# On the Effect of Friction on Tibiofemoral Joint Kinematics
## 1 Introduction
The first mathematical relationship to computing friction force was formulated by a French engineer, [PERSON] (1736-1806), stating that friction is independent of the magnitude of relative velocity between sliding bodies [21]. In his formulation, the friction force is set equal to the applied load multiplied by a coefficient that is the so-called Coulomb coefficient, still being used by both academia and industrial engineers. However, such a model cannot address some underlying physics associated with the friction phenomenon. In the last decades, a great deal of research has been devoted to advancing phenomenological friction models. Such models have aimed at addressing the distinction between the static and kinematic friction, friction phases (stick, stick-slip, and pure slip), velocity-dependent property of friction, break-away force (pre-sliding displacement), and frictional lag [24; 25]. In addition to the variety of friction models that address the above phenomena, the friction of UHMWPE, of which the tibial insert of total knee arthroplasty is made, demonstrates a dependency on contact pressure. In 2001, [PERSON] et al., performed a series of experimental tests using a ball-and-socket test rig and found that friction decreases with increasing contact stress [26]. Following their work, [PERSON] in 2006 reported the same finding while performing pin-on-disc friction tests [27]. Both the above research studies found a significant correlation of the friction coefficient to the maximum contact pressure and reported associated empirical friction equations. In 2017, [PERSON] et al., used a ball-on-disc test setup to obtain the friction coefficient of total knee arthroplasty during a walking gait cycle. Subsequently, they proposed a regression model to represent experimental data mathematically [28]. In 2018, [PERSON] et al., continued their research work while conducting a series of experiments to develop prediction models of the friction coefficient between the metallic ball and UHMWPE disk lubricated with diluted bovine serum [29]. Regression parameters for a suggested model that fits well in experimental data were determined for the stance and swing phases separately. They later successfully proposed a formulation that represents the experimental data well to address the behavior of the friction coefficient over the transition from the mixed to the full-film lubrication regime during a walking gait cycle [30]. The fitted model relates the friction coefficient to both contact stress and relative velocity. As the frictional events occur at the atomic level and are of multi-physics nature owing to interactions of chemical, electromagnetic, and mechanical processes, some frictional phenomena are yet to be understood and mathematically formulated despite great achievements so far [19; 31]. Friction models are to be integrated into algorithms responsible for modeling dynamics of the knee joint. [PERSON] and [PERSON] (1997) investigated the dynamics of total knee replacement under multiple friction coefficients while they used the Coulomb law to compute friction forces [32]. Much of other research work studying the tribology of TKA commonly assumed a fixed friction coefficient, i.e., 0.04, for the whole numerical simulation. [PERSON] and [PERSON] reported that the friction coefficient in artificial human joints is in the range of 0.03-0.10 [33]. However, other studies obtained higher ranges of the friction coefficient for artificial human joints, for example, 0.05-0.33 [27], 0.06-0.25 [26], and 0.06-0.17 [28; 29]. Focusing on the dynamic modeling of total knee arthroplasty (TKA), some investigations have aimed at developing anatomy-based dynamic models to consider the kinetic and kinematic behavior of TKAs like quasi-static models [34; 35; 36; 37; 38; 39; 40; 41], 2D dynamic approaches [42; 43; 44; 45; 46; 47; 48], and sophisticated 3D mathematical dynamic solutions [49; 50; 51]. One of the very first successful attempts to determine the 3D dynamic solution of the knee joint was carried out by [PERSON] and [PERSON] [49].