The Problem of MotorControl319 1 3 2(x 1 , x 2 )Figure 12.2Athree-jointed arm with joint angles 1 , 2 , and 3 , which are
associated with spatial position (x 1 , x 2 ) of the end-effector.
Figure 12.3Possible solutions to a control problem. (a) Open-loop
solution with an internal model Tˆ–^1 of the (inverse) transformation;
(b)closed-loop solution, with C as controller; (c) combination of open-loop
and closed-loop solution.practice (e.g., Metz, 1970) and in tasks requiring high preci-
sion. Even when no net forces result from cocontractions,
they modulate the mechanical characteristics of the joint like
friction.
Saying that joint movement results from the net force of
opposing muscles (in addition to passive forces) is not the
whole story. More precisely, joint rotation results from the
torque, which again is related to the net force in a fairly com-
plicated way, with the relation being dependent on the joint
angle. Even with the movement of the joint, the sequence of
transformations from muscle activation to movement has not
yet reached its end, because in general the goals for our
movements are not defined in terms of joint angles.
Figure 12.2 illustrates a three-jointed arm with the end-
effector pointing to a target. The goals of many movements
are defined in terms of reaching for some spatial target; for
other movements, as in catching a ball, there are temporal tar-
gets in addition; for still other movements, as in writing,
goals are defined in terms of movement traces (or paths).
From Figure12.2 it is apparent that a particular configuration
of joint angles is associated with a particular spatial position
of the end-effector.
Thus far I have sketched the transformation of muscle ac-
tivation to the spatial position of an end-effector like the tip
of the index finger. The purpose was to give some impression
of the complexity of this transformation without going into
too much detail. Sometimes different components of the
transformation are discussed separately, in particular the
kinematic transformation (from joint angles to end-effector
positions) and the dynamic transformation (from torques to
movements of the joints). As a more general term, I shall use
motor transformationto refer to the total transformation or
some part of it.
Given the complexity and the time-varying characteristics
of the motor transformation, one may wonder that humans—
at least after the first few months of their life—are able to
produce purposeful movements at all, and not only random-
appearing ones. This requires that humans be able to deter-
mine the pattern of muscular activity that is required to
produce a particular movement of a particular end-effector.
The very fact that humans can produce purposeful move-
ments indicates that nature has solved this core problem of
motor control; what remains for the movement scientist is to
gain an understanding of what the solution is.An Outline of Possible SolutionsThe core problem of motor control can be stated in a very
simple and general way. Let Tbe a transformation of an input
signalxinto an output signal y. For example, yshall be a
particular time-varying position of an end-effector, and xa
vector that captures time-varying muscle activity. Then the
general problem of control, and that of motor control in par-
ticular, is to determine an input signal xsuch that the output
signalybecomes identical to the desired output signal y*.
The problem is solved when the inverse of the transformation
Tcan be determined, such that T–^1 T =1. Thus, control re-
quires the inversion of a transformation, and there are two
fundamentally different ways to achieve this (see Jordan,
1996, for a detailed discussion).
Figure 12.3a illustrates an open-loop solution which re-
quires an internal model T^–^1 of the transformation, or, more
precisely, of its inverse. There are different ways of imple-
menting such a model formally (e.g., Jordan, 1996). Of