Lecture Notes in Computer Science

Reflex Contributions to the Directional Tuning  

of Arm Stiffness 

Gary Liaw

1

, David W. Franklin

1,2,3

, Etienne Burdet

4

, Abdelhamid Kadi-allah

4

,  

and Mitsuo Kawato

3 

1

 National Institute of Information and Communications Technology, 2-2-2 Hikaridai, 

Keihanna Science City, Kyoto, 619-0288, Japan 

2

 Department of Engineering, University of Cambridge, Cambridge, United Kingdom 

3

 ATR Computational Neuroscience Laboratories, Keihanna Science City, Kyoto, Japan 

4

 Department of Bioengineering, Imperial College London, London, United Kingdom 

Abstract. It has been shown that during arm movement, humans selectively 

change the endpoint stiffness of their arm to compensate for the instability in an 

unstable environment. When the direction of the instability is rotated with 

respect to the direction of movement, it was found that humans modify the 

antisymmetric component of their endpoint stiffness. The antisymmetric 

component of stiffness arises due to reflex responses suggesting that the 

subjects may have tuned their reflex responses as part of the feedforward 

adaptive control. The goal of this study was to examine whether the CNS 

modulates the gain of the reflex response for selective tuning of endpoint 

impedance. Subjects performed reaching movements in three unstable force 

fields produced by a robotic manipulandum, each field differing only in the 

rotational component. After subjects had learned to compensate for the field, 

allowing them to make unperturbed movements to the target, the endpoint 

stiffness of the arm was estimated in the middle of the movements. At the same 

time electromyographic activity (EMG) of six arm muscles was recorded. 

Analysis of the EMG revealed differences across force fields in the reflex gain 

of these muscles consistent with stiffness changes. This study suggests that the 

CNS modulates the reflex gain as part of the adaptive feedforward command in 

which the endpoint impedance is selectively tuned to overcome environmental 

instability. 

Keywords:  Mechanical impedance, limb stiffness, internal model, stretch 

reflex, impedance control, co-contraction, electromyography. 

1   Introduction 

In everyday life, we perform activities in our environment that require us to interact 

with different objects such as tools. These interactions with tools are often inherently 

unstable [1]. For example, when using a screwdriver, the direction of the force 

applied needs to be parallel to the screwdriver in order to maintain contact with the 

screw. However this is further complicated as the force applied by the user is 

susceptible to fluctuations due to inherent signal dependent noise [2]. These variations 

914 

G. Liaw et al. 

in the direction of the applied force could produce movements that may cause the 

screwdriver to slip, and lose contact with the screw. In order to successfully perform 

this and other unstable tasks, the neuromuscular system must overcome such 

instability.  

Viscoeleastic properties of muscle play an important role in motor control, as it reacts 

without delay to disturbances caused by instabilities. The greater the viscoelastic 

impedance of the arm, the better it can resist perturbations that affect the intended 

position or trajectory. The ability to control viscoelastic impedance is therefore 

important, especially in unstable environments [3] or unpredictable circumstances [4]. 

Hogan first suggested that impedance can be selectively controlled by the central 

nervous system (CNS) [5] and evidence for this selective control was later demonstrated 

by others [3,6,7].  

In the latter studies, point-to-point arm reaching movements were used as a 

platform to investigate the level of sophistication of impedance control by the CNS. It 

was demonstrated that arm stiffness could be modulated by the CNS for magnitude 

[6] as well as direction [7] of the environmental instability. It was suggested that such 

modulation was facilitated by feedforward changes in muscle activation, specifically 

by modifying the activations in three muscle groups: the shoulder muscles, the elbow 

muscles and the biarticular muscles.  

Endpoint stiffness can be decomposed into two components: a symmetric 

component and an antisymmetric component [8]. The symmetric component can be 

produced by a combination of passive, intrinsic, or symmetric reflexive stiffness, 

while the antisymmetric component is thought to be mainly due to heteronymous 

reflexive responses [5]. This means that a large antisymmetric component in the 

endpoint stiffness can be interpreted as evidence of contribution from reflexes. Our 

previous study [7] found that adaptation to environmental instability in different 

directions resulted in changes to the antisymmetric component of endpoint stiffness. 

The present study aims at solidifying those results by varying only the cross terms of 

the environmental instability. We will investigate whether the CNS modulates the 

reflex contributions to stiffness by modifying the reflex gain as part of the impedance 

controller. 

2   Methods 

Four healthy, right-handed subjects (three male, one female) participated in the study. 

The experiment was approved by the institutional ethics committee and subjects gave 

informed consent. 

