A Hybrid Systems Model for Simple Manipulation and Self-Manipulation Systems

02/05/2015 ∙ by Aaron M. Johnson, et al. ∙ Carnegie Mellon University 0

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.

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1 Introduction

Simple models of complex robot–world interactions are key to understanding, implementing and generalizing behaviors as well as identifying and composing their reusable constituents to generate new behaviors Full and Koditschek (1999). There is strong appeal to using familiar physical simplifications such as rigid bodies, plastic impacts, persistent contact, Coulomb friction, and massless limbs in building up simple robotics models. Their coarse approximation to the underlying physical processes of interest are widely understood to offer the right combination of analytical tractability and physical realism in isolation. However, it is also widely understood that such individually useful simplifications can introduce catastrophic side-effects when combined (e.g. in Painlevé (1895), Keller (1986), Mason and Wang (1988), Dupont and Yamajako (1994), Trinkle et al. (1997), Chatterjee (1999) and others, as discussed in Section 1.2).

In this paper we assemble a framework of reasonable physical assumptions and accompanying mechanics to develop a formalism for combining them at will in the construction of a simple hybrid system model for contact robotics that yields a provably consistent111 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. and empirically useful approximation to many behavioral settings of interest. As an example of the value of such mathematical models, new work Brill et al. (2015) uses the formal properties of our self-manipulation model to develop rigorous correctness (or, non-existence) proofs for desirable robot behaviors – in that case, gap crossing and ledge mounting. However, while the primary goal of this paper is not numerical analysis, simulation does provide a useful way to visualize key features of the model and the utility of some of the simplifying assumptions. Numerical results obtained through a custom Mathematica222Wolfram Mathematica 9, Numerical integration uses the NDSolve command, event detection uses the WhenEvent command. http://www.wolfram.com/mathematica/ simulation are used throughout the paper to illustrate key concepts, and to suggest the fidelity to physical settings of interest.

For example, our model generates simulations333For this simulation the middle and rear legs are used with a maximum current limit of A, a pseudo-impulse (defined in Section 2.7) magnitude of (hand selected to give the qualitatively best overall results), relative leg timing of (i.e., the middle legs are started s before the rear legs), and once a leg has lifted off the ground it is slowly rotated upwards out of the way. Remaining model parameters are as listed in (Johnson and Koditschek 2013a, Sec. III, Appendix G). of the leaping behavior depicted in Figure 1 that recreate the empirical results of Johnson and Koditschek (2013b) qualitatively (i.e., predicts the same salient features though not necessarily the same metric results), yet enjoys a combination of mathematical properties that we believe will provide a foundation for reasoning about and thereby generalizing the platform design and control strategies that gave rise to such behaviors. Of course, physical fidelity is not mathematically demonstrable and the relevance of the modeling choices we propose (i.e., the empirical sway of this formally self-consistent model) can only be established over the long run in practice by the breadth of physical phenomena they usefully approximate, regardless of the simplification and ease of analysis they afford.

t=0.20t=0.14t=0.18t=0.16t=0.12t=0.10t=0.08t=0.06t=0.04t=0.02t=0.0t=-0.02Scale:
Figure 1: Keyframes from RHex simulation leaping onto a 20cm ledge. Blue arrows show contact forces while the red arrow shows body velocity.

The paper is structured as follows. This section finishes with a summary of contributions, followed by a discussion of their relation to prior work. Section 2 introduces the various simplifying physical modeling assumptions and draws out some of the mathematical consequences bearing on their relationships to alternative formulations and to each other. Section 3 assembles from these pieces a formal hybrid dynamical system model and proves its consistency. Section 4 reviews the scope of physical settings admitted by our assumptions and discusses the most delicate aspects of their interplay with our formal results, providing additional examples that help give a broader context for the applicability of the theory. Section 5 concludes with some final thoughts on the implications of this work and future directions. An extensive Appendix works through the details of selected proofs and provides additional background material.

1.1 Contributions of the Paper

This paper extends a framework for manipulation Murray et al. (1994) and self-manipulation Johnson and Koditschek (2013a) modeling into a formal hybrid dynamical systems specification whose discrete modes are indexed by the active contact constraint set in a manner guaranteed to produce a unique execution from every initial condition under mild conditions on the motor feedback control laws. The foundation on which we rest this physically simple and mathematically tractable modeling framework arises from Assumptions A1A12, introduced in Section 2 and discussed further in Section 4.1, comprising various familiar phenomenological representations and physically natural hypotheses, including: rigid bodies (A1), massless limbs (A4), plastic impact (A8), and static friction (A12). It is known that in general these properties are not mutually consistent, however we formally demonstrate that the particular set of assumptions included here provides a well defined, deterministic, and computationally well-behaved model. The physical fidelity may, in some important applications contexts that we point out, necessarily remain something of a leap of faith relative to the still incomplete state of the theory of rigid body mechanics. To the best of our knowledge this is the first time any succinctly stated list of physical assumptions about rigid body mechanics has been shown to yield a consistent hybrid dynamical system with unique and globally defined executions.

Our central technical contribution is the derivation of a consistent extension of Lagrangian dynamics, Newtonian impact laws, and complementarity contact conditions to systems that have certain rank deficiencies in their inertia tensor that agrees with (i.e., when rank is restored, maintains equivalence to) the nonsingular case (Lemmas 

4, 5, & 8 and Theorems 12). The possibly massless dynamics motivate a reformulation of complementarity as a logical equivalence (Lemma 9) so that its unique solvability (for both force–acceleration and impulse–velocity complementarity problems, Assumptions A9 & A10, respectively) can be shown to imply a unique partition of the guard set (i.e., those states which are to undergo a mode transition) into disjoint components labeled deterministically by the destination mode of the transition (Theorem 5). These conditions are expressed in terms of a higher order scalar relation (, Definition 1), and we exhibit certain properties of this relation that clarify its role in determining the guard set (Lemmas 13).

Even without the introduction of massless limbs there exist many opportunities for repeated (and even Zeno444An 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).) discrete transitions that seem unlikely to add much physical insight (and, speaking practically, generally degrade the numerical performance of simulations based upon this model). Hence, to resolve the qualitative problem of spurious transitions at arbitrarily low velocities (Lemma 10), we introduce a new pseudo-impulse, which acts on the discrete transitional logic (rather than the continuous dynamics), imposing an implicit bound on contact velocity below which such contacts persist (Theorem 3), precluding certain Zeno phenomena (Theorem 11).

As a structure to combine these physical models and assumptions, this work presents the formal definition of the self-manipulation hybrid system in Definition 5 (along with Definitions 24), and the formal demonstration of its consistency (including that it is deterministic and non-blocking, Theorems 49 and Lemmas 67), incorporating a well-behaved notion of completion in case of a Zeno execution (Definition 6, Theorem 10, Corollary 1) by adapting to this more elementary setting the measure theoretic arguments of Ballard (2000).

