求解非光滑轨迹的自适应网格配点优化方法

摘要

The deficiencies of the traditional collocation method,such as the large number of meshes and collocation points,may cause the nonlinear equations to be computationally expensive and result in huge discrete errors.Due to these drawbacks of the traditional collocation method,an adaptive mesh refinement and collocation optimization method is proposed.Firstly,the optimization problem is transformed into the nonlinear programming problem in diffe-rent meshes.After implementing the Lagrange interpolation method to solve the optimization problem in these meshes,the absolute and relative errors are then obtained.Secondly,according to the magnitude of the trajectory curvature,the positions of non-smooth meshes are determined.Afterwards,the number of meshes and points are further determined by the relationship between the maximum relative error and the allowable error.Meanwhile,the number of collocation points is increased to enhance the solution accuracy in the case where the smooth meshes cannot satisfy the tolerance error.Finally,several numerical simulations are conducted to compare the proposed method with some other currently existing methods.The simulation results show that the collocation points and the number of meshes of the proposed method are less than those of the traditional adaptive mesh refinement methods.Moreover,the efficiency of the proposed method is higher than that of the traditional methods.%针对传统配点法求解非光滑最优控制问题时,网格和配点数目多、非线性方程计算量大、误差大等问题,提出了一种自适应网格和配点优化方法.首先将求解最优轨迹优化问题划分为不同网格中的非线性规划问题,在网格内采用拉格朗日插值多项式对优化问题解逼近,求解出绝对误差和相对误差.然后根据轨迹曲率的大小确定出不光滑处的网格,并根据最大相对误差和容许误差的关系进一步确定网格划分的位置和加密数量;同时,对不满足容忍误差的光滑区间自适应增加配点以提高其求解精度.最后通过仿真实例与传统自适应网格优化方法进行了对比,结果表明,在相同计算精度情况下,本文所需的配点数目和网格数量较少,求解效率较高.