On the regularization of impact without collision: the Painlevé paradox and compliance

We consider the problem of a rigid body, subject to a unilateral constraint, in the presence of Coulomb friction. We regularize the problem by assuming compliance (with both stiffness and damping) at the point of contact, for a general class of normal reaction forces. Using a rigorous mathematical approach, we recover impact without collision (IWC) in both the inconsistent and the indeterminate Painlevé paradoxes, in the latter case giving an exact formula for conditions that separate IWC and lift-off. We solve the problem for arbitrary values of the compliance damping and give explicit asymptotic expressions in the limiting cases of small and large damping, all for a large class of rigid bodies.


Introduction
In mechanics, in problems with unilateral constraints in the presence of friction, the rigid-body assumption can result in the governing equations having multiple solutions (the indeterminate case) or no solutions (the inconsistent case). The classical example of Painlevé [1][2][3], consisting of a slender rod slipping 1 along a rough surface (figure 1), is the simplest and most studied example of these phenomena, now known collectively as Painlevé paradoxes [5][6][7][8]. Such paradoxes can occur at physically realistic parameter values in many important engineering systems [9][10][11][12][13][14][15]. 1 We prefer to avoid describing this phase of the motion as sliding because we will be using ideas from piecewise smooth systems [4], where sliding has exactly the opposite meaning.   When a system has no consistent solution, it cannot remain in that state. Lecornu [16] proposed a jump in vertical velocity to escape an inconsistent, horizontal velocity, state. This jump has been called impact without collision (IWC) [17], tangential impact [18] or dynamic jamming [13]. Experimental evidence of IWC is given in [15]. IWC can be incorporated into the rigid-body formulation [19,20] by considering the equations of motion in terms of the normal impulse, rather than time.
Génot & Brogliato [17] considered the dynamics around a critical point, corresponding to zero vertical acceleration of the end of the rod. They proved that, when starting in a consistent state, the rod must stop slipping before reaching the critical point. In particular, paradoxical situations cannot be reached after a period of slipping.
One way to address the Painlevé paradox is to regularize the rigid-body formalism. Physically, this often corresponds to assuming some sort of compliance at the contact point A, typically thought of as a spring, with stiffness (and sometimes damping) that tends to the rigid body model in a suitable limit. Mathematically, very little rigorous work has been done on how IWC and Painlevé paradoxes can be regularized. Dupont & Yamajako [21] treated the problem as a slow-fast system, as we will do. They explored the fast time-scale dynamics, which is unstable for the Painlevé paradoxes. Song et al. [22] established conditions under which these dynamics can be stabilized. Le Suan An [23] considered a system with bilateral constraints and showed qualitatively the presence of a regularized IWC as a jump in vertical velocity from a compliance model with diverging stiffness. Zhao et al. [24] considered the example in figure 1 and regularized the equations by assuming a compliance that consisted of an undamped spring. They estimated, as a function of the stiffness, the orders of magnitude of the time taken in each phase of the (regularized) IWC. Another type of regularization was considered by Neimark & Smirnova [25], who assumed that the normal and tangential reactions took (different) finite times to adjust.
In this paper, we present the first rigorous analysis of the regularized rigid-body formalism, in the presence of compliance with both stiffness and damping. We recover IWC in both the inconsistent and the indeterminate cases, and in the latter case, we present a formula for conditions that separate IWC and lift-off. We solve the problem for arbitrary values of the compliance damping and give explicit asymptotic expressions in the limiting cases of small and large damping. Our results apply directly to a general class of rigid bodies. Our approach is similar to that used in [26,27] to understand the forward problem in piecewise smooth (PWS) systems in the presence of a twofold.
The paper is organized as follows. In §2, we introduce the problem, outline some of the main results known to date and include compliance. In §3, we give a summary of our main results, theorems 3.1 and 3.2, before presenting their derivation in § §4 and 5. We discuss our results in §6 and outline our conclusion in §7.

