EQUILIBRIUM POINT CONTROL CANNOT BE FALSIFIED BY RECONSTRUCTING EQUILIBRIUM POINT
TRAJECTORIES
D.A. Kistemaker, A.J. (Knoek) van Soest and M.F. Bobbert
Institute for Fundamental and Clinical Human Movement Sciences,
Vrije Universiteit, Amsterdam, The Netherlands; e mail: [email protected].
INTRODUCTION
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 the
specification of EP-trajectories, time series of arm
configurations in which internal and external forces are in
equilibrium; if the arm is not in a specified EP, it is driven
towards this EP by muscle forces arising due to reflexes and
muscle mechanics. EP-control has been refuted mainly after
researchers claimed that EP-trajectories underlying
movements of subjects were complex [1]. These researchers
had used a KBI-approach, which involves applying force
perturbations during movements of subjects and fitting a
second order stiffness-damping-inertia (KBI) model to the
kinematic responses. In this study we examined the validity
of the KBI-approach with the help of an EP-controlled
musculoskeletal model (MSM) of the arm that was shown
previously to reproduce very well experimentally observed
maximally fast elbow flexion and extensions movements [2].
METHODS
The 2D MSM of the arm (Fig.
1) was actuated by four Hilltype muscles consisting of a
contractile element (CE), a
series elastic element and a
parallel
elastic
element.
Activation
dynamics
was
modeled to describe the relation
between muscle stimulation and
active state. Muscle spindles Figure 1. Schematic
were assumed to linearly feed drawing of the arm
back the time-delayed CE model. ϕe = elbow angle.
length and contraction velocity.We used our EP-controlled
MSM to simulate fast elbow extension movements that were
either unperturbed or perturbed at one of three different
instants. In line with the literature [1,3], the kinematic
responses to the perturbations were reproduced as good as
possible using a KBI-model by optimization of its stiffness
(K) and damping (B). Subsequently, the parameters of the
KBI-model were used to reconstruct the EP-trajectory that
had served as an input for the EP-controller of our MSM.
RESULTS AND DISCUSSION
The responses of the MSM resembled those observed
experimentally [3] and were very similar to the responses
simulated with the optimized KBI-model (Fig. 2A, 2B).
However, the stiffness values estimated using the KBIapproach were substantially higher than the ‘true’ stiffness
of the MSM (Fig. 2B). The estimated stiffness was found to
depend not only on the ‘true’ stiffness of the MSM but also
on all other dynamical parameters of the MSM, whose
relative contribution depended on the nature of the
perturbation. This is because the actual musculoskeletal
system is of higher order than the KBI-model. It was found
that reconstructed EP-trajectories were in agreement with
those presented in the literature [1]. However, they did not
resemble the simple EP-trajectories that had been used to
generate the movement of the model (Fig. 2C). This is
because in an EP-controlled system, the difference between
EP and actual position does not directly determine the
muscle moments, as commonly assumed in reconstructing
EP trajectories [1], but determines the muscle stimulations
[2]. In order to adequately reconstruct the EP-trajectory it is
necessary to estimate the individual muscle stimulations. As
this cannot be done in an in vivo experiment, experimental
determination of EP-trajectories is doomed to fail.
Figure 2. A: Simulations of unperturbed (φU) and perturbed
movements with stiffness (Kilf) set to 16 Nm·rad-1 at three
different perturbation onsets: 100° (φEARLY), 80° (φMID) or
60° (φEND). B: Responses of the MSM (ΦA) and optimized
KBI-model (ΦKBI). Indicated are also Kilf and estimated
stiffness (KEND). Only results for perturbation onset LATE
were plotted. C: ‘True’ EP-trajectory (EPtrue) and EPtrajectories reconstructed using the KBI-approach as
proposed in the literature [1] (EPG&K) and with damping
relative to the reference velocity (EPref vel).
CONCLUSIONS
Even though this study does not show that people control
their movements according to EP-control, it does show that
the refutation of EP-control on the basis of reconstructing
EP-trajectories was unjust.
REFERENCES
1. Gomi and Kawato, Science, 5:117-20, 1996
2. Kistemaker DA, et al. J Neurophysiol, 95:2898-3012,
2006.
3. Popescu et al. Exp Brain Res, 52:17-28, 2003.