A hybrid systems model for simple manipulation and self-manipulation systems

Abstract

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...

Tail-assisted rigid and compliant legged leaping

“…However, there is biological precedent for dynamic tails acting through ground interaction [17], and we believe this paper presents the first robotic example of a tail used solely in this way. The hybrid self-manipulation model [5] of the tailed Jerboa is complicated enough that a formal inquiry (such as presented in Section II for the three-contact GRC) far exceeds our present scope. We are, however, able to check numerically whether these physically simplified, mathematically tractable models are sufficiently accurate to lend believable design insight.…”

“…Namely, the GRC is an abstract simplicial complex (see e.g.. [20, Sec. III.1]) whose cells represent contact modes [5], i.e., collections of active contact constraints. For the purposes of this paper, the vertices correspond to modes that are characterized by all but one active contact constraint, the edges are characterized by all but two constraints, and so on.…”

“…Specifically, our main contributions toward the goal of understanding and achieving repeatable non-steady leaping across diverse legged platforms include: (a) drawing from a new hybrid systems self-manipulation model [5] empiricallyvalidated analytical insights concerning the interaction of mechanical design with behavioral dexterity (Section II); (b) marshaling empirical evidence to suggest the efficacy Fig. 1.…”

“…The first use of its simple 3-contact GRC, [1], Section II-A, is to achieve result (a), summarized above. Namely, the GRC organizes our analytical investigation of the "tailless" machine's (relatively) simple associated self-manipulation hybrid system [5,6] dynamics, Section II-B, leading up to the key formal conclusion of the paper (Prop. 1): a centered hip, rigid leg version of the mechanism cannot achieve a level leap with any control policy (Section II-C).…”

“…1): a centered hip, rigid leg version of the mechanism cannot achieve a level leap with any control policy (Section II-C). To avoid this problem, Sections II-D introduces a novel way to use compliance in the legs [7], and hybrid self-manipulation [5] simulations of a compliant tailless Jerboa in Section II-E indicate this more complicated machine can achieve level leaps that outperform a rigid legand further outperforms even a dedicated (prismatic shankenergized) vertical hopper (Fig. 3).…”