Nonlinear optimal control problems for trajectory planning with obstacle avoidance present several challenges. While general-purpose optimizers and dynamic programming methods struggle when adopted separately, their combination enabled by a penalty approach was found capable of handling highly nonlinear systems while overcoming the curse of dimensionality. Nevertheless, using dynamic programming with a fixed state space discretization limits the set of reachable solutions, hindering convergence or requiring enormous memory resources for uniformly spaced grids. In this work we solve this issue by incorporating an adaptive refinement of the state space grid, splitting cells where needed to better capture the problem structure while requiring less discretization points overall. Numerical results on a space manipulator demonstrate the improved robustness and efficiency of the combined method with respect to the single components.

翻译：轨迹规划中涉及避障的非线性最优控制问题具有诸多挑战。当通用优化器和动态规划方法单独使用时均存在局限性，但通过罚函数方法将二者结合，既能处理高度非线性系统，又能克服维度灾难。然而，采用固定状态空间离散化的动态规划会限制可达解集，导致收敛困难或需要为均匀网格分配巨大的存储资源。本文通过引入状态空间网格的自适应细化技术解决该问题：在需要捕捉问题结构的区域对网格单元进行分裂，同时整体减少离散化点数量。空间机械臂的数值结果表明，相较于单独使用各组件，所提组合方法在鲁棒性和效率方面均得到显著提升。