## 2 Structure and Principle of the Seat Suspension System
### Design of the Multi-Dimensional Seat Platform
#### 2.3.1 Kinematics Analysis of the Stewart Platform  
For kinetic analysis, two coordinate systems were defined. The (_P-xyz_) was set as the moving coordinate system, and the (_B-xyz_) was set as the static coordinate system. The origin of the static coordinate system (\(\mathcal{O}_{\text{b}}\)) was set to be located in the center of the triangular outer circle formed by three intersection points that were connected on the lower platform (base platform). The origin of the moving coordinate system (\(\mathcal{O}_{\text{p}}\)) was set to be located in the center of the outer circle of the triangle formed by the three intersections that were connected in turn on the upper platform (load platform), as shown in Figure 10.  
Figure 8: Current errors using the BP networks.  
Figure 9: The configuration and structure of the cube Stewart platform.  
In Figure 10, \(t\) represents the vector from point \(O_{\rm b}\) to point \(O_{\rm p}\), \(p_{i}\) is the connection vector from \(O_{\rm P}\) to each leg of the upper platform, and \(b_{i}\) is the connection vector from \(O_{\rm b}\) to each leg of the base platform.  
The Jacobi matrix is related to the sliding velocity \(v_{i}\) of each leg and the velocity vector \(\chi=(v^{T},\omega^{T})\), while the position of the upper platform is a function of \((x,y,z,\alpha,\beta,\gamma)\), as follows [23]:  
\[\omega=\dot{\theta},\ \ \ v=\dot{s},\ \ \ \chi=(v^{T}\omega^{T})^{T} \tag{2}\]  
The sliding speed of each leg is shown as follows:  
\[v_{i}=s_{i}\times(v+\omega\times p_{i})=s_{i}\times v-s_{i}\times p_{i}\times\omega \tag{3}\]  
where \(s\) is the linear displacement of the center position of the upper platform, \(\theta\) is the angular displacement of the upper platform, \(p_{i}\) is the connection vector from \(O_{\rm P}\) to each leg of the upper platform, \(v\) is the linear velocity of the center position of the upper platform, \(\omega\) is the angular velocity of the upper platform (in this article, a dot above a variable indicates the time derivative), and \(s_{i}\) is the displacement of each leg. Then, an antisymmetric matrix can be used to express a cross product:  
\[p_{i}\times\omega=\widetilde{p}_{i}\omega \tag{4}\]  
This equation forms the \(i\)-column of the Jacobian matrix:  
\[\dot{S}_{i}=J\chi=\left(\begin{array}{cc}\ldots&\ldots\\ \dot{I}_{i}^{T}-\dot{I}_{i}^{T}\widetilde{p}_{i}\\ \ldots&\ldots\end{array}\right)\left(\begin{array}{c}v\\ \omega\end{array}\right) \tag{5}\]  
If \(v\) and \(\omega\) are expressed in the upper platform reference system, then the following applies:  
\[I_{i}^{T}=\frac{1}{\dot{I}_{i}}\Big{[}(t-b_{i})^{T}R+p_{i}^{T}\Big{]} \tag{6}\]  
\[-\dot{I}_{i}^{T}\widetilde{p}_{i}=-\frac{1}{\dot{I}_{i}}(t-b_{i})R\widetilde{ p}_{i} \tag{7}\]  
where \(I_{i}\) is the length vector of each leg, and \(R\) is the rotation matrix of the two coordinate systems of the upper and lower platforms. Since the \(x\) and \(y\) coordinates of the origin of the two coordinate systems of the lower platform and the upper platform are the same, the rotation matrix \(R\) of the two coordinate systems is a third-order unit matrix.  
Figure 10: Vectorial representation of the cube Stewart platform. (**a**) Axonometric view of cube Stewart platform; (**b**) top view of cube Stewart platform.