Modeling and analysis of multiple impacts in multibody systems under unilateral and bilateral constrains based on linear projection operators

Abstract

This paper presents a unifying dynamics formulation for nonsmooth multibody systems (MBSs) subject to changing topology and multiple impacts based on a linear projection operator. An oblique projection matrix ubiquitously yields all characteristic variables of such systems as follows: (i) the constrained acceleration before jump discontinuity from the projection of unconstrained acceleration, (ii) post-impact velocity from the projection of pre-impact velocity, (iii) impulse during impact from the projection of pre-impact momentum, (iv) generalized constraint force from the projection of generalized input force, and (v) post-impact kinetic energy from pre-impact kinetic energy based on projected inertia matrix. All solutions are presented in closed-form with elegant geometrical interpretations. The formulation is general enough to be applicable to MBSs subject to simultaneous multiple impacts with nonidentical restitution coefficients, changing topology, i.e., unilateral constraints becoming inactive or vice versa, or even when the overall constraint Jacobian becomes singular. Not only do the solutions always exist regardless of the constraint condition, but also the condition number for a generalized constraint inertia matrix is minimized in order to reduce numerical sensitivity in computation of the projection matrix to roundoff errors. The model is proven to be energetically consistent if a global restitution coefficient is assumed. In the case of nonidentical restitution coefficients, the set of energetically consistent restitution matrices is characterized by using a linear matrix inequality (LMI).

Appendices

Appendix A: Time-derivative of a projection matrix

The Tikhonov regularization theorem [42] describes the pseudo-inverse as the following limit:

$$ \boldsymbol{A}^{+} = \lim_{\epsilon \rightarrow 0} \boldsymbol{A}^{T}\bigl(\boldsymbol{A} \boldsymbol{A} ^{T} + \epsilon \boldsymbol{I}\bigr)^{-1} . $$

(57)

By differentiation of the above expression, one can verify that the time-derivative of the pseudo-inverse can be written in the following form:

$$\begin{aligned} \frac{d}{dt} \boldsymbol{A}^{+} & =\lim _{\epsilon \rightarrow 0} \dot{\boldsymbol{A}} ^{T}\bigl(\boldsymbol{A} \boldsymbol{A}^{T} + \epsilon \boldsymbol{I}\bigr)^{-1} - \boldsymbol{A}^{T} \bigl(\boldsymbol{A} \boldsymbol{A}^{T} + \epsilon \boldsymbol{I}\bigr)^{-1} \bigl[\dot{\boldsymbol{A}} \boldsymbol{A}^{T} + \boldsymbol{A} \dot{\boldsymbol{A}}^{T} \bigr] \bigl(\boldsymbol{A} \boldsymbol{A}^{T} + \epsilon \boldsymbol{I} \bigr)^{-1} \\ & = - \boldsymbol{A}^{+} \dot{\boldsymbol{A}} \boldsymbol{A}^{+} + \lim_{\epsilon \rightarrow 0} \dot{ \boldsymbol{A}}^{T}\bigl(\boldsymbol{A} \boldsymbol{A}^{T} + \epsilon \boldsymbol{I}\bigr)^{-1} - \boldsymbol{A}^{+} \boldsymbol{A} \dot{\boldsymbol{A}}^{T}\bigl(\boldsymbol{A} \boldsymbol{A}^{T} + \epsilon \boldsymbol{I}\bigr)^{-1} \\ &= -\boldsymbol{A}^{+} \dot{\boldsymbol{A}} \boldsymbol{A}^{+} + \lim_{\epsilon \rightarrow 0} \boldsymbol{P} \dot{ \boldsymbol{A}}^{T} \bigl(\boldsymbol{A} \boldsymbol{A}^{T} + \epsilon \boldsymbol{I}\bigr)^{-1}. \end{aligned}$$

(58)

