Page 1

doi: 10.1152/jn.00287.2007

98:1075-1082, 2007. First published 5 July 2007;

J Neurophysiol

Dinant A. Kistemaker, Arthur (Knoek) J. Van Soest and Maarten F. Bobbert

Trajectories

Experimental Reconstruction of Equilibrium Point

Equilibrium Point Control Cannot be Refuted by

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Equilibrium Point Control Cannot be Refuted by Experimental Reconstruction

of Equilibrium Point Trajectories

Dinant A. Kistemaker, Arthur (Knoek) J. Van Soest, and Maarten F. Bobbert

Institute for Fundamental and Clinical Human Movement Sciences, IFKB, Vrije Universiteit, Amsterdam, The Netherlands

Submitted 14 March 2007; accepted in final form 6 June 2007

Kistemaker DA, Van Soest AJ, Bobbert MF. Equilibrium point

control cannot be refuted by experimental reconstruction of equilib-

rium point trajectories. J Neurophysiol 98: 1075–1082, 2007. First

published July 5, 2007; doi:10.1152/jn.00287.2007. In the literature, it

has been hotly debated whether the brain uses internal models or

equilibrium point (EP) control to generate arm movements. EP control

involves specification of EP trajectories, time series of arm configu-

rations in which internal forces and external forces are in equilibrium;

if the arm is not in a specified EP, it is driven toward this EP by

muscle forces arising due to central drive, reflexes, and muscle

mechanics. EP control has been refuted by researchers claiming that

EP trajectories underlying movements of subjects were complex.

These researchers used an approach that involves applying force

perturbations during movements of subjects and fitting a mass-spring-

damper model to the kinematic responses, and then reconstructing the

EP trajectory using the estimated stiffness, damping, and measured

kinematics. In this study, we examined the validity of this approach

using an EP-controlled musculoskeletal model of the arm. We used

the latter model to simulate unperturbed and perturbed maximally fast

movements and optimized the parameter values of a mass-spring-

damper model to make it reproduce as best as possible the kinematic

responses. It was shown that estimated stiffness not only depended on

the “true” stiffness of the musculoskeletal model but on all of its

dynamical parameters. Furthermore it was shown that reconstructed

EP trajectories were in agreement with those presented in the litera-

ture but did not resemble the simple EP trajectories that had been used

to generate the movement of the model. It was concluded that the

refutation of EP control on the basis of results obtained with mass-

spring-damper models was unjust.

I N T R O D U C T I O N

Theories proposed for the control of goal-directed (arm)

movements come in two types: internal model (IM) control

theories and equilibrium-point (EP) control theories. IM con-

trol theories rely on internal models of the dynamics of the

musculoskeletal system to generate the muscle stimulation

patterns (e.g., Kawato 1999; Mehta and Schaal 2002; Schweig-

hofer et al. 1998; Shidara et al. 1993; Todorov and Jordan

2002; see for a comprehensive overview of the different types

of IM controllers Wolpert et al. 1998). EP control involves the

specification of an arm configuration in which internal forces

and external forces are at equilibrium, or an EP trajectory, i.e.,

a time series of such configurations (e.g., Feldman et al. 1990;

Gribble et al. 1998; McIntyre and Bizzi 1996). According to

EP control, muscle forces are not explicitly computed but

rather arise when the limb is not in the specified equilibrium

configuration, due to central drive, reflexes and muscle me-

chanics. At least for single-joint movements, under EP control,

there is no need for an internal dynamics model of the mus-

culoskeletal system; only a mapping from the neural inputs to

the muscles to the equilibrium arm configurations and stiffness

is required (Kistemaker et al. 2006).