Classical Painlevé problem
Consider a rigid rod AB, slipping on a rough horizontal surface, as depicted in figure 1.
The rod has mass m, length 2l, the moment of inertia of the rod about its centre of mass S is given by I and its centre of mass coincides with its centre of gravity. The point S has coordinates (X, Y) relative to an inertial frame of reference (x, y) fixed in the rough surface. The rod makes an angle θ with respect to the horizontal, with θ increasing in a clockwise direction. At A, the rod experiences a contact force (−F T , F N ), which opposes the motion. The dynamics of the rod is then governed by the following equations: where g is the acceleration due to gravity, plus the unilateral constraint y ≥ 0. The coordinates (X, Y) and (x, y) are related geometrically as follows: We now adopt the scalings (X, For a uniform rod, I = 1 3 ml 2 , and so α = 3 in this case. Then for general α, (2.1) and (2.2) can be combined to become, on dropping the tildes, To proceed, we need to determine the relationship between F N and F T . We assume Coulomb friction between the rod and the surface. Hence, whenẋ = 0, we set where μ is the coefficient of friction. By substituting (2.4) into (2.3), we obtain two sets of governing equations for the motion, depending on the sign ofẋ, as follows: where the variables v, w, φ denote velocities in the x, y, θ directions, respectively, and for the configuration in figure 1. The suffices ± correspond toẋ = v ≷ 0, respectively. Suppose F N is known. Then system (2.5) is a Filippov system [4]. Hence, we obtain a welldefined forward flow whenẋ = v = 0 and Remark 2.2. Our results hold for mechanical systems with different q ± , p ± and c ± in (2.6) and even dependency on several angles θ ∈ T d , e.g. the two-link mechanism of Zhao et al. [15]. As expected, S w and S φ in (2.9) are independent of μ, even for general q ± , p ± and c ± .
To solve (2.5) and (2.8), we need to determine F N . The constraint-based method leads to the Painlevé paradox. The compliance-based method is the subject of this paper.

(a) Constraint-based method
In order that the constraint y = 0 be maintained,ÿ(=ẇ) and F N form a complementarity pair given byẇ ≥ 0, F N ≥ 0, F N ·ẇ = 0. (2.10) Note that F N ≥ 0 since the rough surface can only push, not pull, the rod. Then for general motion of the rod, F N and y satisfy the complementarity conditions In other words, at most one of F N and y can be positive.
it. The rigid body equations (2.1) are unable to resolve the dynamics in the third and fourth quadrants. So, we regularize these equations using compliance.

(b) Compliance-based method
We assume that there is compliance at the point A between the rod and the surface, when they are in contact (figure 1). Following [21,28], we assume that there are small excursions into y < 0. Then we require that the nonnegative normal force F N (y, w) is a PWS function of (y, w): where the operation [·] is defined by the last equality and f (y, w) is assumed to be a smooth function of (y, w) satisfying ∂ y f < 0, ∂ w f < 0. The quantities −∂ y f (0, 0) and −∂ w f (0, 0) represent a (scaled) spring constant and damping coefficient, respectively. We are interested in the case when the compliance is very large, so we introduce a small parameter as follows: This choice of scaling [21,28] ensures that the critical damping coefficient (δ crit = 2 in the classical Painlevé problem) is independent of . Our analysis can handle any f of the form f (y, w) But, to obtain our quantitative results, we truncate (2.17) and consider the linear function so that In what follows, the first equation in (2.5) will play no role, so we drop it from now on. Then we combine the remaining five equations in (2.5) with (2.15) and (2.16) to give the following set of governing equations that we will use in the sequel: For > 0, this is a well-defined Filippov system. The slipping region (2.7) and the Filippov vector-field (2.8) are obtained by replacing F N in these expressions with the square bracket −1 [− −1 y − δw] (see also lemma 4.10).

Main results
We now present the main results of our paper, theorems 3.1 and 3.2. Theorem 3.1 shows that, if the rod starts in the fourth quadrant of figure 2, it undergoes (regularized) IWC for a time of O( ln −1 ). The same theorem also gives expressions for the resulting vertical velocity of the rod in terms of the compliance damping and initial horizontal velocity and orientation of the rod.

Theorem 3.1. Consider an initial condition
The function e(δ, θ 0 ), given in (4.30), is smooth and monotonic in δ and has the following asymptotic expansions: Theorem 3.2 is similar to theorem 3.1, but now the rod starts in the first quadrant of figure 2. This theorem also gives an exact formula for initial conditions that separate (regularized) IWC and lift-off.

Theorem 3.2. Consider an initial condition
The limit → 0 shown using (a) the (w,ŷ, v)-variables and (b) a projection onto the (w,ŷ)-plane. The slipping compression phase, shown in red, whereŷ, w and v > 0 all decrease, is described geometrically by an unstable manifold γ u (4.5) of a critical set C, given in (4.4). It ends on the switching manifold Σ. The subsequent sticking phase (in blue) is described by Filippov [4]. It ends along Γ 0 . From there, the lift-off phase (in green) occurs and we return toŷ = 0. In both figures, the grey region is whereF N > 0.
Remark 3.3. These two theorems have not appeared before in the literature. In the rigid-body limit ( → 0), we recover IWC in both cases. Previous authors have not carried out the 'very difficult' calculation [28], performed numerical calculations [6,21] or given a range of estimates for the time of (regularized) IWC in the absence of damping [24]. We give exact and asymptotic expressions for key quantities as well as providing a geometric interpretation of our results, for a large class of rigid bodies, in the presence of a large class of normal forces, as well as giving a precise estimate for the time of (regularized) IWC, all in the presence of both stiffness and damping. Note that we are not attempting to describe all the dynamics around P. There is a canard connecting the third quadrant with the first, and the analysis of it is exceedingly complicated [29] due to fast oscillatory terms. Instead, we follow [24] and consider that the rod dynamics starts in a configuration with p + (θ 0 ) < 0.

