Trajectory optimization problems for legged robots are commonly formulated with fixed contact schedules. These multi-phase Hybrid Trajectory Optimization (HTO) methods result in locally optimal trajectories, but the result depends heavily upon the predefined contact mode sequence. Contact-Implicit Optimization (CIO) offers a potential solution to this issue by allowing the contact mode to be determined throughout the trajectory by the optimization solver. However, CIO suffers from long solve times and convergence issues. This work combines the benefits of these two methods into one algorithm: Staged Contact Optimization (SCO). SCO tightens constraints on contact in stages, eventually fixing them to allow robust and fast convergence to a feasible solution. Results on a planar biped and spatial quadruped demonstrate speed and optimality improvements over CIO and HTO. These properties make SCO well suited for offline trajectory generation or as an effective tool for exploring the dynamic capabilities of a robot.

翻译：针对腿式机器人的轨迹优化问题通常采用固定接触时序的方式进行建模。此类多相混合轨迹优化（HTO）方法虽能生成局部最优轨迹，但优化结果严重依赖于预设的接触模式序列。接触隐式优化（CIO）通过允许优化求解器在轨迹规划过程中自主确定接触模式，为此问题提供了潜在解决方案。然而，CIO方法存在求解时间长、收敛性差等问题。本研究将两种方法的优势融合为单一算法：阶段接触优化（SCO）。该方法分阶段逐步收紧接触约束，最终固定接触模式以实现快速稳定的可行解收敛。在平面双足机器人与空间四足机器人上的实验结果表明，该方法在求解速度与优化质量上均优于CIO与HTO。这些特性使SCO既适用于离线轨迹生成，也可作为探索机器人动态能力的有效工具。