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This paper introduces a framework for representing, generating, and then tuning gaits of legged robots. We introduce a convenient parametrization of gait generators as dynamical systems possessing designer specified stable limit cycles over an appropriate torus. This parametrization affords a continuous selection of operation within a coordination design plane, inspired by biology, spanned by axes that determine the mix of “feedforward/feedback” and “centralized/decentralized” control. Tuning the gait generator parameters through repeated physical experiments with our robot hexapod, RHex, determines the appropriate operating point – the mix of feedback and degree of control decentralization – to achieve significantly increased performance relative to the centralized feedforward operating point that has governed its previous behavior. The present preliminary experiments with these new gaits suggest that they may permit for the first time locomotion over extremely rough terrain that is almost as reliable, rapid, and energy efficient as the very fastest or most efficient outcomes centralized feedforward gaits can achieve on level ground.

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.

We report on a continuous time full body state estimator for a hexapod robot operating in the dynamical regime (entailing a significant aerial phase) on level ground that combines a conventional rate gyro with a novel leg strain based body pose estimator. We implement this estimation procedure on the robot RHex and evaluate its performance using a visual ground truth measurement system. As an independent assessment of our estimator’s quality we also compare its odometry performance to sensorless averaged open loop distance-per-stride estimates.

This paper addresses problems of robot navigation with nonholonomic motion constraints and perceptual cues arising from onboard visual servoing in partially engineered environments. We focus on a unicycle motion model and a variety of artificial beacon constellations motivated by relevance to the autonomous hexapod, RHex. We propose a general hybrid procedure that adapts to the constrained motion setting the standard feedback controller arising from a navigation function in the fully actuated case by switching back and forth between moving “down” and “across” the associated gradient field toward the stable manifold it induces in the constrained dynamics. Guaranteed to avoid obstacles in all cases, we provide some reasonably general sufficient conditions under which the new procedure guarantees convergence to the goal. Simulations are provided for perceptual models previously introduced by other authors.

We report on the design and analysis of a controller that can achieve dynamical self-righting of our hexapedal robot, RHex. Motivated by the initial success of an empirically tuned controller, we present a feedback controller based on a saggital plane model of the robot. We also extend this controller to develop a hybrid pumping strategy that overcomes actuator torque limitations, resulting in robust flipping behavior over a wide range of surfaces. We present simulations and experiments to validate the model and characterize the performance of the new controller.

In this paper, we report on the design of a model-based controller that can achieve dynamical self-righting of a hexapod robot. Extending on our earlier work in this domain, we introduce a tractable multi-point contact model with Coulomb friction. We contrast the singularities inherent to the new model with other available methods and show that for our specific application, it yields dynamics which are well-defined. We then present a feedback controller that achieves “maximal” performance under morphological and actuation constraints, while ensuring the validity of the model by staying away from singularities. Finally, through systematic experiments, we demonstrate that our controller is capable of robust flipping behavior.

For more information: Kod*Lab

Gait parameter adaptation on a physical robot is an error-prone, tedious and time-consuming process. In this paper we present a system for gait adaptation in our RHex series of hexapedal robots that renders this arduous process nearly autonomous. The robot adapts its gait parameters by recourse to a modified version of Nelder-Mead descent while managing its self-experiments and measuring the outcome by visual servoing within a partially engineered environment. The resulting performance gains extend considerably beyond what we have managed with hand tuning. For example, the hest hand tuned alternating tripod gaits never exceeded 0.8 m/s nor achieved specific resistance helow 2.0. In contrast, Nelder-Mead based tuning has yielded alternating tripod gaits at 2.7 m/s (well over 5 body lengths per second) and reduced specific resistance to 0.6 while requiring little human intervention at low and moderate speeds. Comparable gains have been achieved on the much larger ruggedized version of this machine.

We report on a continuous time odometry scheme for a walking hexapod robot built upon a previously developed leg-strain based body pose estimator. We implement this estimation procedure and odometry scheme on the robot RHex and evaluate its performance at widely varying speeds and over different ground conditions by means of a 6 degree of freedom vision based ground truth measurement system (GTMS). We also compare the performance to that of sensorless odometry schemes — both legged as well as on a wheeled version of the robot — using GTMS measurements of elapsed distance.

For more information: Kod*Lab

This paper addresses the problem of registering the hexapedal robot RHex, relative to a known set of beacons, by real-time visual servoing. A suitably constructed navigation function represents the task, in the sense that for a completely actuated machine in the horizontal plane, the gradient dynamics guarantee convergence to the visually cued goal without ever losing sight of the beacons that define it. Since the horizontal plane behavior of RHex can be represented as a unicycle, feeding back the navigation function gradient avoids loss of beacons, but does not yield an asymptotically stable goal. We address new problems arising from the configuration of the beacons and present preliminary experimental results that illustrate the discrepancies between the idealized and physical robot actuation capabilities.

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’s running behaviors relative to the initial prototype open-loop algorithms.