the summed EMG activity and stiffness, respectively, whichwere time-averaged over the entire movement duration. C, D, and E show means andstandard deviations of the deviations that were time averaged over the entire
movement duration for EMG activity, torque, and position, respectively. F shows themean and standard deviation of the end-point deviation. G shows the mean andstandard deviation of the end-point error.
Fig. 11 Plots of correlation coefficients between the stiffness and variabilities of fivequantities for all subjects. A is for the voluntary co-contraction control experimentand B is for the target-size experiment. The circles show extension coefficients andthe squares show flexion coefficients. Coefficients from the same subject areconnected by a thin line. The bold line with asterisks shows the correlation
coefficients that were calculated from all trials of all subjects for both flexion and48

extension data. The EMG deviation (leftmost) shows the correlation coefficients
between EMG activity and EMG variability. The torque deviation (second from left)
shows the correlation coefficients between stiffness and torque variability. The
position deviation (middle) shows the correlation coefficients between stiffness andposition variability. The end-point deviation (second from right) shows thecorrelation coefficients between stiffness and end-point deviation. The end-point
error (rightmost) shows the correlation coefficients between stiffness and end-point
error. The dark shaded regions show the range of statistical insignificance, outside ofwhich the correlation coefficients computed from all trials of all subjects are
statistically significantly different from zero. The light shaded region is theinsignificance range for an individual subject.

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