81 research outputs found

    Parallel Composition of Templates for Tail-Energized Planar Hopping

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    We have built a 4DOF tailed monoped that hops along a boom permitting free sagittal plane motion. This underactuated platform is powered by a hip motor that adjusts leg touchdown angle in flight and balance in stance, along with a tail motor that adjusts body shape in flight and drives energy into the passive leg shank spring during stance. The motor control signals arise from the application in parallel of four simple, completely decoupled 1DOF feedback laws that provably stabilize in isolation four corresponding 1DOF abstract reference plants. Each of these abstract 1DOF closedloopdynamicsrepresentssomesimplebutcrucialspecific component of the locomotion task at hand. We present a partial proof of correctness for this parallel composition of “template” reference systems along with data from the physical platform suggesting these templates are “anchored” as evidenced by the correspondence of their characteristic motions with a suitably transformed image of traces from the physical platform. For more information: http://kodlab.seas.upenn.edu/Avik/ICRA201

    The Penn Jerboa: A Platform for Exploring Parallel Composition of Templates

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    We have built a 12DOF, passive-compliant legged, tailed biped actuated by four brushless DC motors. We anticipate that this machine will achieve varied modes of quasistatic and dynamic balance, enabling a broad range of locomotion tasks including sitting, standing, walking, hopping, running, turning, leaping, and more. Achieving this diversity of behavior with a single under-actuated body, requires a correspondingly diverse array of controllers, motivating our interest in compositional techniques that promote mixing and reuse of a relatively few base constituents to achieve a combinatorially growing array of available choices. Here we report on the development of one important example of such a behavioral programming method, the construction of a novel monopedal sagittal plane hopping gait through parallel composition of four decoupled 1DOF base controllers. For this example behavior, the legs are locked in phase and the body is fastened to a boom to restrict motion to the sagittal plane. The platform's locomotion is powered by the hip motor that adjusts leg touchdown angle in flight and balance in stance, along with a tail motor that adjusts body shape in flight and drives energy into the passive leg shank spring during stance. The motor control signals arise from the application in parallel of four simple, completely decoupled 1DOF feedback laws that provably stabilize in isolation four corresponding 1DOF abstract reference plants. Each of these abstract 1DOF closed loop dynamics represents some simple but crucial specific component of the locomotion task at hand. We present a partial proof of correctness for this parallel composition of template reference systems along with data from the physical platform suggesting these templates are anchored as evidenced by the correspondence of their characteristic motions with a suitably transformed image of traces from the physical platform.Comment: Technical Report to Accompany: A. De and D. Koditschek, "Parallel composition of templates for tail-energized planar hopping," in 2015 IEEE International Conference on Robotics and Automation (ICRA), May 2015. v2: Used plain latex article, correct gap radius and specific force/torque number

    Averaged Anchoring of Decoupled Templates in a Tail-Energized Monoped

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    We refine and advance a notion of parallel composition to achieve for the first time a stability proof and empirical demonstration of a steady-state gait on a highly coupled 3DOF legged platform controlled by two simple (decoupled) feedback laws that provably stabilize in isolation two simple 1DOF mechanical subsystems. Specifically, we stabilize a limit cycle on a tailed monoped to excite sustained sagittal plane translational hopping energized by tail-pumping during stance. The constituent subsystems for which the controllers are nominally designed are: (i) a purely vertical bouncing mass (controlled by injecting energy into its springy shaft); and (ii) a purely tangential rimless wheel (controlled by adjusting the inter-spoke stepping angle).We introduce the use of averaging methods in legged locomotion to prove that this “parallel composition” of independent 1DOF controllers achieves an asymptotically stable closed-loop hybrid limit cycle for a dynamical system that approximates the 3DOF stance mechanics of our physical tailed monoped.We present experimental data demonstrating stability and close agreement between the motion of the physical hopping machine and numerical simulations of the (mathematically tractable) approximating model. More information: http://kodlab.seas.upenn.edu/Avik/AveragingTSLI

    Modular Hopping and Running via Parallel Composition

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    Though multi-functional robot hardware has been created, the complexity in its functionality has been constrained by a lack of algorithms that appropriately manage flexible and autonomous reconfiguration of interconnections to physical and behavioral components. Raibert pioneered a paradigm for the synthesis of planar hopping using a composition of ``parts\u27\u27: controlled vertical hopping, controlled forward speed, and controlled body attitude. Such reduced degree-of-freedom compositions also seem to appear in running animals across several orders of magnitude of scale. Dynamical systems theory can offer a formal representation of such reductions in terms of ``anchored templates,\u27\u27 respecting which Raibert\u27s empirical synthesis (and the animals\u27 empirical performance) can be posed as a parallel composition. However, the orthodox notion (attracting invariant submanifold with restriction dynamics conjugate to a template system) has only been formally synthesized in a few isolated instances in engineering (juggling, brachiating, hexapedal running robots, etc.) and formally observed in biology only in similarly limited contexts. In order to bring Raibert\u27s 1980\u27s work into the 21st century and out of the laboratory, we design a new family of one-, two-, and four-legged robots with high power density, transparency, and control bandwidth. On these platforms, we demonstrate a growing collection of {\{body, behavior}\} pairs that successfully embody dynamical running / hopping ``gaits\u27\u27 specified using compositions of a few templates, with few parameters and a great deal of empirical robustness. We aim for and report substantial advances toward a formal notion of parallel composition---embodied behaviors that are correct by design even in the presence of nefarious coupling and perturbation---using a new analytical tool (hybrid dynamical averaging). With ideas of verifiable behavioral modularity and a firm understanding of the hardware tools required to implement them, we are closer to identifying the components required to flexibly program the exchange of work between machines and their environment. Knowing how to combine and sequence stable basins to solve arbitrarily complex tasks will result in improved foundations for robotics as it goes from ad-hoc practice to science (with predictive theories) in the next few decades

    Frontal plane stabilization and hopping with a 2DOF tail

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    The Jerboa, a tailed bipedal robot with two hip-actuated, passive-compliant legs and a doubly actuated tail, has been shown both formally and empirically to exhibit a variety of stable hopping and running gaits in the sagittal plane. In this paper we take the first steps toward operating Jerboa as a fully spatial machine by addressing the predominant mode of destabilization away from the sagittal plane: body roll. We develop a provably stable controller for underactuated aerial stabilization of the coupled body roll and tail angles, that uses just the tail torques. We show that this controller is successful at reliably reorienting the Jerboa body in roughly 150 ms of freefall from a large set of initial conditions. This controller also enables (and appears intuitively to be crucial for) sustained empirically stable hopping in the frontal plane by virtue of its substantial robustness against destabilizing perturbations and calibration errors. The controller as well as the analysis methods developed here are applicable to any robotic platform with a similar doubly-actuated spherical tail joint

    Tail-Assisted Rigid and Compliant Legged Leaping

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    This paper explores the design space of simple legged robots capable of leaping culminating in new behaviors for the Penn Jerboa, an underactuated, dynamically dexterous robot. Using a combination of formal reasoning and physical intuition, we analyze and test successively more capable leaping behaviors through successively more complicated body mechanics. The final version of this machine studied here bounds up a ledge 1.5 times its hip height and crosses a gap 2 times its body length, exceeding in this last regard the mark set by the far more mature RHex hexapod. Theoretical contributions include a non-existence proof of a useful class of leaps for a stripped-down initial version of the new machine, setting in motion the sequence of improvements leading to the final resulting performance. Conceptual contributions include a growing understanding of the Ground Reaction Complex as an effective abstraction for classifying and generating transitional contact behaviors in robotics
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