1.2 Relation to Prior Literature

This paper aims to promote simplified physics based models of robotic systems for purposes of analysis. Doing so entails integrating results and ideas that have developed somewhat independently across several different longstanding technical fields. For surveys of some of these ideas (with a focus on numerical considerations), see e.g. Brogliato et al. (2002), Gilardi and Sharf (2002).

1.2.1 Numerical Simulation Methods

While this paper is focused on a model for analysis and not simulation, it is informative to consider how other simulation strategies compare. The model developed here generates trajectories from the flow of hybrid dynamical systems defined by differential–algebraic equations (DAEs) between discrete transitions and so, in the language of (Brogliato et al. 2002, Sec. 6.3), simulations of these trajectories could be obtained555E.g. using the algorithm proposed in Burden, Gonzalez, Vasudevan, Bajcsy and Sastry (2015) for hybrid control systems. via an event-driven scheme, as opposed to a penalized-constraint/continuous-contact scheme, or a time-stepping scheme.

Event-driven schemes have a long history, e.g. Wehage and Haug (1982), Pfeiffer and Glocker (1996), (Brogliato et al. 2002, Sec. 6.7), and include the hybrid dynamical systems formulations outlined in the next section. Typically they entail alternating between integration of smooth dynamics involving (usually) finite forces from contacts and the discontinuous handling of constraint addition or deletion (the “events”). Here, we extend these methods and codify the event-driven scheme in terms of a formal hybrid dynamical system. In contrast, some event-driven schemes formulate the contact dynamics as always consisting of impulses, e.g. Mirtich and Canny (1995). These impulse-based simulations combine both smooth and discontinuous contact interactions into impulses, with a continuous-time ballistic trajectory in between events.

Time-stepping schemes, which also account for contact interactions only using impulses by integrating applied forces over small time steps, are numerically efficient especially for systems with large numbers of constraints, see e.g. Stewart and Trinkle (1996) Anitescu and Potra (1997), or (Brogliato et al. 2002, Sec. 7). These models can be relaxed (by allowing contact forces and impulses to arise even before contact occurs) to enable efficient numerical simulation and motion synthesis Drumwright and Shell (2011), Todorov et al. (2012). These methods allow contact constraints to be added or removed at any time step, but only once per time step. Furthermore, no distinction between continuous contact forces and discontinuous impulses is made. In this way these methods relax the requirements of the Principle of Constraints, i.e. that, “Constraints shall be maintained by forces, so long as this is possible; otherwise, and only otherwise, by impulses” (Kilmister and Reeve 1966, p. 79) (as noted e.g. in (Stewart and Trinkle 1996, Section 1)). Their advantage in avoiding many of the well explored physical paradoxes of rigid body mechanics (including Zeno 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.

1.2.2 Hybrid Dynamical Systems

This 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 from the more general class of hybrid automata considered in (Lygeros et al. 2003, Def. II.1) to facilitate connections with the broader hybrid systems literature. Our self-manipulation system is closely related to the -dimensional hybrid system of (Simic et al. 2005, Def. 2.1), the simple hybrid system of (Or and Ames 2011, Def. 1), and hybrid dynamical system of (Burden, Revzen and Sastry 2015, Def. 1) as we require: (i) multiple disjoint domains of varying dimension, disallowed by Simic et al. (2005), Or and Ames (2011); (ii) guards with arbitrary codimension, disallowed by Burden, Revzen and Sastry (2015); and we desire (iii) more analytical and geometric structure than is provided by the general framework in Lygeros et al. (2003), specifically domains that are differentiable manifolds and guards that are sub-analytic. Note that (i) is precluded in Simic et al. (2005), Or and Ames (2011) only for notational expediency since any multitude of domains may be embedded as disjoint submanifolds of a high-dimensional Euclidean space. The condition (ii) is excluded by Burden, Revzen and Sastry (2015) since it is generally incompatible with the results contained therein.

One property of hybrid systems that is crucial to establish for the present setting is that the guards are disjoint, i.e. no state is a member of two distinct guards, so there is no ambiguity as to which reset map to apply. This key property yields the proof that the model is deterministic (Lygeros et al. 2003, Def. III.2). Furthermore the system is set up such that every point on the boundary of the domain where the flow points outward is a member of a guard, thus guaranteeing that the system is non-blocking (Lygeros et al. 2003, Def. III.1), i.e. the execution continues for infinite time.

The self-manipulation hybrid system developed in this paper uses the active contact constraints to define the discrete state or status (that we will call the mode), as in e.g. Hurmuzlu and Marghitu (1994), Brogliato et al. (2002). However even when starting with a simple Lagrangian hybrid system without modes for every contact condition it appears to be useful to add such states to allow executions to be completed beyond a so-called “Zeno equilibrium” Ames et al. (2006), Or and Ames (2011). Furthermore, the pseudo-impulse we introduce avoids certain Zeno executions by allowing the system to remain in a constrained mode after finitely many transitions, in a manner analogous to but formally distinct from the truncation proposed in Ames et al. (2006), Or and Ames (2011).

1.2.3 Consistent Complementarity

Any formulation that allows for persistent contact through impact must determine which contacts to make active and which to remove666The removal ends up being the harder question, as “there is no problem in deciding when and which constraint to add to the active set since there is a constraint function to base the decision on. The problem of dropping constraints is more delicate…” (Lötstedt 1982, p. 283).. When there is no impulse (i.e., no constraint to add, but one or more constraints have violated the unilateral constraint cone), the removal process is called force–acceleration complementarity, as it is commonly modeled by a complementarity problem involving contact force and separating acceleration, e.g. (Trinkle et al. 1997, Eqn. 12), (Brogliato et al. 2002, Eqn. 10), where in the simplest case of a single contact point with zero or negative contact force it is simply removed. This complementarity problem framework can introduce paradoxical consequences in certain physical problem settings, for example in taking the rigid limit of a deformable body Chatterjee (1999). It can also be computationally efficient to relax the hard constraints of the complementarity conditions, resulting in a convex optimization problem Todorov (2011), Drumwright and Shell (2011).

An impulse induced from one or more contact constraints becoming active will generally necessitate the removal of other constraints, specifically, those that would require a negative impulse to remain. When invoked as a modeling principle, this impulse–velocity complementarity precludes a simultaneous impulse and separation velocity at a particular contact, e.g. (Lötstedt 1982, Eqn. 2.10b), (Brogliato et al. 2002, Eqn. 9). Imposing this modeling discipline affords the well established benefit of yielding a unique post-collision state for collisions modelled as plastic frictionless impacts Ingleton (1966), Cottle (1968), van der Schaft and Schumacher (1998), Heemels et al. (2000). Unfortunately further generalizations can lead to inconsistencies and ambiguities Chatterjee (1999), Hurmuzlu and Marghitu (1994), Ivanov (1995), Seghete and Murphey (2010). The existence and uniqueness of a solution must therefore be separately established in each physical circumstance that includes friction – or merely be assumed.

