To generate reliable motion for legged robots through trajectory optimization, it is crucial to simultaneously compute the robot's path and contact sequence, as well as accurately consider the dynamics in the problem formulation. In this paper, we present a phase-based trajectory optimization that ensures the feasibility of translational dynamics and friction cone constraints throughout the entire trajectory. Specifically, our approach leverages the superposition properties of linear differential equations to decouple the translational dynamics for each contact point, which operates under different phase sequences. Furthermore, we utilize the differentiation matrix of B{\'e}zier polynomials to derive an analytical relationship between the robot's position and force, thereby ensuring the consistent satisfaction of translational dynamics. Additionally, by exploiting the convex closure property of B{\'e}zier polynomials, our method ensures compliance with friction cone constraints. Using the aforementioned approach, the proposed trajectory optimization framework can generate dynamically reliable motions with various gait sequences for legged robots. We validate our framework using a quadruped robot model, focusing on the feasibility of dynamics and motion generation.

翻译：为了通过轨迹优化为足式机器人生成可靠的运动，必须同时计算机器人的路径与接触序列，并在问题建模中精确考虑动力学因素。本文提出一种基于相位的轨迹优化方法，确保在整个轨迹中平移动力学与摩擦锥约束的可行性。具体而言，我们的方法利用线性微分方程的叠加特性，将每个接触点的平移动力学解耦，这些接触点在不同相位序列下运行。此外，我们利用Bézier多项式的微分矩阵推导出机器人位置与力之间的解析关系，从而确保平移动力学的一致性满足。同时，通过利用Bézier多项式的凸包性质，我们的方法保证了摩擦锥约束的符合性。采用上述方法，所提出的轨迹优化框架能够为足式机器人生成具有多种步态序列的动态可靠运动。我们使用四足机器人模型验证了该框架，重点关注动力学可行性及运动生成能力。