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.本论文针对机械设计与机器人学领域中机构的尺寸综合难题，通过构建基于数值代数几何（numerical algebraic geometry）的优化框架展开研究。本研究旨在全局优化由设计需求衍生出的非线性目标函数，这需要求解所有临界点，其中包括全局最小值点。为此，我们在数值延拓技术中引入了创新性启发式算法，我们将其命名为“随机单值性回路（random monodromy loops）”，该算法可实现非线性系统临界点的逼近与根求解。该方法不仅适用于机构设计领域，还可推广至诸多应用场景，包括机器人系统运动分析以及多领域中的非线性系统问题。为克服现有技术固有的可扩展性瓶颈，我们提出了基于单值性（monodromy）的迭代根累积策略——单值性是描述参数化系统中根演化的核心现象。本研究方法以随机过程为核心特色，可通过概率方式获取解，并借助统计模型评估解的可行性。通过多个实例验证，我们证明了该算法在求解高维非线性目标函数临界点方面的有效性。能够全面求解非凸优化问题的能力，在科学与工程领域具有深远的应用价值。本研究开创性地提出了一种用于可视化高维损失曲面的全新方法，我们将其命名为“鞍点图（saddle graphs）”，该方法有助于更深入地理解优化函数的特性。此外，我们利用流形学习（manifold learning）技术对函数曲面进行重构与可视化，其中包括机器学习损失函数的曲面。基于上述研究基础，我们针对多类应用场景中的平面与空间机构设计与分析开发了定制化工具，应用场景包括仿人手指、足式机器人、物料搬运夹爪、可展开机构以及绳索驱动系统。值得注意的是，本方法能够有效规避局部极小值陷阱，从而可以更全面地探索潜在设计方案。综上，本论文在优化框架的构建方面取得了重要进展，其研究成果可覆盖多个领域，为未来的研究与应用提供了极具潜力的发展方向。