2.1   Apparatus  

Subjects were seated in a chair with their shoulder held against the back of the chair by 

seatbelts to restrict trunk movement. The height of the chair was adjusted such that the 

workspace was at the subject’s shoulder level. A custom-molded thermoplastic cuff was 

used to restrict the subject’s wrist motion and a horizontal beam was secured to the 

subject’s forearm for support. The subject’s right arm, along with the cuff and the beam 

were coupled to the parallel-link direct drive air-magnet floating manipulandum (PFM) 

(Fig 1). Details of the PFM and setup can be found elsewhere [9]. 

 

Reflex Contributions to the Directional Tuning of Arm Stiffness 

915 

Subjects performed point-to-point reaching movements in the horizontal plane 

from a start target to an end target. The start and end targets were marked with a circle 

of diameters 2.5cm and 3cm, respectively, and located 31cm and 56cm, respectively, 

from and directly in front of the shoulder joint. The subject’s arm was restricted to 

keep the movement to two degrees of freedom. Subjects were asked to perform the 

movements in 600ms ± 100ms. A computer monitor provided feedback to the subjects 

on whether the previous movement was successful or not. During force field trials, 

safety boundaries were set up 5cm from the center on either side, beyond which the 

force field would revert to a null field. A successful movement was one where the 

subject completed the movement by reaching the target within the desired duration 

range, and did not exceed the safety boundaries. An opaque table above the 

workspace blocked the subject's view of the manipulandum and the subject's arm. The 

position of the hand and the start and end target circles were projected onto the 

surface of the table to provide subjects with visual feedback.   

PFM

target

start

y

x

[0,0]

 

Fig. 1. Experimental Setup. The subject’s right arm was attached to the handle of the PFM with 

a cuff, restricting wrist motion. Reaching movement were performed from a start position  

[0, 31]cm to a target position [0, 56]cm relative to the shoulder joint. The PFM either applied a 

computer-controlled force field or controlled displacements for stiffness estimation. 

2.2   Force Fields 

Subjects performed point-to-point reaching movements in three different unstable 

divergent force fields as well as a null field. The three force fields exerted a force 

away from the center line (y-axis) of the workspace that is proportional to the size of 

the deviation from the center line; the subjects experienced no force on the hand if 

there is no deviation. The three force fields differed in direction and magnitude.  

As a baseline for adaptation, a simple divergent force field (DF), oriented 

perpendicular to the direction of movement, was simulated. The equation was: 

200

0

0

0

x

y

F

x

F

y

⎡ ⎤ ⎡

⎤ ⎡ ⎤

=

⋅

⎢ ⎥ ⎢

⎥ ⎢ ⎥

⎣

⎦ ⎣ ⎦

⎣ ⎦

 

(1) 

916 

G. Liaw et al. 

where  F

x

  and  F

y

 are the forces applied by the PFM in the x and y directions, 

respectively. The other two divergent force fields simply varied in terms of the cross 

term linking an error in x with the force produced in y. Each of these rotated the 

forces either in the clockwise direction (CW DF) or counter clockwise direction 

(CCW DF). The equations of the force fields were:   

CW DF

; CCW DF

200

0

200

0

: : 

600

0

600

0

x

x

y

y

F

F

x

x

F

F

y

y

⎡ ⎤

⎡ ⎤

⎡

⎤ ⎡ ⎤

⎡

⎤ ⎡ ⎤

=

⋅

=

⋅

⎢ ⎥

⎢ ⎥

⎢

⎥ ⎢ ⎥

⎢

⎥ ⎢ ⎥

−

⎣

⎦ ⎣ ⎦

⎣

⎦ ⎣ ⎦

⎣ ⎦

⎣ ⎦

 

(2) 

In order to compensate for the imposed force fields, it has been shown that subjects 

increase the limb stiffness to compensate for the instability of the environment [3,6]. 

In order to increase one cross term (eg. K

yx

) without modifying the other one (eg. 

K

xy

), it has been suggested that subjects change the reflex gain [7]. Therefore, these 

fields were designed to examine if changes in the reflex gain occur as part of the 

impedance controller which compensates for the environment.  

2.3   Protocol 

All subjects practiced making movements in the NF on at least one day prior to the 

experiment. These training trials were used to accustom the subjects to the equipment 

and to the movement speed and accuracy requirements. Each force field was learned 

on a different day. Subjects first learned the DF, after which their endpoint stiffness 

was estimated. The order of training in the other two rotated DF force fields was 

randomized across subjects. For each force field, three procedures were conducted on 

the same day: learning, stiffness estimation, and after effects trials. During learning of 

the DF, subjects first completed 30 successful movements in the NF, after which the 

DF was activated, although no information was provided to the subjects about when 

activation of the DF would occur. Subjects then made movements in the DF until 100 

successful trials were completed. For learning the rotated DF, they first performed 