Although EP control is parsimonious and allows for a

natural integration of the control of posture and the control of

movement (Ostry and Feldman 2003), it has been rejected by

many authors after Gomi and Kawato (1996, 1997) had esti-

mated joint stiffness during fast arm movements of human

subjects. Gomi and Kawato (1996, 1997, see also Katayama

and Kawato 1993; Popescu et al. 2003) argued that under EP

control, the net moments driving the arm are the product of the

stiffness and the difference between the actual movement

trajectory and the equilibrium-point trajectory. To reconstruct

the EP trajectory, stiffness K (and damping B) were first

estimated in these studies in vivo by subjecting the human

controlled musculoskeletal system to perturbations. The pa-

rameter values K (stiffness), B (damping), and I (inertia) of the

second-order mass-spring-damper model (that will be referred

to as the KBI-model from here on) are optimized to achieve a

best fit between the experimentally observed perturbation re-

sponses and the KBI-model’s responses. Then estimated stiff-

ness, damping, and inertia and the measured kinematics are

used to calculate the EPs. Gomi and Kawato (1996, 1997)

reconstructed EP trajectories using such a KBI approach and

concluded that the EP trajectories were not “simple,” i.e., they

did not resemble the actual movement trajectories. A typical

reconstructed EP trajectory first led the actual trajectory to

generate the accelerating moment and then fell behind the

actual trajectory to generate the decelerating moment. Further-

more, it had a velocity profile with multiple peaks, which was

very different from the actual velocity profile with only one

distinct peak. Obviously the calculation of complicated EP

trajectories by the CNS would require a model of the dynamics

of the musculoskeletal system and would therefore obliterate

the computational attractiveness of EP control (e.g., Gomi and

Kawato 1996; Wolpert and Ghahramani 2000).

In a recent study (Kistemaker et al. 2006), we used a

musculoskeletal model of the arm to explore the feasibility of

EP control for fast arm movements. This model contains a

substantial amount of biological detail and in particular con-

tains the elements and characteristics that make life difficult for

any control theory (admittedly, the model does not contain

individual motor units, but given the size principle this does

not make it fundamentally easier to control). With an EP

Address for reprint requests and other correspondence: D. A. Kistemaker,

Institute for Fundamental and Clinical Human Movement Sciences, Vrije

Universiteit, van der Boechorststraat 9, 1081 BT Amsterdam, The Netherlands

(E-mail: d.kistemaker@fbw.vu.nl).

The costs of publication of this article were defrayed in part by the payment

of page charges. The article must therefore be hereby marked “advertisement”

in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

J Neurophysiol 98: 1075–1082, 2007.

First published July 5, 2007; doi:10.1152/jn.00287.2007.

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controller that combined open- and closed-loop control and

using simple constant-speed EP trajectories, we could make

our musculoskeletal model reproduce very well experimentally

observed maximally fast elbow flexion and extension move-

ments. According to the arguments of Gomi and Kawato

(1996, 1997), fast movements can only be generated using

simple EP trajectories if the stiffness is high. However, the

stiffness in our model, more precisely the intrinsic low-fre-

quency elbow joint stiffness (Kilf; defined here as the change in

steady-state elbow joint moment per unit change in steady-state

joint angle), fell low in the range of experimentally estimated

stiffness values reported in the literature (Kistemaker et al.

2007). This calls for a revisiting of the estimation of stiffness

and the reconstruction of equilibrium trajectories.

The purpose of the present study was to examine the validity

of using KBI models for estimation of stiffness and damping of

the musculoskeletal system and for reconstruction of EP tra-

jectories. It needs no argument that this can only be done if the

true stiffness and damping of the musculoskeletal system and

the true EP trajectory are known. For that reason, we employed

a modeling and simulation approach, using the EP-controlled

musculoskeletal model that has recently been shown to pro-

duce realistic single-joint movements (see Kistemaker et al.

2006). We first used the model to simulate unperturbed and

perturbed movements at three different levels of open-loop

intrinsic stiffness (Kilf). Subsequently, we optimized the pa-

rameter values of a KBI model to make it reproduce as best as

possible the responses of our controlled musculoskeletal model

to perturbations, and we compared the parameter values so

obtained to the “true” parameters of the controlled musculo-

skeletal model. Second, we used the estimated stiffness and

damping values to reconstruct the EP trajectory and compared

this trajectory with the EP trajectory that had served as an input

for the EP controller.

M E T H O D S

Outline of the study

Simulations were carried out with a musculoskeletal model of

the arm controlled by a hybrid open- and closed-loop EP-control-

ler. In the present study, we used the musculoskeletal model and

the EP controller with simple constant-speed EP trajectories to

simulate elbow extension movements from 120 to 60° at three

different levels of intrinsic low-frequency joint stiffness (i.e.,

Kilf? 16, 10 and 5 Nm?rad?1), obtained by manipulating the level

of co-contraction/open loop neural input to the muscle (STIMopen).

