# Development and Validation of a System for Game Control Based on a Force Plate
## 3 Mathematical Model of the System
### System Calibration and Game Control  
Before using the system for game control based on a force plate, an individualized calibration of the system needs to be conducted for each player. The goal of the calibration process is to adapt the system to the player so that the player can have the best possible experience controlling the game. The calibration procedure of the system for game control based on a force plate is shown in Figure 4.  
The calibration procedure is conducted in three steps. In the first step of the calibration (Step 1 in Figure 4), the player standing on the force plate must maintain an upright and still standing during the ground reaction force measurements. After 20 s, measuring is complete, and the player's mass is calculated according to Equation (7) (\(\tilde{z}_{CoM}\approx 0\) m/s\({}^{2}\)):  
\[m_{H}=\frac{\frac{1}{N}\sum\limits_{i=1}^{N}F_{zi}}{g}, \tag{8}\]  
where \(N\) is the number of measured instances of the \(F_{z}\) force on the load cells. The position vector of the initial player's CoP \({}^{O}\textbf{h}\) is calculated as the mean value of the CoP measurements according to Equations (4) and (5) during the 20 s of measuring as follows:  
\[{}^{O}\textbf{h}=\left[ \begin{array}{c}x_{CoP0}\\ y_{CoP0}\end{array} \right]=\left[ \begin{array}{c}\frac{1}{N}\sum\limits_{i =1}^{N}x_{CoPi}\\ \frac{1}{N}\sum\limits_{i=1}^{N}y_{CoPi}\end{array} \right] \tag{9}\]  
The result of the first calibration step is the player's mass \(m_{H}\) and the initial player CoP position \({}^{O}\textbf{h}\) in the local frame {H}. In the second step of the calibration process, the player moves his or her CoM so as to reach the maximum CoP deviations along the \(x\) and \(y\) axes on the force plate in the local frame {H}. While conducting the second step (Step 2 in Figure 4), which takes 20 s, the player receives visual feedback on the position of his or her CoP in the PC application. The result of the second step is the identification of the player's work plane. Searching for the player's work plane is shown in Figure 5. The maximum player CoP deviations during the execution of the second calibration step along the \(x\) and \(y\) axes are as follows:  
* \(\text{max}(^{H}x_{CoP})\): the maximum distance of the player's CoP position from the origin along the positive \(x\) axis;
* \(\text{max}(^{H}-x_{CoP})\): the maximum distance of the player's CoP position from the origin along the negative \(x\) axis;
* \(\text{max}(^{H}y_{CoP})\): the maximum distance of the player's CoP position from the origin along the positive \(y\) axis;  
Figure 4: Calibration procedure of the system for game control based on a force plate. \(\max(^{H}-y_{CoP})\): the maximum distance of the player's CoP position from the origin along the negative \(y\) axis. The smallest maximum distances of the player's CoP position along the positive and negative \(x\) and \(y\) axes represent the borders of the player's work plane (see Figure 5) such that the player is able to reach every point of the work plane, (i.e., every point on the game screen). Based on the example in Figure 5, the smallest maximum distance from the origin along the positive and negative \(x\) axis is \(\max(^{H}x_{CoP})\). The position of the CoP in the local frame {H} needs to be transferred into the local frame {G}, which is the local frame that controls the game. The player's CoP in the local frame {G} can be calculated as follows:  
\[{}^{G}\textbf{CoP}=\left[ \begin{array}{cc}{}^{G}x_{CoP}\\ {}^{G}y_{CoP}\end{array} \right]=\left[ \begin{array}{cc}{}^{\alpha}x&0\\ 0&{}^{\alpha}y\end{array}