On the other hand, using (58) in the time-derivative of the expression of the projection matrix $\boldsymbol{P}= \boldsymbol{I} - \boldsymbol{A}^{+} \boldsymbol{A}$ yields

$$\begin{aligned} \dot{\boldsymbol{P}} =& - \frac{\mathrm{d}}{\mathrm{d}t} \boldsymbol{A}^{+} \boldsymbol{A} - \boldsymbol{A}^{+} \dot{ \boldsymbol{A}} \\ =& \boldsymbol{A}^{+} \dot{\boldsymbol{A}} \boldsymbol{A}^{+} \boldsymbol{A} + \lim_{\epsilon \rightarrow 0} \boldsymbol{P} \dot{ \boldsymbol{A}}^{T} \bigl(\boldsymbol{A} \boldsymbol{A} ^{T} + \epsilon \boldsymbol{I}\bigr)^{-1} \boldsymbol{A} - \boldsymbol{A}^{+} \dot{\boldsymbol{A}} \\ =& \boldsymbol{A}^{+} \dot{\boldsymbol{A}} (\boldsymbol{I} - \boldsymbol{P}) + \boldsymbol{P} \dot{\boldsymbol{A}} ^{T} \boldsymbol{A}^{+T} - \boldsymbol{A}^{+} \dot{\boldsymbol{A}} \\ =& \boldsymbol{\varLambda }+ \boldsymbol{\varLambda }^{T}, \end{aligned}$$

where $\boldsymbol{\varLambda }=- \boldsymbol{A}^{+} \dot{\boldsymbol{A}} \boldsymbol{P}$. Note that identity $\boldsymbol{P} \boldsymbol{A}^{+} = \boldsymbol{A}^{+T} \boldsymbol{P} = \boldsymbol{0}$ implies $\boldsymbol{\varLambda }^{T} \boldsymbol{P} = \boldsymbol{0}$ and hence one can conclude $\ddot{\boldsymbol{q}}_{\perp }= \dot{\boldsymbol{P}} \dot{\boldsymbol{q}}= \boldsymbol{\varLambda } \dot{\boldsymbol{q}} = \boldsymbol{\varOmega }\dot{\boldsymbol{q}}$ [43].

Appendix B: Properties of $\boldsymbol{M}_{c}$

Consider a nonzero vector $\boldsymbol{a} \in \mathbb{R}^{n}$ and its orthogonal decomposition components $\boldsymbol{a}_{\parallel }=\boldsymbol{P} \boldsymbol{a}$ and $\boldsymbol{a}_{\perp }=(\boldsymbol{I} - \boldsymbol{P}) \boldsymbol{a}$. Then, one can say that

$$ \boldsymbol{a}^{T} {\boldsymbol{M}}_{c} \boldsymbol{a} = \boldsymbol{a}^{T}_{\parallel } \boldsymbol{M} \boldsymbol{a}_{\parallel } + \nu \| \boldsymbol{a}_{\perp } \|^{2} >0. $$

(59)

Notice that both terms $\boldsymbol{a}^{T}_{\parallel } \boldsymbol{M} \boldsymbol{a}_{\parallel }>0$ and $\| \boldsymbol{a}_{\perp } \|^{2}>0$ are positive semi-definite. Moreover, for a given nonzero vector $\boldsymbol{a}$, if $\boldsymbol{a}_{\perp }= \boldsymbol{0}$ then $\boldsymbol{a}^{T}_{\parallel } \neq 0$, and vice versa. This means that the sum of the two orthogonal terms must be positive definite, and so must be the constraint inertia matrix ${\boldsymbol{M}}_{c}$.

By definition we have

$$\begin{aligned} \boldsymbol{\varOmega }\dot{\boldsymbol{q}} = (\boldsymbol{I} - \boldsymbol{P}) \boldsymbol{\varOmega } \dot{\boldsymbol{q}.} \end{aligned}$$

Since $\boldsymbol{M}_{c}$ is always invertible, i.e., $\boldsymbol{M}_{c}^{-1}\boldsymbol{M} _{c} = \boldsymbol{I}$, the above equation can be equivalently written as

$$\begin{aligned} \boldsymbol{\varOmega }\dot{\boldsymbol{q}} & = \boldsymbol{M}_{c}^{-1} \boldsymbol{M}_{c} (\boldsymbol{I} - \boldsymbol{P}) \boldsymbol{ \varOmega }\dot{\boldsymbol{q}} \\ & = \boldsymbol{M}_{c}^{-1} \nu (\boldsymbol{I} - \boldsymbol{P}) \boldsymbol{\varOmega }\dot{\boldsymbol{q}} \\ & = \nu \boldsymbol{M}_{c}^{-1}\boldsymbol{\varOmega }\dot{ \boldsymbol{q},} \end{aligned}$$

which proves (9e).