For each stiffness level, we simulated the responses to moment

perturbations at three different onsets depending on the elbow joint

position. In line with the approach advocated in the literature (e.g.,

Bennet et al. 1992; Gomi and Kawato 1996, 1997; Gomi and Osu

1998; Popescu et al. 2003), these responses were approximated by

a KBI model; the parameter values of the KBI model were

optimized to achieve a best fit between the responses of the

controlled musculoskeletal model to the perturbations and the KBI

model’s responses to these same perturbations. Subsequently, the

parameter values of the KBI model so obtained were used in an

attempt to reconstruct the EP trajectory that had served as an input

for the EP-controlled musculoskeletal model. In the following text,

the musculoskeletal model, the EP controller and the simulation

procedures are described in detail.

Musculoskeletal model of the arm

The musculoskeletal model of the arm used in this study (Fig. 1)

has been described in full detail elsewhere (Kistemaker et al. 2006;

see also Van Soest and Bobbert 1993). It consisted of three rigid

segments (representing forearm, upper arm, and shoulder blade),

interconnected by two hinge joints (representing glenohumeral

joint and elbow joint). Only elbow flexion/extension movements in

the horizontal plane were allowed. The lower arm was actuated by

four lumped muscles: a monoarticular elbow flexor (MEF; repre-

senting m. brachioradialis, m. brachialis, m. pronator teres, m.

extensor carpi radialis), a monoarticular elbow extensor (MEE;

representing m. triceps brachii caput laterale, m. triceps brachii

caput mediale, m. anconeus, m. extensor carpi ulnaris), a biartic-

ular elbow flexor (BEF; representing m. biceps brachii caput

longum and caput breve) and a biarticular elbow extensor (BEE; m.

triceps brachii caput longum). The muscles were modeled as

Hill-type units consisting of a contractile element (CE), a parallel

elastic element (PE), and a series elastic element (SE). Activation

dynamics, describing the relation between neural input to the

muscle and active state was modeled according to Hatze (1981; see

also Kistemaker et al. 2005). In line with Hatze’s description, the

neural input to the muscle will be termed STIM throughout this

study, and active state will be referred to as q. A variable step-size

ODE solver based on the Runge-Kutta (4, 5) formula was used to

numerically solve the differential equations of the musculoskeletal

model.

EP controller

The EP trajectory was defined as a “ramp” trajectory from the

initial (120°) to the final (60°) position. From the EP trajectory,

desired CE length (?), and desired CE contraction velocity (?˙) were

derived, as outlined in the following text. The total neural input to the

muscle (STIMh) generated by the hybrid open- and closed-loop EP

controller equaled

FIG. 1.

model consisted of three rigid segments interconnected by 2 hinges, actuated

by 4 Hill-type muscles. ?e? elbow angle and ?s? shoulder angle. The model

was constrained to move only in the elbow joint and in the horizontal plane.

The upper arm and shoulder angle were incorporated because the forearm was

actuated by both mono- and biarticular muscles, with the lengths of the

biarticular muscles depending on both shoulder and elbow joint angle. In the

simulations, the upper arm was prevented from moving by setting its initial

angular velocity to 0 and by adding an external moment on the upper arm such

that the angular acceleration of the upper arm was 0 at all times.

Schematic drawing of the musculoskeletal model of the arm. The

1076D. A. KISTEMAKER, A. J. VAN SOEST, AND M. F. BOBBERT

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Page 4

STIMh?t? ? ?STIMopen?t? ? STIMclosed?t??

with

STIMclosed?t? ? kp? ???t? ? lCE?t ? ??? ? kd??˙?t? ? vCE?t ? ???

STIMhis calculated by:

STIMh?t? ? ?STIMopen?t? ? kp? ???t? ? lCE?t ? ??? ? kd? ??˙?t? ? vCE?t ? ????0

1

(1)

The expression {x}0

to 1 and negative values were set to 0.

