Many forms of locomotion, both natural and artificial, are dominated by viscous friction in the sense that without power expenditure they quickly come to a standstill. From geometric mechanics, it is known that for swimming at the "Stokesian" (viscous; zero Reynolds number) limit, the motion is governed by a reduced order "connection" model that describes how body shape change produces motion for the body frame with respect to the world. In the "perturbed Stokes regime" where inertial forces are still dominated by viscosity, but are not negligible (low Reynolds number), we show that motion is still governed by a functional relationship between shape velocity and body velocity, but this function is no longer linear in shape change rate. We derive this model using results from singular perturbation theory, and the theory of noncompact normally hyperbolic invariant manifolds (NHIMs).Using the theoretical properties of this reduced-order model, we develop an algorithm that estimates an approximation to the dynamics near a cyclic body shape change (a "gait") directly from observational data of shape and body motion. This extends our previous work which assumed kinematic "connection" models. To compare the old and new algorithms, we analyze simulated swimmers over a range of inertia to damping ratios. Our new class of models performs well on the Stokesian regime, and over several orders of magnitude outside it into the perturbed Stokes regime, where it gives significantly improved prediction accuracy compared to previous work.In addition to algorithmic improvements, we thereby present a new class of models that is of independent interest. Their application to data-driven modeling improves our ability to study the optimality of animal gaits, and our ability to use hardware-in-the-loop optimization to produce gaits for robots.

We consider C 1 dynamical systems having a globally attracting hyperbolic fixed point or periodic orbit and prove existence and uniqueness results for C k,α loc globally defined linearizing semiconjugacies, of which Koopman eigenfunctions are a special case. Our main results both generalize and sharpen Sternberg's C k linearization theorem for hyperbolic sinks, and in particular our corollaries include uniqueness statements for Sternberg linearizations and Floquet normal forms. Additional corollaries include existence and uniqueness results for C k,α loc Koopman eigenfunctions, including a complete classification of C ∞ eigenfunctions assuming a C ∞ dynamical system with semisimple and nonresonant linearization. We give an intrinsic definition of "principal Koopman eigenfunctions" which generalizes the definition of Mohr and Mezić for linear systems, and which includes the notions of "isostables" and "isostable coordinates" appearing in work by Ermentrout, Mauroy, Mezić, Moehlis, Wilson, and others. Our main results yield existence and uniqueness theorems for the principal eigenfunctions and isostable coordinates and also show, e.g., that the (a priori non-unique) "pullback algebra" defined in [MM16b] is unique under certain conditions. We also discuss the limit used to define the "faster" isostable coordinates in [WE18, MWMM19] in light of our main results.

We study global properties of the global (center-)stable manifold of a normally attracting invariant manifold (NAIM), the special case of a normally hyperbolic invariant manifold (NHIM) with empty unstable bundle. We restrict our attention to continuous-time dynamical systems, or flows. We show that the global stable foliation of a NAIM has the structure of a topological disk bundle, and that similar statements hold for inflowing NAIMs and for general compact NHIMs. Furthermore, the global stable foliation has a C k disk bundle structure if the local stable foliation is assumed C k . We then show that the dynamics restricted to the stable manifold of a compact inflowing NAIM are globally topologically conjugate to the linearized transverse dynamics at the NAIM. Moreover, we give conditions ensuring the existence of a global C k linearizing conjugacy. We also prove a C k global linearization result for inflowing NAIMs; we believe that even the local version of this result is new, and may be useful in applications to slow-fast systems. We illustrate the theory by giving applications to geometric singular perturbation theory in the case of an attracting critical manifold: we show that the domain of the Fenichel Normal Form can be extended to the entire global stable manifold, and under additional nonresonance assumptions we derive a smooth global linear normal form. * No academic affiliation (jaap@jaapeldering.nl)

Legged locomotion is a challenging regime both for experimental analysis and for robot design. From biology, we know that legged animals can perform spectacular feats which our machines can only surpass on some specially controlled surfaces such as roads. We present a concise review of the theoretical underpinnings of Data Driven Floquet Analysis (DDFA), an approach for empirical modeling of rhythmic dynamical systems. We provide a review of recent and classical results which justify its use in the analysis of legged systems. LOCOMOTION AS AN OSCILLATORLocomotion is a process whereby the body moves through space. Up to changes of shape the body is related to its history by a continuous trajectory in the space of body frame position and orientation -the Lie group SE(3). The mathematical structure representing such a system 1 is that of a "principal fiber bundle" -a product-like construction pairing the shape-space of the body S, with SE(3). Together these are the body's "configuration space" Q. When locomoting, bodies exert forces on a medium in their environment to produce the reaction forces that move them. Mechanics dictates a relationship -the "connection" -between shape velocity and configuration velocity, i.e. between TS and TQ. This highly abstracted view is sometimes used in the undulatory locomotion literature, 1 where the arbitrariness of body frame choice makes its explicit treatment necessary. However, it remains a valid representation for virtually all self-propelling bodies. In an ideal world, we would be able to predict robust and reliable estimates of the connection for any body of interest, in any medium. Such an estimate would be a complete and self-contained representation of how that body could move through the medium.Due to the daunting complexity of body-medium interactions most work on locomotion is focused on simplified models. Low dimensional, sprung mass models have captured salient features of the dynamics of many legged locomotion systems. 2 These models can be organized using the "templates and anchors" hypotheses (TAH): 3 "Animals have many DOF, but move 'as if ' they have only a few. Animals limit pose to a behaviorally relevant family of postures". Template-and-Anchor is a relationship between models of locomotion -one being more elaborate, the other more parsimonious. Both are assumed to be good long-term predictors of the motion, making the template a dimensionally reduced model of the anchor.At this point most treatments of locomotion in biomechanics and robotics would proceed to discuss the structure of various templates, and the predictions obtained from simulation of high-dimensional anchored models. Such approaches are model-driven -the discussion is focused primarily on models, their justification from assumptions about the physics and biology, and the derivaiton of governing parameters for the models. In our data-driven approach the focus is on large information-rich datasets. We used first principles to define a broad class of models, sidestepping the issue of assumptions ...

No abstract