The Coordination of Arm Movements: An Experimentally Confirmed Mathematical

This paper presents studies of the coordination of voluntary human arm movements. A mathematical model is formulated which is shown to predict both the qualitative features and the quantitative details observed experimentally in planar, multijoint arm movements. Coordination is modeled mathematically by defining an objective function, a measure of performance for any possible movement. The unique trajectory which yields the best performance is determined using dynamic optimization theory. In the work presented here, the objective function is the square of the magnitude of jerk (rate of change of acceleration) of the hand integrated over the entire movement. This is equivalent to assuming that a major goal of motor coordination is the production of the smoothest possible movement of the hand. Experimental observations of human subjects performing voluntary unconstrained movements in a horizontal plane are presented. They confirm the following predictions of the mathematical model: unconstrained point-to-point motions are approximately straight with bell-shaped tangential velocity profiles; curved motions (through an intermediate point or around an obstacle) have portions of low curvature joined by portions of high curvature; at points of high curvature, the tangential velocity is reduced; the durations of the lowcurvature portions are approximately equal. The theoretical analysis is based solely on the kinematics of movement independent of the dynamics of the musculoskeletal system and is successful only when formulated in terms of the motion of the hand in extracorporal space. The Received July 13, 1984, Revised January 7, 1985; Accepted January 14, 1985 This paper describes research performed In the Department of Psychology and the Artificial Intelligence Laboratory at the Massachusetts Institute of Technology. The research was supported in part by National Institute of Neurological Dtsease and Stroke Research Grant NS09343, National institute of Arthritis, Metabolism, and Digestive Diseases Grant AM26710, National Eye lnstltute Grant EY02621, and the Advanced Research Projects Agency of the Department of Defense under Office of Naval Research Contract N00014-80-C-0505. T. F. was supported by the Whttaker Health Sciences Fund and by the Bantrell Fellowship in Neurosciences. We wish to acknowledge the contributions of Drs. William Abend, Emilio Bizzi, Pierro Morasso, and John Hollerbach who assisted this project in several ways, both through their insightful comments and by making their experlmental data available to us. Recently, a few studies of the kinematic and dynamic aspects of multijoint human and monkey arm movements have been conducted. The objective of these studies was to identify common kinematic features or stereotyped patterns of muscle activation characterizing these movements (Georgopoulos et al., 1981; Soechting and Lacquaniti, 1981; Morasso, 1981; Abend et al., 1982; Hollerbach and Flash, 1982). The planning and control of the kinematic aspects of arm movements is termed trajectory formation. The term trajectory refers to the configuration of the arm in space and to the speed of movement as the hand moves from its initial to its final position. Some investigators (Greene, 1972; Saltzman, 1979; Soechting and Lacquaniti, 1981) have argued that trajectories are planned in joint variables. It has been claimed that the CNS uses a strategy of maintaining constant ratios between angular velocities of the joints in order to bring about a reduction in the complexity of the control problem by reducing the number of degrees of freedom. In contrast to this view, other investigators have argued that simplicity of motor control is achieved by planning hand trajectories in extracorporal space; joint rotations are then tailored to produce these desired hand movements (Lashley, 1951; Bernstein, 1967). Recently, this view has gained support from studies of planar, unconstrained human and monkey movements (Georgopoulos, 1981; Morasso, 1981; Abend et al., 1982). When moving the hand between pairs of targets, subjects tended to generate roughly straight hand trajectories with single-peaked, bell-shaped speed profiles; this behavior was independent of the part of the work-space in which the movement was performed. Because the common invariant features of these movements were only evident in the extracorporal coordinates of the hand, these results are a strong indication that planning takes place in terms of hand trajectories rather than joint rotations. Present address: Department of Applied Mathematics, The Weizman Can this conclusion be generalized to more complex movements? Institute of Science, Rehovot 76100, Israel. When subjects were instructed to generate curved movements, the 3 To whom correspondence should be addressed. single-peaked hand speed profile was not preserved. Although the implications with respect to movement organization are discussed. How is movement control organized? Which variable(s) are controlled? These questions have become a growing concern of motor neurophysiologists (Granit, 1981; Stein, 1982). Although investigations have traditionally focused on single muscle contractions or single-joint movements, these systems cannot reveal the problems confronted by the central nervous system in the control of normal multijoint movements. Even a two-joint motion is vastly more complicated than a single-joint motion; in moving from one point to another, on what basis does the central nervous system select one specific trajectory from the infinite number possible? In what coordinate frame is the trajectory planned? However, these complexities offer new research opportunities; investigations of multijoint movements may provide considerable insight into the strategies employed by the central nervous system in the control of skilled activities.