Proof of theorem 3.1: impact without collision in the inconsistent case
The proof of theorem 3.1 is divided into three phases, illustrated in figure 3. These phases are a generalization of the phases of IWC in its rigid-body formulation [15].
-Slipping compression ( §4b): During this phase, y, w and v all decrease. The dynamics follow an unstable manifold γ u of a set of critical points C, given in (4.4) below, as → 0. Along γ u the normal force F N = O( −1 ) and v will therefore quickly decrease to 0. Mathematically, this part is complicated by the fact that the initial condition (3.1) belongs to the critical set C as → 0.
(a) Slow-fast setting: initial scaling Before we consider the first phase of IWC, we apply the scaling also used in [21,28], which brings the two terms in (2.19) to the same order. Now let Equations (2.20) then readŷ with respect to the fast time τ = −1 t, where () = d/dτ . This is a slow-fast system in non-standard form [28]. Only θ is truly slow whereas (ŷ, w, φ, v) are all fast. But the set of critical points for = 0 is just three-dimensional. System (4.3) is PWS [26,27]. We now show that (4.3) + contains stable and unstable manifolds γ s,u when the equivalent rigid-body equations exhibit a Painlevé paradox, when p + (θ 0 ) < 0. The saddle structure of C within the fourth quadrant has been recognized before [21][22][23].
Following the initial scaling (4.1) of this section, we now consider the three phases of IWC.
as → 0. (i) Proof of proposition 4.4 We prove proposition 4.4 using Fenichel's normal form theory [33]. But since (4. with v > 0 is PWS, care must be taken. There are at least two ways to proceed. One way is to consider the smooth system (4.3)F N =−ŷ−δw , then rectify C by straightening out its stable and unstable manifolds. Then, (4.3)F N =−ŷ−δw will be a standard slow-fast system to which Fenichel's normal form theory applies. Subsequently, one would then have to ensure that conclusions based on the smooth (4.3)F N =−ŷ−δw also extend to the PWS system (4.3)F N =[−ŷ−δw] . One way to do this is to consider the following scaling κ 1 :ŷ = r 1ŷ1 , w = r 1 w 1 , = r 1 , (4.10) zooming in on C atŷ = 0, w = 0. In terms of the original variables, y = 2ŷ 1 , w = w 1 . Both the scalings (ŷ, w) and (ŷ 1 , w 1 ) have appeared in the literature [6,21,28].

Linearization around M 2 gives only three non-zero eigenvalues
Proof. The first two statements follow from straightforward calculation. For γ u 2 (θ 0 , φ 0 , v 0 ), we restrict to the invariant set 2 = 0, w 2 = −λ + and solve the resulting reduced system.
Proof. By Fenichel's normal form theory, we can straighten out stable and unstable fibres. Lemma 4.9. For ν and ρ sufficiently small, then within U 2 there exists a smooth transformation Proof. Replace r 2 by νr 2 in (4.19) and consider ν small. Using = 2 r 2 , this brings the system into a classical slow-fast system for 0 < ν 1, where ( 2 , w 2 , r 2 ) are fast variables while (θ, φ, v) are slow. In particular, 2 = r 2 = 0, w 2 = −λ + is a saddle-type slow manifold for ν small. The system is therefore amenable to Fenichel's normal form theory [33]. The result then follows by returning to the original r 2 and using φ =φ + O(r 2 ) together with r 2 2 = in the w 2 equation.
(iv) Completing the proof of proposition 4.4 To complete the proof of proposition 4.4, we then return to (4.3) using (4.11) and integrate initial conditions