110 movements in the DF before making movements in the rotated DF until 100 

successful trials were achieved. There was a short break before the stiffness 

estimation. Details of the method and analysis for the stiffness estimation can be 

found in [6,7,10]. Briefly, subjects first completed 20 successful trials in whichever 

force field was being tested. Following this, an additional 160 successful trials were 

completed in the force field of which 80 trials were randomly selected for stiffness 

estimation. During these trials, a 300ms displacement is applied near the midpoint of 

the trajectory. The displacement is divided up into three periods: 100ms ramp up, 

100ms hold, and a 100ms ramp down period. The average force and displacement 

measured during the final 80 ms of the hold period were used to estimate the 2 by 2 

endpoint stiffness matrix (

K

) by linear regression. Finally, after effects were recorded 

by randomly interspersing 20 null field trials with 80 force field trials. On these null 

field trials, subjects expect to be moving in the force field but no errors will be 

induced by the force fields. This allows for a clear picture of the feedforward control 

during the movements.  

On each experimental day, electromyographic activity of six arm muscles was 

measured. Surface EMG was recorded from two monoarticular shoulder muscles: 

pectoralis major and posterior deltoid; two biarticular muscles: biceps brachii and 

 

Reflex Contributions to the Directional Tuning of Arm Stiffness 

917 

long head of the triceps; and two monoarticular elbow muscles: brachioradialis, and 

lateral head of the triceps. EMG was recorded using pairs of silver-silver chloride 

surface electrodes. The EMG signals were Butterworth band-pass filtered between 25 

Hz and 1.0 kHz and sampled at 2.0 kHz.  

S1

S2

S3

S4

100 N/m

A

B

C

D

-50

0

50

0

0.2

0.4

0.6

0.8

1

0

0.5

1

1.5

2

2.5

x 10

5

Orientation

Shape

Size

DF   CW   CCW

DF   CW   CCW

DF   CW   CCW

 

Fig. 2. Endpoint Stiffness. A. Endpoint stiffness was adapted differently in each force field. 

The stiffness ellipses for each of the four subjects participating in the study are shown for each 

of the three unstable force fields (DF: filled ellipse; CW DF: solid line; CCW DF: dotted line). 

The direction of the increased stiffness for each unstable force field is similar across all of the 

subjects. These directions are close to the directions of instability in the environment. B. The 

orientation of the endpoint stiffness ellipses. C. The shape of the endpoint stiffness ellipses. D. 

The size of the endpoint stiffness ellipses. Relative to the DF stiffness, the stiffness measured in 

the two rotated divergent force fields is larger and modified in orientation. 

3   Results 

Subjects were able to adapt to all force fields and make smooth straight movements to 

the final target. While the onset of the force fields caused disturbed trajectories in 

early learning, subjects were able to reduce these errors gradually as learning 

progressed. After learning was completed, stiffness measurements were performed in 

each of the three unstable force fields. 

3.1   Endpoint Stiffness 

The endpoint stiffness can be represented in terms of the eigenvalues and 

eigenvectors of the stiffness matrix using an ellipse. Singular value decomposition of 

the stiffness matrix K was used to find the eigenvalues [11]. Subjects modified the 

endpoint stiffness of their arms in different directions according to the different 

environments in which they were moving (Fig 2). Relative to the baseline stiffness in 

918 

G. Liaw et al. 

the 200 DF, the orientation of the endpoint stiffness in the two rotated DF fields had 

been changed. In each case the orientation occurred in the direction in which the 

environmental stiffness had been rotated. While the shape was fairly consistent across 

all fields, and more anisotropic than seen in null fields [7], the size of the stiffness 

was generally larger in the rotated DF fields. 

The endpoint stiffness can also be examined in terms of the four components of the 

stiffness matrix (Fig 3). For both the CW DF and the CCW DF, the K

xx

 and K

yy

 terms 

were increased relative to the baseline DF level. However, the biggest difference in 

terms of the adaptation to the two force fields occurred in the K

yx

 term. For the CW 

DF, this term was decreased relative to the baseline, whereas in the CCW DF this 

term had been increased. In contrast, the K

xy

 term was maintained close to zero in 

both cases. Analysis of the effect of these terms was performed (Fig 3C). The change 

in the cross stiffness terms (K

xy

 and K

yx

) was such that it compensates for the 

environmental instability with an oppositely directed force.  

0

500

1000

0

500

1000

Kxx Kxy Kyx Kyy

Kxx Kxy Kyx Kyy

A

B

C

force produced

by Kxy

force 

produced

by Kyx

perturbations

in x & y axes

CW DF

CCW DF

CW DF

CCW DF

 

Fig. 3. Changes in the stiffness relative to the DF stiffness. A,B. Mean endpoint stiffness across 

subjects. Each panel shows the mean values for one of the rotated DF fields relative to the 

stiffness in the baseline DF field (light grey bars). C. Effect of the change in cross terms of the 

endpoint stiffness matrix (K

xy

 and K

yx

). For each of the rotated divergent force fields,  

the resulting change in the cross terms relative to the baseline DF stiffness was calculated. The 

force (solid line) resulting from a 1 cm displacement (dotted line) was calculated for each of  

the ΔK

xy

 and ΔK

yx

 terms. The K

xy

 terms after adaptation produced only small changes in force. 

