# Linearized Friction

Coulomb friction instantaneously maximizes the dissipation of kinetic energy between two objects in contact.

### Mathematical Model

For a single contact point, this physical phenomenon can be modeled by the following optimization problem,

\[\begin{align*} \underset{b}{\text{minimize}} & \quad v^T b \\ \text{subject to} & \quad \|b\|_2 \leq \mu \gamma, \end{align*}\]

where $v \in \mathbf{R}^{2}$ is the tangential velocity at the contact point, $b \in \mathbf{R}^2$ is the friction force, and $\mu \in \mathbf{R}_{+}$ is the coefficient of friction between the two objects.

### Linearized Model

This above problem is naturally a convex second-order cone program, and can be efficiently and reliably solved. However, classically, an approximate version:

\[\begin{align*} \underset{\beta}{\text{minimize}} & \quad [v^T -v^T] \beta, \\ \text{subject to} & \quad \beta^T \mathbf{1} \leq \mu \gamma, \\ & \quad \beta \geq 0, \end{align*}\]

which satisfies the LCP formulation, is instead solved. Here, the friction cone is linearized and the friction vector, $\beta \in \mathbf{R}^{4}$, is correspondingly overparameterized and subject to additional non-negative constraints.

The optimality conditions of the above problem and constraints used in the LCP are:

\[\begin{align*} [v^T -v^T]^T + \psi \mathbf{1} - \eta &= 0, \\ \mu \gamma -\beta^T \textbf{1} & \geq 0,\\ \psi \cdot (\mu \gamma - \beta^T \textbf{1}) &= 0, \\ \beta \circ \eta &= 0, \\ \beta, \psi, \eta &\geq 0, \end{align*}\]

where $\psi \in \mathbf{R}$ and $\eta \in \mathbf{R}^{4}$ are the dual variables associated with the friction cone and positivity constraints, respectively, and $\textbf{1}$ is a vector of ones.