Chapter 2. Prehension

122 THE PHASES OF PREHENSION
where F is force, K is stiffness of the spring, and ax is the change in
displacement that occurs when the force is applied. The agonist and
antagonist muscles can be viewed as a pair of opposing springs whose
equilibrium states determines the joint angle. Movement occurs by
varying the equilibrium point over time, changing the relative tensions
of the two opposing springs. An equilibrium configuration is defined
for a given value of muscle activation as that position where the forces
of opposing muscles generate equal and opposite torques about the
joints (Hogan 1985). Once the equilbrium point is set, an invariant
characteristic curve is established, and movement occurs to bring the
muscle into the new equilbrium point. The trajectory of the limb fol-
lows an invariant path that brings the limb quickly into equilibrium,
and no specific dynamics computation is made then by the CNS. An
advantage of this method of control is that the muscles-as-springs au-
tomatically generate restoring forces to keep the arm on a specified
virtual trajectory when small perturbations occur, avoiding the need
for an active feedback controller to correct small errors. The issue of
motor equivalence is simplified, as is the degrees of freedom problem,
and variability, since speed is controlled independently of resting
length.
For the mass-spring model, a controller is not needed for generat-
ing joint torques over time (Step 3 in Figure 5.3) because they can be
generated by properties of the muscles themselves. The controller
changes the virtual position of the arm (i.e., the equilibrium point
between agonist/antagonist muscles) by adjusting some control
parameter. Feldman (1986) argued that the resting length threshold
between agonist/antagonist is selected as the control variable. In
contrast, Bizzi (1980) argued that the equilibrium point is controlled
by setting the alpha motoneuron activity to the agonisdantagonist
muscles. There is evidence for both. For example, Bizzi, Dev,
Morasso and Polit (1978), trained primates to perform pointing tasks.
After training, they were deafferented. Under conditions where
unknown loads were added to the arm, there was consistent over or
under shooting. When the load was removed, the correct position was
attained.
In summary, kinematic transformations have been observed in be-
havioral data that suggests that the CNS uses linear approximations for
performing the transformation. There is little variability in the angular
motion between the shoulder and elbow joints (Soechting &
Lacquaniti, 1981). A directional population vector has been observed
*A system is in equilibrium when there are no net forces and torques acting on it.