The Problem of Motor Control 321Figure 12.4 (a) A complex movement pattern produced under the instruc-
tion to maintain a constant velocity; (b) circles produced under the instruc-
tion of a constant velocity (left) and a time-varying velocity (right). These
examples are taken from Derwort (1938). Recordings were made with a light
placed on the index finger. The shutter of a camera was opened about
60 times per second; distance between dots thus represents distance covered
in 160 s, smaller distances indicating smaller, and larger distances higher,
velocity.the movement toward the bulb is more gradual and extended
in time (Marteniuk, MacKenzie, Jeannerod, Athènes, &
Dugas, 1987). Another task constraint has been reported re-
cently: The time it takes to move a mug to the mouth depends
in a particular way on the diameter of the mug and the dis-
tance from the level of water to the edge (Latash & Jaric, in
press). Such task constraints are at least to some degree re-
flected by our everyday experience.
A second type of constraints, which are taken into account
when movement trajectories are indeterminate, is of a more
organismic nature and related to the costs of movements.
Although the general notion of cost minimization—as far as
this is possible with the given task constraints—has a high
degree of plausibility, it poses more of a problem than a
solution. There are many different kinds of costs that can po-
tentially be minimized. For example, Nelson (1983) analyzed
the consequences of minimizing five different kinds of costs
for the trajectories of movements aimed at a target. Other
criteria have been added (e.g., Cruse, 1986; Cruse & Brüwer,
1987; Rosenbaum, Slotta, Vaughan, & Plamondon, 1991;
Rosenbaum, Vaughan, Barnes, & Jorgensen, 1992; Uno,
Kawato, & Suzuki, 1989), and perhaps any list will be in-
complete.
A fairly general principle seems to be that movement tra-
jectories are selected by the criterion of smoothness. Al-
though in principle smoothness can be defined in different
ways, one of the possible criteria is minimization of jerk, that
is, minimization of the integral of the squared third derivative
of end-effector position with respect to time (Flash & Hogan,
1985). The principle can be extended and used to model com-
plex movement patterns, as in handwriting (cf. Teulings,
1996). In addition, for drawing-like movements, it produces
a particular relation between curvature and tangential veloc-
ity, which is known as the two-thirds power law (Viviani &
Flash, 1995). Basically, with a larger radius of curvature,
velocity tends to be higher than with a smaller radius of cur-
vature even when the instruction is to maintain a constant ve-
locity (Figure 12.4). The dependency of velocity on curvature
is particularly conspicuous in drawing ellipses for which
the radius of curvature varies continuously. Although the re-
verse relation has received less attention, variations of veloc-
ity do also induce variations of curvature; for example, when
one attempts to draw circles with a pattern of smaller-higher-
smaller-higher velocity within each cycle, the result is likely
to be ellipses (Derwort, 1938).
Indeterminateness does exist even when the goal of a
movement specifies a trajectory of the end-effector in every
detail. Of course, in such cases the movement trajectory is
not indeterminate, but the input to the motor transformation
is. The origin of the indeterminateness is apparent from
Figure 12.2, where the target position is specified in terms of
two spatial dimensions, but it can be reached with different
configurations of three joints. More generally, the output of
the motor transformation has a lower dimensionality than the
input, so that the inversion of the motor transformation has no
unique solution. The problem of how to deal with the many
dimensions of the input is often called the degrees-of-
freedom problem. A consequence is motor equivalence: The
same movement can be performed in many different ways.
Again, cost minimization can be considered as a way to
reach a unique solution (cf. Cruse, 1986; Cruse & Brüwer,
1987; Rosenbaum et al., 1991). Another possibility is the
freezing of degrees of freedom. For example, in handwriting
adults mainly use the wrist and the fingers, and hardly or not
at all the elbow and the shoulder joints. When one observes
preschoolers at their first attempts to write (which might not
be the appropriate term for the result, but perhaps for the in-
tention), one can notice that the wrist and fingers are largely
immobilized, and that mainly the more proximal joints,
which are closer to the trunk, are used (Blöte & Dijkstra,
1989). This can also be observed when adult right-handers
write with their left hand (Newell & van Emmerik, 1989).
Finally, the high dimensionality of an input vector can be
reduced to a small number of degrees of freedom by way of
introducing covariations. A somewhat trivial example again
can be seen in handwriting: With a normal tripod grip, thumb,
index finger, and middle finger are mechanically coupled
(because of holding the pen) and can no longer be moved
independently.
Motor equivalence implies not only the existence of
criteria for selecting one of the many options, but also that
different options can be chosen in case that it is desirable or
necessary. For example, when one asks people to tap with
their index finger as rapidly as possible, and to do so as long