In this paper, a new balancing approach called “optimal balancing” is presented for open-chain robot manipulators based on open-loop optimal control. In fact, an optimal trajectory planning problem is outlined in which states, controls and the values of counterweights must be determined simultaneously in order to minimize the given performance index for a predefined point-to-point task. Optimal balancing method can be propounded beside the other methods such as unbalancing, static balancing and adaptive balancing, with this superiority that the objective criterion value obtained of proposed method is very lower than the objective criterion value obtained of other methods. For this purpose, the optimal control problem is extended to the case where the performance index, the differential constraints and the prescribed final conditions contain parameters. Using the fundamental theorem of calculus of variations, the necessary conditions for optimality are derived which lead to the optimality conditions associated with the Pontryagin's minimum principle and an additional condition associated with the constant parameters. By developing the obtained optimality conditions for the two-link manipulator, a two-point boundary value problem is achieved which can be solved with bvp4c command in MATLAB®. The obtained results show that optimal balancing in comparison with the previous methods can reduce the performance index significantly. This method can be easily applied to the more complicated manipulator such as a three degrees of freedom articulated manipulator.
