Optimization based motion planning provides a useful modeling framework through various costs and constraints. Using Graph of Convex Sets (GCS) for trajectory optimization gives guarantees of feasibility and optimality by representing configuration space as the finite union of convex sets. Nonlinear parametrizations can be used to extend this technique to handle cases such as kinematic loops, but this distorts distances, such that solving with convex objectives will yield paths that are suboptimal in the original space. We present a method to extend GCS to nonconvex objectives, allowing us to "undistort" the optimization landscape while maintaining feasibility guarantees. We demonstrate our method's efficacy on three different robotic planning domains: a bimanual robot moving an object with both arms, the set of 3D rotations using Euler angles, and a rational parametrization of kinematics that enables certifying regions as collision free. Across the board, our method significantly improves path length and trajectory duration with only a minimal increase in runtime. Website: https://shrutigarg914.github.io/pgd-gcs-results/

翻译：基于优化的运动规划通过多种成本与约束条件提供了有效的建模框架。利用凸集图进行轨迹优化，通过将构型空间表示为凸集的有限并集，可获得可行性与最优性保证。非线性参数化可用于扩展该技术以处理运动学闭环等情况，但这会导致距离失真，使得采用凸目标函数求解时将在原始空间中获得次优路径。我们提出一种扩展凸集图至非凸目标函数的方法，能够在保持可行性保证的同时实现优化空间的"无失真化"。我们在三个机器人规划领域验证了本方法的有效性：双臂机器人协同搬运物体、采用欧拉角的三维旋转集合，以及可实现碰撞区域认证的运动学有理参数化。在所有案例中，本方法在仅轻微增加运行时间的前提下，显著提升了路径长度与轨迹持续时间的优化效果。项目网站：https://shrutigarg914.github.io/pgd-gcs-results/