Modeling three-dimensional surface-to-surface rigid contact and impact

Abstract

This work presents a rigid body framework for analyzing three-dimensional surface contacts and impacts as a simultaneous multi-point impact problem with friction. A method is developed to address the indeterminacy issue typically associated with multi-point contact and impact analysis. This is accomplished using the constraints on impulses and contact forces defined by the Coulomb friction law and rigid body constraints. The proposed approach relies on a global interpretation of Stronge’s energetic coefficient of restitution (ECOR) to maintain energetic consistency. A key aspect of this work involves addressing the three-dimensionality of this problem, which requires a numerical integration in the impulse domain to address the slip/no-slip behavior in the tangential plane of the impact. This work also models the transition to contact after a series of impacts, and proposes a method for enforcing frictional contact constraints. Several examples of simulation results using the proposed method are presented here.

Appendices

Appendix A: Velocity projection method

In order to demonstrate how a rigid body constraint expressed in terms of velocity can be projected in terms of impulses. Let us consider a two-point contact involving the points $i$ and $j$. Let the velocity-level rigid body constraints between the two points be expressed as

$$ {\mathbf{w}} {\boldsymbol{\vartheta }} = (v_{x_{i}}-v_{x_{j}})\eta _{x} + (v_{y_{i}}-v_{y_{j}})\eta _{y} + (v_{z_{i}}-v_{z_{j}})\eta _{z} = 0 . $$

(69)

Then the velocity vector may be rewritten as

$$\mathit{\vartheta }=\left[\begin{array}{c}{v}_{x_j}-(v_{y_i}-v_{y_j})\frac{{\eta }_y}{{\eta }_x}-(v_{z_i}-v_{z_j})\frac{{\eta }_z}{{\eta }_x}\\ v_{y_i}\\ v_{z_i}\\ v_{x_j}\\ v_{y_j}\\ v_{z_j}\end{array}\right]=\underset{P}{\underbrace{\left[\begin{array}{ccccc}-\alpha & -\beta & 1& \alpha & \beta \\ 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}}{\mathit{\vartheta }}_s,$$

(70)

where ${\boldsymbol{\vartheta }}_{s}$ are independent velocities, $P$ is a matrix representing the rigid body constraint, and $\alpha =\frac{\eta _{y}}{\eta _{x}}$ and $\beta = \frac{\eta _{z}}{\eta _{x}}$. This may be also expressed as

$$ {\boldsymbol{\vartheta }} = J \dot{\mathbf{q}} = P {\boldsymbol{\vartheta }} _{s} . $$

(71)

This yields

$$ {\boldsymbol{\vartheta }}_{s} = P^{+} J \dot{\mathbf{q}} , $$

(72)

where $P^{+} = {(P^{T} P)}^{-1} P^{T}$ is the generalized inverse of the matrix $P$. As shown in [88], the virtual work due to the contact forces yields the relation

$$ {\mathbf{F}}^{T} {\boldsymbol{\vartheta }} = {\boldsymbol{\varGamma }} ^{T} \dot{\mathbf{q}} . $$

(73)

Let $\mathbf{F}_{s}$ be a vector containing the constrained force terms corresponding to the independent velocities ${\boldsymbol{\vartheta }} _{s}$. Enforcement of the velocity level rigid body constraints shouldn’t cause any change in the virtual work of the system. Hence,

$$ {\mathbf{F}_{s}}^{T} {\boldsymbol{\vartheta }}_{s} = {\boldsymbol{\varGamma }} ^{T} \dot{\mathbf{q}} . $$

(74)

Now substituting the relation ${\boldsymbol{\varGamma }} = J^{T} \mathbf{F}$ and (72) into (74) yields

$$ {\mathbf{F}_{s}}^{T} P^{+} J \dot{\mathbf{q}} = { \bigl( J^{T} \mathbf{F} \bigr)}^{T} \dot{\mathbf{q}} . $$

(75)

This yields the constrained force relationships

$$ {\mathbf{F}} = (P^{+})^{T} {\mathbf{F}}_{s}, \qquad \mathbf{F}_{s} = P^{T} \mathbf{F}. $$

