# Deep Learning Methods to Analyze the Forces and Torques in Joints Motion
## 2 Data and Modeling Details
### A Mechanical-Based Force and Torque Analysis  
In the context of mechanical modeling, the human body (Figure 2a) in motion is represented as a simplified mechanism comprising interconnected rods and pivots. As shown in Figure 2b, the two shins (feet included) are represented by rods AB and EF, while the two thighs are represented by rods BC and DE. In the simplification, the pelvis, torso, head, and arms are collectively represented as a single component, denoted as CD. This simplification allows for a more streamlined analysis of the mechanical interactions between these five segments, which are interconnected at their respective joints, denoted as A, B, C, D, E and F. The mass and inertia of each component are considered to be proportional to a player's height. As shown in Figure 2b, the simplified model for a lunging player is based on two approximations. First, we consider the player's motion to be two-dimensional in a vertical plane. Second, since we focus on the significant mechanical impacts of lunging (or abrupt stopping), we consider that the friction between the propelling foot and the ground is large enough to prevent relative motion. In the lunging in Figure 2c, the five rods can be picked out and perform force analysis. In mechanical derivations, each rod keeps stationary in its own reference frame with the origin attached to the left end of the rod. Thus, rod AB lies stationary in its reference frame attached to point A. There are four forces exerted to rot AB, including three real forces, \(\mathbf{f}_{A}\), \(\mathbf{f}_{B}\) and \(\mathbf{G}_{AB}\) and a virtual force \(\mathbf{F}_{AB}\). Thus, the net force is 0:  
\[\mathbf{f}_{A}+\mathbf{f}_{B}+\mathbf{G}_{AB}+\mathbf{F}_{AB}=\mathbf{0} \tag{2}\]  
Bold letters represent vectors. \(\mathbf{f}_{A}\), \(\mathbf{f}_{B}\) are the forces at joint A and B. \(\mathbf{G}_{AB}\) is the gravity of rod AB. \(\mathbf{F}_{AB}\) is the inertia force of rod AB due to the selection of the non-inertia reference frame. For the calculation of the moment of the rod AB, point A is taken as the reference point. Thus, \(\mathbf{f}_{A}\) has no contribution to the total torque. Besides the three torques from \(\mathbf{f}_{B}\), \(\mathbf{G}_{AB}\) and \(\mathbf{F}_{AB}\), there are two driving torques, \(T_{A}\) and \(T_{B}\), for the contracting/stretching of the ankle and the knee and a virtual torque \(M_{AB}\) due to the selection of the reference frame with respect to point A. The six torques build a rotational equilibrium. \[T_{f_{B}}+T_{G_{AB}}+T_{F_{AB}}+T_{A}+T_{B}+M_{AB}=0. \tag{3}\]  
Figure 2: (**a**) A snapshot with players attacking each other with critical points identified by the Regional Convolutional Neural Network algorithm. (**b**) The lower body of each player can be simplified as a joined stick model for mechanical analysis. Letters A, B, C, D, E and F are the joint points. (**c**) Associated force diagrams of each stick with forces denoted. The virtual forces are denoted as dashed arrows. The directions of forces are not the actual ones and should be solved dynamically in each snapshot. The inertia force and the inertia torque can be calculated as  
\[\mathbf{F}_{AB} = -m_{AB}\bigg{(}\mathbf{a}_{A}+\frac{1}{2}\alpha_{AB}\mathbf{Q} \mathbf{r}_{AB}-\frac{1}{2}\omega_{AB}^{2}\mathbf{r}_{AB}\bigg{)}\, \tag{4}\] \[M_{AB} = -I_{A,AB}\alpha_{AB}-\frac{1}{2}m_{AB}\mathbf{a}_{A}^{\top} \mathbf{Q}\mathbf{r}_{AB}\, \tag{5}\]  
where \(\mathbf{a}_{A}\) is the linear acceleration with respect to the ground. \(\alpha_{AB}\) and \(\omega_{AB}\) are the angular acceleration and angular velocity of rod AB with respect to point A. \(m_{AB}\) is the mass of rod AB while \(I_{A,AB}\) is the inertia of rod AB with respect to point A. The inertias and masses are calculated by assuming that the player's body is the average body defined in China National Standard [19] (details in Appendix C). Additionally, to make the forces and torques comparable across players, all players are assumed to be 80 kg in weight and 1.8 m in height. \(\mathbf{Q}=