From definition we have $\boldsymbol{M}_{c} \boldsymbol{P} = \boldsymbol{M}$, and hence $\boldsymbol{M} \boldsymbol{M}_{c}^{-1} = \boldsymbol{M}_{c} \boldsymbol{P} \boldsymbol{M}_{c}^{-1} = \boldsymbol{M} _{c} \boldsymbol{M}_{c}^{-1} \boldsymbol{P} = \boldsymbol{P}$, which proves relationship (9d).

Now, consider the characteristic equation of the constraint mass matrix

$$\begin{aligned} \bigl( \boldsymbol{P} \boldsymbol{M} \boldsymbol{P} + \nu (\boldsymbol{I} - \boldsymbol{P}) \bigr) \boldsymbol{x} - \lambda \boldsymbol{x} =0. \end{aligned}$$

Clearly, $\lambda =\nu$ is the eigenvalue for all orthogonal eigenvectors which span $\mathcal{N}^{\perp }(\boldsymbol{A})$ because $\lambda =\nu$ means $(\boldsymbol{P} \boldsymbol{M} \boldsymbol{P} - \boldsymbol{P})\boldsymbol{x} =\boldsymbol{0} \quad \forall \boldsymbol{x} \in {\mathcal{N}}^{\perp }(\boldsymbol{A})$. The remaining set of orthogonal eigenvectors must lie in $\mathcal{N}(\boldsymbol{A})$ that correspond to the nonzero eigenvalues of $\boldsymbol{P} \boldsymbol{M} \boldsymbol{P}$:

$$\begin{aligned} \boldsymbol{P} \boldsymbol{M} \boldsymbol{P} \boldsymbol{x} - \lambda \boldsymbol{x} =\boldsymbol{0} \qquad \lambda \neq 0 \quad \forall \boldsymbol{x} \in {\mathcal{N}}. \end{aligned}$$

Therefore, the set of all eigenvalues of the positive definite matrix ${\boldsymbol{M}}_{c}$ is the union of the above sets corresponding to the eigenvectors in $\mathcal{N}$ and $\mathcal{N}^{\perp }$, i.e.,

$$ \lambda ({\boldsymbol{M}}_{c})=: \bigl\{ \underbrace{ \nu , \dots , \nu }_{r}, \; \underbrace{ \lambda _{\stackrel{\mathrm{min}}{\neq 0}} ( \boldsymbol{P} \boldsymbol{M} \boldsymbol{P}), \dots , \lambda _{\mathrm{max}}( \boldsymbol{P} \boldsymbol{M} \boldsymbol{P})} _{n-r} \bigr\} , $$

(60)

where $\{ \lambda _{\stackrel{\mathrm{min}}{\neq 0}}(\boldsymbol{P} \boldsymbol{M} \boldsymbol{P}), \dots , \lambda _{\mathrm{max}}(\boldsymbol{P} \boldsymbol{M} \boldsymbol{P}) \}$ are all nonzero eigenvalues of $\boldsymbol{P} \boldsymbol{M} \boldsymbol{P}$. According to (60), the condition number of $\boldsymbol{M}_{c}$, which is simply the ratio of the largest to smallest eigenvalues, is

$$ \mbox{cond}({\boldsymbol{M}}_{c}) = \frac{\max (\nu , \lambda _{\mathrm{max}}( \boldsymbol{P} \boldsymbol{M} \boldsymbol{P}) )}{\min (\nu , \lambda _{\stackrel{\mathrm{min}}{\neq 0} }(\boldsymbol{P} \boldsymbol{M} \boldsymbol{P}))} . $$

(61)

Clearly, the RHS of (61) is at its minimum if $\nu$ is selected to be within the lower- and upper-bounds defined in (9f).

By virtue of (60), the Singular Value Decomposition of $\boldsymbol{M}_{c}$ takes the form

$$ \boldsymbol{M}_{c} = \begin{bmatrix} \boldsymbol{V}_{1} & \boldsymbol{V}_{2} \end{bmatrix} \begin{bmatrix} \nu \boldsymbol{I} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{\varSigma } \end{bmatrix} \begin{bmatrix} \boldsymbol{V}_{1}^{T} \\ \boldsymbol{V}_{2}^{T} \end{bmatrix} , $$

