# Estimation of Tibiofemoral and Patellofemoral Joint Forces during Squatting and Kneeling
## 2 Materials and Methods
### Analytical Model  
A 3-D model of the lower limb consisting of the three segments namely foot, shank, and thigh was configured for calculation of the knee joint forces arising during kneeling and squatting tasks (Figure 2). The sagittal plane was defined by the centers of the ankle, knee, and hip joints. The static knee joint axis was aligned with the line passing through the most prominent point of the lateral and the medial epicondyles. The midpoint between the most prominent point of the lateral and the medial epicondyles is referred to as the knee joint center. The ankle joint center was defined by the midpoint between the malleoli, and the hip joint center was derived by a procedure based on the geometry of the pelvic iliac spines according to the _Plug-in Gait_ module (version 1.7, Oxford Metrics Group, Oxford, UK). The sagittal planes of the ankle and knee joints were defined by the planes orthogonal to the medial-lateral axes between the malleoli and the epicondyles, respectively, fixed at the respective joint center. The mass and inertia properties were derived with respect to the segment length and body mass from the data of [PERSON] [25]. The model accounts for different external contact forces, such as the ground reaction force on the foot and, in the case of kneeling, on the proximal part of the shank. The resultant force due to thigh-calf contact during deep squatting or deep kneeling was also considered and applied to the shank-foot system. Besides the static knee joint center, the instantaneous axis of rotation of the knee joint was derived from the guiding of the cruciate ligaments during knee flexion according to a crossed four-bar linkage [26]. All further kinetic calculations were based on this instantaneous center of rotation. The orientation of the patellar ligament and the quadriceps tendon in the sagittal plane of the knee joint were derived by linearization and enhanced by linear extrapolation from the model of [PERSON] et al. [27] and [PERSON]'s data [28] (Figure 3). The variation in the moment arm length of the patellar ligament with respect to the knee joint flexion was derived from the data from [PERSON] et al. [29] by piecewise linear approximation (Figure 4). The hamstring muscle group was introduced as being representative of all the knee flexors with a constant moment arm length (\(r_{\rm{ht}}\) = 35 mm) acting parallel to the femoral longitudinal axis [20; 29]. Figure 2: Segments model of the lower limb and applied forces–foot ground contact (\(F_{\rm{f}}\)), knee ground contact (\(F_{\rm{t}}\)), thigh-calf contact (\(F_{\rm{tc}}\)), patellar ligament (\(F_{\rm{pl}}\)), quadriceps tendon (\(F_{\rm{qt}}\)), hamstrings tendon (\(F_{\rm{ht}}\)), tibiofemoral joint (\(F_{\rm{tf}}\)), and patellofemoral joint (\(F_{\rm{pf}}\)). The inertial and gravitational forces are not explicitly depicted. The local segment coordinate systems of the tibia and the femur are indicated by (\(x_{\rm{t}}\), \(y_{\rm{t}}\), \(z_{\rm{t}}\)) and (\(x_{\rm{f}}\), \(y_{\rm{t}}\), \(z_{\rm{t}}\)), respectively. The knee joint moment was calculated by considering the moments of the ground reaction forces, of the thigh-calf contact force, of the gravitational and inertial forces of the shank, and of the foot. The knee-ground contact force was treated as a simple two case scenario due to the absence of further information on the areal force distribution. The knee-ground contact force was either applied entirely to the shank where its moment, with respect to the knee joint center, acted in the flexion direction; or it was applied entirely to the patella in terms of a worst-case assumption. Assuming no antagonistic co-activation of the muscles, the flexion/extension knee joint moment was attributed to the patellar ligament, or the hamstring muscles, depending upon its sign.