(76)

Therefore, from (76), a projection of rigid body constraints onto the force space can be obtained as

$$ \bigl( I - { \bigl( P^{+} \bigr)}^{T} P^{T} \bigr) \mathbf{F} = \mathbf{0} . $$

(77)

Note that in (77), ${ ( P^{+} )}^{T} P ^{T} \neq I$. The matrix ${ ( P^{+} )}^{T} P^{T}$ projects the forces $\mathbf{F}$ to a space orthogonal to the velocity-level constraint such that it is equal to $\mathbf{F}$. Now using the coefficients of $P$ from (70), the matrix $I - { ( P^{+} )}^{T} P^{T}$ can be symbolically obtained as

$$I-{\left(P^{+}\right)}^T{P}^T=\frac{1}{2({\alpha }^2+{\beta }^2+1)}\left[\begin{array}{cccccc}1& \alpha & \beta & -1& -\alpha & -\beta \\ \alpha & {\alpha }^2& \alpha \beta & -\alpha & -{\alpha }^2& -\alpha \beta \\ \beta & \alpha \beta & {\beta }^2& -\beta & -\alpha \beta & -{\beta }^2\\ -1& -\alpha & -\beta & 1& \alpha & \beta \\ -\alpha & -{\alpha }^2& -\alpha \beta & \alpha & {\alpha }^2& \alpha \beta \\ -\beta & -\alpha \beta & -{\beta }^2& \beta & \alpha \beta & {\beta }^2\end{array}\right].$$

(78)

It can be seen that all equations obtained by substituting (78) into (77) yield the same force constraint:

$$ f_{x_{i}} + \alpha f_{y_{i}} + \beta f_{z_{i}} - f_{x_{j}} - \alpha f _{y_{j}} - \beta f_{z_{j}} = 0. $$

(79)

Now substituting back $\alpha = \frac{\eta _{y}}{\eta _{x}}$ and $\beta = \frac{\eta _{z}}{\eta _{x}}$ and rearranging gives the rigid body constraint

$$ (f_{x_{i}}-f_{x_{j}})\eta _{x} + (f_{y_{i}}-f_{y_{j}}) \eta _{y} + (f _{z_{i}}-f_{z_{j}})\eta _{z} = \mathbf{w} {\mathbf{F}} = 0, $$

(80)

or in terms of differential impulses,

$$ {\mathbf{w}} d\mathbf{p} = 0. $$

(81)

Appendix B: Finding multi-polynomial roots using Macaulay matrix

In the literature, one can find two main approaches for solving multi-polynomial system of equations. These are either homotopy-continuation based approaches [89,90,91,92,93] or elimination-theory based approaches [94,95,96,97]. The homotopy-continuation based methods essentially transform the multi-polynomial root problem into an ordinary differential equation initial value problem, by harnessing some topological properties. The initial values here are some known roots of a polynomial system of the same degree. The idea is to perform numerical integration to reach the solution set for the desired multi-polynomial system The elimination based approaches, on the other hand, make use of the algebraic structure of the multi-polynomial system to simplify the problem in terms of an independent variable. These approaches may be further classified in to two categories which either use (i) Gröbner basis [94,95,96, 98] or (ii) resultants [95, 97, 99,100,101,102]. The Gröbner basis of a multi-polynomial system is analogous to the row-reduced echelon obtained for linear systems after performing the Gaussian elimination process. If there exist finitely many solutions to any given system, then the Gröbner basis for this system would yield a unique set of equations (known as ideals) such that it contains at least one univariate equation. Thus, the root-finding methods which use Gröbner basis first find the roots in terms of a single variable and perform repeated substitution to obtain all roots [94, 96, 98]. One of the drawbacks of Gröbner basis based approaches is that they require a great deal of symbolic manipulation, which makes this process computationally expensive [100].

