高效计算时间最优轨迹的牛顿-共轭梯度增广拉格朗日方法

基于牛顿-共轭梯度(Newton-CG)增广拉格朗日算法, 给出了一种计算机数控(CNC)系统时间最优轨迹规划问题的高效求解方法. 通过非线性变量代换, 时间最优轨迹规划问题被表述为一个固定时间域的凸最优控制问题. 基于扩展极大值原理, 证明了弦误差与分轴加速度约束的时间最优轨迹具有bang-bang的约束结构. 基于控制向量参数化方法, 问题被转化为具有无穷维约束的半无穷规划问题. 通过构造拉格朗日函数, 约束优化问题转化为一系列无约束问题. 由于问题凸性, 故迭代求解采用高效的线搜索Newton-CG方法. 通过求解给定测试路径的时间最优轨迹规划问题, 验证了所提方法的有效性.

Newton-CG Augmented Lagrangian Approach for Efficient Computation of Time Optimal Trajectory

A Newton-CG(conjugate gradient) augmented Lagrangian approach was proposed for solving the time optimal trajectory planning problem of computer numerical control (CNC) systems. By using nonlinear variable substitution, time optimal trajectory planning problem is formulated as a time-independent convex optimal control problem. Then based on the extended Pontryagin maximum principle, detailed proofs are provided to show that the optimal control of the chord error and axis acceleration constrained problem has “bang bang” structure. Based on control vector parameterization (CVP) method, the resulted optimal control problem is further converted into a semi-infinite programming problem with infinite dimension constraints. Augmented Lagrangian functions are constructed to convert the constrained optimization problem into a series of non constrained optimization subproblems. An iteration process is performed by using the linear search Newton-CG method. The results of the time optimal trajectory planning for test paths demonstrate the effectiveness of the approach.