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

Jie Ye,Huibin Lin,Kegang Zhao,Yanwei Liu,Xiangdong Huang

doi:10.1177/1687814017741737

Abstract

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.

Highlights

Electric vehicle (EV) is equipped with only a single-speed transmission for the wide torque characteristics of the traction motor and the strict restrictions on the size, mass, and cost of the transmission system.[1]
For a better comparison of #1 and #2 methods, the objective function, path constraints, and point constraints of the optimal control problem (OCP) in the torque phase and the inertia phase are established based on the same principle proposed in section ‘‘Multi-stage global trajectory optimization method.’’ While the constraint of the overall shift duration are acted on the durations of the torque phase and the inertia phase for the switching moment of the two phases is unable to be determined before the trajectory optimization
The comparison indicates that the proposed global trajectory optimization method could coordinate the shift qualities in the torque phase and the inertia phase synthetically and be more adaptable under pretty strict path and point constraints

Summary

Introduction

Electric vehicle (EV) is equipped with only a single-speed transmission for the wide torque characteristics of the traction motor and the strict restrictions on the size, mass, and cost of the transmission system.[1]. As presented in the introduction, the trajectory optimization problem of the overlapping shift process, which includes multiple variable phases, is more reasonable to be regarded as an integral multi-stage global OCP to get the best compromise of the shift qualities of the torque phase and inertia phase. One is caused by the continuous change in the motor torque or the friction clutch torque in process, whereas the other is caused by the sudden transition from dynamic friction condition to static friction condition of the friction clutch at the speed synchronization moment (i.e. the end of the inertia phase) It is suggested in Lu et al.[30] and Zhao et al.[31] that the maximum amplitude of the jerk, rather than the integral of the quadratic item of the jerk, is usually used as the evaluation index of the shift smoothness. In order to ensure the motor torque within the torque capacity of the motor, the inequality path constraint C3 on the state variable x3 is introduced and presented as

Objective function where

#2 Method

Findings

Conclusion