求解轨迹优化问题的局部配点法的稀疏性研究

摘要

直接配点法通过对控制变量和状态变量都进行离散将轨迹优化问题转化为非线性规划(NLP)问题.为了提高NLP的求解效率,需要利用其偏导数的稀疏特性并建立偏导数的高效计算方法.本文研究了局部配点法离散得到的NLP的一阶偏导数的稀疏特性,建立了一阶偏导数的高效计算方法.推导了NLP的目标函数梯度和约束雅克比矩阵的数学表达式,得到了NLP偏导数的稀疏型,并且将NLP的偏导数分解为原始轨迹优化问题的偏导数.由于原始轨迹优化问题的约束和变量的数量远少于NLP的约束和变量的数量,从而显著减小了NLP的一阶偏导数的计算量.含有离散气动力和推力数据的仿真算例验证了本文方法的有效性.仿真结果表明,与有限差分法直接计算NLP的偏导数相比,本文方法能够将优化耗时减小至4％以内,随着离散节点数目的增加,计算效率的提升更为显著.%In a direct local collocation method,a trajectory optimization problem is transcribed into a nonlinear programming (NLP) problem.Solving this NLP as efficiently as possible requires that the sparsity of the NLP derivatives should be exploited and the derivatives should be efficiently calculated.In this paper,a computational efficient method is developed for computing the first derivatives of the NLP functions arising from a local discretization of a trajectory optimization problem.Specifically,the expressions are derived for the NLP objective function gradient and constraint Jacobian.It is shown that the NLP derivatives can be reduced to the first derivatives of the functions in the trajectory optimization problem.As a result,the method derived in this paper reduces significantly the amount of computation required to obtain the first-derivatives required by a NLP solver.The approach derived in this paper is demonstrated by an example with discrete aerodynamic data and thrust data where it is forund that the time required to solve the NLP is reduced to less than 4％ compared with the direct differentiation of the NLP functions using a finite difference method,and the efficiency improvement is more significant as the number of the grid points increases.