# On the Effect of Friction on Tibiofemoral Joint Kinematics
## 2 An Overview on Friction Models
This modification, which lets the new model be valid for all \(\mathbf{v}_{t}\) from a computational viewpoint, is given by  
\[\mathbf{F}=-\mu_{c}\|\mathbf{F}_{n}\|\frac{\mathbf{v}_{t}}{\|\mathbf{v}_{t}\| }\text{tanh}(k\|\mathbf{v}_{t}\|) \tag{3}\]in which \(k\) is a coefficient that determines how fast the hyperbolic tangent function changes from near 0 to near +1, which can also be represented by \(\frac{\|\mathbf{v}_{t}\|}{v_{0}}\) where \(v_{0}\) depicts a chosen velocity at which one expects the tanh function to take a value close to its maximum. Another model uses a linear friction\(-\)velocity relationship at the proximity of zero velocity to smooth out the discontinuity, which can be expressed as follows [53]:  
\[\mathbf{F}=-\mu_{c}\|\mathbf{F}_{n}\|\frac{\mathbf{v}_{t}}{\|\mathbf{v}_{t}\|} \text{min}(k_{sat}\|\mathbf{v}_{t}\|,1) \tag{4}\]  
in which \(k_{sat}\) is the slope of linear function between zero velocity and \(1/\|\mathbf{v}_{t}\|\) to eliminate the difficulty in determining the friction force at zero sliding speed. Obviously, the friction force at zero relative velocity between the interacting bodies is not zero and is equal to the tangential applied force. Hence, the main drawback of the modified Coulomb friction laws explained above is that they assume zero friction at zero relative velocity, which does not make sense from a physics viewpoint. With the study of lubrication by [PERSON] and [PERSON], and the classical paper on theoretical modeling of fluid-film lubrication, which [PERSON] published [54], the following expression for the viscous friction can be written  
\[\mathbf{F}=-k_{v}\mathbf{v}_{t} \tag{5}\]  
in which \(k_{v}\) is the viscous coefficient. In full-film contacts, the viscous model can be assumed a good representation of behavior of the system. Moreover, the model can act well enough in other contact conditions provided that the viscous coefficient is tuned. Sometimes a better fit to experimental data can be achieved by assuming a nonlinear term of velocity in Equation (5). As such, Equation (5) can be recast as follows [24]:  
\[\mathbf{F}=-k_{v}\|\mathbf{v}_{t}\|^{\delta_{v}}\frac{\mathbf{v}_{t}}{\| \mathbf{v}_{t}\|} \tag{6}\]  
where the power \(\delta_{v}\) depends on the geometry of bodies in relative motion. The viscous friction term can be superimposed on the other friction models such as the Coulomb friction [53], which results in, for example, the following formulation. \[\mathbf{F}=\left\{ \begin{array}{cc}-\mu_{c}\|\mathbf{F}_{n}\|\frac{\mathbf{ v}_{t}}{\|\mathbf{v}_{t}\|}-k_{v}\mathbf{v}_{t}&\|\mathbf{v}_{t}\|>0\\ -\mathbf{F}_{e}:\text{max}(\|\mathbf{F}_{e}\|)=\mu_{c}\|\mathbf{F}_{n}\|&\| \mathbf{v}_{t}\|=0\end{array} \right. \tag{7}\]  
The Coulomb friction law, and its modified versions, do not explain the negative damping characteristic of friction [11], which was observed by [PERSON] as a continuous nonlinear velocity-dependent phenomenon called the Stribeck effect [55].This dependency of friction on the relative sliding velocity was also observed by [PERSON] [56] and [PERSON] [57] experimentally. The general form of the [PERSON] friction model can be written as follows:  
\[\mathbf{F}=\left\{ \begin{array}{cc}-\mu(\|\mathbf{v}_{t}\|)\|\mathbf{F}_{n} \|\frac{\mathbf{v}_{t}}{\|\mathbf{v}_{t}\|}&\|\mathbf{v}_{t}\|>0\\ -\mathbf{F}_{e}:\text{max}(\|\mathbf{F}_{e}\|)=\mu_{s}\|\mathbf{F}_{n}\|&\| \mathbf{v}_{t}\|=0\end{array} \right. \tag{8}\]  
where \(\mu(\|\mathbf{v}_{t}\|)\) is a friction coefficient function for which multiple formulations are available in the literature [58] and one of the common forms of which is given below. \[\mu(\|\mathbf{v}_{t}\|)=\mu_{c}+(\mu_{s}-\mu_{c})e^{-(\frac{\|\mathbf{v}_{t} \|}{v_{Q}})^{\delta_{Q}}} \tag{9}\]  
Here, \(v_{Q}\) is the Stribeck velocity, and \(\delta_{Q}\) is an exponent that depends on the geometry of the application. It is worth noting that \(\mu_{c}\) depicts both the Coulomb friction coefficient and kinetic friction coefficient in this paper while being recognizable according to the context.