Another class of elimination based root-finding methods use resultants for simultaneously calculating all possible roots for a given system of polynomials. Resultants were originally proposed for two univariate polynomial systems, and was defined as the determinant of what is known as the Sylvester matrix. The Sylvester matrix contains the coefficients of the two polynomials. If there exists a common root for any, two univariate polynomial system, the resultant of these polynomials must be zero [100, 102]. Resultants for two polynomial systems can be also formulated as the determinant of Bezout–Cayley matrix [102]. In addition to determining the existence of a common root for univariate bi-polynomial systems, resultants may be also used calculating the common roots of two multivariate polynomials. This is usually done by forming a Sylvester matrix while treating one of the two variables (known as the hidden variable) as part of the coefficients of the polynomial. This yields the resultant to be a function of the hidden variable included as coefficient. The values of this hidden variable at the common roots are then calculated by setting the resultant to zero, and solving for the unknown. Subsequently, the other variables may be treated as the hidden variable to find their values at the common roots. This method of calculating the common roots of a two-polynomial system may be extended to a more general multivariate polynomial system by repeated resultant computation by selecting any two polynomials at a time. However, such a method yields misleading roots, since any two polynomials picked from a set of polynomials may not necessarily have a common root. A more standard way of extending the resultant formulation for multi-polynomial system is expressed as the ratio of the determinants of two matrices, which are known as the Macaulay matrices. Similar to the Sylvester resultants, the Macaulay resultants can be used to determine the existence of any common roots in the multi-polynomial system. The Macaulay matrix may be also used to calculate the roots of a multi-polynomial system. This can be done by calculating what is known as the U-resultant of the multi-polynomial system. The U-resultant can be calculated by augmenting the given polynomial system with a symbolic coefficient linear equation, and taking the determinant of its Macaulay matrix. The symbolic expression given by the U-resultant can then be used to calculate all roots of the system. Despite the simplicity of the U-resultant technique, the symbolic computation involved makes this method computationally expensive.

This work uses a more popular and comparatively inexpensive method for solving the multi-polynomial system in (45). This method extends the idea of hiding variable using the Macaulay matrix form, to transform the root-finding problem to a polynomial eigenvalue problem (PEP). However, this transformation can only be performed for a system with the same number of variables as equations. Note that the system in (45) is an over-determined system of polynomials with the same degree. Hence, before transforming (45) into a PEP, one needs to find the minimal basis set of equations. This can be done by simply performing a Gauss–Jordan elimination on the coefficients of the system. Let the reduced set of equations, after performing the G–J elimination on the coefficients of (45), be

(82)

The reduced system of polynomials (82) has 3 equations and 3 unknowns, and has roots identical to (45). Now the first step in transformation of (82) to PEP is to formulate the Macaulay matrix by hiding one of the variables into the coefficient field, or in other words, treating one of the variables as part of the coefficient of the system. Let us choose $u_{1}$ as the hidden variable. Then the multi-polynomial system in (82) is given by

(83)

Equations (83) are defined using the variables $u_{2}$ and $u_{3}$. Let the degrees of the polynomials , , and be $d_{1}$, $d_{2}$, and $d_{3}$, respectively. These degrees are counted while disregarding $u_{1}$ as a variable. The method of hidden variable only works for homogeneous polynomials, so (83) needs to be homogenized by adding an extra variable $u_{4}$. The homogenized set of equations are given by

(84)

where

Note that the polynomial system given in (45) is already homogeneous, so the homogenization has no effect on the original polynomial system. The total degree $d$ of the multi-polynomial system is calculated using the degrees of each polynomial $d_{i}$ in the system as

$$ d = \sum^{n=3}_{i=1} (d_{i} -1) + 1. $$

(85)