The STIMopenis a vector that refers to the open-loop part of the

neural inputs to the modeled muscles [STIMMEE_open; STIMMEF_open;

STIMBEF_open; STIMBEE_open], i.e., the neural inputs to the muscles

that, in the absence of external forces, bring the arm to the desired EP

on the basis of intrinsic muscle properties even if no neural feedback

is operative. In a previous study, it was shown that 1) EPs could be set

open-loop over the whole physiological range of motion, 2) each EP

could be set with numerous different STIMopen, with each yielding a

particular Kilf(Kistemaker et al. 2007). Kilfwas defined as the change

in steady-state elbow joint moment per unit change in steady-state

joint angle. ? is a vector with the desired steady state CE lengths of

the modeled muscles [?MEE; ?MEF; ?BEE; ?BEF] and ?˙(t) is the time

derivative of ?(t). kpand kddenote the feedback constants (see

following text). Feedback of CE length (lCE) and CE contraction

velocity (vCE) was assumed to be linear and delayed by 25 ms; the

time delay in the feedback loop (?) was implemented using a fifth-

order Pade ´ approximation (Golub and Van Loan 1989). The procedure

of calculating STIMopenand ? for a given EP at a desired level of Kilf

involved the following steps: 1) calculating muscle-tendon complex

lengths (lMTC) in the EP using the relation between lMTCand shoulder

and elbow angle (as described in Kistemaker et al. 2006). 2) Calcu-

lating isometric CE forces (FCE_isom) for all muscles such that: a) the

sum of the moments exerted by these muscles relative to the elbow

joint axis equals zero (?MMUS? 0); b) the sum of the low-frequency

stiffnesses of the muscle-tendon complexes (under open loop control)

at the EP considered resulted in the desired low-frequency elbow joint

stiffness Kilf(i.e., the partial derivative of MMUSwith respect to the

elbow joint angle at the EP considered); and c) sum of FCE_isomis as

small as possible. And 3) calculating the steady-state CE lengths at the

EP considered from FCE_isom.

Criterion c in step 2 was incorporated because several STIMopen

vectors exist that yield a certain level of Kilfin the desired EP. To deal

with this indeterminacy, the STIMopenwas selected yielding the

smallest amount of total isometric CE force which is likely to

minimize metabolic demands. Steps 1–3 are repeated for the whole EP

trajectory in time steps of 0.01 s for all three desired stiffness values,

i.e., 16, 10, and 5 Nm?rad?1. ?˙, the vector of the desired CE

contraction velocities, was calculated by dividing the difference be-

tween two successive ?s by 0.01 s. All in all this yields three different

time histories of STIMopen(t), ?(t), and ?˙(t).

To facilitate the comparison between true and reconstructed EP

trajectory, continuous control was used throughout this study, even

though in a previous study, it was shown that when open-loop

components were sent out intermittently, maximal attainable move-

ment speed increased (Kistemaker et al. 2006).

1means that values of x higher than 1 were set

Optimization of feedback gains and duration of

the EP trajectory

For each Kilflevel, we used a grid search to find a combination of

duration of the EP trajectory, kpand kdthat led to a movement with

the highest ? ˙maxand with an RMS error of ?2° in the elbow angle

time history. This RMS error value was based on a previous study in

which similar simulation results were compared with experimental

data (Kistemaker et al. 2006).

Perturbations and fitting procedure

Moment perturbations were applied at the instant that the elbow

joint reached one of three different angles: early (100°), mid (80°),

and end (60°). Following Popescu et al. (2003), we used “square-

wave” moment perturbations: the perturbing moment instantaneously

switched from 0 to 20 Nm to stay there for 15 ms, subsequently

switched instantaneously to ?20 Nm to stay there for 15 ms, and then

switched back to zero. The responses of the musculoskeletal system

(?A(t)) to the perturbations were obtained by subtracting the per-

turbed movements (?P(t)) from the unperturbed movements (?U(t))

?A?t? ? ?U?t? ? ?p?t?

