## 3 Results and Discussion
### Completely Static Operative Mode of the Test Rig  
#### 3.4.1 Setting of a Frame of Reference Fixed to the Vehicle  
The vehicle frame of reference \(x_{t}\)-\(y_{t}\)-\(z_{t}\)-\(O_{t}\) can be assumed oriented as follows (Figure 6):  
* \(x_{t}\) axis: aligned with the tractor median axis, contained in the tractor support plane;
* \(y_{t}\) axle: aligned with the rear axle of the tractor, passing through the center of the rear tyre traces and, hence, contained in the tractor support plane; the point of the passage of the \(y_{t}\) axle can be identified by taking the center of the rear wheel hub as reference (see Figure 6);
* \(z_{t}\) axis: perpendicular to the other two axles, and therefore also perpendicular to the support plane of the tractor;
* Origin \(O_{t}\): positioned at the center of the tractor rear axle, contained in the support plane of the tractor. Figure 5: Farm tractor with different widths at the front and rear tracks: in this case \(L\) and \(R\) are the extremal points and \(2\ell\) the length (i.e., the vehicle width) of the segment perpendicular to the longitudinal axis in correspondence of the projection of the center of gravity (\(CoG^{\prime}\)). 4.2. Positioning of the _CoG_ Projection in the Support Plane Frame of Reference According to the Forces Measured by the Load Cells of the Four Quadrants  
It is important to note that the four-quadrant system is not able to give directly the absolute position of the _CoG_ in the frame of reference of the turntable: it gives the relative position of the _CoG_ in the reference frame of the tractor under test. Indeed, the quadrants are large enough to tolerate angular and axial misalignments in tractor positioning and, considering the large variety of the dimensions of tractors, there is no way (i.e., no technical feature) to force the tractor reference system to be perfectly aligned with the platform reference system (Figure 7). If the experimenters have the care to place a single wheel of the vehicle to be tested on each quadrant, at a fixed \(\alpha\), the weights supported by each quadrant do not vary whatever the misalignment of the vehicle. Each quadrant (numbered from 1 to 4 counter-clockwise in the same way as the quadrants of the Cartesian plane; see Figure 8a) is supported by a load cell that measures only the component (\(W_{1},\ldots\ W_{4}\)) of the binding reaction perpendicular to the support plane, whatever the angle of inclination of it. The load cells are placed symmetrically with respect to the lines of subdivision of the  
Figure 6. Placement of the tractor frame of reference. Figure 7. Axial (**a,b**) and angular (**c,d**) misalignments of a tractor frame of reference with respect to the turntable frame of reference. Please note that each of these scenarios will result in the same weight distribution on the quadrants. support plane into quadrants (coinciding with the \(x_{SP}\) and \(y_{SP}\) axes of the reference frame fixed to the support platform). For the situation represented in Figure 8c, it follows (NB: \(W_{tot}=P\) only if \(\alpha=0\)):  
\[\left\{ \begin{array}{l}W_{FR}=W_{1}\\ W_{FL}=W_{2}\\ W_{RR}=W_{4}\\ W_{RL}=W_{3}\\ W_{tot}=\sum_{i=1\text{ to }4}W_{i}=mg\cdot\cos\alpha=P\cdot\cos\alpha\end{array} \right. \tag{10}\]  
Given the main characteristic dimensions of the vehicle (the wheelbase \(WB\), the front track width \(FT\), the rear track width \(RT\); see Figure 8b), the \(CoG^{\prime}\) position on the support plane in the vehicle reference frame is given by the following equations (\(a>0\) if \(CoG^{\prime}\) is on the positive half-plane of \(x_{t}\), i.e., always, \(b>0\) if \(CoG^{\prime}\) is on the positive half-plane of \(y_{t}\); see Figure 8b):  
\[\left(CoG^{\prime}\right)_{t}\left\{ \begin{array}{l}a=\frac{WB\cdot\left(W_{ FR}+W_{FL}\right)}{W_{tot}}\\ b=\frac{FT\cdot\left(W_{FL}-W_{FB}\right)+RT\cdot\left(W_{RL}-W_{BR}\right)}{2 \cdot W_{tot}}\end{array} \right.