Trajectory optimization is a fundamental tool for generating efficient, dynamically consistent periodic motions, commonly referred to as gaits, in legged locomotion. This paper explores the indirect method of periodic trajectory optimization, emphasizing its application in creating optimal gaits for legged systems and contrasting it with the more common direct method. While the direct method offers implementation flexibility, it is limited by its need for an input space parameterization. In contrast, the indirect method achieves higher accuracy by computing the control input from states and costates 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 optimal gait libraries using single shooting and numerical continuation; (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 higher accuracy of the indirect method; and (5) demonstrating applicability to the more complex legged robot RABBIT, with ten dynamic states and four inputs. These results establish a systematic indirect optimal control framework for synthesizing families of optimal periodic trajectories in legged locomotion.

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