Iterative Learning Control for Optimal Trajectory in Robotic Manipulator

In order to control a robotic manipulator, the following three computational problems at different levels must be solved: (1) the determination of a desired trajectory in the task-oriented coordinates, (2) the transformation of its coordinates to the joint-angle coordinates (Inverse kinematics) and (3) the generation of motor command (Inverse dynamics). The above first problem (trajectory determination) is mainly discussed in this paper. In our previous work, we proposed a mathematical model for formation of trajectories in human arm movements, and then confirmed that hand trajectories predicted by this model were in good agreement with the experimental data. Our model was based on minimization of rate of change of motor command. In the work presented here, we aim to generate the smooth trajectory of a multi-joint manipulator like that of human arm. Therefore, in order to determine a desired trajectory of the multi-joint manipulator, we define the following objective function: the square of the magnitude of rate of change of torque integrated over the entire movement. Since this objective function critically depends on the dynamics of the manipulator system, it is very difficult to determine the unique trajectory which yields the best performance. We overcome this difficulty by developping an iterative learning control in which the optimal trajectory and the associated motor command (torque) are simultaneously computed. Mathematically, a multipoint boundary-value problem for nonlinear differential equations provided by dynamic optimization theory is solved by using a Newton-like method in a function space. In our method, the optimal trajectory and the associated torque can be simultaneously obtained without solving inverse kinematics and inverse dynamics, by only repetitively moving the manipulator and measuring its trajectories.