A complementarity-based rolling friction model for rigid contacts

Abstract

In this work (also, preprint ANL/MCS-P3020-0812, Argonne National Laboratory) we introduce a complementarity-based rolling friction model to characterize dissipative phenomena at the interface between moving parts. Since the formulation is based on differential inclusions, the model fits well in the context of nonsmooth dynamics, and it does not require short integration timesteps. The method encompasses a rolling resistance limit for static cases, similar to what happens for sliding friction; this is a simple yet efficient approach to problems involving transitions from rolling to resting, and vice-versa. We propose a convex relaxation of the formulation in order to achieve algorithmic robustness and stability; moreover, we show the side effects of the convexification. A natural application of the model is the dynamics of granular materials, because of the high computational efficiency and the need for only a small set of parameters. In particular, when used as a micromechanical model for rolling resistance between granular particles, the model can provide an alternative way to capture the effect of irregular shapes. Other applications can be related to real-time simulations of rolling parts in bearing and guideways, as shown in examples.

Appendices

Appendix A: Kinematics of rolling in three-dimensional space

Let $\mathbf {\omega}_{A,W}^{(W)}$ and $\mathbf {\omega}_{B,W}^{(W)}$ denote the angular velocities of two bodies A and B, relative to the absolute reference frame W and expressed in the basis of the frame (W). We assume A and B to be rigid or with negligible deformations. Introducing the rotation matrix R _A,W ∈SO(ℝ,3) that represents the rotation of A respect to W, we have $\mathbf {\omega}_{A,W}^{(W)} = {R}_{A,W} \mathbf {\omega}_{A,W}^{(A)}$ and $\mathbf {\omega}_{B,W}^{(W)} = {R}_{B,W} \mathbf {\omega}_{B,W}^{(A)}$.

Let the unimodular quaternion $\mathbf {\varepsilon }_{A,W} \in \mathbb{H}_{1}$ represent the rotation of the frame A respect to absolute frame W. We recall that, for unimodular quaternions, the inverse $\mathbf {\varepsilon }^{-1}$ is also the conjugate $\mathbf {\varepsilon }^{*}$. We also recall that it is possible to compute R from $\mathbf {\varepsilon }$ and vice versa.

Thanks to a property of quaternion algebra [13] the relative rotation of two references is

By performing differentiation respect to time, we get

From the result in [27], the quaternion derivative can be transformed in angular velocity, using pure quaternions:

(47)

(48)

Since $( \mathbf {\varepsilon }_{1}^{*} \mathbf {\varepsilon }_{2})^{*} = \mathbf {\varepsilon }_{2}^{*} \mathbf {\varepsilon }_{1}$, and remembering that $\mathbf {\varepsilon } \mathbf {\varepsilon }^{*} =1$, we can develop Eq. (48) into

(49)

The product $\dot{ \mathbf {\varepsilon }}_{B,W} \mathbf {\varepsilon }_{B,W}^{*}$ in the second term of the summation can be replaced with the pure quaternion $\frac{1}{2}[0, \mathbf {\omega}_{B,W}^{(W)}]$ using Eq. (47). Also, the first term can be premultiplied by $\mathbf {\varepsilon }_{A,W}^{*} \mathbf {\varepsilon }_{A,W} =1$, becoming $2 \mathbf {\varepsilon }_{A,W}^{*} \mathbf {\varepsilon }_{A,W} \dot{ \mathbf {\varepsilon }}_{A,W}^{*} \mathbf {\varepsilon }_{A,W}$; here the product between the second and third quaternion can be replaced with the pure quaternion $\frac{1}{2}[0, \mathbf {\omega}_{A,W}^{(W)}]^{*}$, again using Eq. (47). Thus we have

(50)

A rotation in 3D space of the vector part of a pure quaternion can be obtained with unitary quaternions, that is, $[0,\mathbf {v}^{(W)}]= \mathbf {\varepsilon }_{A,W} [0,\mathbf {v}^{(A)}] \mathbf {\varepsilon }_{A,W}^{*}$.

Hence, recalling that $[0, \mathbf {\omega}_{A,W}^{(W)}]^{*}= -[0, \mathbf {\omega}_{A,W}^{(W)}]$ by the property of conjugate quaternions, we can rewrite Eq. (50) and obtain the expected result for relative angular velocity $\mathbf {\omega}_{r}$:

$$ \everymath{\displaystyle }\begin{array}{@{}rll} \bigl[0,\mathbf {\omega}_{BA}^{(A)}\bigr] &=& - \bigl[0, \mathbf { \omega}_{A,W}^{(A)}\bigr] + \bigl[0, \mathbf { \omega}_{B,W}^{(A)}\bigr] \cr\noalign{\vspace{3pt}} \mathbf {\omega}_{BA}^{(A)} &=& \mathbf {\omega}_{B,W}^{(A)} - \mathbf {\omega}_{A,W}^{(A)}. \end{array} $$

(51)

Appendix B: Formulation of D vectors

We assume that the vector of generalized velocities v contains the speeds of the centers of mass of the bodies $\dot{\mathbf {x}}^{(W)}$, expressed in absolute coordinates (W) and the angular velocities $\mathbf {\omega}^{(i)}$ expressed in the local coordinates of the ith body, as $\mathbf {v}= [ \dot{\mathbf {x}}^{(W)}_{1}, \mathbf {\omega}^{(1)}_{1}, \dot{\mathbf {x}}^{(W)}_{2}, \mathbf {\omega}^{(2)}_{2}, \ldots ]^{T}$.

