We present a homotopic approach to solving challenging, optimization-based motion planning problems. The approach uses Homotopy Optimization, which, unlike standard continuation methods for solving homotopy problems, solves a sequence of constrained optimization problems rather than a sequence of nonlinear systems of equations. The insight behind our proposed algorithm is formulating the discovery of this sequence of optimization problems as a search problem in a multidimensional homotopy parameter space. Our proposed algorithm, the Probabilistic Homotopy Optimization algorithm, switches between solve and sample phases, using solutions to easy problems as initial guesses to more challenging problems. We analyze how our algorithm performs in the presence of common challenges to homotopy methods, such as bifurcation, folding, and disconnectedness of the homotopy solution manifold. Finally, we demonstrate its utility via a case study on two dynamic motion planning problems: the cart-pole and the MIT Humanoid.

翻译：本文提出了一种同伦方法，用于解决具有挑战性的基于优化的运动规划问题。该方法采用同伦优化技术，与解决同伦问题的标准延拓法不同，它求解的是一系列约束优化问题，而非非线性方程组序列。我们提出算法的核心思想在于：将这一系列优化问题的发现过程，构建为在多维同伦参数空间中的搜索问题。我们提出的概率同伦优化算法在求解与采样阶段之间切换，利用简单问题的解作为更具挑战性问题的初始猜测。我们分析了算法在面对同伦方法常见挑战（如分岔、折叠及同伦解流形的不连通性）时的表现。最后，通过两个动态运动规划案例研究（倒立摆系统与MIT人形机器人）展示了该算法的实用性。