The Geometry of Optimal Gaits for Inertia-Dominated Kinematic Systems
Isolated mechanical systems-e.g., those floating in space, in free-fall, or on a frictionless surface-are able to achieve net rotation by cyclically changing their shape, even if they have no net angular momentum. Similarly, swimmers immersed in "perfect fluids" are able to use cyclic shap...
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Published in | IEEE transactions on robotics Vol. 38; no. 5; pp. 3279 - 3299 |
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Main Authors | , , , , , , , |
Format | Journal Article |
Language | English |
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New York
IEEE
01.10.2022
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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Abstract | Isolated mechanical systems-e.g., those floating in space, in free-fall, or on a frictionless surface-are able to achieve net rotation by cyclically changing their shape, even if they have no net angular momentum. Similarly, swimmers immersed in "perfect fluids" are able to use cyclic shape changes to both translate and rotate even if the swimmer-fluid system has no net linear or angular momentum. Finally, systems fully constrained by direct nonholonomic constraints (e.g., passive wheels) can push against these constraints to move through the world. Previous work has demonstrated that the displacement induced by these shape changes corresponds to the amount of constraint curvature that the gaits enclose. Properly assessing or optimizing the utility of a gait also requires considering the time or resources required to execute it: A gait that produces a small displacement per cycle, but that can be executed in a short time, may produce a faster average velocity than a gait that produces more displacement, but takes longer to complete a cycle at the same instantaneous effort. In this paper, we consider gaits under two instantaneous measures of effort. For each of these costs, we demonstrate that fixing the average instantaneous cost to a unit value allows us to transform the effort costs into time-to-execute costs for any given gait cycle. We then illustrate how the interaction between the constraint curvature and these costs leads to characteristic geometries for optimal cycles, in which the gait trajectories resemble elastic hoops distended from within by internal pressures. |
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AbstractList | Isolated mechanical systems—e.g., those floating in space, in free-fall, or on a frictionless surface—are able to achieve net rotation by cyclically changing their shape, even if they have no net angular momentum. Similarly, swimmers immersed in “perfect fluids” are able to use cyclic shape changes to both translate and rotate even if the swimmer-fluid system has no net linear or angular momentum. Finally, systems fully constrained by direct nonholonomic constraints (e.g., passive wheels) can push against these constraints to move through the world. Previous work has demonstrated that the displacement induced by these shape changes corresponds to the amount of constraint curvature that the gaits enclose. Properly assessing or optimizing the utility of a gait also requires considering the time or resources required to execute it: A gait that produces a small displacement per cycle, but that can be executed in a short time, may produce a faster average velocity than a gait that produces more displacement, but takes longer to complete a cycle at the same instantaneous effort. In this paper, we consider gaits under two instantaneous measures of effort. For each of these costs, we demonstrate that fixing the average instantaneous cost to a unit value allows us to transform the effort costs into time-to-execute costs for any given gait cycle. We then illustrate how the interaction between the constraint curvature and these costs leads to characteristic geometries for optimal cycles, in which the gait trajectories resemble elastic hoops distended from within by internal pressures. |
Author | Brock, Zachary Fu, Ruijie Faraji, Hossein Hatton, Ross L. Chen, Shuoqi Choset, Howie Ramasamy, Suresh Justus, Nathan |
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Snippet | Isolated mechanical systems-e.g., those floating in space, in free-fall, or on a frictionless surface-are able to achieve net rotation by cyclically changing... Isolated mechanical systems—e.g., those floating in space, in free-fall, or on a frictionless surface—are able to achieve net rotation by cyclically changing... |
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SubjectTerms | Angular momentum Constraints Cost function Costs Curvature Displacement Dynamics Gait Geometric mechanics Internal pressure Jacobian matrices Kinematics Kinetic energy lie brackets locomotion Mechanical systems Optimization Shape Swimming Trajectory |
Title | The Geometry of Optimal Gaits for Inertia-Dominated Kinematic Systems |
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