Given a contact between a pair of two rigid bodies A and B, we define the positions of the two contact points with respect to the centers of mass, expressed in the coordinate systems of the two bodies, as $\mathbf {s}_{A}^{(A)}$ and $\mathbf {s}_{B}^{(B)}$. The absolute rotations of the coordinate systems of the bodies are ${R}_{A}^{(W)}, {R}_{B}^{(W)} \in \mathsf{SO}(\mathbb{R},3)$ and the absolute rotation of the contact plane is ${R}_{P}^{(W)} \in \mathsf{SO}(\mathbb{R},3) = [ \mathbf {n}, \mathbf {u}, \mathbf {w} ]$. Thus, the vectors $\mathbf {D}_{\gamma_{n}}$, $\mathbf {D}_{\gamma_{u}}$, $\mathbf {D}_{\gamma_{w}}$ can be computed as ${D}_{\gamma}= [ \mathbf {D}_{\gamma_{n}}, \mathbf {D}_{\gamma_{u}}, \mathbf {D}_{\gamma_{w}} ] \in \mathbb{R}^{3 \times m_{v}}$,

(52)

where ${ \tilde {s}}$ is the skew symmetric matrix such that ${ \tilde {s}}\mathbf {x} = \mathbf {s} \wedge \mathbf {x}$.

Similarly, recalling the result in Eq. (51), one can compute the vectors $\mathbf {D}_{\tau_{n}}$, $\mathbf {D}_{\tau_{u}}$, $\mathbf {D}_{\tau_{w}}$ as ${D}_{\tau}= [ \mathbf {D}_{\tau_{n}}, \mathbf {D}_{\tau_{u}}, \mathbf {D}_{\tau_{w}} ] \in \mathbb{R}^{3 \times m_{v}}$:

(53)

We remark that, because of the extreme sparsity of (52) and (53), only the following four 3×6 matrices need to be stored per each contact

(54)

(55)

(56)

(57)

Here we considered B as the reference body: otherwise, if A were the reference for contact coordinates, signs should be swapped in all terms in Eqs. (52)–(57).

Appendix C: Computing projections on intersections of cones

We describe the procedure to compute the Euclidean projection of a point x on an intersection of circular cones that have one common component (in the case studied here, that component is the normal force). We assume that a generic point x is structured as follows:

(58)

and that the m circular cones are second-order cones defined by

where μ _i >0, i=1,2,…,m. We are interested in computing the projection of a vector x on ⋂K _i , that is,

For example, in the case treated in this work, we are interested in simultaneous modeling of sliding, rolling, and spinning friction in three dimensional configurations. That is, we have three cones, m=3 and x is a six-dimensional vector, x=(γ _n ,γ _u ,γ _w ,τ _u ,τ _w ,τ _n ). The mapping (58) is the following: x ₀=γ _n , l ₁=(γ _u ,γ _w ), l ₂=(τ _u ,τ _w ), l ₃=τ _n . The friction coefficients are μ ₁=μ, μ ₂=ρ, μ ₃=σ.

The crucial observation that simplifies the computation of the projection is that the component $\tilde{l}_{i}$ of the projection $\tilde{x}$ must be collinear with l _i . Indeed, if this is not the case, then rotating $\tilde{l}_{i}$ over l _i will preserve feasibility but will necessarily reduce $\| x-\tilde{x} \|$, a contradiction. Therefore, there exists t _i such that $\tilde{l}_{i}=t_{i} l_{i}$. The optimization that defines the projection then becomes

We have normalized the component of y in terms of y ₀ to allow for the range of t _i to be the same. For a given y ₀, the optimal t _i , which we denote by t _i (y ₀), is easy to compute. Indeed we obtain the following

Substituting t _i for the optimal values t _i (y ₀) in the optimization problem, we obtain that the problem is equivalent to

Here I is the indicator function of a set. It is immediately apparent that this function is piecewise quadratic and that it is convex. Indeed, convexity follows from the fact that each term function is convex, the first term as a quadratic, and the other terms as their graphs are the union of a parabola with a flat line.

To find its optimum, we can do the following.

• 1. Define and order the breakpoints 0, and $\frac{\| l_{i} \|}{\mu_{i}}$, with i=1,2,…,m. Successive breakpoints define a piece.

• 2. On each piece find the minimum of the quadratic function.

• 3. Compute the overall minimum, which is the lowest value of all such minima.

Once $\tilde{x}_{0}=y_{0}$ is determined, t _i (y ₀) is computed, and the other components of the projection are computed as $\tilde{l}_{i}=t_{i}(\tilde{x}_{0})\frac{\mu_{i} \tilde{x}_{0}}{\| l_{i} \|}$.

For a large number of breakpoints we can exploit convexity of ψ, by noting that we can evaluate the function at the breakpoints, and find the minimum value. Then, by convexity, the overall minimum must occur in a segment that neighbors the breakpoint with the minimum value. Hence, one minimizes the quadratic only in those intervals.

To summarize:

• 1. Define and order the breakpoints 0, and $\frac{\| l_{i} \|}{\mu_{i}}$, with i=1,2,…,m. Successive breakpoints define a piece. We assume without loss of generality that the labels have been permuted so that the natural order has the breakpoints in increasing order, that is, i<j⇒ $\frac{\| l_{i} \|}{\mu_{i}} < \frac{\| l_{j} \|}{\mu_{j}}$. If two breakpoints have the same value, we delete their index.

• 2. Enumerate the objective function ψ at the breakpoints, and find the i for which $\psi(\frac{\| l_{i} \|}{\mu_{i}}) \leq \psi(\frac{\| l_{j} \|}{\mu_{j}})$, ∀j. If there is one such i, the overall minimum is on a neighboring segment; if there are two, it is on the segment in between (there cannot be three different indices, since the function is not piecewise constant).

• 3. Minimize the piecewise quadratic on either the one or two segments identified, and report the result.

For a small number of breakpoints (i.e., the number of cones m is small), it is not likely that this reduced method would practically be much faster than comprehensive enumeration.