(62)

where matrix $\boldsymbol{\varSigma }=\mbox{diag}\{ \lambda _{\stackrel{\mathrm{min}}{\neq 0}}(\boldsymbol{P} \boldsymbol{M} \boldsymbol{P}), \dots ,\lambda _{\mathrm{max}}(\boldsymbol{P} \boldsymbol{M} \boldsymbol{P})\}$ contains the nonzero singular values, $\boldsymbol{V}=[\boldsymbol{V}_{1} \;\; \boldsymbol{V}_{2}]$ is a unitary matrix so that $\mbox{span}(\boldsymbol{V}_{1}) \equiv {\mathcal{N}} ^{\perp }$ and $\mbox{span}(\boldsymbol{V}_{2}) \equiv {\mathcal{N}}$, i.e., $\boldsymbol{P} = \boldsymbol{V}_{2} \boldsymbol{V}_{2}^{T}$, $\boldsymbol{V}_{2}^{T} \boldsymbol{V}_{2} = \boldsymbol{I}$, and $\boldsymbol{V}_{1}^{T} \boldsymbol{V}_{2} =\boldsymbol{0}$. Thus

$$\begin{aligned} \boldsymbol{M}_{c}^{-1} \boldsymbol{P} &= \begin{bmatrix} \boldsymbol{V}_{1} & \boldsymbol{V}_{2} \end{bmatrix} \begin{bmatrix} \nu ^{-1} \boldsymbol{I} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{\varSigma }^{-1} \end{bmatrix} \begin{bmatrix} \boldsymbol{V}_{1}^{T} \\ \boldsymbol{V}_{2}^{T} \end{bmatrix} \boldsymbol{V}_{2} \boldsymbol{V}_{2}^{T} \\ &= \boldsymbol{V}_{2} \boldsymbol{\varSigma }^{-1} \boldsymbol{V}_{2}^{T} = \boldsymbol{M}_{o}^{+}. \end{aligned}$$

Appendix C: Properties of $\boldsymbol{S}$

Using (9b) in the following derivations yields

$$\begin{aligned} \boldsymbol{S} \boldsymbol{P} &= \boldsymbol{P} - \boldsymbol{M}_{c}^{-1} \boldsymbol{P} \boldsymbol{M} \boldsymbol{P} \\ &= \boldsymbol{P} - \boldsymbol{M}_{c}^{-1} \boldsymbol{M}_{c} \boldsymbol{P} \\ & = \boldsymbol{P} - \boldsymbol{P} = \boldsymbol{0}, \end{aligned}$$

which proves identity (11a). It follows that

$$\begin{aligned} (\boldsymbol{I} - \boldsymbol{P}) \boldsymbol{S}^{T} = \boldsymbol{S}^{T} - \boldsymbol{P} \boldsymbol{S}^{T} = \boldsymbol{S}^{T}. \end{aligned}$$

On the other hand, by definition we have

$$\begin{aligned} (\boldsymbol{I} - \boldsymbol{P}) \boldsymbol{S} &= \boldsymbol{I} - \boldsymbol{P} - \boldsymbol{M}_{c}^{-1} \boldsymbol{P} \boldsymbol{M} + \boldsymbol{P} \boldsymbol{M}_{c}^{-1} \boldsymbol{P} \boldsymbol{M} \\ &= \boldsymbol{I} - \boldsymbol{P} - \boldsymbol{M}_{c}^{-1} \boldsymbol{P} \boldsymbol{M} + \boldsymbol{M}_{c}^{-1} \boldsymbol{P}^{2} \boldsymbol{M} \\ & = \boldsymbol{I} - \boldsymbol{P}, \end{aligned}$$

which proves identity (11c).

Finally,

$$\begin{aligned} \boldsymbol{M} \boldsymbol{S} &= \boldsymbol{M} - \boldsymbol{M} \boldsymbol{M}_{c}^{-1} \boldsymbol{P} \boldsymbol{M} \\ &= \boldsymbol{M} - \boldsymbol{M} \boldsymbol{P} \boldsymbol{M}_{c}^{-1} \boldsymbol{M} \\ &= \bigl(\boldsymbol{I} - \boldsymbol{M} \boldsymbol{P} \boldsymbol{M}_{c}^{-1} \bigr) \boldsymbol{M} \\ & = \boldsymbol{S}^{T} \boldsymbol{M}. \end{aligned}$$

On the other hand, using the above result in the following derivation yields

$$ \boldsymbol{S}^{T} \boldsymbol{M} \boldsymbol{S} = \boldsymbol{S}^{T} \boldsymbol{S}^{T} \boldsymbol{M} = \boldsymbol{S}^{T} \boldsymbol{M}, $$

which proves identity (11d).