## 3 Levelling Strategy and Analysis  
### Mathematical Modelling of the Mobile Platform  
A plane is determined by two intersecting lines to establish the schematic diagram of the picking platform, as shown in Figure 3[10]. The coordinate system OXYZ is established with the platform center as the coordinate origin O. The vertical upwards direction of the center point is the positive direction of the \(Z\)-axis, the horizontal rightward direction is the positive direction of the \(X\)-axis, and the horizontal forward direction is the positive direction of the \(Y\)-axis. The nine-axis inclination sensor is installed in the center of the platform, and the two single-axis inclination sensors are installed on the \(X\)-axis and \(Y\)-axis at the same distance from the nine-axis sensor, which are used to measure the inclination of the \(X\)-axis and \(Y\)-axis. The fixed-point method is used to calculate the telescopic length of the electric push rod. When the inclination sensor outputs zero \(X\)- and \(Y\)-axis inclination relative to the center point, the platform is horizontal. It is assumed that the distance  
Figure 2: Working principle diagram of levelling. between two adjacent outriggers is equal and is level with the fixed-point method, and the fixed point is P\({}_{1}\). Three inclination sensors installed in the center of the mobile platform fuse the inclination information of the central point of the picking platform relative to the \(X\)-axis and \(Y\)-axis. After fusion, the angle of the \(X\)-axis inclination is \(\alpha\). When the \(Y\)-axis is tilted after fusion, the angle is \(\beta\). When the platform is tilted, the transformation matrices of the mobile platform around the \(X\)- and \(Y\)-axes are, respectively:  
\[\mathrm{R_{X}}= \begin{pmatrix}1&0&0\\ 0&\cos\alpha&-\sin\alpha\\ 0&\sin\alpha&\cos\alpha\end{pmatrix} \mathrm{R_{Y}}= \begin{pmatrix}\cos\beta &0&\sin\beta\\ 1&0&1\\ -\sin\beta&0&\cos\beta\end{pmatrix}  \tag{1}\]  
After the mobile platform rotates around the \(X\)- and \(Y\)-axis in space, the following is obtained:  
\[\mathrm{R_{1}}=\mathrm{R_{X}}\mathrm{R_{Y}}= \begin{pmatrix}\cos\beta&0&\sin \beta\\ \sin\alpha\sin\beta&\cos\alpha&-\sin\alpha\cos\beta\\ -\cos\alpha\sin\beta&\sin\beta&\cos\alpha\cos\beta\end{pmatrix}  \tag{2}\]  
When the mobile platform is not tilted, the coordinates of support points P\({}_{1}\), P\({}_{2}\), P\({}_{3}\), and P\({}_{4}\) are:  
\[ \begin{cases}\mathrm{P_{1}}=\begin{pmatrix}-\frac{\mathrm{a}}{2},-\frac{ \mathrm{a}}{2},0\end{pmatrix}^{\mathrm{T}}&\mathrm{P_{2}}=\begin{pmatrix}- \frac{\mathrm{a}}{2},\frac{\mathrm{a}}{2},0\end{pmatrix}^{\mathrm{T}}\\ \mathrm{P_{3}}=\begin{pmatrix}\frac{\mathrm{a}}{2},\frac{\mathrm{a}}{2},0 \end{pmatrix}^{\mathrm{T}}&\mathrm{P_{4}}=\begin{pmatrix}\frac{\mathrm{a}}{2},-\frac{\mathrm{a}}{2},0\end{pmatrix}^{\mathrm{T}}\end{cases}  \tag{3}\]  
After the mobile platform rotates using rotation matrix R\({}_{1}\), the coordinates of support points P\({}_{1}\), P\({}_{2}\), P\({}_{3}\), and P\({}_{4}\) are transformed into:  
\[ \begin{cases}\mathrm{P_{1}}=\begin{pmatrix}-\frac{\mathrm{a}}{2}\cos\beta,- \frac{\mathrm{a}}{2}(\sin\sin\beta+\cos\alpha),\frac{\mathrm{a}}{2}(\cos\alpha \cos\beta-\sin\alpha)\end{pmatrix}^{\mathrm{T}}\\ \mathrm{P_{2}}=\begin{pmatrix}-\frac{\mathrm{a}}{2}\cos\beta,\frac{\mathrm{a}}{2 }(\cos\alpha-\sin\alpha\sin\beta),\frac{\mathrm{a}}{2}(\cos\alpha\cos\beta+ \sin\alpha)\end{pmatrix}^{\mathrm{T}}\\ \mathrm{P_{3}}=\begin{pmatrix}\frac{\mathrm{a}}{2}\cos\beta,\frac{\mathrm{a}}{2 }(\sin\alpha\sin\beta+\cos\alpha),\frac{\mathrm{a}}{2}(\sin\alpha-\cos\alpha \cos\beta)\end{pmatrix}^{\mathrm{T}}\\ \mathrm{P_{4}}=\begin{pmatrix}\frac{\mathrm{a}}{2}\cos\beta,\frac{\mathrm{a}}{2 }(\sin\alpha\sin\beta-\cos\alpha),\frac{\mathrm{a}}{2}(-\cos\alpha\cos\beta- \sin\alpha)\end{pmatrix}^{\mathrm{T}}\end{cases}  \tag{4}\]  
Then, the height of each of the four electric push rods after rotation is:  
\[