# On the Effect of Friction on Tibiofemoral Joint Kinematics
## 2 An Overview on Friction Models  
Friction is generally described as the resistance to motion when two surfaces slide against each other. Friction between solid bodies is an extremely complicated physical phenomenon encompassing elastic and plastic deformations of the articulating bodies, chemical reactions, interactions with wear particles, micro-fractures and cracks, excitation of electrons, and the transfer of particles from one body to another. However, it is possible to represent this complex phenomenon using either simple friction laws with a couple of friction parameters or more sophisticated friction models that commonly require determining a larger number of parameters experimentally. One of the first and simplest friction laws is the Coulomb friction law. [PERSON] (1736-1806) determined that the frictional force between two bodies pressed together with normal force \(\|\mathbf{F}_{n}\|\) can be calculated by multiplying the normal force and friction coefficient that can be computed empirically [21]. The relation between the applied load and relative velocity has the following mathematical form:  
\[\mathbf{F}=\left\{ \begin{array}{cc}-\mu_{c}\|\mathbf{F}_{n}\|\frac{\mathbf{ v}_{t}}{\|\mathbf{v}_{t}\|}&\|\mathbf{v}_{t}\|>0\\ -\mathbf{F}_{e}:\max(\|\mathbf{F}_{e}\|)=\mu_{c}\|\mathbf{F}_{n}\|&\|\mathbf{ v}_{t}\|=0\end{array} \right. \tag{1}\]  
where \(\mathbf{F}\) is the friction force, \(\mathbf{v}_{t}\) the tangential sliding speed, \(\mu_{c}\) is the Coulomb friction coefficient, and finally \(\mathbf{F}_{e}\) is external, tangential force. Using this model to deal with friction in dynamic systems leads to a computational burden due to the discontinuity at zero velocity. The friction force can reach any value between \(-\mu_{c}\|\mathbf{F}_{n}\|\) and \(\mu_{c}\|\mathbf{F}_{n}\|\) due to a sudden alteration in the velocity direction at zero speed. From a physical viewpoint, knowing the external force, imposed on the sliding bodies, can resolve this issue as the friction load is set equal to the externally applied force according to [PERSON]'s first law. Although obtaining the external load is experimentally possible at zero velocity, this is not straightforward to determine in numerical simulations. It is worth noting that the Coulomb friction model only needs one input parameter, i.e., \(\mu_{c}\), which is why both academic and industrial communities commonly use this model for systems that do not require very precise quantifications. Another issue with the Coulomb friction law is that it does not capture the distinction between static and kinematic friction. It is well-known that the external force required to make bodies in contact move from their rest status is higher than that when they are in relative motion. Therefore, the Coulomb friction law is modified by replacing the Coulomb coefficient in Equation (1) with a static friction coefficient, i.e., \(\mu_{s}\) (\(\mu_{s}>\mu_{c}\)), which is sometimes called Coulomb friction with stiction. \[\mathbf{F}=\left\{ \begin{array}{cc}-\mu_{c}\|\mathbf{F}_{n}\|\frac{\mathbf{ v}_{t}}{\|\mathbf{v}_{t}\|}&\|\mathbf{v}_{t}\|>0\\ -\mathbf{F}_{e}:\max(\|\mathbf{F}_{e}\|)=\mu_{s}\|\mathbf{F}_{n}\|&\|\mathbf{ v}_{t}\|=0\end{array} \right. \tag{2}\]  
The discontinuity issue at zero velocity is still present in this model, making it hard to detect stick and stick-slip phases of friction between bodies in contact. Moreover, the transition from the stick friction phase to the slip one is not smooth and continuous. To resolve these issues, smooth and continuous Coulomb friction models were developed to prevent the computational burden caused by the force discontinuity by introducing a smooth curve around zero velocity. To this end, several continuous functions have been adopted in the literature, like linear, exponential, and trigonometric function types. One such model is considered here that can be obtained by multiplying the hyperbolic tangent function of velocity to the Coulomb friction model, Equation (1) [53].