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 ofconstraint curvaturethat 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.

In the literature, researchers studying the lateral dynamics of tractor-trailer systems have each developed their own reduced-order lateral dynamic model, each with their own assumptions, state representations, and derivation methods. Little to no work has been performed to compare the accuracy of these models to each other, nor to validate their results against high-fidelity multi-body simulations or real truck data. The purpose of this paper is to identify several reduced-order lateral dynamic models for tractor-trailer systems present in the current literature, simulate their estimated states through common driving maneuvers, and then compare their estimates against reference data from a high-fidelity multi-body dynamic model. From this comparison, we identify the reduced-order model that maintains the least error from the reference data as the best representation of a real tractor-trailer system. The results of our comparison will be useful to researchers interested in using a reduced-order dynamic model from the literature as the basis for articulation angle estimation, model-predictive control, or adaptive control algorithms for autonomous tractor-semitrailer systems.