# On the Effect of Friction on Tibiofemoral Joint Kinematics
## 5 Dynamics Solver
### Dynamics Modeling  
The governing equations of the system motion are derived based on the free body diagram of the tibofemoral joint represented in the second right-hand panel of Figure 1. The resulting forces and moments applied to the femoral bone due to the existence of all surrounding soft tissues such as muscles and hip-joint reactions are computed by means of musculoskeletal modeling. Such loadings and moments are moved to the origin of the femur coordinate system and written in the global coordinate system, illustrated in the second right-hand panel of Figure 1, as \(\mathbf{F}_{ext}\) and \(\mathbf{M}_{ext}\). The loads and moments owing to the contact, friction, and ligaments are also determined and summed up to those external ones. The governing equations of the translational motion of the femoral component, therefore, can be formulated using [PERSON]'s second law of motion as follows [69; 82]:  
\[M\ddot{\mathbf{X}}=\mathbf{F}_{T},\mathbf{F}_{T}=\mathbf{F}_{ext}+\mathbf{F}_{ cont}+\mathbf{F}_{\mu}+\mathbf{F}_{lig} \tag{25}\]  
in which the mass of the femoral bone is designated by \(M\). The translational acceleration vector of the femur is also depicted by \(\ddot{\mathbf{X}}\). Furthermore, \(\mathbf{F}_{T}\) stands for the vectorial summation of forces imposed on the bone, which is mathematically formulated in Equation (25). In this equation, the external forces are depoicted by \(\mathbf{F}_{ext}\), resultant contact forces \(\mathbf{F}_{cont}\), friction force \(\mathbf{F}_{\mu}\), and finally ligaments loads are designated by \(\mathbf{F}_{lig}\). In this solution setup, three coordinate systems are defined, one of which is the global coordinate system (\(XYZ\)) attached to the underneath of the tibial insert and on the top of the tibial tray. It is worth mentioning that the tibia is considered stationary, while the femur has six degrees of freedom to translate and rotate. One degree of freedom of this dynamic system is, later, constrained while knowing the flexion-extension angle of the knee joint over a level walking gait. The other coordinate system is a local system (\(xyz\)) attached to the STL center of the femoral component, rotating and translating with the femur. However, the third coordinate system, \(\zeta\eta\zeta\), is considered with the same origin as the femoral coordinate system, although it does not rotate with the femur and it has bases aligned with those of the global coordinate system. The angular-momentum equation can now be written in the body coordinate system of the femur, which governs the rotational motion of the femur as follows [69]:  
\[\mathbf{M}_{\mathcal{C}}^{*}=\mathbf{I}^{\mathbf{C}*}\dot{\mathbf{\omega}}^{*}+ \mathbf{\tilde{\omega}}^{*}\mathbf{I}^{\mathbf{C}*}\mathbf{\omega}^{*} \tag{26}\]  
where \(\mathbf{M}_{\mathcal{C}}^{*}\) is the summation of not only external moments but also those resulted from the contact, friction and ligaments, which are computed with respect to the body coordinate system of the femoral part (\(xyz\)). Compared to the computation of moments in the coordinate system \(xyz\), it is much easier to obtain moments in the third coordinate system discussed above. The summation of moments in the third coordinate system is designated by \(\mathbf{M}_{\mathcal{C}}\) and is related to \(\mathbf{M}_{\mathcal{C}}^{*}\) by the following relationship:  
\[\mathbf{M}_{\mathcal{C}}^{*}=\mathbf{R}^{T}\mathbf{M}_{\mathcal{C}} \tag{27}\]  
in which \(\mathbf{R}^{T}\) is the transpose of the rotation tensor of the femur, \(\mathbf{I}^{\mathbf{C}*}\) and \(\mathbf{\omega}^{*}\) appeared in Equation (26) and stand for the tensor of inertia and the angular velocity vector of the femoral component in the body coordinate system, respectively. In the dynamic model developed in this study, the rotation tensor is defined based on the Euler angles of the \(z\)-\(x\)-\(y\) sequence, given by the array of Euler angles \(q^{T}=\{\phi,\,\theta,\psi\}\) to characterize the rotation of the femur as well as the location of nodes on the femoral-bearing surface.