In this paper, we study trajectory planning and control in voluntary, human arm movements. When a hand is moved to a target, the central nervous system must select one specific trajectory among an infinite number of possible trajectories that lead to the target position. First, we discuss what criterion is adopted for trajectory determination. Several researchers measured the hand trajectories of skilled movements and found common invariant features. For example, when moving the hand between a pair of targets, subjects tended to generate roughly straight hand paths with bell-shaped speed profiles. On the basis of these observations and dynamic optimization theory, we propose a mathematical model which accounts for formation of hand trajectories. This model is formulated by defining an objective function, a measure of performance for any possible movement: square of the rate of change of torque integrated over the entire movement. That is, the objective function CT is defined as follows: (formula; see text) We overcome this difficult by developing an iterative scheme, with which the optimal trajectory and the associated motor command are simultaneously computed. To evaluate our model, human hand trajectories were experimentally measured under various behavioral situations. These results supported the idea that the human hand trajectory is planned and controlled in accordance with the minimum torque-change criterion.

Quantitative examinations of internal representations for arm trajectory planning: minimum commanded torque change model. A number of invariant features of multijoint planar reaching movements have been observed in measured hand trajectories. These features include roughly straight hand paths and bell-shaped speed profiles where the trajectory curvatures between transverse and radial movements have been found to be different. For quantitative and statistical investigations, we obtained a large amount of trajectory data within a wide range of the workspace in the horizontal and sagittal planes (400 trajectories for each subject). A pair of movements within the horizontal and sagittal planes was set to be equivalent in the elbow and shoulder flexion/extension. The trajectory curvatures of the corresponding pair in these planes were almost the same. Moreover, these curvatures can be accurately reproduced with a linear regression from the summation of rotations in the elbow and shoulder joints. This means that trajectory curvatures systematically depend on the movement location and direction represented in the intrinsic body coordinates. We then examined the following four candidates as planning spaces and the four corresponding computational models for trajectory planning. The candidates were as follows: the minimum hand jerk model in an extrinsic-kinematic space, the minimum angle jerk model in an intrinsic-kinematic space, the minimum torque change model in an intrinsic-dynamic-mechanical space, and the minimum commanded torque change model in an intrinsic-dynamic-neural space. The minimum commanded torque change model, which is proposed here as a computable version of the minimum motor command change model, reproduced actual trajectories best for curvature, position, velocity, acceleration, and torque. The model's prediction that the longer the duration of the movement the larger the trajectory curvature was also confirmed. Movements passing through via-points in the horizontal plane were also measured, and they converged to those predicted by the minimum commanded torque change model with training. Our results indicated that the brain may plan, and learn to plan, the optimal trajectory in the intrinsic coordinates considering arm and muscle dynamics and using representations for motor commands controlling muscle tensions.

No abstract

We proposed that the trajectory followed by human subject arms tended to minimize the time integral of the square of the rate of change of torque (Uno et al. 1987). This minimum torque-change model predicted and reproduced human multi-joint movement data quite well (Uno et al. 1989). Here, we propose a neural network model for trajectory formation based on the minimum torque-change criterion. Basic ideas of information representation and algorithm are (i) spatial representation of time, (ii) learning of forward dynamics and kinetics model and (iii) relaxation computation based on the acquired model. The model can resolve ill-posed inverse kinematics and inverse dynamics problems for redundant controlled object as well as ill-posed trajectory formation problems. By computer simulation, we show that the model can produce a multi-joint arm trajectory while avoiding obstacles or passing through viapoints.