320 Motor Control
course, such an internal model is not necessarily a kind of en-
tity, located in some part of the brain, but it can result from
the activity of a network that is distributed widely across both
central and more peripheral levels of the motor system (e.g.,
Kalveram, 1991).
Figure 12.3b illustrates a closed-loop solution for which
no internal model T^–^1 is required. Instead, the inversion of
the transformation results from the structure of the loop.
(This is shown formally by Jordan, 1996.) Intuitively this be-
comes clear from the following consideration. A closed-loop
system can reduce the deviation between the output yand the
desired output y. To the extent that this is successful, yand
y become similar. This then implies, because y=T(x), that
xapproximatesT–^1 (y*).
For some years, open-loop and closed-loop models of
motor control were contrasted (cf. Stelmach, 1982). How-
ever, by now it is clear that nature combines both types of
solution, roughly in a way illustrated in Figure 12.3c. This
combination maintains the advantages of both types of solu-
tion and avoids the disadvantages of each of them. In addi-
tion, the combination exhibits some characteristics that
match characteristics of human movements (Cruse, Dean,
Heuer, & Schmidt, 1990).
The disadvantage of an open-loop solution is its limited
precision. The motor transformation is complex, and it has
time-varying characteristics. When we use tools or operate
machines, there are additional transformations that must be
taken into account, like the transformation of a steering-wheel
rotation into a change of the direction in which a vehicle is
heading. Thus, internal models of inverse transformations can
only be approximations. The disadvantage of a closed-loop
system is that it involves time delays and can become insta-
ble, in particular when the gain is high. On the other hand, a
high gain is desirable to improve accuracy. When both sys-
tems are combined, open-loop control will serve to approxi-
mate the desired output; closed-loop control is suited to
reducing the remaining deviation even when the gain is rela-
tively small, which serves to avoid instabilities.
There are two different types of procedure to determine
whether a control system is closed-loop or open-loop. The
first is to cut the potential feedback loop, and the second is to
distort the potential feedback signal. Both manipulations
should have essentially no effect when the control system
is open-loop, but strong effects when the control system is
closed-loop; with eliminated feedback, the closed-loop sys-
tem should produce no change of the output signal or only
random changes, and with distorted feedback the output
should be distorted. Human movements are often little af-
fected by elimination of feedback, but strongly affected by its
distortion. Such results do not give a clear answer with
respect to the dichotomy of open-loop versus closed-loop
control, but they conform to expectations based on the com-
bined control modes (Cruse et al., 1990).Indeterminateness of the SolutionsTypically movements are not fully determined by their goals.
An example is reaching, with the goal being defined in terms
of a spatial target position. Thus, only the endpoint of the
movement is specified by the goal, but not its time-course. In
spite of this indeterminateness a solution is reached, which
takes additional task constraints as well as organismic con-
straints into account.
Perhaps the most extensively studied task constraint is the
size of the spatial target, which affects movement duration
and the shape of velocity-time curves (e.g., MacKenzie,
Marteniuk, Dugas, Liske, & Eickmeier, 1987). Basically, for
smaller targets humans choose to produce slower move-
ments. The relation of movement time not only to target
width, but also to the distance of the target from the start po-
sition, is of a particular kind known as Fitts’ law. The early
1950s, when Fitts (1954) first described the relation, saw the
rise of information theory in psychology. Thus, the relation
was formulated in terms of information measures, and the
tradition has left it in that form. Fitts’ law states that move-
ment time is a linear function of the index of difficulty, which
is defined as log 2 (2AW),Abeing the movement amplitude
andWthe width of the target.
Fitts’ law describes a particular kind of speed-accuracy
trade-off: Faster movements have a larger scatter of their
end-positions than slower movements, so when a small scat-
ter is required because the target is small, slower movements
have to be chosen. The law is astonishingly robust (cf. Keele,
1986), and it has given rise to various theoretical accounts
(Crossman & Goodeve, 1963/1983; Fitts, 1954; Meyer,
Abrams, Kornblum, Wright, & Smith, 1988), but also to al-
ternative formulations (cf. Plamondon & Alimi, 1997) and to
contrasting observations (e.g., Schmidt, Zelaznik, Hawkins,
Frank, & Quinn, 1979), in particular for situations that re-
quire a certain movement duration, rather than reading a
spatial target of a particular width. (Wright & Meyer, 1983;
Zelaznik, Mone, McCabe, & Thaman, 1988).
Although they have received much less attention, other
task constraints than target size affect the chosen movement
trajectory. For example, it makes a difference whether the
spatial target has to be hit or whether an object in the same
position has to be grasped, and in the latter case it makes a
difference whether the object is a tennis ball or a light bulb.
The movement to the light bulb takes more time than the
movement to the tennis ball; in particular, the deceleration of