# On the Effect of Friction on Tibiofemoral Joint Kinematics
## 2 An Overview on Friction Models
Friction parameters in Equation (9) can be obtained by performing the friction test for motions with constant velocity. Although the Stribeck friction model is velocity-dependent, this model can be considered a steady friction model. Moreover, the reason for decreasing friction with increasing speed in dry sliding metallic bodies was experimentally investigated and was found to be due to the material softening as a result of high temperatures generated at the contact surfaces [59; 60]. Moreover, the friction in a lubricated contact decreases with increasing the relative velocity due to a decrease in the number of surface asperities being in contact until full-film lubrication is obtained. Afterward, the friction can gain a value, and either be constant, increase, or decrease with increasing the sliding speed due to viscous and thermal effects. In the case of an increase in friction when full-film lubrication is built, one can add the viscous friction presented in Equation (5) to the relationship Equation (9) to mimic the intuitive physics. The Stribeck model can provide a good representation of the friction between sliding surfaces. The discontinuity problem of the Coulomb friction model is a problem also observed in the Stribeck model. To cope with this issue, the same procedures taken for the Coulomb friction, like using a hyperbolic tangent function, are applicable here as well. [PERSON] and [PERSON] suggested a formulation, Equation (10), which is quite different from that in Equation (9), while resolving the aforementioned discontinuity issue [61]. \[\mu(\|\mathbf{v}_{t}\|)=\left\{ \begin{array}{cc}-\frac{c_{s}}{v_{0}^{2}}( \|\mathbf{v}_{t}\|-v_{0})^{2}+c_{s},&\|\mathbf{v}_{t}\|<v_{0}\\ \mu_{c}+(\mu_{s}-\mu_{c})\exp(-\xi(\|\mathbf{v}_{t}\|-v_{0}))&\|\mathbf{v}_{t} \|\geq v_{0}\end{array} \right. \tag{10}\]  
The first part of the above friction coefficient function uses a near-zero continuous curve to avoid divergence of the numerical model. The second term is the Stribeck friction relation. In addition, \(\xi>0\) is the negative slope of the sliding state [62]. After the friction coefficient starts from zero, it increases to peak friction, which [PERSON] and [PERSON] [61] referred to as the static friction coefficient, \(\mu_{s}\). The friction coefficient then reduces with increasing tangential velocity until the friction finally reaches a steady state, i.e., \(\mu_{c}\). As was discussed previously, the aforementioned models are not efficient to determine friction magnitude when the velocity is zero nor to manage the stiction phenomenon. [PERSON] introduced a concept of the dead-velocity band of the zero proximity, i.e., \(\|\mathbf{v}_{t}\|<v_{D}\), to cope with the difficulties encountered to detect zero velocity and prevent switching between multi-state relationships to transit between stick, stick-slip and pure slip phases of the friction [63]. Within the abovementioned velocity interval, the speed is assumed to be null. The friction phase also is stiction, where the friction force can be equal to either tangential external force or static friction. For the velocities outside of that dead band, the friction force can get the form of Coulomb friction, Stibeck friction, and so forth. The following equation can express this idea. \[\mathbf{F}=\left\{ \begin{array}{cc}-\text{min}(\|\mathbf{F}_{e}\|,\ \|\mathbf{F}_{s}\|)\frac{\mathbf{F}_{e}}{\|\mathbf{F}_{e}\|}&\|\mathbf{v}_{t} \|<v_{D}\\ \mathbf{F}(\mathbf{v}_{t}),&\|\mathbf{v}_{t}\|\geq v_{D}\end{array} \right. \tag{11}\]  
However, the concept [PERSON] used to introduce this friction model does not comply with friction physics. Moreover, this model requires the external load as an input, which is often not available explicitly and is an internal state of the dynamical system. In addition to the phenomenological friction models, there are empirical friction models proposed for UHMWPE in the literature. [PERSON] et al. [26] reported the following formula derived from experimental data, showing a significant correlation to the contact stress.