Coulomb friction instantaneously maximizes the dissipation of kinetic energy between two objects in contact.
For a single contact point, this physical phenomenon can be modeled by the following optimization problem,
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.
This above problem is naturally a convex second-order cone program, and can be efficiently and reliably solved. However, classically, an approximate version:
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:
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.