Massless legs introduce new problems into the complementarity problem. The massless leg condition in general, as introduced in (Johnson and Koditschek 2013a, Assumption C.6) and also used in countless prior works, e.g. Blickhan (1989), Kajita et al. (1992), Holmes et al. (2006), allows for the neglect of certain states deemed inconsequential to the dynamics of interest when unconstrained (of course, the appropriateness of this neglect is task dependent rather than in any way intrinsic to the underlying physics, c.f. (Johnson and Koditschek 2013a, Sec. IV.C.5) or Balasubramanian et al. (2008)). Indeed a massless leg that is not touching the ground is unconstrained and its position can be taken as arbitrary (or regarded as evolving according to dynamics sufficiently decoupled as to be considered independent), as used in the behavior analysis in (Johnson and Koditschek 2013a, Sec. IV.C.3). However the complementarity condition as used in e.g. (Lötstedt 1982, Eqn. 2.10b), (Brogliato et al. 2002, Eqn. 9), and listed in (51)–(52) is ill-posed in the absence of mass since there is no well-defined separation velocity or acceleration, nor anything precluding all massless contact points from always separating (at least for the dynamic model of interest here, as opposed to a quasistatic model, Trinkle and Zeng (1995)). Instead here we reformulate the complementarity condition as (46)–(47) to not depend on the separation velocity.

1.2.4 Impact Mechanics

The usual Newtonian impact law (as in e.g. (Chatterjee and Ruina 1998, Eqn. 3), (Featherstone 2008, Eqn. 11.65) and many others) can be thought of as a mass-orthogonal projection onto the constraint manifold as used in e.g. augmented Lagrangian techniques (Bayo and Ledesma 1996, Eqn. 25). More generally, Moreau (1985) showed that impact problems can be modeled using measure differential inclusions. The algebraic plastic impact law involves inversion of the inertia tensor, which precludes the possibility of massless limbs and necessitates the reformulation given in this paper. Even if there are no truly massless links, a nearly massless body segment yields a poorly-conditioned inertia tensor (Johnson 2014, Sec. 5.1.1), leading to similar formulations as the one presented here in (Westervelt et al. 2003, Eqn. 9) or, for continuous time dynamics, (Holmes et al. 2006, Sec. 4.3) (Featherstone 2008, Eqn. 3.17).

In this paper we restrict our attention to systems modeled as exhibiting only perfectly plastic impact (perfectly inelastic impact). In the elastic impact case, it is necessary to consider the relative stiffness of contact points; depending on the restitution law invoked, multiple outcomes are consistent with the constitutive assumptions Hurmuzlu and Marghitu (1994), Chatterjee and Ruina (1998). Though it is possible to bypass this technical obstacle by introducing an additional constitutive hypothesis, e.g. (Ballard 2000, H3 in Sec. 3.3), it remains to be validated (either theoretically or experimentally) that such assumptions accurately represent the physical system’s behavior. Plastic impact avoids these inconsistencies, but more importantly we claim plastic impact provides a more useful model of the robotic systems of interest. Elastic impact is clearly needed in some robotics applications such as juggling Buehler et al. (1994), Schaal and Atkeson (1993), tapping Huang and Mason (1998) or ping-pong Andersson (1989), but plastic impact, where there is no restitution and therefore no separation velocity after impact, is a more desirable model for most forms of locomotion (when it is important to keep feet on the ground) Westervelt et al. (2003), Chatterjee et al. (2002) and manipulation (when it is important to keep fingers on the object) Chatterjee et al. (2002), Wang and Mason (1987).

The new pseudo-impulse presented here, in addition to the Zeno results mentioned above, eliminates other evidently unwanted transitions by allowing the continuous-time forces to play a role in the impact process which is primarily “logical” (as opposed to energetic). This role may be best summarized by comparison to the most common alternatives. For example, instead of introducing a variable coefficient of restitution Quinn (2005) (which our plastic impacts of interest already eliminate), the pseudo-impulse is not applied to the continuous (energetic) system directly but instead used to regularize the complementarity driven hybrid switching logic. Or as a second point of comparison, rather than introducing a fixed dead zone in impact energy Pagilla and Yu (2001) or velocity (Brogliato et al. 2002, Sec. 6.4), the magnitude of the effect on our model’s hybrid logic is not fixed but rather scales with the continuous time forces. An effect similar to this pseudo-impulse condition is also introduced by time-stepping simulations Stewart and Trinkle (1996), Anitescu and Potra (1997), which, true to their name, always consider forces over small but finite time-steps. Under such schemes the magnitude of this effect is not a fixed, independent, user-imposed parameter since it must remain proportional to the duration of each time-step. Our preference for the independent, fixed choice reflects both mathematical convenience (the clearly defined hybrid dynamical system with its formal properties) as well as our taste in preferring to work with robotics models targeted for specific physical environments and settings.

1.2.5 The Effect of Friction Models

While this paper focuses on the impact problem, which friction greatly complicates Keller (1986), Wang and Mason (1987), McGeer and Palmer (1989), Wang and Kumar (1994), Trinkle et al. (1997), even simulating continuous-time dynamics of rigid bodies with friction can be difficult (formally -hard Baraff (1991)) due to the possibility of “jamming” events Mason and Wang (1988), Dupont and Yamajako (1994), first attributed to Painlevé (1895). In this paper, following the model from Johnson and Koditschek (2013a), strong assumptions about frictional contact avoid these issues and enable integration of the dynamics as a differential–algebraic equation (DAE). As noted above, an alternative method to numerically solving these problems is the time-stepping approaches pursued in, e.g., Stewart and Trinkle (1996), Anitescu and Potra (1997), which resolve these issues by allowing for impulses at any time step. To resolve these issues in more general extensions of the system presented here (in particular those that are not well modeled by the frictional assumption, A12), the hybrid dynamical system could similarly be extended by allowing impulses at times without collisions, with such jamming events considered with additional guards and reset maps. We refer the interested reader to “Is Painlevé a real obstacle?” (Brogliato et al. 2002, Sec. 8.1) for further discussion of these issues.

