The coordination of movement: optimal feedback control and beyond

Abstract

Optimal control theory and its more recent extension, optimal feedback control theory, provide valuable insights into the flexible and task-dependent control of movements. Here, we focus on the problem of coordination, defined as movements that involve multiple effectors (muscles, joints or limbs). Optimal control theory makes quantitative predictions concerning the distribution of work across multiple effectors. Optimal feedback control theory further predicts variation in feedback control with changes in task demands and the correlation structure between different effectors. We highlight two crucial areas of research, hierarchical control and the problem of movement initiation, that need to be developed for an optimal feedback control theory framework to characterise movement coordination more fully and to serve as a basis for studying the neural mechanisms involved in voluntary motor control.

Figure 1

Pulling direction of the right extensor carpi ulnaris (ECU), extensor carpi radialis brevis (ECRB), extensor carpi radialis longus (ECRL), flexor carpi ulnaris (FCU) and flexor carpi radialis (FCR) in a midrange wrist position. The coloured circles indicate the normalised muscle activation for each movement direction based on a minimisation of the cost function [17]. The tuning function for each muscle, as well as the deviation of the preferred direction from the pulling direction, matches well with empirical results.

Figure 2

Task-dependent feedback control during a bimanual task. (a) In the two-cursor task, a force field applied to the left hand is corrected by the action of the left hand alone. (b) In the one-cursor task, part of the correction is performed by the right hand. (c) The task dependent component q(x) of the cost function comprises the distance between the position of the left hand (p_L) and its goal (g_L) and the distance between the right hand (p_R) and its goal (G_R). Minimisation of this cost function results in independent control gains (L) for the two hands. (d) The cost function for the one-cursor task predicts feedback control in which motor commands for the left hand (u_L) depend on the state of both the left hand and right hands (x̂_L and x̂_R, respectively). Reproduced with permission from Ref. [32].

Figure 3

Structured variability induced by task-dependent feedback gains. (a) Correlations of horizontal endpoint position of the left (x) and right (y) hands are found in the one-cursor task (red line and dots) but not in the two-cursor task (blue line and dots). In the one-cursor task, variability along the task-redundant dimension (distance between hands, left up–right down diagonal) is not corrected. (b) The negative correlation develops during the movement, indicating that it arises from a feedback control law rather than from correlations in the initial motor commands [32].

Figure 4

Coordination between effectors based on higher-level state estimates. (a) Control signals to the left hand (u₁) depend on a state estimate of that hand (x̂₁) and of the controlled object (x̂_C). The higher-level state is estimated through a dynamic forward model based on information from the effector (dashed curve). (b) In the one-cursor task, both hands influence the state estimate for the common cursor and, thus, the motor commands to the two hands become coordinated. (c) During throwing, the opening of the fingers to release the ball (y axis) is not invariant across slow and fast throws, when plotted against the shoulder azimuth. (d) Hand opening is invariant across throwing speeds only when plotted against the angular position of the hand in space. Reproduced with permission from Ref. [59].