Using Lie group symmetries for fast corrective motion planning

Abstract

In this paper we develop an algorithmic framework allowing for fast and elegant path correction exploiting Lie group symmetries and operating without the need for explicit control strategies such as cross-track regulation. These systems occur across the gamut of robotics, notably in locomotion, be it ground, underwater, airborne, or surgical domains. Instead of reintegrating an entire trajectory, the method selectively alters small key segments of an initial trajectory in a consistent way so as to transform it via symmetry operations. The algorithm is formulated for arbitrary Lie groups and applied in the context of the special Euclidean group and subgroups thereof. A sampling-based motion planner is developed that uses this method to create paths for underactuated systems with differential constraints. It is also shown how the path correction method acts as a controller within a feedback control loop for real-time path correction. These approaches are demonstrated for ground vehicles in the plane and for flexible bevel tip needle steering in space. The results show that using symmetry-based path correction for motion planning provides a prudent and simple, yet computationally tractable, integrated planning and control strategy.