98:1075-1082, 2007. First published 5 July 2007;
Dinant A. Kistemaker, Arthur (Knoek) J. Van Soest and Maarten F. Bobbert
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
The costs of publication of this article were defrayed in part by the payment
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J Neurophysiol 98: 1075–1082, 2007.
First published July 5, 2007; doi:10.1152/jn.00287.2007.
1075 0022-3077/07 $8.00 Copyright © 2007 The American Physiological Societywww.jn.org
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