(2)

In line with the literature (e.g., Popescu et al. 2003), the values for

stiffness (K) and damping (B) of a second-order linear KBI model

were optimized such that during the first 0.05 s after perturbation

onset, the sum of the squared difference between ?A(t) and simulated

response of the KBI model (?KBI(t)) was minimized. The value for

inertia (I) was set to be identical to that of the musculoskeletal model

(i.e., 0.078 kg ? m2). The optimal values for K and B were identified

using a Nelder-Mead simplex search method (Lagarias et al. 1998).

This approach will henceforth be referred to as the KBI-approach to

estimating stiffness.

Reconstruction of the EP trajectory

The EP trajectory (?eq(t)) was reconstructed according to the

approach proposed by Gomi and Kawato (1996, 1997). First, the

controlled musculoskeletal system was simplified to a second-order

linear model

MKBI?t? ? Mnet?t? ? I ? ? ¨U?t? ? K?t? ? ??eq?t? ? ?U?t?? ? B?t? ? ? ? ? ˙U?t??

(3)

where MKBIis the moment delivered by the KBI model and K(t), B(t),

and I are the stiffness, damping and inertia parameters of the KBI

model. K(t) and B(t) were obtained by linear interpolation of K and B

estimated at positions early, mid, and end. From Eq. 3, the EP

trajectory ?eq(t) can be reconstructed

?eq?t? ? ?K?t? ? ?U?t? ? B?t? ? ? ˙U?t? ? I ? ? ¨U?t?? ? K?t??1

(4)

It might be argued that in reconstructing ?eq(t) one should not use

the absolute velocity, but rather a reference velocity because in the

control scheme of the EP-controller damping is also relative to a

reference velocity. That is, instead of Eq. 3 one should use

Mnet?t? ? I ? ?U?t? ? K?t? ? ??eq?t? ? ?U?t?? ? B?t? ? ?? ˙eq?t? ? ? ˙U?t??

(5)

For sufficiently small ?t (0.001 s was used throughout this study),

? ˙eq(t) can adequately be described by

? ˙eq?t? ? ??eq?t? ? ?eq?t ? ?t?? ? ?t?1

(6)

so that the expression for calculating ?eq(t) becomes

?eq?t? ?K?t? ? ?U?t? ? B?t? ? ? ˙U?t? ? B?t? ? ?eq?t ? ?t? ? ?t?1? ? ¨U?t? ? I

K?t? ? B?t? ? ?t?1

(7)

At t ? 0, the system is in equilibrium, hence ?eq(0) ? ?U(0) and

? ˙eq(0) ? 0. This approach, either as described by Eqs. 4 or 7, will

henceforth be referred to as the KBI approach to estimating EP

trajectories.

R E S U L T S

Figures 2A and 3 show the perturbed and unperturbed

simulated movements for all three stiffness conditions for all

three perturbation instants (see also Table 1). The unperturbed

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movements produced by the musculoskeletal model were qual-

itatively similar to experimental data reported in a previous

study (Kistemaker et al. 2006). In that study, it was shown that

the EP controller was capable of making the musculoskeletal

model reproduce experimentally observed kinematic data. Fig-

ure 4 shows the perturbation responses of the model and

experimentally observed responses during similar single-joint

arm movements by Popescu et al. (2003). The responses of the

musculoskeletal model to the perturbations were somewhat

faster than responses observed in human subjects. This is due

to the fact that in our simulations the perturbing moment was

a square wave, whereas in the study of Popescu et al. (2003),

it was intended to be a square wave but in reality increased

more gradually. Nonetheless, the responses of the musculo-

skeletal model to the perturbations resemble those of human

subjects (Popescu et al. 2003; see Fig. 4). These results lend

further support for our contention that the controlled muscu-

loskeletal model captures the salient features of the real sys-

tem.

Figure 2 (B–D) shows the elbow angle responses to the

moment perturbations, obtained by subtracting the perturbed

from the unperturbed movement as well as the simulated

responses of the KBI model, for the 16 Nm ? rad?1condition.

The simulated responses when Kilfwas set to 10 and 5 Nm ?

rad?1were quantitatively similar (data not shown). As can be

appreciated from Fig. 2, the responses of the KBI model with

optimized parameter values were very similar to those of the

musculoskeletal model. This finding is in accordance with

previous studies in which it was found that the response of the

musculoskeletal system to small perturbations can be ade-

quately approximated with a second order KBI model (e.g.,

Agarwal and Gottlieb 1977; Hunter and Kearney 1982; Win-

ters and Stark 1988). Table 2 presents the values of K and B

estimated using the KBI approach for the different stiffness

conditions as well as the true stiffness values of the musculo-

skeletal model (Kilf). For all stiffness conditions and all per-

turbation onsets, the estimated stiffness values were substan-

tially higher than the true stiffness values.

