Causal infinitesimal modeling of nonlinear impulsive systems and safe path planning in robotics

Abstract

Modeling mechanical systems is a mathematical abstraction of describing the relation between the cause (e.g. a force and inertia) and the resulting motions. Depending on the assumptions simplifying the model, and the mathematical framework chosen to describe the model, different aspects of the behavior of systems can be analyzed. The objective of this thesis is to create a new mathematical framework which respects the microscopic feature (e.g. impulsive contact force modeling) and the macroscopic behavior (e.g. safe trajectory planning). In the first part, the microscopic modeling of an impact phase in mechanical systems, which interact with rigid environments, is considered. A new generalized function theory, entitled as Krylov generalized function (KGF) theory, is developed within the framework provided by nonstandard analysis (NSA). In KGF theory, a singular function is point-wise well defined, and it is used to model the singular contact force to generate the instantaneous jumps in velocity domain. In the second part, the macroscopic view in motion planning of mechanical systems is considered in the sense that the control is used to avoid a collision (microscopic phase) with the environment while driving the system to the desired configuration. The objective of the second part is to find an analytic framework to either generate safe trajectories for point mass robots or to find analytic conditions capturing the collision between rectangular bodies in planar space and cuboid bodies in three-dimensional space. Inspired by algebraic and geometric properties of polynomials, a new paradigm of considering the position of point robots, and approximating the boundary or surface of the full body robot is proposed and used in optimal path planning problem.