Stability Analysis of Legged Locomotion Models by Symmetry-Factored Return Maps

We present a new stability analysis for hybrid legged locomotion systems based on the “symmetric” factorization of return maps.We apply this analysis to two-degrees-of-freedom (2DoF) and threedegrees- of-freedom (3DoF) models of the spring loaded inverted pendulum (SLIP) with different leg recirculation strategies. Despite the non-integrability of the SLIP dynamics, we obtain a necessary condition for asymptotic stability (and a sufficient condition for instability) at a fixed point, formulated as an exact algebraic expression in the physical parameters. We use this expression to characterize analytically the sensory cost and stabilizing benefit of various feedback schemes previously proposed for the 2DoF SLIP model, posited as a low-dimensional representation of running.We apply the result as well to a 3DoF SLIP model that will be treated at greater length in a companion paper as a descriptive model for the robot RHex

Template Based Control of Hexapedal Running

In this paper, we introduce a hexapedal locomotion controller that simulation evidence suggests will be capable of driving our RHex robot at speeds exceeding five body lengths per second with reliable stability and rapid maneuverability. We use a low dimensional passively compliant biped as a template -- a control target for the alternating tripod gait of the physical machine. We impose upon the physical machine an approrimate inverse dynamics within-stride controller designed to force the true high dimensional system dynamics down onto the lower dimensional subspace corresponding to the template. Numerical simulations suggest the presence of asymptotically stable mnning gaits with large basins of attraction. Moreover, this controller improves substantially the maneuverability and dynamic range of RHex\u27s running behaviors relative to the initial prototype open-loop algorithms

Free-Standing Leaping Experiments with a Power-Autonomous, Elastic-Spined Quadruped

We document initial experiments with Canid, a freestanding, power-autonomous quadrupedal robot equipped with a parallel actuated elastic spine. Research into robotic bounding and galloping platforms holds scientific and engineering interest because it can both probe biological hypotheses regarding bounding and galloping mammals and also provide the engineering community with a new class of agile, efficient and rapidly-locomoting legged robots. We detail the design features of Canid that promote our goals of agile operation in a relatively cheap, conventionally prototyped, commercial off-the-shelf actuated platform. We introduce new measurement methodology aimed at capturing our robot’s “body energy” during real time operation as a means of quantifying its potential for agile behavior. Finally, we present joint motor, inertial and motion capture data taken from Canid’s initial leaps into highly energetic regimes exhibiting large accelerations that illustrate the use of this measure and suggest its future potential as a platform for developing efficient, stable, hence useful bounding gaits. For more information: Kod*La

Reduced order modeling of legged locomotion in the horizontal plane

Sprawled-posture insects exhibit a remarkable ability to rapidly run in a stable manner over complex terrain, a feat that has yet to be equaled by man-made legged robots. Recent experimental results suggest that these insects may employ a hierarchical control structure, in which locomotion performance emerges from a combination of insect structure and leg function, which is only augmented by neural reflexes in the presence of large or persistent disturbances. In this study, we utilize a previously developed reduced order model for lateral plane locomotion and extend it by developing leg positioning and actuation strategies that are consistent with insect leg function evidenced in experiments. We examine improvements in locomotion performance obtained by utilizing these strategies in terms of gait stability and robustness to external perturbations, such as those that might be encountered when running over rough terrain. Leg positioning strategies examined include leg angle control based upon previous leg angles and leg recirculation strategies that prescribe, in a feedforward manner, the motion of the leg as it swings from its lift-off to touch-down positions. Both these protocols provide control authority in re-orienting system momentum and are shown to improve gait stability and robustness to external perturbations. While the original reduced order template models leg dynamics by an energy-conserving spring, we incorporate energetic variations through leg actuation that varies the force-free leg length during the stance phase while preserving qualitatively correct force and velocity profiles. In contrast to the partially asymptotically stable gaits identified in previous analyses, incorporating leg actuation in conjunction with the leg positioning strategies produces completely asymptotically stable gaits in body coordinates. Locomotion performance of the resulting model is subsequently analyzed by computing the basin of stability of our model in response to energetic perturbations of the type experimentally implemented on running, sprawled-posture insects

Towards Reactive Control of Transitional Legged Robot Maneuvers

We propose the idea of a discrete navigation problem – consisting of controlling the state of a discrete-time control system to reach a goal set while in the interim avoiding a set of obstacle states – to approximate a simplified class of transitional legged robotic tasks such as leaping which have no well established mathematical description that lends itself to synthesis. The control relation given in Theorem 1 is (assuming a task solution exists) necessary and sufficient to solve a discrete navigation problem in a minimum number of steps, and is well suited to computation when a legged system’s continuous-time within-stride controller anchors sufficiently simple stance mechanics. We demonstrate the efficacy of this control technique on a physical hopping robot affixed to a boom to reactively leap over an obstacle with a running start, controlling in continuous time during stance to exhibit a linear stance map

