Event--Selected Vector Field Discontinuities Yield Piecewise--Differentiable Flows

Burden, Samuel A.; Sastry, S. Shankar; Koditschek, Daniel E.; Revzen, Shai

doi:10.1137/15m1016588

Cited by 27 publications

(42 citation statements)

References 43 publications

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“…5 In this context, "neutral" means 9 x`" 9 x, where 9 x`refers to the fore-aft speed at the subsequent touchdown event. 6 We suspect that the less restrictive Geyer approximation [27] is sufficient, but leave this generalization to future work. x`´9 x produced by the MBHop model for a range of fore-aft speed 9…”

Section: B Controlled Fore-aft Speed (1dof)mentioning

confidence: 99%

“…Observation (d) implies that higher authority sensorimotor control activity ought to target continuous phases of the locomotion cycle, leaving the transition event interventions to more passive and mechanical sources of regulation [6]. In sum, these observations motivate the search for modular, reduced order representations of locomotion task constituents that are specialized to couple selected actuation affordances to particular DOFs at particular phases of the locomotion cycle.…”

Section: Introductionmentioning

confidence: 99%

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Parallel composition of templates for tail-energized planar hopping

Koditschek

2015

2015 IEEE International Conference on Robotics and Automation (ICRA)

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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. Abstract-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 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.

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Section: B Controlled Fore-aft Speed (1dof)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Parallel composition of templates for tail-energized planar hopping

Koditschek

2015

2015 IEEE International Conference on Robotics and Automation (ICRA)

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“…is a Lipschitz (global) homeomorphism. Moreover, if any of (1)- (3) holds, then f is a local P C 1 homeomorphism at x 0 .…”

Section: Example 24 Letmentioning

confidence: 99%

“…We point out that P C 1 or, more generally, P C r maps naturally appear in many contexts, see e.g., [16,Ch. 1], in particular in applied dynamical systems see, e.g., [3] and the references therein.…”

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confidence: 99%

Local inversion of a class of piecewise regular maps

Poggiolini¹,

Spadini²

2018

Communications on Pure &Amp; Applied Analysis

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This paper provides sufficient conditions for any map L, that is strongly piecewise linear relatively to a decomposition of R k in admissible cones, to be invertible. Namely, via a degree theory argument, we show that when there are at most four convex pieces (or three pieces with at most a non convex one), the map is invertible. Examples show that the result cannot be plainly extended to a greater number of pieces. Our result is obtained by studying the structure of strongly piecewise linear maps. We then extend the results to the P C 1 case.

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“…The literature on hybrid systems provides little treatment of oscillators with intersecting or tangent guards, except for one notable exception. 27 In recent work, 28 we have developed a framework for the analysis of a broad class of such systems -"Event Selected C r (Hybrid Systems)" or ESS hereon. A key insight regarding ESS is that flows of ESS are topologically conjugate to flow-boxes (see Theorem 8).…”

Section: Hybrid Oscillator Theorymentioning

confidence: 99%

Data driven models of legged locomotion

Revzen

Kvalheim

2015

SPIE Proceedings

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Legged locomotion is a challenging regime both for experimental analysis and for robot design. From biology, we know that legged animals can perform spectacular feats which our machines can only surpass on some specially controlled surfaces such as roads. We present a concise review of the theoretical underpinnings of Data Driven Floquet Analysis (DDFA), an approach for empirical modeling of rhythmic dynamical systems. We provide a review of recent and classical results which justify its use in the analysis of legged systems. LOCOMOTION AS AN OSCILLATORLocomotion is a process whereby the body moves through space. Up to changes of shape the body is related to its history by a continuous trajectory in the space of body frame position and orientation -the Lie group SE(3). The mathematical structure representing such a system 1 is that of a "principal fiber bundle" -a product-like construction pairing the shape-space of the body S, with SE(3). Together these are the body's "configuration space" Q. When locomoting, bodies exert forces on a medium in their environment to produce the reaction forces that move them. Mechanics dictates a relationship -the "connection" -between shape velocity and configuration velocity, i.e. between TS and TQ. This highly abstracted view is sometimes used in the undulatory locomotion literature, 1 where the arbitrariness of body frame choice makes its explicit treatment necessary. However, it remains a valid representation for virtually all self-propelling bodies. In an ideal world, we would be able to predict robust and reliable estimates of the connection for any body of interest, in any medium. Such an estimate would be a complete and self-contained representation of how that body could move through the medium.Due to the daunting complexity of body-medium interactions most work on locomotion is focused on simplified models. Low dimensional, sprung mass models have captured salient features of the dynamics of many legged locomotion systems. 2 These models can be organized using the "templates and anchors" hypotheses (TAH): 3 "Animals have many DOF, but move 'as if ' they have only a few. Animals limit pose to a behaviorally relevant family of postures". Template-and-Anchor is a relationship between models of locomotion -one being more elaborate, the other more parsimonious. Both are assumed to be good long-term predictors of the motion, making the template a dimensionally reduced model of the anchor.At this point most treatments of locomotion in biomechanics and robotics would proceed to discuss the structure of various templates, and the predictions obtained from simulation of high-dimensional anchored models. Such approaches are model-driven -the discussion is focused primarily on models, their justification from assumptions about the physics and biology, and the derivaiton of governing parameters for the models. In our data-driven approach the focus is on large information-rich datasets. We used first principles to define a broad class of models, sidestepping the issue of assumptions ...

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Event--Selected Vector Field Discontinuities Yield Piecewise--Differentiable Flows

Cited by 27 publications

References 43 publications

Parallel composition of templates for tail-energized planar hopping

Parallel composition of templates for tail-energized planar hopping

Local inversion of a class of piecewise regular maps

Data driven models of legged locomotion

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