# Friction Modeling

Friction is an inevitable component of almost any mechanical system. The complete elimination of friction tends to be extremely costly and challenging to implement. Thus, the accurate modeling of friction is typically necessary for effective control design.

## Basic Friction Models

In many systems, the friction is acceptably representable with simply a static and a viscous friction component. In these cases, the friction coefficients are often considered as constants due to uniform loading in the case of static friction and linear representability of the viscous friction.

### Simple Static Friction

The simplified force due to static friction in a case where the normal force is constant may be represented by a discontinuous sign function with an amplitude equal to the measured constant static friction (F_{sc}). As the friction acts against the direction of travel, the force is thus negative.

${\displaystyle F_{s}=-F_{sc}\ sign\left({\frac {dx}{dt}}\right)}$

Due to the discrete nature of most simulations, the discontinuous function may cause numerical issues in the solver, thus a hyperbolic tangent function often offers better results. The width of the hyperbolic tangent behavior may be set by the magnitude of the variable ${\displaystyle w}$.

${\displaystyle F_{s}=-F_{sc}\ tanh\left({\frac {3}{w}}{\frac {dx}{dt}}\right)}$

### Simple Viscous Friction

${\displaystyle F_{v}=-B_{v}{\dot {x}}}$