Simple models of legged locomotion based on compliant limb behavior = Grundmodelle pedaler Lokomotion basierend auf nachgiebigem Beinverhalten

In der vorliegenden Dissertation werden einfache Modelle zur Beinlokomotion unter der gemeinsamen Hypothese entwickelt, dass die beiden grundlegenden und als verschieden angesehenen Gangarten Gehen und Rennen auf ein allgemeines Konzept zurückgeführt werden können, welches in den Standphasen allein auf nachgiebigem Beinverhalten beruht. Hierbei wird auf der Ebene der mechanischen Beschreibung der Gangarten nachgiebiges Beinverhalten mittels des vom Rennen bekannten Masse-Feder-Modells abstrahiert. Zunächst wird eine vergleichsweise einfache, analytische Näherungslösung desselben identifiziert; in einem weiteren Schritt wird die charakteristische Geschwindigkeit des Gangartwechsels aus federartigem Beinverhalten erklärt; und schließlich wird ein zweibeiniges Masse-Feder-Modell für Gehen vorgeschlagen, welches die beobachteten Bodenreaktionskräfte dieser Gangart beschreibt. Auf der Ebene der neuromechanischen Beschreibung wird aufgezeigt, wie das mit einer mechanischen Feder abstrahierte Beinverhalten durch eine positive Rückkopplung der Muskelkraft dezentral und autonom innerhalb des Muskelskelettapparats erzeugt werden kann. Schließlich werden die Einzelergebnisse der Arbeit zusammengefasst, wobei die beiden fundamentalen Gangarten Gehen und Rennen innerhalb des zweibeinigen Masse-Feder-Modells vereinigt werden und die Bedeutung dieses, auf nachgiebigem Beinverhalten beruhenden Zusammenschlusses sowohl für die biomechanische und motorische Grundlagenforschung als auch für Anwendungen in der Robotik, Rehabilitation und Prothetik erörtert wird

Design of a Bio-Inspired Dynamical Vertical Climbing Robot

This paper reviews a template for dynamical climbing originating in biology, explores its stability properties in a numerical model, and presents empirical data from a physical prototype as evidence of the feasibility of adapting the dynamics of the template to robot that runs vertically upward. The recently proposed pendulous climbing model abstracts remarkable similarities in dynamic wall scaling behavior exhibited by radically different animal species. The present paper’s first contribution summarizes a numerical study of this model to hypothesize that these animals’ apparently wasteful commitments to lateral oscillations may be justified by a significant gain in the dynamical stability and, hence, the robustness of their resulting climbing capability. The paper’s second contribution documents the design and offers preliminary empirical data arising from a physical instantiation of this model. Notwithstanding the substantial differences between the proposed bio-inspired template and this physical manifestation, initial data suggest the mechanical climber may be capable of reproducing both the motions and ground reaction forces characteristic of dynamical climbing animals. Even without proper tuning, the robot’s steady state trajectories manifest a substantial exchange of kinetic and potential energy, resulting in vertical speeds of 0.30 m/s (0.75 bl/s) and claiming its place as the first bio-inspired dynamical legged climbing platform

A Bioinspired Dynamical Vertical Climbing Robot

This paper describes the inspiration, design, analysis, implementation of and experimentation with the first dynamical vertical climbing robot. Biologists have proposed a pendulous climbing model that abstracts remarkable similarities in dynamic wall scaling behavior exhibited by radically different animal species. We study numerically a version of that pendulous climbing template dynamically re-scaled for applicability to utilitarian payloads with conventional electronics and actuation. This simulation study reveals that the incorporation of passive compliance can compensate for an artifact’s poorer power density and scale disadvantages relative to biology. However the introduction of additional dynamical elements raises new concerns about stability regarding both the power stroke and limb coordination that we allay via mathematical analysis of further simplified models. Combining these numerical and analytical insights into a series of design prototypes, we document the correspondence of the various models to the variously scaled platforms and report that our approximately two kilogram platform climbs dynamically at vertical speeds up to 1.5 bodylengths per second. In particular, the final 2.6 kg final prototype climbs at an average steady state speed of 0.66 m/s against gravity on a carpeted vertical wall, in rough agreement with our various models’ predictions