Coordinates transformation and learning control for visually-guided voluntary movement with iteration: a Newton-like method in a function space.

In order to control visually-guided voluntary movements, the central nervous system (CNS) must solve the following three computational problems at different levels: (1) determination of a desired trajectory in the visual coordinates, (2) transformation of the coordinates of the desired trajectory to the body coordinates and (3) generation of motor command. In this paper, the second and the third problems are treated at computational, representational and hardware levels of Marr. We first study the problems at the computational level, and then propose an iterative learning scheme as a possible algorithm. This is a trial and error type learning such as repetitive training of golf swing. The amount of motor command needed to coordinate activities of many muscles is not determined at once, but in a step-wise, trial and error fashion in the course of a set of repetitions. Actually, the motor command in the (n + 1)-th iteration is a sum of the motor command in the n-th iteration plus two modification terms which are, respectively, proportional to acceleration and speed errors between the desired trajectory and the realized trajectory in the n-th iteration. We mathematically formulate this iterative learning control as a Newton-like method in functional spaces and prove its convergence under appropriate mathematical conditions with use of dynamical system theory and functional analysis. Computer simulations of this iterative learning control of a robotic manipulator in the body or visual coordinates are shown. Finally, we propose that areas 2, 5, and 7 of the sensory association cortex are possible sites of this learning control. Further we propose neural network model which acquires transformation matrices from acceleration or velocity to motor command, which are used in these schemes.