揭示蛇形机器人最优步态族的几何结构，提升操控与机动性。

Motion planning for locomotion systems typically requires translating high-level rigid-body tasks into low-level joint trajectories-a process that is straightforward for car-like robots with fixed, unbounded actuation inputs but more challenging for systems like snake robots, where the mapping depends on the current configuration and is constrained by joint limits. In this paper, we focus on generating continuous families of optimal gaits-collections of gaits parameterized by step size or steering rate-to enhance controllability and maneuverability. We uncover the underlying geometric structure of these optimal gait families and propose methods for constructing them using both global and local search strategies, where the local method and the global method compensate each other. The global search approach is robust to nonsmooth behavior, albeit yielding reduced-order solutions, while the local search provides higher accuracy but can be unstable near nonsmooth regions. To demonstrate our framework, we generate optimal gait families for viscous and perfect-fluid three-link swimmers. This work lays a foundation for integrating low-level joint controllers with higher-level motion planners in complex locomotion systems.