Base constraint function (2.1)
Velocity constraint function (2.1)
Force constraint function (8)
differentiable function (2.1)
Solution to the predicate (32)
Coriolis forces (12)
Force–acceleration predicate (38)
Contact constraints (2.1)
Set of active contact constraints (2.1)
Complementarity scope (33)
Identity matrix, of dimension (2.1)
Impulse–velocity predicate (50)
Set of all contact constraints (2.1)
Inertia tensor (2.3)
Constrained inverse inertia tensor (8)
Potential forces (e.g. gravity) (12)
New touchdown predicate (22)
Impulse in state space (2.5)
Impulses in constraint space (25), (56)
Pseudo-impulse predicate (58)
Continuous state (2.1)
Continuous state and velocity (2.1)
Touchdown predicate (21)
Unilateral constraint cone (2.1)
Corresponding normal constraint (2)
Small time duration of impact (56)
Instantaneous change in velocity (2.5)
Lagrange multipliers (13)
Constrained contact inertia tensor (8)
External forces and torques (12)
  Trending negative/positive (Def. 1)
Table 1: Key symbols used throughout this paper, in addition to (Johnson and Koditschek 2013a, Table 1), with section or equation number of introduction marked. See also Table 2 for symbols introduced in Section 3.

2 Modeling Assumptions

The continuous Lagrangian dynamics of self-manipulation is specified in Johnson and Koditschek (2013a) using the notation and terminology of Murray et al. (1994) and summarized in Section 2.1. We continue to work within that framework here and briefly list the subtle differences between these two classes of systems in Section 2.2. However the impulsive dynamics (instantaneous changes in velocity when a new contact is added) were not specified in either, and so we will introduce a plastic impact model in Section 2.5 and explore the induced complementarity conditions in Section 2.6. In addition, will make explicit how the massless leg (Section 2.3) and frictional assumptions (Section 2.8) made in Johnson and Koditschek (2013a) affect both the continuous time (Section 2.4) and impulsive dynamics, leading to a new formulation for the dynamics that is equivalent to the usual formulation when there are no massless links. Finally, Section 2.7 introduces a new pseudo-impulse that eliminates certain Zeno executions and related chattering behavior.

2.1 Setup and Notation

The notation used in this paper is chosen to be consistent with (Johnson and Koditschek 2013a, Table I) (and agreeing where possible with Murray et al. (1994)) or is defined as it is used and summarized in Table 1. The base component of the state is denoted, , while the full state is, , and this state completely describes the motion of interest, as,

Assumption A1 (Rigid Bodies).

The robot is made up of a finite number of rigid bodies whose configuration lies in a connected complete Riemannian manifold .

Since the configuration spaces of many extant robots are not linear (due e.g. to rotary joints, rigid body rotations, or constrained mechanisms), it is most natural to invoke the general framework of differentiable manifolds to model the state space. For concreteness we will often consider the case where consists of joint angles and the special Euclidean group of dimension , but our formal results will be stated for an arbitrary connected complete Riemannian manifold . We recognize that this generality necessitates mathematical formalisms and notation that are not uniformly adopted in the robotics community (exceptions such as Murray et al. (1994) notwithstanding); we aim whenever possible to translate unfamiliar objects into standard terminology and provide a terse overview of the background material needed to parse the more general case in Appendix C.

We are concerned with sets of contact constraints (e.g., ) that we shall call modes or contact modes hereinafter, subsets of indices whose particular elements (e.g., ) index the contact constraints that prevail at some instant (Johnson and Koditschek 2013a, Sec. II.C) (Murray et al. 1994, Sec. 5.2.1). In addition to contact with the robot’s environment, contact constraints may include cases of self-contact as well as joint limits. The universe of all possible constraint indices from which these subsets are taken will be denoted , partitioned by those that are in the normal (non-penetrating) direction and those that are in tangential (non-sliding) direction. Similarly, for any set of constraints specified by mode , define the subsets and , where clearly and .

Contact constraints in the normal direction777Note that normal direction constraints for non-adhesive contact will be unilateral, although within a contact mode they can be considered bilateral until the constraint force is violated (e.g. (Lötstedt 1982, Sec. 4))., , specify a holonomic constraint of the form for (and whose corresponding velocity constraint is equivalent to the Jacobian , (Johnson and Koditschek 2013a, Eqn. 11)), while those in the tangential direction, , specify a nonholonomic constraint of the form where again . For a given contact mode , the space of constrained positions is a manifold of dimension (i.e., the number of constraints in ).

In the interest of notational clarity, we will generally express functional dependence on contact modes via subscript, e.g., , and when it is clear from context, we will further suppress the subscript, e.g. . For example, and used extensively throughout this paper, fixing an ordering on we can obtain the velocity constraints active in mode , , as a selection of rows from the set of all velocity constraints , i.e.,888However, note that most functions of the mode are not a simple projections, and so e.g. , defined in (8), , but rather is as defined in (8), i.e. constructed with the corresponding .

(1)

where

is the Boolean projection matrix formed by the rows of canonical unit vectors associated with the elements in the index set

. Similarly for a single constraint , .

We make the following assumption on the combined maps,

Assumption A2 (Simple Constraints).

All constraints are independent, that is for all contact modes , the maps and are constant rank.

We refer the reader to Appendix C for the definition of rank of a map; in coordinates, this condition states that the gradient vectors of each coordinate of the respective maps are linearly independent at every point. If this condition failed to hold, the configuration space could possess singularities that could preclude existence and/or uniqueness of trajectories for the mechanical system. Note that this precludes the possibility of redundant constraints, though there are methods of resolving such redundancies, e.g. in Greenfield et al. (2005). In particular, this requirement will be met if and are constant rank999This stronger assumption would not be true if there were two parallel constraints that, due to geometry, could not simultaneously be active, in which case the original requirement must be checked for all ..

We note that there is an assignment,

(2)

of contacts to normal contacts such that maps tangential contacts to the corresponding normal contact (where is the appropriate identity matrix). Note that for each and , is orthogonal to .

It is well established that the motion of mutually constrained rigid bodies can be effectively modeled using polynomial maps Wampler and Sommese (2013), hence imposing contact constraints arising from their interaction with the piecewise polynomial representations of the environment (commonly adopted by the sensory community Lalonde et al. (2007)) leads to,

Assumption A3 (Analytic Constraints).

All constraints are analytic functions, that is for all contact modes , the maps and are .

Given an analytic vector field subject to an analytic constraint, as shown in Lemma 3 it is possible to determine whether the constraint remains active over a nonzero time horizon by evaluating Lie derivatives at a single instant in time. If either the vector field or constraint were merely smooth, the differential equation determined by the vector field would, in general, need to be solved over a nonzero time horizon to determine whether the constraint remained active.

Assumption A4 (Persistent Contact).

Contact with the world occurs through a finite number of active constraints indexed by that apply continuous time forces. Furthermore, contact persists until the next event (e.g. touchdown or liftoff).

