# Two-Dimensional Mechanical Model of Human Stability in External Force-Caused Fall
## 4 Mathematical Model  
The current configuration of the considered biomechanical system is uniquely determined by 13 angles \(\varphi_{i}\) that are measured counterclockwise in the sagittal plane (i.e., YZ-plane). A rule was adopted in the paper that the directed angle \(\varphi_{i}\) is measured from the horizontal axis of the own local system, which is parallel to the Z axis, to the symmetry axis of the given segment. The time derivatives \(\varphi_{i}\) of the angles \(\varphi_{i}\) stand for the angular velocities of the segments, and the second derivatives \(\ddot{\varphi_{i}}\) are their angular accelerations. The Newton-Euler approach requires the introduction of quantities defining the kinematic state of the centres of mass of each body segment. Let \(y_{i}\) and \(z_{i}\) be the Cartesian coordinates of the mass centre \(C_{i}\) within the global coordinate system OXYZ. Differentiating the coordinates over time, we get the Cartesian components of the velocities of the mass centres. Differentiating twice yields the Cartesian components of the accelerations of the mass centres. Since the dimensions of model parts are described by angles and distances from the centre of mass of the current segment, the initial state of the examined person does not affect the results. Within the framework of the two-dimensional problem, we write three equations of motion for each of the parts of the biomechanical model. The first two of them result from the first Euler's law. The third equation, resulting from the second Euler's law, concerns the angular momentum around the mass centre. For feet whose free-body diagram is shown in Figure 4, the equations are as follows:  
\[m_{i}\ddot{z}_{i}=F_{z_{i}}+R_{z_{i}}, \tag{1}\]  
\[m_{i}\ddot{y}_{i}=F_{y_{i}}+R_{y_{i}}-Q_{i}, \tag{2}\]  
\[I_{i}\ddot{\varphi}_{i}=M_{i}+F_{y_{i}}l_{i}\cos\varphi_{i}-F_{z_{i}}l_{i}\sin \varphi_{i}+R_{y_{i}}l_{R_{i}}\cos\varphi_{R_{i}}-R_{z_{i}}l_{R_{i}}\sin \varphi_{R_{i}}, \tag{3}\]  
where \(i=1\) for the right foot and \(i=10\) for the left foot. The free-body diagrams for shanks, thighs, arms, and forearms differ only in symbol indices. Therefore, we present all these diagrams only once in Figure 5.  
Figure 4: Foot model sketch (identical for left and right foot). The symbols stand for: _i_—segment number, which takes the value of 1 or 11 (according to Figure 3), _l_—the distance between the centre of mass of the foot (\(C_{i}\)) and the ankle joint, _l_\({}_{R_{i}}\)– the distance between the centre \(C_{i}\) and the centre of pressure _COP_, \(\varphi_{R_{i}}\)—the directed angle that defines the location of the segment joining the centres of mass and pressure, and _R_\({}_{y_{i}}\), _R_\({}_{z_{i}}\)—components of the reaction forces. According to Figure 5, the equations of motion for the limbs segments have the following form:  
\[m_{i}\ddot{z}_{i}=-F_{z_{i-1}}+F_{z_{i}}, \tag{4}\]  
\[m_{i}\ddot{y}_{i}=-F_{y_{i-1}}+F_{y_{i}}-Q_{i}, \tag{5}\]  
\[I_{i}\ddot{\varphi}_{i}=M_{i}-M_{i-1}-F_{z_{i}}d_{i}\sin\varphi_{i}+F_{y_{i}}d_ {i}\cos\varphi_{i}-F_{z_{i-1}}(l_{i}-d_{i})\sin\varphi_{i}+F_{y_{i-1}}(l_{i}-d _{i})\cos\varphi_{i}, \tag{6}\]  
where \(i\in\{2,3,5,6,8,9,12,13\}\). Hands, as peripheral parts of the model, are characterized by a simple load scheme shown in Figure 6.  
The equations of motion for both segments representing hands take the following form:  
\[m_{i}\ddot{z}_{i}=F_{z_{i}}, \tag{7}\]  
\[m_{i}\ddot{y}_{i}=F_{y_{i}}+R_{y_{i}}-Q_{i}, \tag{2}\]0  
\[m_{i}\ddot{y}_{i}=F_{y_{i}}+R_{y_{i}}-Q_{i}, \tag{2}\]1  
\[m_{i}\ddot{y}_{i}=F_{y_{i}}+R_{y_{i}}-Q_{i}, \tag{2}\]2 Free-body diagram representative for the segments standing for shanks, thighs, arms, and forearms. The symbols used mean: \(i\)—segment number, which takes one of the following values: 2, 3, 5, 6, 8, 9, 12, 13 (according to Figure 3), \(l_{i}\)—length of the \(ith\) segment, and \(d_{i}\)—distance from the end of the \(ith\) segment, which connects to the \(i\)+1 segment, to the segment’s centre of mass (\(C_{i}\)).