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Kinetic quasi-velocities in unilaterally constrained Lagrangian mechanics with impacts and friction

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Abstract

Quasi-velocities computed with the kinetic metric of a Lagrangian system are introduced, and the quasi-Lagrange equations are derived with and without friction. This is shown to be very well suited to systems subject to unilateral constraints (hence varying topology) and impacts. Energetical consistency of a generalized kinematic impact law is carefully studied, both in the frictionless and the frictional cases. Some results concerning the existence and uniqueness of solutions to the so-called contact linear complementarity problem, when friction is present, are provided.

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Notes

  1. This is the set denoted \(\mathcal{T}_{C}^{\perp}\) in [35, Eq. (5.11)].

  2. Notice that the assumption that the constraints are functionally independent, guarantees that the norms \(\|\nabla h_{i}(q)\|_{\tiny{M^{-1}}}\) never vanish, so \(\operatorname {diag}(\|\nabla h_{i}(q)\|_{\tiny{M^{-1}}})\) is positive definite.

  3. The Delassus’ operator is sometimes called the fundamental matrix [12].

  4. Clearly, if the vectors t q,i are chosen mutually orthogonal then D t(q)=I.

  5. For simplicity of notation the dependence of p′ on q is not recalled.

  6. This is equivalently stated as \(\dot{q}(t^{-}) \in - T_{\varPhi_{u}}(q(t))\) [69].

  7. \(x^{T}\mathcal{E}_{\mathrm{nn}}^{T}G^{-1}(q)x=x^{T} (\mathcal{E}_{\mathrm{nn}}^{T}G^{-1}(q))^{T}x=x^{T}G^{-1}(q)\mathcal{E}_{\mathrm{nn}}x\) for any vector \(x \in \mathbb {R}^{p}\).

  8. Necessary and sufficient conditions for co-positivity of A=A T are a 11⩾0, a 22⩾0, \(a_{12}+\sqrt{a_{11}a_{22}} \geqslant 0\) [39].

  9. Recall that q is a constant during the shock, so that all the matrices that depend on q only are also constant. Thus, P and H may depend on q as well.

  10. The idea of doing such a change of coordinates was first introduced in [17] in the context of maximal monotone differential inclusions.

  11. Also called an M-matrix [25, p. 222].

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  1. Bipop team-project, ZIRST Montbonnot, INRIA, 655 avenue de l’Europe, 38334, Saint-Ismier cedex, France

    Bernard Brogliato

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Appendices

Appendix A: Theorem 2.11 in [23]

We give here just an excerpt of this result, and two easy corollaries of it.

Theorem 3

Let \(M \in \mathbb {R}^{n \times n}\) be a positive definite matrix. Then every matrix

$$A \in \biggl\{A \mid \biggl\| \biggl(\frac{M+M^{T}}{2} \biggr)^{-1} \biggr\|_{2} \|M-A\|_{2} < 1 \biggr\} $$

is positive definite.

Corollary 2

Let D=P+N, where D, P, and N are n×n real matrices, and P>0, not necessarily symmetric. If

$$ \|N\|_{2} < \frac{1}{\| (\frac{P+P^{T}}{2} )^{-1}\|_{2}} $$
(88)

then D>0.

Proof

Follows from Theorem 3 with \(A \stackrel{\Delta}{=}D\) and \(M\stackrel{\Delta}{=} P\). □

Corollary 3

Let \(M \in \mathbb {R}^{n \times n}\) be a symmetric positive definite matrix. Let A=BM for some matrix B. If

$$\|M^{-1}\|_{2} \|M\|_{2} \|I-B\|_{2} < 1 $$

then A is positive definite.

Proof

Since M=M T applying Theorem 3 gives that ∥M −1∥2∥M−BM∥2<1 guarantees that A>0. Now from [7, Proposition 9.3.5] one has ∥M−BM∥2⩽∥M∥2∥I−B∥2: the result follows. □

Appendix B: Theorem 3.8.6 in [25]

Theorem 4

Let \(M \in \mathbb {R}^{n \times n}\) be a co-positive matrix, and let \(q \in \mathbb {R}^{n}\) be given. If the implication

$$\bigl[v \geqslant 0, Mv \geqslant 0, v^{T}Mv=0\bigr] \quad \Rightarrow \quad \bigl[v^{T}q \geqslant 0\bigr] $$

is valid, then the LCP: 0⩽λ⊥Mλ+q⩾0 has a solution.

The implication is equivalently rewritten as \(q \in Q_{M}^{*}\).

Appendix C: Propositions 8.1.2 and 9.3.5 in [7]

Proposition 16

Let A and B be n×n symmetric real matrices, and S be an m×n real matrix. Then: (xi) if A⩽B then SAS T⩽SBS T; (xiii) if SAS T⩽SBS T and rank(S)=n, then A⩽B.

Proposition 17

Let A be an n×m and B an m×l matrices. If p∈[1,2], then ∥AB∥ p ⩽∥A∥ p ∥B∥ p .

Appendix D: Proof of (59)

Let us start from (58). Since R has full rank and using \(P\bar{G}(q)=\varGamma^{T}\) \((\Leftrightarrow \varGamma R^{-1}=\bar{G}^{T}R)\), this is equivalent to

$$ z-Rv\bigl(t^{-}\bigr) \in -R^{-1}\varGamma^{T} N_{W(q)}\bigl(\varGamma R^{-1}\bigl(z+R\varLambda v \bigl(t^{-}\bigr)\bigr)\bigr) $$
(89)

Let y=z+RΛv(t −). In view of the definition of \(\bar{W}(q)\), and since \(N_{\bar{W}(q)}(\cdot)=\partial \psi_{\bar{W}(q)}(\cdot)\), one has \(N_{\bar{W}(q)}(y) =\partial \psi_{\bar{W}(q)}(y)=(\varGamma R^{-1})^{T} \partial \psi_{W(q)}(\varGamma R^{-1}y)\), where the chain rule of convex analysis has been used. The expression in (60) is then a consequence of the fact that \(x-y \in -N_{K}(x) \Leftrightarrow x =\operatorname {proj}[K;y]\) for any vectors x, y, and convex set K of appropriate dimensions.

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Brogliato, B. Kinetic quasi-velocities in unilaterally constrained Lagrangian mechanics with impacts and friction. Multibody Syst Dyn 32, 175–216 (2014). https://doi.org/10.1007/s11044-013-9392-5

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