This assumption is related to the Principle of Constraints, as discussed in Section 1.2.1. Its adoption partitions trajectories so that at all times between instantaneous touchdown or liftoff events there persists a well-defined set of active constraints (enabling the systematic a priori enumeration and analysis of these constraint sets and their sequences, e.g. Johnson and Koditschek (2013b)). This contrasts with simulations generated by time-stepping algorithms, wherein contact Stewart and Trinkle (1996) or interpenetration Anitescu and Potra (1997) are resolved only at multiples of the timestep, and no distinction between forces and impulse are made (indeed this relaxation is what enables the efficient and consistent simulation in such formulations).

The impact problem can be summarized as determining which constraints to add or remove from the active set. The active set continues to constrain the system so long as the unilateral constraint cone (Johnson and Koditschek 2013a, Eqn. 7) is positive, , where is the vector of Lagrange multipliers (constraint forces) (Johnson and Koditschek 2013a, Eqn. 33). Included in is both the non-attachment condition that normal direction forces are positive as well as the friction cone that relates the magnitude of the normal and tangential components.

In the complementarity problems, the following definition simplifies statements involving higher-order derivatives of the state that seem to arise unavoidably (as stated in (van der Schaft and Schumacher 1998, Sec. 3), (Heemels et al. 2000, Sec. 1), formalizing the concepts represented in e.g. (Featherstone 2008, Fig. 11.4), (Siciliano and Khatib 2008, Sec. 27.2)),

Definition 1.

Given a smooth function defined over a smooth manifold , a point , and a smooth vector field , we say that is trending negative with respect to the vector field at , denoted , (or if the context specifies ), if,

(3)

where is the Lie derivative101010See e.g. (Lee 2012, Ch. 9), and note our convention that, . of with respect to the vector field . Similarly, we say that is trending positive at , denoted , when . We say that is identically zero at , denoted , when . Finally, we say that is trending non-negative at , denoted , when or , and that is trending non-positive at , denoted , when or .

We refer the reader to Appendix C for the definition of a vector field ; in the case where , the tangent bundle can be canonically identified with to obtain a more familiar function

that determines an ordinary differential equation

.

That is, if and only if the following vector,

(4)

is lexicographically smaller than zero (Bertsimas and Tsitsiklis 1997, Def. 3.5). As an example of when these properties are important, consider the examples in Figure 2. In each case, the initial configuration (taken as the bottom point of the circle) is , and the initial velocity is . Assume that the particle has unit mass (), and that there are no non-contact forces (, , , as defined in Section 2.4). In all cases the particle is touching the constraint () but has no impacting or separating velocity (), so there is no impulse (as defined in Section 2.5). Furthermore in c) and d) there is no impacting or separating acceleration (). However in a) and c) the constraint function is trending positive, , while in b) and d) the constraint function is trending negative .

(a)(b)(c)(d)
Figure 2: Four examples of a planar point particle () with a single constraint (), defined as (a) , (b) , (c) , and (d) . Note that if the particle velocity is directed to the right (), as illustrated, then: the constraint function is trending positive () in (a) and (c); the constraint function is trending negative () in (b) and (d).

Furthermore, we will make use of the following properties of this trending relation,

Lemma 1.

The closure of or is , while the closure of or is .

This is easy to see as for any vector field.

Lemma 2.

Given a smooth vector field, , a point in a smooth boundaryless manifold, , and a smooth positive function, , any other smooth function, , is trending negative if and only if its product with has the same property, i.e.,

(5)

See Appendix A for a proof.

Lemma 3.

Let be a function and be a vector field over a boundaryless manifold , and let denote an integral curve for through . Then is trending positive at with respect to , , if and only if there exists such that,

(6)

The requirement that the manifold be boundaryless is introduced to simplify the statement of this Lemma; the Lemma clearly applies to the interior of a manifold with corners (which is, after all, simply a manifold without boundary) (Joyce 2012, Def. 2.1).

To see that the lemma is true, note that if is an integral curve for such that is positive for sufficiently small, then since is analytic we conclude (3) is satisfied. The other direction follows easily by contradiction using the mean value Theorem. We note that this is not true if or are merely . Also note that the conditions of the lemma do not imply that for two reasons: 1) it is possible that , and 2) even for , grazing contact would handled incorrectly (consider the horizontal vector field in the plane, , and the function at the origin).

Lemma 3 implies a computationally efficient way to test these trending conditions is to simply integrate a flow until it reaches a zero crossing.

2.2 Manipulation and Self-Manipulation

This section will briefly summarize the self-manipulation formalism introduced in Johnson and Koditschek (2013a), as it relates to manipulation, e.g. as presented in Murray et al. (1994). Each defines a number of frames on the robot and its environment – the palm frame, the object frame, the contact frame, etc. In an effort to keep the problems as similar as possible, the following conventions were adopted in Johnson and Koditschek (2013a),

  • In self-manipulation, the robot is the object being manipulated and so to properly consider the forces and torques on the object the robot’s palm frame, , and the object frame, , are chosen to be coincident, (Johnson and Koditschek 2013a, Sec. II-B).

  • Thus motions that would, in a manipulation problem, move an object to the right will really move the robot to the left, and so the self-manipulation grasp map (a component of ) is a reflection of the manipulation grasp map, , (Johnson and Koditschek 2013a, Eqn. 15).

  • By convention the contact frame is defined at any point of contact with the -axis pointing into the object (away from the finger tip), (Murray et al. 1994, Sec. 5.2.1). In self-manipulation the convention of (Johnson and Koditschek 2013a, Sec. II-C) is to keep the contact frame consistent with respect to the legs, and so the -axis points away from the robot and into the ground. This results in a unilateral constraint cone, , that is negative, (Johnson and Koditschek 2013a, Eqn. 76, 78).

  • Since the palm reference frame is accelerating with respect to the world, the inertia tensor, , (Johnson and Koditschek 2013a, Eqn. 26), and by extension the Coriolis terms, , (Johnson and Koditschek 2013a, Eqn. 30), are more coupled and lack the block diagonal structure present in manipulation problems, (Murray et al. 1994, Eqn. 6.24).

It should be no surprise that the problem formulations are structurally equivalent since the underlying kinematics and dynamics are indifferent to the problem category. However owing to the notational differences summarized above, through the remainder of this paper we choose to write out the problems in terms of a self-manipulation system, with the understanding that the results contained herein apply equally well to manipulation systems once these transformations are incorporated.

