Near-Optimal Motion Planning Algorithms Via A Topological and Geometric Perspective

Motion planning is a fundamental problem in robotics, which involves finding a path for an autonomous system, such as a robot, from a given source to a destination while avoiding collisions with obstacles. The properties of the planning space heavily influence the performance of existing motion planning algorithms, which can pose significant challenges in handling complex regions, such as narrow passages or cluttered environments, even for simple objects. The problem of motion planning becomes deterministic if the details of the space are fully known, which is often difficult to achieve in constantly changing environments. Sampling-based algorithms are widely used among motion planning paradigms because they capture the topology of space into a roadmap. These planners have successfully solved high-dimensional planning problems with a probabilistic-complete guarantee, i.e., it guarantees to find a path if one exists as the number of vertices goes to infinity. Despite their progress, these methods have failed to optimize the sub-region information of the environment for reuse by other planners. This results in re-planning overhead at each execution, affecting the performance complexity for computation time and memory space usage. In this research, we address the problem by focusing on the theoretical foundation of the algorithmic approach that leverages the strengths of sampling-based motion planners and the Topological Data Analysis methods to extract intricate properties of the environment. The work contributes a novel algorithm to overcome the performance shortcomings of existing motion planners by capturing and preserving the essential topological and geometric features to generate a homotopy-equivalent roadmap of the environment. This roadmap provides a mathematically rich representation of the environment, including an approximate measure of the collision-free space. In addition, the roadmap graph vertices sampled close to the obstacles exhibit advantages when navigating through narrow passages and cluttered environments, making obstacle-avoidance path planning significantly more efficient. The application of the proposed algorithms solves motion planning problems, such as sub-optimal planning, diverse path planning, and fault-tolerant planning, by demonstrating the improvement in computational performance and path quality. Furthermore, we explore the potential of these algorithms in solving computational biology problems, particularly in finding optimal binding positions for protein-ligand or protein-protein interactions. Overall, our work contributes a new way to classify routes in higher dimensional space and shows promising results for high-dimensional robots, such as articulated linkage robots. The findings of this research provide a comprehensive solution to motion planning problems and offer a new perspective on solving computational biology problems