Minimum acceleration
criterion with constraints implies bang-bang control as an underlying principle
for optimal trajectories of arm reaching movements
Shay Ben-Itzhak and Amir Karniel
Accepted for publication in Neural Computation, April, 2007
Abstract - Rapid
arm reaching movements serve as an excellent test bed for any theory about trajectory
formation. How are these movements
planned? A minimum acceleration
criterion has been examined in the past and the solution obtained, based on the
Euler-Poisson equation, failed to predict that the hand would begin and end the
movement at rest (i.e. with zero acceleration).
Therefore this criterion was rejected in favor of the minimum jerk which
was proved to be successful in describing many features of human movements. This paper follows an alternative approach,
and solves the minimum acceleration problem with constraints using Pontryagin's
minimum principle. We use the minimum
principle to obtain minimum acceleration trajectories and use the jerk as a
control signal. In order to find a
solution that does not include non-physiological impulse functions, constraints
on the maximum and minimum jerk values are assumed. The analytical solution provides a three phase
piecewise constant jerk signal (bang-bang control) where the magnitude of the
jerk and the two switching times depend on the magnitude of the maximum and
minimum available jerk values. This
result fits the observed trajectories of reaching movements and takes into
account both the extrinsic coordinates and the muscle limitations in a single
framework. The minimum acceleration with
constraints principle is discussed as a unifying approach for many observations
about the neural control of movements.