2.3 Massless Considerations

To properly define the dynamics of a partially massless system, consider a parametrized family of singular semi-Riemannian metrics,

(7)

such that is the (possibly) degenerate inertia tensor for the system (Johnson and Koditschek 2013a, Eqn. 26) and may be singular, while assigns a small mass and inertia to any putatively massless links such that is full-rank for all (for our present purposes, it is sufficient to use a limiting model such as rather than some more specific physically motivated one). We invoke the general definition of Riemmanian metric here since it provides the coordinate-invariant formulation of the familiar mass or inertia matrix associated with a collection of rigid bodies, and refer the reader to Lee (1997) for a formal definition and Section 2.4 for additional details. The dynamics of the system in contact mode can be expressed (as shown below) using the inverse of the following block matrix containing , defining111111 Note that (Johnson and Koditschek 2013a, Eqn. 40) used the notation while in this paper we will use to signify the slight difference in definition used here, and to avoid confusion with the pullback of , usually noted as , but which happens to be . , , and as,

(8)
(9)

From this definition, note that the following properties hold,

(10)
(11)

To ensure that the inverse of the matrix in (9) (sometimes called the “Lagrangian matrix of coefficients”, e.g. (Papalambros and Wilde 2000, Eqn. 7.79), and sometimes used in robotics for numerical reasons, e.g. (Holmes et al. 2006, Sec. 4.3)) is well-defined, we will require some modeling assumptions on the nature of the massless appendages. Thus if the inverse exists, this -parametrized curve takes its image in (the group of invertible matrices over ) within which matrix inversion is a continuous operation, hence the limit commutes with the inverse operation, and is a well defined smooth curve defined over all .

To meet this requirement, massless appendages will be allowed here only in a limited form,

Assumption A5 (Constrained Massless Limbs).

For all limbs in contact with the world, any rank deficiencies of the inertia tensor  (Johnson and Koditschek 2013a, Eqn. 26) are “corrected” by velocity constraints  sufficient to guarantee that any remaining allowed physical movement excites some associated kinetic energy, that is, the block matrix in (9) is invertible.

If the “rank correction” condition in this assumption were violated, then it would not be possible in general to determine the system’s instantaneous acceleration solely from the internal, applied, and Coriolis forces; it could happen that either no accelerations are consistent with the net forces, or an infinite set of accelerations are. This condition admits its most physically straightforward expression via the requirement that the inertia tensor is nonsingular when written with respect to generalized or reduced coordinates, (i.e., any local chart arising from an implicit function solution to the constraint equation (Johnson and Koditschek 2013a, Eqn. 10)). However, for purposes of this paper, we find it more useful to work with the Lagrange-d’Alembert formulation of the constrained dynamics, (Johnson and Koditschek 2013a, Eqn. 33), hence, we translate that natural assumption into more formal algebraic terms governing the relationship between the lifted (velocity) constraints, (Johnson and Koditschek 2013a, Eqn. 11), and the overall inertia tensor  as follows,

Lemma 4.

The matrix , (9), is invertible if and only if the inertia tensor expressed in generalized or reduced coordinates, (Johnson and Koditschek 2013a, Eqn. 36), is invertible (Johnson and Koditschek 2013a, Sec. II.K, Assumption A.4).

as shown in Appendix B.1. See Section 4.2 for a discussion of physical scenarios that meet this requirement.

When not constrained on the ground, any such massless links or limbs must then be removed from consideration as mechanical degrees-of-freedom: since they are massless, when unconstrained, the associated joints can be considered to have arbitrary configuration. Their evolution is instead treated according to the principle,

Assumption A6 (Unconstrained Massless Limbs).

For all limbs not in contact with the world, any components of the state that do not excite some kinetic energy must be removed from the usual dynamics and instead considered to evolve in isolation according to some independent, decoupled dynamics.121212 That is, in contact mode , the configuration manifold decomposes as a product of manifolds , where corresponds to a subset of the system coordinates such that the matrix in (9) is nonsingular, and corresponds to the remaining coordinates. The dynamics for the coordinates of is given by some vector field . Here we have written the dynamics as a second order vector field so that the dynamics of the full system may be written in a notationally consistent manner. This is not required; regardless of how the dynamics are defined for these coordinates, there will be no coupling of energy with the rest of the system through the inertia tensor.

In the same vein as the remark following Assumption A5 (Constrained Massless Limbs), we observe that it is not possible to uniquely determine accelerations of unconstrained massless limbs due to corresponding degeneracy in the inertia tensor. Excluding such limb states from the coupled Lagrangian mechanics governing the remaining body and limb segments will enable us in the sequel to specify a differential-algebraic equation that admits unique solutions. As the dynamics of the excluded states do not affect those of the remaining states, for the rest of this section we will abuse notation and suppress the subscript from the state space , so that unless stated otherwise we are concerned with only the “active” component of the decomposed state space for the mode of interest. See also Section 4.4 for a discussion of Zeno (Def. 6) considerations with massless legs.

2.4 Continuous Dynamics

With this notation, the continuous-time dynamics of (Johnson and Koditschek 2013a, Eqn. 33) in contact mode can be expressed as,

(12)
(13)

where is the applied forces, is the centripetal and Coriolis forces, and is the nonlinear and gravitational forces (Johnson and Koditschek 2013a, Eqn. 30, 31).

When , (7), is invertible (including, possibly, even for ), it is easy to verify the equivalences (and dropping for now the subscripted contact mode ),

(14)
(15)
(16)

as shown in Appendix B, (114). Note that constructions such as these are commonly used in robotics when is invertible, e.g. (Khatib 1983, Eqns. 45–46) and many others (where their has the opposite sign of our and their corresponds to , although note that the definition (15) is exact and not defined as a minimal-energy pseudo-inverse).

Lemma 5.

When is invertible, the dynamics (12) and (13) are equivalent to the more common expression (as stated e.g. in the last equations of (Johnson and Koditschek 2013a, Appendix D), or (Murray et al. 1994, Eqn. 6.5, 6.6)),

(17)
(18)

The claim follows directly from substituting (14)–(16), the explicit solution to (8) when is invertible, into (12)–(13), as worked out in Appendix B.2.

Whether is invertible or not, we require,

Assumption A7 (Lagrangian Dynamics).

In each contact mode , the time evolution of the active coordinates of the system are governed by Lagrangian dynamics, and the applied forces are such that the vector field defined by (12) for coordinates in and12 for coordinates in is forward complete, i.e. the maximal integral curve through any initial condition is defined for all positive time.

Recall from the rigid body and unconstrained massless assumptions (A1A6) that the configuration space, , is a manifold without boundary. Thus the major obstacle to verifying Assumption A7 lies in preventing finite-time “escape” from the state space , e.g. because the velocity grows without bound or there are “open edges” in the configuration manifold (i.e. the manifold is not compact). If the configuration manifold were compact, then it would suffice to impose a global bound on the magnitude of the vector field in (12). If the configuration space were instead Euclidean, , then it would suffice to impose a global Lipschitz continuity condition on the vector field in (12). We note that configuration obstacles such as joint limits or self-intersections are treated as constraints in Section 3, and hence pose no obstacle to satisfying the above boundarylessness and completeness conditions on the configuration space.