The next step is to take the set of all monomials of degree $d$ using the variables $u_{2}$, $u_{3}$, and $u_{4}$ and partition them into $n=3$ subsets. Let the set of all monomials of degree $d$ be represented as $\mathbf{u}^{\alpha }= {u_{2}}^{\alpha _{2}} {u_{3}}^{\alpha _{3}} {u_{4}}^{\alpha _{4}}$ such that $|\alpha | = \sum_{i=2}^{n+1=4} \alpha _{i} = d$. The set of monomials $\mathbf{u}^{\alpha }$ is then partitioned into $n=3$ subsets as

$$ \begin{aligned} S_{1} & = \bigl\lbrace {\mathbf{u}}^{\alpha }: {u_{2}}^{d_{1}} \text{ divides } {\mathbf{u}}^{\alpha } \bigr\rbrace , \\ S_{2} & = \bigl\lbrace {\mathbf{u}}^{\alpha }: {u_{2}}^{d_{1}} \text{ doesn't divide } {\mathbf{u}}^{\alpha }, \text{ but } {u_{3}}^{d_{2}} \text{ does} \bigr\rbrace ,\\ S_{3} & = \bigl\lbrace {\mathbf{u}}^{\alpha }: {u_{2}}^{d_{1}} , {u_{3}}^{d_{2}} \text{ doesn't divide } {\mathbf{u}}^{\alpha }, \text{ but } {u_{4}}^{d_{3}} \text{ does} \bigr\rbrace . \end{aligned} $$

(86)

Now a new set of polynomials equivalent to (84) can be created using these partitions:

(87)

Note that the functions have the hidden variable $u_{1}$ included in their coefficient field. Hence, the homogeneous system of polynomials in (87) also contains the hidden variable $u_{1}$. Now, the polynomials in (87) can be dehomogenized by setting the additional variable $u_{4} = 1$. Thus after dehomogenization (87) may be represented as

$$ \mathcal{M}(u_{1}) {\boldsymbol{\nu }} = \mathbf{0}, $$

(88)

where $\mathcal{M}(u_{1})$ is the Macaulay matrix for the multi-polynomial system (45) with the variable $u_{1}$ hidden in the coefficient field, such that the elements of $\mathcal{M}(u_{1})$ are univariate functions of $u_{1}$; ${\boldsymbol{\nu }}$ is a vector containing all possible monomials in variables $u_{2}$ and $u_{3}$. Since the elements of $\mathcal{M}(u _{1})$ contain powers of $u_{1}$, they may be also represented as a matrix polynomial. Thus, (88) can be posed as a polynomial eigenvalue problem (PEP),

$$ \bigl( \lambda ^{k} \mathcal{M}_{k} + \lambda ^{k-1} \mathcal{M}_{k-1} + \cdots + \lambda \mathcal{M}_{1} + \mathcal{M}_{0} \bigr) {\boldsymbol{\nu }} = 0, $$

(89)

where $k$ is the highest power of $u_{1}$ in $\mathcal{M}(u_{1})$. Note that $\mathcal{M}_{k}$ are matrices with numeric elements. PEPs such as (89) can be easily solved by transforming the problem into a generalized eigenvalue problem. However, this work uses a built-in command in MATLAB called polyeig() to obtain the solutions of PEPs. Solving the PEP in (89) yields the eigenvalues $\lambda$ and eigenvectors ${\boldsymbol{\nu }}$. The eigenvalues $\lambda$ take the values of the hidden variable $u_{1}$ at the roots of the multi-polynomial system (45). The eigenvectors ${\boldsymbol{\nu }}$ would contain the variables $u_{2}$ and $u_{3}$, which can be selected for corresponding values of $u_{1}= \lambda$.

Now it is noteworthy that a solution of PEPs such as (89), by transformation to generalized eigenvalue problem, generates matrices of very large dimensions for large values of $n$ and $d$. The eigenvalue problems for very large matrices are usually difficult to compute and can be numerically unstable. However, in the context of this work, the eigenvalue problems encountered involve only moderately large dimensional matrices (since $n=3$ and $d = \sum^{n=3}_{i=1} (d_{i} -1) + 1 = 4$) that are typically easy to compute. The accuracy of the solution can usually be improved by a method known as “root polishing”, which involves using the solutions from the eigenvalue problem as initial guesses and performing Newton–Raphson iterations [99]. However, in this work, this wasn’t necessary due to the relatively low dimension of the PEP.