The ΔK

yx

 produced opposite effects in each of the CW DF and CCW DF fields. These forces 

were directed to resist the oppositely directed forces from the divergent force field. 

3.2   Electromyographic Activity 

The large difference between the two stiffness terms (K

yx

 and K

xy

) (Fig 3) cannot 

normally be produced by pure co-contraction of opposing muscles [5, 7]. This 

difference, produced by modifying the antisymmetric part of the stiffness matrix, 

therefore suggests that subjects may have modulated part of their stiffness by  

 

 

Reflex Contributions to the Directional Tuning of Arm Stiffness 

919 

Pectoralis Major

+x

+y

-y

-x

-0.4

0

0.4

-0.4

0

0.4

-0.4

0

0.4

-0.4

0

0.4

-0.4

0

0.4

-0.4

0

0.4

-0.4

0

0.4

-0.4

0

0.4

DF CW CCW

contrib. to

stiffness

[95-175 ms]

long latency

reflex

[60-125 ms]

reflex response scaled

for background activity

 

Fig. 4. The long latency reflex response of the Pectoralis Major changes depending on the force 

field. During the stiffness measurement the hand was moved by 8 mm in each of eight 

directions as illustrated by the arrow insert in the middle of the figure. The reflex response 

scaled by the background activity in unperturbed trials is shown for two intervals: a long 

latency reflex interval (60-125 ms) and the interval which could contribute to the stiffness 

measurement (95-175 ms). The response in each force field is plotted as separate bars: DF 

(left), CW DF (center) and CCW DF (right), with error bars denoting standard error of the 

mean. 

modifying the reflex gain. We investigated this by analyzing the reflex responses 

produced by the perturbations applied to measure stiffness. The integrated EMG was 

calculated during two periods after the perturbation: (60-125 ms) and (95-175 ms). 

The second interval was chosen as the period of muscle activation which would 

contribute to the stiffness estimate. This interval was determined by the fact that the 

stiffness was estimated between 120-200 ms after the onset of the perturbation and 

that the force produced by muscle activity (EMG) in the arm is time delayed by 

approximately 25 ms [12]. The integrated EMG was subtracted by the EMG normally 

present during this time in the movement (from unperturbed trials) and divided by the 

same number, producing a gain value which can be compared across the three fields. 

920 

G. Liaw et al. 

Posterior Deltoid

+x

+y

-y

-x

0

0.5

1

0

0.5

1

0

0.5

1

0

0.5

1

0

0.5

1

0

0.5

1

0

0.5

1

0

0.5

1

DF CW CCW

contrib. to

stiffness

[95-175 ms]

long latency

reflex

[60-125 ms]

reflex response scaled

for background activity

 

Fig. 5. The long latency reflex response of the Posterior Deltoid changes depending on the 

force field. During the stiffness measurement the hand was moved by 8 mm in each of eight 

directions as illustrated by the arrow insert in the middle of the figure. The reflex response 

scaled by the background activity in unperturbed trials is shown for two intervals: a long 

latency reflex interval (60-125 ms) and the interval which could contribute to the stiffness 

measurement (95-175 ms). The response in each force field is plotted as separate bars: DF 

(left), CW DF (center) and CCW DF (right), with error bars denoting standard error of the 

mean. 

The reflex responses in the Pectoralis Major (Fig 4) demonstrate large differences 

in the long latency reflex responses depending on the force field to which the subject 

has adapted. For perturbations in the +x, +x –y, and –y directions, the reflex response 

has been inhibited for the CW DF field, and perhaps slightly enhanced in the CCW 

DF. For oppositely directed perturbations the opposite effect is seen. The CW DF 

shows an excitatory reflex response whereas the CCW DF is again inhibited. 

Similarly, the reflex responses in the Posterior Deltoid (Fig 5) show the same 

effect. The response of long latency reflex depends on the force field to which the 

subject has adapted. For perturbations in the +x, +x –y, and –y directions, the reflex 

response is excitatory for the CW DF field. For oppositely directed perturbations the 

 

 

Reflex Contributions to the Directional Tuning of Arm Stiffness 

921 

opposite effect is seen. The CW DF shows an inhibited reflex response to the 

perturbations whereas the CCW DF is unaffected compared to the baseline field. 

Clearly changes in the reflex gain as part of the adaptation to the environment have 

occurred. Similar changes in the reflex responses were seen in all six arm muscles that 

were recorded. These reflex responses were modified appropriately to compensate for 

the destabilizing effects of the force fields.