The difference between estimated and true stiffness values

was greatest near the end of the movement; estimated stiffness

was on average almost tree times as high as the true stiffness.

This result can be understood from the series connection

between an elastic SE and visco-elastic CE. The slope of the

force-velocity relationship, i.e., the damping, becomes less

with increasing speed. Near the end phase of the movement,

vCEwas lower than at the early and mid phases, and hence CE

damping was higher. With higher damping, the moment per-

turbation gave rise to a smaller change in CE length, and as the

change in SE length depends only on the magnitude of the

force step, the total muscle length change was smaller in the

end phase. Consequently, the stiffness, being the change in

muscle force (moment) divided by the change in length (joint

angle), estimated using the KBI approach is higher in the end

phase than in the other phases.

The estimated values for K and B were used to reconstruct

EP trajectories according to the method proposed by Gomi and

0 0.10.2 0.30.40.5 0.6

60

80

100

120

time [s]

Angle [deg]

Kinematics, stiffness 16 Nm⋅rad−1

EARLY

MID

END

A

ϕU

ϕEARLY

ϕMID

ϕEND

0.20.25 0.3

55

70

END

0 0.025

∆ time [s]

0.05

0

1

2

3

4

EARLY

∆ Angle [deg]

Kilf = 16.0

KEARLY = 22.7

B

ΦA

ΦKBI

00.025

∆ time [s]

0.05

0

1

2

3

4

MID

Kilf = 16.0

KMID = 23.4

C

ΦA

ΦKBI

00.025

∆ time [s]

0.05

0

1

2

3

4

END

Kilf = 16.0

KEND = 45.0

D

ΦA

ΦKBI

FIG. 2.

low-frequency stiffness (Kilf) set to 16 Nm ? rad?1at 3 different perturbation

onsets: the instant the elbow joint reached an angle of either 100° (?EARLY),

80° (?MID), or 60° (?END). B–D: responses (unperturbed minus perturbed

movement) of the musculoskeletal model (?A) and responses simulated using

a KBI-model (?KBI). Indicated are also the “true” (Kilf) and estimated stiffness

values (KEARLY, KMID, and KENDrespectively, see also Table 2).

A: simulated unperturbed (?U) and perturbed movements with

0 0.1 0.2 0.30.40.50.6

60

80

100

120

time [s]

Angle [deg]

Kinematics, stiffness 10 Nm⋅rad−1

EARLY

MID

END

A

ϕU

ϕEARLY

ϕMID

ϕEND

0.220.27 0.32

55

70

END

0 0.10.20.30.40.5 0.6

60

80

100

120

time [s]

Angle [deg]

Kinematics, stiffness 5 Nm⋅rad−1

EARLY

MID

END

B

ϕU

ϕEARLY

ϕMID

ϕEND

0.250.30.35

55

70

END

FIG. 3.

frequency stiffness (Kilf) set to 10 Nm ? rad?1(A) and set to 5 Nm ? rad?1(B)

at 3 different perturbation onsets. Perturbation responses of the model at these

stiffness levels (data not shown) were qualitatively similar to the responses

obtained when the stiffness was set to a level of 16 Nm?rad?1(see Fig. 2, B–D).

Simulated unperturbed (?U) and perturbed movements with low-

TABLE 1.

perturbations started (? ˙EARLY, ? ˙MIDand ? ˙END, respectively) and

maximal angular velocity (? ˙MAX) for each of the three

stiffness conditions

Angular velocities at the instant that the moment

Condition

? ˙EARLY

? ˙MID

? ˙END

? ˙MAX

16 Nm ? rad?1

10 Nm ? rad?1

5 Nm ? rad?1

?417

?374

?339

?463

?417

?376

?112

?96

?73

?474

?430

?387

All values for angular velocities are in deg?s?1.

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