Fast trajectory replanning using Laplacian mesh optimization

Adjusting to new situations by changing the shape of a prerecorded trajectory is an important aspect for robot manipulation in a constrained environment. For being recognized as a distinctive trajectory, the goal of any trajectory modification is to keep local and global properties as similar as possible compared to the reference trajectory. This paper presents a framework that can alter the shape of a trajectory by defining the position of a set of sampling points while maintaining local properties in a least-squares manner. The method consists of a three-staged approach first modifying the global shape of the trajectory and subsequently taking local features into account. Inspired by mesh processing used for 3D surface editing, differential coordinates based on the discretized Laplacian operator are used for measuring and maintaining local trajectory properties when deforming the trajectory. Last, a post-processing step based on a relaxed 'as-rigid-as-possible' principle allows local deformations and length modifications of the trajectory for a better tradeoff between preserving local and global properties. Experiments verifying the applicability of the proposed algorithm are conducted using a 7-DoF anthropomorphic arm following a previously recorded and modified trajectory.