>[!quote] In a Nutshell >Approaches to model the friction term of a rigid body moving in euclidean space or for [[Robotic Joints|robot joints]] in [[Generalized Coordinates, Configuration or Joint Space|configuration space]]. The following formulations are based on the #Robotics notation in [[- Robotics, Dynamics and Control -]] and therefore relate to robot joint torques. For the euclidean case, simply switch out all torques with force components in $3$-dimensional space. --- #### Coulomb Friction Simplest model, only uses a single parameter for statis and dynamic friction. - **Static friction (stiction):** When the velocity is zero, the friction force can take any value within a bounded interval to prevent motion. - **Dynamic (kinetic) friction:** When the velocity is nonzero, the friction force has a constant magnitude and opposes the direction of motion. >[!info] Coulomb Friction Model >Mathematically, the torque resulting from friction in a joint based on the Coulomb model is expressed as$\boldsymbol{\tau}_\text{fric,comp,i}(\dot{q}_i)=\cases{c^{+}_\text{Coulomb,i}, \quad \dot{q}_i >0 \\ c^{-}_\text{Coulomb,i}, \quad \dot{q}_i <0}.$ --- #### Coulomb Friction with Viscous Friction Adds a linear proportionality to the velocity, resulting in a damping term. Especially useful in systems involving fluids. >[!info] Coulomb Model with Viscous Friction >Mathematically, it extends the above to$\boldsymbol{\tau}_\text{fric,comp,i}(\dot{q}_i)=\cases{c^{+}_\text{Coulomb,i}+c^{+}_\text{visc,i}\cdot \dot{q}_i, \quad \dot{q}_i >0 \\ c^{-}_\text{Coulomb,i}+c^{-}_\text{visc,i}\cdot \dot{q}_i, \quad \dot{q}_i <0}.$ --- #### Stribeck Friction Accounts for the Stribeck effect, i.e. the observation that friction rapidly decreases when joints begin moving. >[!info] Stribeck Friction Model >Mathematically, this is achieved by adding an exponentially decaying term via $\tau_{\text{stribeck},i}(\dot{q}_i)= \begin{cases} \left[c^{s,+}_\text{Coulomb,i} - c^{\infty,+}_\text{Coulomb,i}\right] \exp\left(-\left|\dfrac{\dot{q}_i}{v_{s,i}}\right|^\delta\right) + c^{\infty,+}_\text{Coulomb,i} + c^{+}_\text{visc,i}\cdot \dot{q}_i, & \dot{q}_i > 0, \\[2mm] \left[c^{s,-}_\text{Coulomb,i} - c^{\infty,-}_\text{Coulomb,i}\right] \exp\left(-\left|\dfrac{\dot{q}_i}{v_{s,i}}\right|^\delta\right) + c^{\infty,-}_\text{Coulomb,i} + c^{-}_\text{visc,i}\cdot \dot{q}_i, & \dot{q}_i < 0, \end{cases},$where the upper index $\infty$ denotes asymptotic friction components, $v_{s,i}$ is a characteristic velocity and $\delta$ is a shaping parameter. --- Potentially add: - LuGre Model - Dahl Model