Date of Award
10-24-2024
Degree Name
Doctor of Philosophy (PhD)
Degree Type
Dissertation
Abstract
Motion Planning is a fundamental problem in robotics that aims to find an optimal trajectory for a system to move on while avoiding obstacles in the environment. Often, a feasible trajectory connecting the start and target point with the shortest length is highly desirable. Additionally, in scenarios such as drone racing or surveillance, topology constraints may arise. At the low level, the LQR or PID controller is utilized to steer the agent to move along the designed trajectory. At a high level, optimization-based, search-based, or sample-based algorithms are utilized to synthesize the feasible trajectory. In this thesis, we deal with optimization-based trajectory synthesizing along with a special type of topology constraint named homotopy and homology class constraints. In the first part of the thesis, we just ignore topology constraints and emphasize how to transform motion planning tasks into optimization tasks while considering start-point end-point constraints and minimal energy or minimal time loss function. Although the loss function is a differential function, the first-order gradient optimization method, such as Adam, shows less ability to find the optimal. However, methods that utilize second-order information, such as the Gaussian-Newton method and the interior point optimizer, can solve the problem quickly and perfectly. The second part of the thesis emphasizes our proposed optimization method for motion planning with homotopy and homology class constraints. We first introduce the Auxiliary Energy Reduction Technique. The hallmark of our approach is that we first introduce virtual control terms to the original system dynamics that ensure that any preset state trajectory is dynamically feasible with respect to the new extended system. We then gradually shift the contribution of the artificial inputs to the actual original inputs, and in the end, the trajectory will be deformed to the one of the same homotopy class that is now also feasible with respect to the original system. However, the aforementioned method suffers from low efficiency when the required homotopy class is complex. Therefore, in the second method, we deal with two-dimensional obstacles by synthesizing auxiliary trajectories for obstacles then synthesizing optimal trajectories for the agent, and then gradually deforming obstacle trajectories to the original ones and keeping the agent’s trajectory optimal, which improves efficiency. To explore the homology class constraints, in the third method, we solve homology class constraints with respect to three-dimensional obstacles by embedding them in two- dimensional. In the fourth method, we combine our method of the third method to extend the second method to deal with special 3-dimensional obstacles with homotopy class constraints.
Language
English (en)
Chair
Shen Zeng
Committee Members
Andrew Clark; Jr-Shin Li; Mohamed Ali Belabbas; Shen Zeng; ShiNung Ching