However, since in examples of interest the configuration space is neither compact nor a vector space (due e.g. to rotary joints, rigid body rotations, or constrained mechanisms), we often require a more general condition. One such condition is obtained from (Ballard 2000, Thm. 10); since we rely on this sufficient condition elsewhere in the paper, we transcribe it explicitly into our notation as follows. When is a complete connected configuration manifold and is a nondegenerate inertia tensor (i.e., at every the coordinate representation of is invertible, thus here precluding the possibility of massless limbs, Assumption A5), we let denote the distance metric induced by the Riemannian metric associated with (Lee 1997, Ch. 6). For any vector we define . For any covector we define , where is the vector obtained by “raising an index” (in coordinates, ) (Lee 1997, Ch. 3).

Lemma 6.

If the ambient configuration space is a complete connected Riemannian manifold, is a nondegenerate inertia tensor, and the magnitude of grows at most linearly with velocity and distance from some (hence any) point in , i.e. if there exists , such that,

(19)

then the flow associated with the vector field (12) is forward complete, i.e. the maximal integral curve through any initial condition is defined for all positive time, and hence Assumption A7 is satisfied.

Proof.

This is simply an application of (Ballard 2000, Thm. 10) in the absence of unilateral constraints. ∎

We expect this condition to be met by any model based on a physical system, and is trivially met if there is a global bound on the magnitude of the applied, , and potential, , forces (whereas, notice, the necessarily unbounded Coriolis and centripetal forces are accounted for by the Lemma and require no further consideration).

Unfortunately this condition assumes that the inertia tensor is nondegenerate, precluding the presence of massless limbs (Assumption A5). Allowing instead for a degenerate inertia tensor but enforcing the unconstrained massless limb assumption (A6), we now describe a set of sufficient conditions that ensure Assumption A7 holds.

Lemma 7.

Suppose that in each contact mode the active constraints are either holonomic or integrable (Murray et al. 1994, Sec. 6.1.1), meaning that there exists a reduced configuration manifold (i.e., generalized coordinates) such that every point in lies in the image of an embedding that is invariant under (12) (Johnson and Koditschek 2013a, Sec. G) and restricted to which the reduced inertia tensor (Johnson and Koditschek 2013a, Eqn. 36) is nondegenerate.

If the hypotheses in Lemma 6 are satisfied for , its reduced inertia tensor, and its reduced dynamics (Johnson and Koditschek 2013a, Eqn. 34), and furthermore the vector field governing unconstrained massless limbs is forward complete and uncoupled from the massive or constrained coordinates, i.e.,

(20)

then Assumption A7 is satisfied.

Proof.

We seek to define a forward-complete flow consistent with the vector field in (12). Let denote the embedding associated with the reduced coordinates (Johnson and Koditschek 2013a, Sec. G). Apply Lemma 6 to the reduced system to obtain a forward-complete flow . Then since (Ballard 2000, Prop. 3) implies maps integral curves from the reduced state space to the original, for all and , defining yields the desired forward-complete flow on . ∎

Lemmas 67 provide sufficient conditions guaranteeing that certain systems with either full rank inertia tensors or only holonomic constraints satisfy Assumption A7 – in the most general case, however, this will remain an assumption. We speculate that it is possible to derive a condition analogous to (19) using concepts from singular Riemannian geometry Hermann (1973) that ensure the existence of a forward-complete flow in the presence of nonintegrable constraints and a singular inertia tensor.

2.5 Impulsive Dynamics

Define the touchdown predicate, , where , as,

(21)

so that is true only at those points where contact should be considered for addition (in a manner to be qualified in Theorem 2 by the impulse–velocity complementarity condition, (50), defined below). Furthermore, define the new touchdown predicate,

(22)

such that is true only at those states where some new constraint is impacting.

At impact into contact mode , any incoming constraint velocity must be eliminated. Here, we assume a Newtonian impact law, e.g. (Chatterjee and Ruina 1998, Eqn. 3) or (Featherstone 2008, Eqn. 11.65), that is,

Assumption A8 (Plastic Impact).

Impacts will be plastic (inelastic), occur instantaneously, and their effect described by an algebraic equation (23), defined below.

In general, , the instantaneous change in velocity from in contact mode before impact to in contact mode after impact, is modeled as, (recall that maps velocities in the contact frames to velocities of the system state). The coefficient of restitution, , may be defined in any of the usual ways, however throughout this paper plastic impact () is assumed. We restrict to plastic impacts as we believe it to be more relevant to most robotics applications, and since ambiguities arise when an elastic impact occurs in a system with multiple active constraints: different choices of impact model can yield distinct post-impact velocities (see Section 1.2.4). For plastic impacts, the post-impact velocity in mode is,

(23)

where the final simplification follows from (11) and matches (Westervelt et al. 2003, Eqn. 9). The body impulse in configuration coordinates is,

(24)

The contact impulse (i.e., the impulse at the contact points that induces the desired change in velocity to agree with the new contact mode ) is,

(25)

where recall that , , , and are functions of the state (which does not change during impact, i.e. ), and the impulses, and , are also functions of the incoming velocity, . The final simplification arises from (11), matches (Westervelt et al. 2003, Eqn. 10), and will be used in Section 2.7.

Lemma 8.

When is invertible, contact impulse (25) into contact mode is equivalent to the non-degenerate plastic impact law,

(26)

as listed e.g. in (Chatterjee and Ruina 1998, Eqn. 3).

As with the proof of Lemma 5, the result may be seen by substituting (15) or (16), the explicit solution to (8) when is invertible, into (25), as worked out in Appendix B.3.

2.6 Complementarity

We now introduce the classical complementarity problems for forces and impulses at the contact points, and provide a reformulation that allows massless limbs. We begin with a general statement of the complementarity property (as in e.g. Ingleton (1966), Cottle (1968), Lötstedt (1982), van der Schaft and Schumacher (1998)), then subsequently specialize in Sections 2.6.1 and 2.6.2 to formulations of force–acceleration and impulse–velocity complementarity. Both versions have the general form of seeking real vectors and such that,

(27)

(where for a vector , ) subject to some problem-specific constraints. While the most general problem is uncoupled, that is and may be chosen arbitrarily so long as they satisfy (27), the cases we consider here are coupled by these problem-specific constraints (Pang et al. 1996, Sec. 3). In the linear complementarity problem (LCP), for instance, the coupling constraint has the form (e.g., (Brogliato et al. 2002, Eqn. 8)). The functional relationships between and for the complementarity problems in this paper will in general be nonlinear (as discussed in the rest of this section). Since the relation of interest will generally be problem-specific and index dependent in an essential way, we introduce temporarily an abstract scalar relation, instead of or and similarly instead of or , whose different instantiations will be prescribed in the force–acceleration and impulse–velocity versions of the problem, respectively.

