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Toward a vocabulary of legged leaping
 in IEEE Intl. Conference on Robotics and Automation, 2013, to Appear. Online: http://kodlab.seas.upenn.edu/Aaron/ICRA2013
"... 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 fro ..."
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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. I.
Automatic identification of dynamic piecewise affine models for a running robot
 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems
, 2013
"... Abstract — This paper presents a simple, datadriven technique for identifying models for the dynamics of legged robots. Piecewise Affine (PWA) models are used to approximate the observed nonlinear system dynamics of a hexapedal millirobot. The high dimension of the state space (16) and very large ..."
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Abstract — This paper presents a simple, datadriven technique for identifying models for the dynamics of legged robots. Piecewise Affine (PWA) models are used to approximate the observed nonlinear system dynamics of a hexapedal millirobot. The high dimension of the state space (16) and very large number of state observations (∼100,000) motivated the use of statistical clustering methods to automatically choose the submodel regions. Comparisons of models with 1 to 50 PWA regions are analyzed with respect to state derivative prediction and forward simulation accuracy. Derivative prediction accuracy was shown to reduce average inaxis absolute error by up to 52 % compared to a null estimator. Simulation results show tracking of state trajectories over one stride length, and the degradation of simulation prediction is analyzed across model complexity and time horizon. We describe metrics for comparing the performance of different model complexities across onestep and simulation predictions. I.
Model Reduction Near Periodic Orbits of Hybrid Dynamical Systems
, 2014
"... We show that, near periodic orbits, a class of hybrid models can be reduced to or approximated by smooth continuous–time dynamical systems. Specifically, near an exponentially stable periodic orbit undergoing isolated transitions in a hybrid dynamical system, nearby executions generically contract s ..."
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We show that, near periodic orbits, a class of hybrid models can be reduced to or approximated by smooth continuous–time dynamical systems. Specifically, near an exponentially stable periodic orbit undergoing isolated transitions in a hybrid dynamical system, nearby executions generically contract superexponentially to a constant– dimensional subsystem. Under a non–degeneracy condition on the rank deficiency of the associated Poincare ́ map, the contraction occurs in finite time regardless of the stability properties of the orbit. Hybrid transitions may be removed from the resulting subsystem via a topological quotient that admits a smooth structure to yield an equivalent smooth dynamical system. We demonstrate reduction of a high–dimensional underactuated mechanical model for terrestrial locomotion, assess structural stability of deadbeat controllers for rhythmic locomotion and manipulation, and derive a normal form for the stability basin of a hybrid oscillator. These applications illustrate the utility of our theoretical results for synthesis and analysis of feedback control laws for rhythmic hybrid behavior.
System Identification of Rhythmic Hybrid Dynamical Systems via Discrete Time Harmonic Transfer Functions
"... Abstract — Few tools exist for identifying the dynamics of rhythmic systems from input–output data. This paper investigates the system identification of stable, rhythmic hybrid dynamical systems, i.e. systems possessing a stable limit cycle but that can be perturbed away from the limit cycle by a s ..."
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Abstract — Few tools exist for identifying the dynamics of rhythmic systems from input–output data. This paper investigates the system identification of stable, rhythmic hybrid dynamical systems, i.e. systems possessing a stable limit cycle but that can be perturbed away from the limit cycle by a set of external inputs, and measured at a set of system outputs. By choosing a set of Poincare ́ sections, we show that such a system can be (locally) approximated as a linear discretetime periodic system. To perform input–output system identification, we transform the system into the frequency domain using discretetime harmonic transfer functions. Using this formulation, we present a set of stimuli and analysis techniques to recover the components of the HTFs nonparametrically. We demonstrate the framework using a hybrid springmass hopper. Finally, we fit a parametric approximation to the fundamental harmonic transfer function and show that the poles coincide with the eigenvalues of the Poincare ́ return map. I.
The Penn Jerboa: A Platform for Exploring Parallel Composition of Templates
"... We have built a 12DOF, passivecompliant 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 ..."
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We have built a 12DOF, passivecompliant 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 underactuated 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
THE ROLE OF SYMMETRY AND DISSIPATION IN BIOLOCOMOTION
"... Abstract. In this paper we illustrate the potential role which relative limit cycles may play in biolocomotion. We do this by describing, in great detail, an elementary example of reduction of a lightly dissipative system modelling crawlingtype locomotion. The symmetry group (R) is the set of trans ..."
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Abstract. In this paper we illustrate the potential role which relative limit cycles may play in biolocomotion. We do this by describing, in great detail, an elementary example of reduction of a lightly dissipative system modelling crawlingtype locomotion. The symmetry group (R) is the set of translations along a onedimensional ground. Given a timeperiodic perturbation, the system will admit a relative limit cycle whereupon each period is related to the previous by a shift along the ground. Generalization to a twodimensional ground is described later in the paper with respect to the symmetry group SE(2). In this case the resulting limit cycles allow the body to turn and translate by a fixed angle with each period of the perturbation. These toy models identify how symmetry reduction and dissipation can conspire to create robust behavior in crawling, and possibly walking, locomotion. 1.
Parameter Identification Near Periodic Orbits of Hybrid Dynamical Systems?
"... Abstract: We present a novel identification framework that enables the use of firstorder methods when estimating model parameters near a periodic orbit of a hybrid dynamical system. The proposed method reduces the space of initial conditions to a smooth manifold that contains the hybrid dynamics ne ..."
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Abstract: We present a novel identification framework that enables the use of firstorder methods when estimating model parameters near a periodic orbit of a hybrid dynamical system. The proposed method reduces the space of initial conditions to a smooth manifold that contains the hybrid dynamics near the periodic orbit while maintaining the parametric dependence of the original hybrid model. Firstorder methods apply on this subsystem to minimize average prediction error, thus identifying parameters for the original hybrid system. We implement the technique and provide simulation results for a hybrid model relevant to terrestrial locomotion.