Time optimal trajectory planning of four bar mechanism using indirect approach

Time optimal trajectory planning of closed chain mechanisms has not been done by indirect method yet. In this paper, this problem is considered for a four bar mechanism and its solution is presented on the base of the indirect solution of optimal control problem. To this end, the additional coordinates are omitted using the holonomic constraints, so the dynamic equation is obtained with respect to only one generalized coordinate. Then the necessary conditions for optimality are derived using Pontryagin's minimum principle by considering the constraint on the applied torque. The obtained equations lead to a two-point boundary value problem (BVP) that its solution is the optimum answer. Unlike the direct methods that result in approximate solution, indirect method leads to an exact solution. But the main challenge in indirect method is solving the BVP. Solving this problem is sensitive to the initial guess. This problem is much more severe for time optimal problem which has a high nonlinear answer in bang-bang form. To overcome this problem an algorithm is proposed to solve the time optimal problem with any desired accuracy, and the initial solution can simply be zero at the start of the algorithm. But in the time optimal trajectory the large jerk is occurred, due to control signals switching. In order to overcome this problem, another algorithm is presented to calculate the minimum time with bounded jerk. Finally, the simulation results show the performance of the proposed method in time optimal trajectory planning.