Solutions to this problem produce a natural bipartition on some index set, , the scope (some subset of the universal scope , to be discussed below), where and . Here, the role of and will be played by physically determined functions of a specified (“incoming”) state, , to yield an “outgoing” bipartition of the indexing scope, . The indexing scope will be a function only of the incoming continuous state, , as defined in (33).

It should now be clear that for this paper the complementarity problem is reduced to finding the unknown bipartition , also known as the mode selection problem van der Schaft and Schumacher (1998), as opposed to finding the values of the two complementarity vectors directly, e.g. Cottle (1968). Namely, given an index set , two functions that map a subset into a Euclidian space with dimension equal to the size of the index set, and a generic relation (to be instantiated as or in the following sections), we require a solution to a set of constraint equations of the form,

(28)
(29)

(where by definition, ). For the complementarity problems of interest in this paper, the equality constraints in (28)–(29) will hold for all arguments by construction (enforced, e.g., by the flow (12) in the force–acceleration version, and by the impact map (25) in the impulse–velocity version).

The complementarity problem as stated thus far is not explicitly coupled (Pang et al. 1996, Sec. 3), i.e. it places no requirements on the relationship between and other than their common dependence on and , which is why existence and uniqueness properties are challenging to define in general. Furthermore, this necessitates the evaluation of both and for constraints that are not in . With the possibly massless limbs in our setting, the evaluation of will not always be possible as the concept of a separation velocity or acceleration is poorly defined (once such a contact point has lifted off the ground the corresponding joints must be dropped from the state according to the unconstrained massless limb assumption, A6). Thus the specifics of in the problems considered in this paper necessitate an alternate formulation that takes advantage of the coupling between and , as the inequality constraints have the property that,

(30)

(the importance of the mode was first noted in (Ingleton 1966, Eqn. 1.7.3)). This suggests the combined expression,

(31)

which is equivalent to (28) & (29),

Lemma 9.

The separate relational statements of the complementarity problem, (28)–(29), are equivalent to a single biconditional statement, (31), provided that the complementary functions and satisfy (30).

Proof.

First note that for it is trivially true that and so (31) simplifies to the first condition of (28). For , the expression in (31) along with the substitution of (30) reduces to the second condition of (29). ∎

Expressing (31) as a predicate ,

We will denote by,

(32)

the implicit function that solves this set of constraints for the unknown required bipartition, where varies with the particular instances as determined by the appropriate form of . Note that while the codomain is , the solution will always be a member of .

The existence of this implicit function (32) (i.e., the existence and uniqueness of a solution, , to the mode selection problem) will in the most general cases have to be an additional assumption131313However note that the remainder of this paper only requires a unique choice of a solution that satisfies the predicate if multiple solutions exist. (see Assumption A9 and A10, below), although the specific complementarity problems in this section (i.e., based on the relationship of the specific functions and used in these cases) in the absence of friction reduce down to the conventional LCP problem and so existence and uniqueness has been proven in e.g. Ingleton (1966), Cottle (1968), Lötstedt (1982), van der Schaft and Schumacher (1998).

The motivating literature and related work discussed in Section 1.2 generally imposes two complementarity conditions on rigid body dynamics models. The force–acceleration (FA) variant of (28)–(29), presented in (35)–(36), stipulates that there cannot be both a continuous time contact force and a separation acceleration at the same contact point, and is widely considered to arise from fundamental physical reasoning. In the present setting, FA complementarity governs exclusively the nature of liftoff events (and extended in Section 2.8 to stick/slip events) wherein the number of active contacts (i.e., cardinality of the mode set) is reduced for reasons discussed in Section 2.6.1. In contrast, during instantaneous impact events the contact forces have no time to perform work. Instead, the impulse–velocity (IV) variant of (28)–(29), presented in (46)–(47), precludes a simultaneous impact-induced contact impulse and separation velocity at the same contact point. This constraint is known not to follow inevitably from the rational mechanics of rigid body models Chatterjee (1999), but represents a commonly exploited algorithmic simplification that we will embrace in this inelastic model at the possible expense of consistency with elastic impact models in the limit. In the present setting, IV complementarity governs exclusively the nature of touchdown events wherein one or more new contacts become active (i.e., cardinality of the mode set is increased) for reasons discussed in Section 2.6.2.

2.6.1 Force–Acceleration (FA) Complementarity

For continuous time contact forces, when is false and therefore when one or more contact constraints violate the unilateral constraint cone141414 Recall from Section 2.2 that in the normal direction is according to the frame conventions of (Johnson and Koditschek 2013a, Eqn. 76, 78). , some constraint will lift off and must be removed from the set of active constraints, resulting in a transition to a new mode. Determining that next mode sets up a complementarity problem over the existing contact mode between the unilateral constraint cone, , if the contact is kept, and the separation acceleration if it is removed (recall that as an active constraint the state velocity is initially ). The full scope of contact constraints that should be considered is the set of all contacts which are “touching”, i.e. those whose normal direction component have zero contact distance and a non-separating velocity151515 Note that thus far only normal direction constraints have been considered, however Section 2.8 will extend this to include tangential (sliding friction) constraints and this scope is defined in this general way in order to apply there as well.,

(33)
(34)

Recall that force–acceleration complementarity only holds when is false and so the final condition will apply here. Furthermore, while the full scope is formally required and does not depend on the active mode, numerically it suffices to check – this reduced scope eliminates the numerical challenge of checking the exact equality of (33). Any constraints that are not in and therefore not algebraically guaranteed to satisfy this equality will, due to numerical error, only be close. This numerical approximation will miss cases such as Figure 2, in particular cases (a) and (d), wherein a constraint that is not in the current active mode () satisfies the scope in (33) (). In all of the cases in Figure 2, the point has zero contact distance () and no relative velocity (). However these examples are not generic as any perturbation in the state or constraint will resolve this problem.

For transition into , consider contact force (13) both in but also in the alternative mode where contact is maintained (the reason for this alternative mode will become clear in Theorem 1),

(35)
(36)

where the identically zero constraints are guaranteed to hold in consequence of the dynamics governing mode , namely, the invariance of the flow ( by (12)) and the Lagrange multipliers ( by (13)). Note the importance here of the trending positive/negative conditions (Definition 1) – in general it is not sufficient to simply check the sign of the contact force but possibly higher derivatives as well. For example, in Figure 2

, cases (c) and (d), assume the particle is sliding along the constraint from left to right. At the moment the particle reaches the origin, the contact force is zero. However in (c) the contact force is trending negative and the constraint should be removed, while in (d) the contact force is trending positive and it should be maintained.

Constraints (35), (36) can be simplified into a form analogous to (30), hence, by Lemma 9, they may be further reduced to,

(37)

or as the predicate ,