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Rigid bodies, plastic impact, persistent contact, Coulomb friction, and massless limbs are ubiquitous simplifications introduced to reduce the complexity of mechanics models despite the obvious physical inaccuracies that each incurs individually. In concert, it is well known that the interaction of such idealized approximations can lead to conflicting and even paradoxical results. As robotics modeling moves from the consideration of isolated behaviors to the analysis of tasks requiring their composition, a mathematically tractable framework for building models that combine these simple approximations yet achieve reliable results is overdue. In this paper we present a formal hybrid dynamical system model that introduces suitably restricted compositions of these familiar abstractions with the guarantee of consistency analogous to global existence and uniqueness in classical dynamical systems. The hybrid system developed here provides a discontinuous but self-consistent approximation to the continuous (though possibly very stiff and fast) dynamics of a physical robot undergoing intermittent impacts. The modeling choices sacrifice some quantitative numerical efficiencies while maintaining qualitatively correct and analytically tractable results with consistency guarantees promoting their use in formal reasoning about mechanism, feedback control, and behavior design in robots that make and break contact with their environment. 1 Here, consistent refers to a combination of properties detailed in Section 3.4 analogous to the guarantee of global existence and uniqueness of solutions for a classical dynamical system. 2 Wolfram Mathematica 9, Numerical integration uses the NDSolve command, event detection uses the WhenEvent command. http://www.wolfram.com/mathematica/ 4 An execution of a hybrid dynamical system exhibits the Zeno phenomena if it undergoes an infinite number of discrete or logical switches in finite time (Definition 6). 5 E.g. using the algorithm proposed in Burden, Gonzalez, Vasudevan, Bajcsy and Sastry (2015) for hybrid control systems.3 phenomena (Drumwright 2010) as well as apparent contradictions between frictional forces and impulses discussed in Section 1.2.5) seems to come at the cost of persistence of contact. In contrast, here, persistence is one of the key simplifying modeling assumptions, expressing our intuitive experience of limbs interacting with the world, enabling some of the other assumptions, and affording our strong formal results. Despite being targeted at a different numerical integration scheme, many of the results in this paper, such as the consistent handling of massless limbs, are potentially applicable to time-stepping schemes. Hybrid Dynamical SystemsThis paper models manipulation and self-manipulation systems using a hybrid systems paradigm that assumes instantaneous transitions. Though we develop our (so-called) self-manipulation hybrid dynamical system for a similar class of mechanical systems as that considered in (van der Schaft and Schumacher 1998, Ex. 3.3), we specialize fr...

This paper introduces self-manipulation as a new formal design methodology for legged robots with varying ground interactions. The term denotes a set of modeling choices that permit a uniform and bodycentric representation of the equations of motion-essentially a guide to the selection and configuration of coordinate frames. We present the hybrid system kinematics, dynamics, and transitions in the form of a consistently structured representation that simplifies and unites the account of these, otherwise bewilderingly diverse differential algebraic equations. Cleaving as closely as possible to the modeling strategies developed within the mature manipulation literature, self-manipulation models can leverage those insights and results where applicable, while clarifying the fundamental differences. Our primary motivation is not to facilitate numerical simulation but rather to promote design insight. We instantiate the abstract formalism for a simplified model of RHex, and illustrate its utility by applying a variety of analytical and computational techniques to derive new results bearing on behaviors, controllers, and platform design. For each example, we present empirical results documenting the specific benefits of the new insight into the robot's transitions from standing to moving in place and to leaping.INDEX TERMS Legged locomotion, Robot control, Robot kinematics, Manipulator dynamics. I. INTRODUCTIONLegged robots, such as the notional mechanism in Fig. 1, will necessarily experience a variety of changing contact conditions as they perform ever more complex tasks requiring greater autonomy on novel, and possibly shifting, terrain. As 310

