Optimal Kinematic Synthesis via Saddle Graphs Computed by Random Monodromy Loops

This dissertation addresses the challenge of dimensional synthesis in mechanisms used in machine design and robotics through the development of an optimization framework leveraging numerical algebraic geometry. Our approach aims to globally optimize nonlinear objective functions derived from design requirements, necessitating the computation of all critical points, including the global minimum. To achieve this, we introduce innovative heuristic algorithms within numerical continuation techniques, which we refer to as “random monodromy loops”, enabling the approximation of critical points and root-finding in nonlinear systems. This method extends beyond mechanisms to various applications, including motion analysis of robotic systems and nonlinear systems in diverse fields.
To overcome scalability limitations inherent in existing techniques, we propose iterative root accumulation strategies utilizing monodromy, a phenomenon governing the evolution of roots in parameterized systems. Our approach, distinguished by employing random processes, probabilistically obtains solutions and employs statistical models to estimate solution feasibility. Through various examples, we demonstrate the algorithm's effectiveness in discovering critical points of high-dimensional nonlinear objective functions.
The ability to comprehensively address non-convex optimization problems has far-reaching implications in scientific and engineering domains. Our research pioneers a novel approach to visualize high-dimensional loss landscapes, termed “saddle graphs”, facilitating a deeper understanding of optimization functions. Further, we employ manifold learning techniques to reconstruct and visualize function landscapes, including those of machine learning loss functions.
Building on this foundation, we develop customized tools tailored for the design and analysis of planar and spatial mechanisms in diverse applications, including humanoid fingers, legged robots, material handling grippers, deployable mechanisms, and cable-driven systems. Notably, our approach mitigates local minima traps, enabling a more expansive exploration of design candidates.
In summary, this thesis presents a significant advancement in optimization framework development, with implications spanning multiple domains and offering promising avenues for future research and application.