The Geometry of Optimal Gait Families for Steering Kinematic Locomoting Systems

TL;DR

The paper addresses the challenge of designing continuous families of optimal gaits for steering kinematic locomoting systems, enabling improved controllability and maneuverability beyond isolated gaits. It builds a geometric framework around the local connection, constraint curvature function, and metric-based costs to map shape-space motions to body displacements and costs. A dual-search strategy is developed: a global brute-force approach to reveal a reduced-order gait family and a local continuation-based method to refine higher-order gait parameters, aided by a null-projection tangential predictor to manage bifurcations. The methods are demonstrated on viscous and inertial three-link swimmers, generating both forward and steering gaits and their baby-step variants, including two-dimensional families parameterized by steering rate and step size. This framework lays the groundwork for integrating low-level joint controllers with high-level motion planners, potentially enhancing robustness and reach in complex locomotion tasks.

Abstract

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.

Figures (11)

• Figure 1: Schematic comparing the command spaces of a differential-drive car laumond1998robot and the Pareto optimal gaits (act as the boundary of the command spaces) of a viscous three-link swimmer hatton2011geometric. The horizontal axis represents forward velocity $v$, and the vertical axis represents angular velocity $\omega$. The gray and red curves respectively depict the boundaries of the command spaces for the differential-drive car and the viscous three-link swimmer, under unit power consumption. For the differential-drive car, this boundary is a circular arc. The arrows on the car indicate the direction of the corresponding wheel's rotation. The arrows on the swimmer indicate the corresponding body motion. Only forward-right-turning motions of viscous swimmers are shown. The velocity norm ${v^2+\omega^2}$ is defined as the control effort for the differential-drive car. (b) Schematic of forward function for the viscous three-link swimmers. The grey and red regions denote the negative and positive areas of the height function, respectively. The grey and red arrow-marked curves are the optimal gait cycles in the $x$ and $\theta$ direction, respectively. The net displacement is the area integral enclosed by each gait. The forward gait encloses the grey region while minimizing the cost (or pathlength). (c) Schematic of an optimal gait family and its Pareto front deng2022enhancing for the viscous three-link swimmer in the average-velocity space. Each axis corresponds to the unit-effort averaged forward or angular velocity, $v$ and $\omega$, produced by each gait. The red dot indicates the turning gait, and the gray dot indicates the forward gait. The black solid and dashed curves connect these gaits on the Pareto front, with the solid portion denoting the front in an overlapping region of the command space.
• Figure 2: An overview of an example articulated swimmer and its optimal gaits. (a) Schematic of a three-link viscous swimmer. (b) Two optimal gaits: The maximum-displacement gait (red dashed line) in the $x$-direction, and the maximum-efficiency gait (red solid line) in the $x$-direction (i.e. the largest displacement in $x$-direction per unit power dissipation). The gait is plotted over the system’s Constraint Curvature Function (CCF) corresponding to $x$-direction movement $D(\mathbf{A})^x$, which gives an approximation of the system’s locomotive behavior. These two gaits are found in tam2007optimalramasamy2019geometry. (c) The maximum-efficiency gait (gray solid line) for turning motion on the turning CCF.
• Figure 3: Example System for Gait and Gait Family Optimization (a) A level set of the CCF is depicted as a circular paraboloid, $1 - \alpha_1^2 - \alpha_2^2$. For simplicity, the gait cycle is restricted to an ellipse, parameterized by $p_1$ (the semi-major axis) and $p_2$ (the square of the semi-minor axis). The unconstrained optimal gait is indicated by the red dotted line. (b) Schematic of unconstrained gait optimization. Each point in the parameter space corresponds to an elliptical gait cycle in the shape space. In the absence of constraints, the optimization proceeds by flowing along the vector field (e.g., the negative gradient of gait efficiency). The black circle represents an optimizer step, and the filled red circle marks the optimal point. (c) Schematic of constrained gait optimization. Here, the optimal points form a continuous curve where a level set of the cost function osculates with a level set of the constraint (net displacement) at the optimal point. Arrows indicate the search direction for solutions. Two solution methods are illustrated: pointwise optimization (with the circle representing an optimization step) and parametric optimization (with the thick red line representing a continuous solution curve—i.e., an optimal gait family). Dashed and solid lines denote level sets of the cost and the constraint, respectively.
• Figure 4: The manifold $\Gamma$ consists of two subsets, $\Gamma_1$ and $\Gamma_2$, each representing all possible gait parameters, though some correspond to qualitatively identical gaits. If $\Gamma_1$ contains qualitatively distinct gaits from $\Gamma_2$, a bifurcation occurs at their intersection. The dashed and solid curves represent different solutions from separate branches. The surface contour color represents the continuation parameter $c$. $\Delta\gamma_{\text{pred}}$ is the tangent predictor, $e$ is the unit vector in the search direction, and $T_{\gamma}\Gamma$ denotes the tangent space of the solution manifold at $\gamma$.
• Figure 5: Translation and rotation gait families of the three-link swimmers under the drag-dominated and inertia-dominated environments. Our algorithm generates a continuous set of gaits producing different levels of displacement, of which we illustrate four each for translation and rotation, with $1, \frac{3}{4}, \frac{2}{4},$ and $\frac{1}{4}$ step size of the maximum efficiency gait. The first column shows the body trajectories of the viscous three-link swimmer when the system executes each gait three times. The second and third columns show the $x$ and $\theta$ baby-step gaits for the viscous and inertial three-link system in isometric embedding coordinates (for which metric pathlength is approximately the same as in-page pathlength), plotted against the corresponding $x$ and $\theta$ CCF contours. A detailed explanation of the isometric embedding space can be found in \ref{['app:gaitcost']} and hatton2017kinematic.

Theorems & Definitions (6)

• Definition 1
• Definition 2: Feasible Point
• Definition 3: Sufficient Condition
• Definition 4: Stationary points
• Definition 5: Regular points
• Theorem 1: Implicit function theorem