Toward a Vocabulary of Legged Leaping AbstractAs dynamic robot behaviors become more capable and well understood, the need arises for a wide variety of equally capable and systematically applicable transitions between them. We use a hybrid systems framework to characterize the dynamic transitions of a planar "legged" rigid body from rest on level ground to a fully aerial state. The various contact conditions fit together to form a topologically regular structure, the "ground reaction complex". The body's actuated dynamics excite multifarious transitions between the cells of this complex, whose regular adjacency relations index naturally the resulting "leaps" (path sequences through the cells from rest to free flight). We exhibit on a RHex robot some of the most interesting "words" formed by these achievable path sequences, documenting unprecedented levels of performance and new application possibilities that illustrate the value of understanding and expressing this vocabulary systematically. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to pubs-permissions@ieee.org. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.This conference paper is available at ScholarlyCommons: http://repository.upenn.edu/ese_papers/662 Toward a Vocabulary of Legged LeapingAaron M. Johnson and D. E. Koditschek Abstract-As dynamic robot behaviors become more capable and well understood, the need arises for a wide variety of equally capable and systematically applicable transitions between them. We use a hybrid systems framework to characterize the dynamic transitions of a planar "legged" rigid body from rest on level ground to a fully aerial state. The various contact conditions fit together to form a topologically regular structure, the "ground reaction complex". The body's actuated dynamics excite multifarious transitions between the cells of this complex, whose regular adjacency relations index naturally the resulting "leaps" (path sequences through the cells from rest to free flight). We exhibit on a RHex robot some of the most interesting "words" formed by these achievable path sequences, documenting unprecedented levels of performance and new application possibilities that illustrate the value of understanding and expressing this vocabulary systematically.

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This paper documents near-autonomous negotiation of synthetic and natural climbing terrain by a rugged legged robot, achieved through sequential composition of appropriate perceptually triggered locomotion primitives. The first, simple composition achieves autonomous uphill climbs in unstructured outdoor terrain while avoiding surrounding obstacles such as trees and bushes. The second, slightly more complex composition achieves autonomous stairwell climbing in a variety of different buildings. In both cases, the intrinsic motor competence of the legged platform requires only small amounts of sensory information to yield near-complete autonomy. Both of these behaviors were developed using X-RHex, a new revision of RHex that is a laboratory on legs, allowing a style of rapid development of sensorimotor tasks with a convenience near to that of conducting experiments on a lab bench. Applications of this work include urban search and rescue as well as reconnaissance operations in which robust yet simple-to-implement autonomy allows a robot access to difficult environments with little burden to a human operator. Abstract -This paper documents near-autonomous negotiation of synthetic and natural climbing terrain by a rugged legged robot, achieved through sequential composition of appropriate perceptually triggered locomotion primitives. The first, simple composition achieves autonomous uphill climbs in unstructured outdoor terrain while avoiding surrounding obstacles such as trees and bushes. The second, slightly more complex composition achieves autonomous stairwell climbing in a variety of different buildings. In both cases, the intrinsic motor competence of the legged platform requires only small amounts of sensory information to yield near-complete autonomy. Both of these behaviors were developed using X-RHex, a new revision of RHex that is a laboratory on legs, allowing a style of rapid development of sensorimotor tasks with a convenience near to that of conducting experiments on a lab bench. Applications of this work include urban search and rescue as well as reconnaissance operations in which robust yet simple-to-implement autonomy allows a robot access to difficult environments with little burden to a human operator.

We examine consequences of the non-Boltzmann nature of probability distributions for one-particle kinetic energy, momentum, and velocity for finite systems of classical hard spheres with constant total energy and nonidentical masses. By comparing two cases, reflecting walls (NVE or microcanonical ensemble) and periodic boundaries (NVEPG or molecular dynamics ensemble), we describe three consequences of the center-of-mass constraint in periodic boundary conditions: the equipartition theorem no longer holds for unequal masses, the ratio of the average relative velocity to the average velocity is increased by a factor of [N/(N-1)]1/2, and the ratio of average collision energy to average kinetic energy is increased by a factor of N/(N-1). Simulations in one, two, and three dimensions confirm the analytic results for arbitrary dimension.

In this paper we explore the design space of tails intended for self-righting a robot's body during free fall. Conservation of total angular momentum imposes a dimensionless index of rotational efficacy upon the robot's kinematic and dynamical parameters whose selection insures that for a given tail rotation, the body rotation will be identical at any size scale. In contrast, the duration of such a body reorientation depends upon the acceleration of the tail relative to the body, and power density of the tail's actuator must increase with size in order to achieve the same maneuver in the same relative time. Assuming a simple controller and powerlimited actuator, we consider maneuverability constraints upon two different types of parametersmorphological and energetic-that can be used for design. We show how these constraints inform contrasting tail design on two robots separated by a four-fold length scale, the 177g Tailbot and the 8.1kg X-RHex Lite (XRL). We compare previously published empirical self-righting behavior of the Tailbot with new, tailed XRL experiments wherein we drop it nose first from a 2.7 body length height and also deliberately run it off an elevated cliff to land safely on its springy legs in both cases. This was supported primarily by the ARL/GDRS RCTA and the NSF CiBER-IGERT under Award DGE-0903711.