(c) Sticking
After the slipping compression phase of the previous section, the rod then sticks on Σ, with (ŷ, w, θ , φ) given by (4.9). This is a corollary of the following lemma.
Then there exists a set of visible folds at Proof. Simple computations, following [4]; see also proposition 2.1.
The forward motion of (4.9) within Σ s ⊂ Σ for 1 is therefore subsequently described by the Filippov vector-field (2.8) in proposition 2.1, (4.27) here written in terms ofŷ and the fast time τ , until sticking ends at the visible fold Γ . Note this always occurs for 0 < 1 sinceŷ = w > 0, for [−ŷ − δw] > 0. We first focus on = 0. From (4.27), θ = θ 0 , a constant, and We now integrate (4.28), using (4.9) for = 0 as initial conditions, given by up until the section Γ 0 :ŷ + δw = 0 shown in figure 3a, where sticking ceases for = 0, by lemma 4.10 and (4.26) =0 . We then obtain a function e(δ, θ 0 ) > 0 in the following proposition, which relates the horizontal velocity at the start of the slipping compression phase v 0 (3.1) with the values of (ŷ, w, φ) on Γ 0 , at the end of the sticking phase.
For 0 < 1, sticking ends along the visible fold at Γ . We therefore perturb from = 0 as follows: Proof. Since the = 0 system is transverse to Γ 0 , we can apply regular perturbation theory and the implicit function theorem to perturb τ s continuously to τ s + o (1). The result then follows.

(d) Lift-off
Beyond Γ we haveF N ≡ 0 and lift-off occurs. For = 0, we haveŷ = w and w = θ = φ = v = 0. By proposition 4.13 and regular perturbation theory, we obtain the desired result in theorem 3.1. In terms of the original (slow) time t, it follows that the time of IWC is of order O( ln −1 ) (recall (4.25)). As → 0, IWC occurs instantaneously.
A similar figure appears in [6]. solution follows a saddle-type slow manifold for an extended period of time.) In theorem 3.2, we consider w 10 < w 1 * . The remainder of the proof of theorem 3.2 on IWC in the indeterminate case then follows the proof of theorem 3.1.
We plot e(0, θ 0 ) in figure 6a for α = 3 and μ = 1.4. Figure 6b shows the graph of e(δ, 1) and e(δ,  (w,ŷ)-plane, together with the theoretical predictions of figure 3b. Note that the numerical and analytical solutions are indistinguishable, in both the sticking and lift-off regimes. The cyan orbit lifts off directly. The purple orbit, being on the other side of the stable manifold of C 1 , follows the unstable manifold (γ u , shown in red) until sticking occurs. Then whenF N = 0 atŷ + δŵ = 0 (dashed line), lift-off occurs almost vertically in the (w,ŷ)-plane. Figure 7c,d shows the vertical velocity w and horizontal velocity v, respectively, for both orbits over the same time interval as figure 7b; note the sharp transition for the purple orbit around t = 0.5, as it undergoes IWC.
In figure 7c, we include two dashed lines w = ev 0 and w = −(p + /q + )v 0 , corresponding to our analytical results (3.3) and (4.9), which also hold for the indeterminate case (from theorem 3.2), in excellent agreement with the numerical results.

Conclusion
We have considered the problem of a rigid body, subject to a unilateral constraint, in the presence of Coulomb friction. Our approach was to regularize the problem by assuming a compliance with stiffness and damping at the point of contact. This leads to a slow-fast system, where the small parameter is the inverse of the square root of the stiffness. Like other authors, we found that the fast time-scale dynamics is unstable. Dupont & Yamajako [21] established conditions in which these dynamics can be stabilized. By contrast, McClamroch [28] established under what conditions the unstable fast time-scale dynamics could be controlled by the slow time-scale dynamics. Other authors have used the initial scaling (4.1), together with the scaling κ 1 to numerically compute stability boundaries [21,28] or phase plane diagrams [6].
The main achievement of this paper is to rigorously derive these, and other, results that have eluded others in simpler settings. For example, the work of Zhao et al. [24] assumes no damping in the compliance and uses formal methods to provide estimates of the times spent in the three phases of IWC. They suggest that their analysis can '· · · roughly explain why the Painlevé paradox can result in [IWC]'. By contrast, we assumed that the compliance has both stiffness and damping, analysed the problem rigorously, derived exact and asymptotic expressions for many important quantities in the problem and showed exactly how and why the Painlevé paradox can result in IWC. There are no existing results comparable to (3.3)-(3.5) for any value of δ.
Our results are presented for arbitrary values of the compliance damping, and we are able to give explicit asymptotic expressions in the limiting cases of small and large damping, all for a large class of rigid bodies, including the case of the classical Painlevé example in figure 1.
Given a general class of rigid body and a general class of normal reaction, we have been able to derive an explicit connection between the initial horizontal velocity of the body and its liftoff vertical velocity, for arbitrary values of the compliance damping, as a function of the initial orientation of the body.