Data driven models of legged locomotion

Abstract

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 ...

Gait modeling and optimization for the perturbed Stokes regime

“…a model valid for small variations in the gait cycle -then local optimality tests can be formulated and evaluated on these models. Such an approach was taken in Bittner et al [2018], which introduced an algorithm informed by both geometric mechanics and data-driven techniques for studying oscillators [Revzen and Guckenheimer, 2008, Revzen, 2009, Revzen and Kvalheim, 2015.…”

Empirical validation of a spined sagittal-plane quadrupedal model

“…This model was sufficiently descriptive as to roughly represent the robot's mass center trajectory during a bounding limit cycle when simulated using a nearly identical controller to that used on the robot, thus making it a potential option for low dimensional representations of core actuation in legged locomotion and motivating a future more formal data-driven analysis along the lines of [20], [21]. Empirical results of a running transition from a bound with a rigid back to a bound with core bending suggest a speed benefit derived from the core actuation, which will be carefully examined in future work.…”

“…We compare this steady-state behavior to a simple reduced-order model of a spined quadruped and show rough correspondence between the two, concluding that the proposed simplified model is sufficiently descriptive to exhibit realistic bounding limit cycles utilizing core actuation and is thus of interest for robotic and biological applications, motivating a more formal data-driven analysis for future work along the lines of [20], [21].…”

A Data-Driven Approach to Connection Modeling

“…For observed systems, it is not always known beforehand what the limit cycle is, what the dynamics of attraction to the limit cycle are, or even how long the period of oscillation is. Data-Driven Floquet Analysis (DDFA) [Revzen, 2009;Revzen and Kvalheim, 2015] enables extraction of these properties. The key elements of this extraction process are specified below.…”

“…n ġn . From oscillator theory [Guckenheimer, 1983;Revzen and Kvalheim, 2015] we know that any exponentially stable oscillator (which we assume this to be) can be parameterized by phase ϕ : R → [0, T ) ⊂ R based on the following rules:…”