Geometric-based Optimization Algorithms for Cable Routing and Branching in Cluttered Environments

The need for designing lighter and more compact systems often leaves limited space for planning routes for the connectors that enable interactions among the system’s components. Finding optimal routes for these connectors in a densely populated environment left behind at the detail design stage has been a challenging problem for decades. A variety of deterministic as well as heuristic methods has been developed to address different instances of this problem. While the focus of the deterministic methods is primarily on the optimality of the final solution, the heuristics offer acceptable solutions, especially for such problems, in a reasonable amount of time without guaranteeing to find optimal solutions. This study is an attempt to furthering the efforts in deterministic optimization methods to tackle the routing problem in two and three dimensions by focusing on the optimality of final solutions. The objective of this research is twofold. First, a mathematical framework is proposed for the optimization of the layout of wiring connectors in planar cluttered environments. The problem looks at finding the optimal tree network that spans multiple components to be connected with the aim of minimizing the overall length of the connectors while maximizing their common length (for maintainability and traceability of connectors). The optimization problem is formulated as a bi-objective problem and two solution methods are proposed: (1) to solve for the optimal locations of a known number of breakouts (where the connectors branch out) using mixed-binary optimization and visibility notion and (2) to find the minimum length tree that spans multiple components of the system and generates the optimal layout using the previously-developed convex hull based routing. The computational performance of these methods in solving a variety of problems is further evaluated. Second, the problem of finding the shortest route connecting two given nodes in a 3D cluttered environment is considered and addressed through deterministically generating a graphical representation of the collision-free space and searching for the shortest path on the found graph. The method is tested on sample workspaces with scattered convex polyhedra and its computational performance is evaluated. The work demonstrates the NP-hardness aspect of the problem which becomes quickly intractable as added components or increase in facets are considered

On-Orbit Manoeuvring Using Superquadric Potential Fields

On-orbit manoeuvring represents an essential process in many space missions such as orbital assembly, servicing and reconfiguration. A new methodology, based on the potential field method along with superquadric repulsive potentials, is discussed in this thesis. The methodology allows motion in a cluttered environment by combining translation and rotation in order to avoid collisions. This combination reduces the manoeuvring cost and duration, while allowing collision avoidance through combinations of rotation and translation. Different attractive potential fields are discussed: parabolic, conic, and a new hyperbolic potential. The superquadric model is used to represent the repulsive potential with several enhancements. These enhancements are: accuracy of separation distance estimation, modifying the model to be suitable for moving obstacles, and adding the effect of obstacle rotation through quaternions. Adding dynamic parameters such as object translational velocity and angular velocity to the potential field can lead to unbounded actuator control force. This problem is overcome in this thesis through combining parabolic and conic functions to form an attractive potential or through using a hyperbolic function. The global stability and convergence of the solution is guaranteed through the appropriate choice of the control laws based on Lyapunov's theorem. Several on-orbit manoeuvring problems are then conducted such as on-orbit assembly using impulsive and continuous strategies, structure disassembly and reconfiguration and free-flyer manoeuvring near a space station. Such examples demonstrate the accuracy and robustness of the method for on-orbit motion planning

An augmented Voronoi roadmap for 3D translational motion planning for a convex polyhedron moving amidst convex polyhedral obstacles

This paper concerns the development of a piecewise linear Voronoi roadmap for translating a convex polyhedron in a three-dimensional (3-D) polyhedral world. In general the Voronoi roadmap is incomplete for motion planning, i.e., it can have several disjoint components in one connected component of free space. An analysis of the roadmap shows that incompleteness is caused by the occurrence of the following simple geometric structure: a polygon in the Voronoi surface containing one or more polygons inside it. We formally bring out the details of this geometric structure and give an efficient augmentation of the roadmap that makes it complete. We show that the roadmap has size $e= O(n^2Q^2l^2)$, where n is the total number of faces on the obstacles, Q is the total number of obstacles and I is the number of faces on the moving object. We also present an algorithm to construct the roadmap in $O((n + Ql)e+Q^2 log Q)$ time