Optimal balancing of planar cable robot in point to point motion using the indirect approach

In this paper, a new balancing approach is presented for planar cable robot. In fact, an optimal trajectory planning problem is defined in which states, controls and the values of counterweights calculated simultaneously in order to minimize the performance objective in a point to point motion. The necessary conditions for optimality are achieved by calculus of variations which leads to the optimality conditions related with the Pontryagin´s minimum principle and an additional condition related with the constant parameters. The optimality conditions are developed for the two cable robot. Thus a two-point boundary value problem is obtained which can be solved with bvp4c command in MATLAB software. The results show that optimal balancing in comparison with the unbalanced and static balanced methods can reduce the performance index significantly. This technique can be applied to the complicated robots such as a spatial cable robot.