Time Optimal and PID Controller for Armed Manipulator Robots

This paper is fourfold. First, three different time-optimal control problems for simulating the manipulator robots with one, two, and three arms are mathematically formulated. Corresponding to the related dynamical systems, the nonlinear system of ordinary differential equations is derived. It has been found that these problems are time-free and have distinct initial and boundary conditions, making them hard to solve. To find the minimum time with different payloads, a successful numerical method based on the finite difference and the three-stage Lobatto formula is proposed. Secondly, the related torquecontrolling problems are simulated, and then for one, two, and three armed manipulator robots, they are solved using the PID controller method. Thirdly, it is shown that, compared to the time-optimal controlling problem, the optimal PID torque controller solution takes more time to do the job than was expected. However, the solution in the PID controller method shows less oscillation than the time-optimal control problem. Fourthly, mathematical theories are used, and the numerical results for both methods and different payloads are compared.