# System for Evaluation and Compensation of Leg Length Discrepancy for Human Body Balancing
## 3 Mathematical Model of the System
### Force Plates  
The study of a human body's ability to maintain balance is often related to the CoP and CoM. While the CoP is a point defined in two dimensions and can be directly calculated with the help of force plates [28], the CoM is a point in three-dimensional space, and for the purposes of this paper, a projection of the CoM in the transverse plane is observed. As explained above, to calculate the CoP and CoM, two force plates, one per left and right leg, are used, each comprising force sensors (Figure 7).  
The left and right legs' CoPs are shown in the local frame of the left and right force plates. The parameters for force measuring in Figure 7 are:  
* \(a\) is the distance between the sensors along the \(x\)-axis;
* \(b\) is the distance between the sensors along the \(y\)-axis; and
* \(l\) is the distance between the origins of the left and right local force plate frame.  
In the local force plate frame, the CoP is defined with the following equation [29; 30]:  
\[\mathbf{CoP}_{j}=\left[\begin{array}{c}CoP_{jx}\\ CoP_{jy}\end{array}\right]=\frac{1}{\sum\limits_{i=1}^{4}F_{ji}}\sum\limits_{i= 1}^{4}\mathbf{e}_{ji}F_{ji} \tag{12}\]  
Figure 7: Left and right leg force plate frames.  
In Equation (12), \(F_{ji}\) (\(j=L,R\); \(i=1,2,3,4\)) is the force of each individual force plate sensor, whereas \(\mathbf{e}_{ji}\) is the position vector of each individual force plate sensor, which can be expressed in the left (\(j=L\)) or right (\(j=R\)) force plate frame as:  
\[\mathbf{e}_{j1}=\left[\begin{array}{c}-\frac{g}{2}\\ -\frac{g}{2}\end{array}\right],\mathbf{e}_{j2}=\left[\begin{array}{c}\frac{ g}{2}\\ -\frac{h}{2}\end{array}\right],\mathbf{e}_{j3}=\left[\begin{array}{c}-\frac{g }{2}\\ \frac{h}{2}\end{array}\right],\mathbf{e}_{j4}=\left[\begin{array}{c}\frac{g}{2 }\\ \frac{h}{2}\end{array}\right] \tag{13}\]  
The coordinates of the left (\(j=L\)) and right (\(j=R\)) force plates' CoPs in the frame {S} can be expressed with the following equations:  
\[{}^{S}\mathbf{CoP}_{j}=\mathbf{CoP}_{j}+{}^{S}\mathbf{c}_{j}, \tag{14}\]  
while, at the same time, the position vectors of the left and right force plate centers in the frame {S} are:  
\[{}^{S}\mathbf{c}_{L}=\left[\begin{array}{cc}-\frac{l}{2}&0\end{array}\right] ^{T},^{S}\mathbf{c}_{R}=\left[\begin{array}{cc}\frac{l}{2}&0\end{array} \right]^{T} \tag{15}\]  
The CoP of the human body can be defined with the CoP of the left and right legs as follows:  
\[\mathbf{CoP}=\frac{{}^{S}\mathbf{CoP}_{L}\sum\limits_{i=1}^{4}F_{Li}+{}^{S} \mathbf{CoP}_{R}\sum\limits_{i=1}^{4}F_{Ri}}{\sum\limits_{i=1}^{4}F_{Li}+\sum \limits_{i=1}^{4}F_{Ri}} \tag{16}\]  
With an ideally static posture of the human body, the CoM projection in the transverse plane is in the same point as the CoP. However, due to the human body swaying, there will be a deviation between the CoM and CoP. Hence, when standing still, instead of equality, the CoM of the human body in the transverse plane is approximated with the CoP [28]:  
\[\mathbf{CoM}=\left[\begin{array}{c}CoM_{x}\\ CoM_{y}\end{array}\right]\approx\mathbf{CoP} \tag{17}\]