Trajectory optimization is an essential tool for generating efficient, dynamically consistent gaits in legged locomotion. This paper explores the indirect method of trajectory optimization, emphasizing its application in creating optimal periodic gaits for legged systems and contrasting it with the more common direct method. While the direct method provides flexibility in implementation, it is limited by its need for an input space parameterization. In contrast, the indirect method improves accuracy by computing the control input from states and costates obtained along the optimal trajectory. In this work, we tackle the convergence challenges associated with indirect shooting methods by utilizing numerical continuation methods. This is particularly useful for the systematic development of gait libraries. Our contributions include: (1) the formalization of a general periodic trajectory optimization problem that extends existing first-order necessary conditions to a broader range of cost functions and operating conditions; (2) a methodology for efficiently generating libraries of optimal trajectories (gaits) utilizing a single shooting approach combined with numerical continuation methods; (3) a novel approach for reconstructing Lagrange multipliers and costates from passive gaits; (4) a comparative analysis of the indirect and direct shooting methods using a compass-gait walker as a case study, demonstrating the improved accuracy of the indirect method in generating optimal gaits; and (5) demonstrating applicability to the more complex legged robot RABBIT, with ten dynamic states and four inputs. The findings underscore the potential of the indirect method for generating families of optimal gaits, thereby advancing the field of trajectory optimization in legged robotics.

翻译：轨迹优化是生成腿部运动高效、动态一致步态的重要工具。本文探索轨迹优化的间接方法，强调其在生成腿部系统最优周期步态中的应用，并与更常见的直接方法进行对比。直接方法在实现上具有灵活性，但受限于其对输入空间参数化的需求。相比之下，间接方法通过从最优轨迹中获取的状态和协态计算控制输入，从而提高了精度。在本工作中，我们利用数值延拓方法解决了间接打靶法相关的收敛挑战。这对于步态库的系统开发尤为有用。我们的贡献包括：(1) 将一般周期轨迹优化问题形式化，将现有的一阶必要条件扩展到更广泛的成本函数和运行条件；(2) 提出一种利用单打靶法结合数值延拓方法高效生成最优轨迹（步态）库的方法；(3) 一种从被动步态重建拉格朗日乘子和协态的新方法；(4) 以罗盘步态步行器为案例，对间接和直接打靶法进行对比分析，证明了间接法在生成最优步态方面具有更高的精度；(5) 展示了该方法在更复杂的腿式机器人RABBIT（具有十个动态状态和四个输入）上的适用性。研究结果强调了间接法在生成最优步态族方面的潜力，从而推进了腿式机器人轨迹优化领域的发展。