Multi-stage global trajectory optimization for the overlapping shift of a seamless two-speed transmission using Legendre pseudo-spectral method

For a seamless two-speed transmission specially equipped in electric vehicles, the global trajectory optimization problem of the overlapping shift process with multiple stages and complex path/point constraints is investigated in this article. The overlapping shift principle and the dynamic models in each phase are also demonstrated in detail. The parameters to evaluate the shift quality are analyzed and then introduced into the objective function and constraints, respectively. The jerks in process and at the key moments have been distinguished and selected as the path and point constraints, respectively. Moreover, considering the joint characteristics of the torque phase and the inertia phase, the trajectory optimization problems in the above two phases are summarized as a multi-stage global trajectory optimization problem. Then, Legendre pseudo-spectral method is used to transfer the multi-stage global trajectory optimization problem into a nonlinear programming problem for numerical solutions. Finally, the effectiveness and feasibility of the multi-stage global trajectory optimization method have been verified through the comparison of the optimal shift trajectories under different conditions with that obtained by the piecewise trajectory optimization method.