## 4 Verification of Spatial Geometric Characteristic  
In Section 3.4, the characteristic of ball positioning in the dynamic equilibrium state is stated. For rigor, the geometric relationship of the space vector is used to examine and verify it. As shown in Figure 5a, any two vectors \(\mathbf{L}_{1}\) and \(\mathbf{L}_{2}\) in space, where \(P_{1}\) and \(P_{2}\) are arbitrary points on the \(\mathbf{L}_{1}\) and \(\mathbf{L}_{2}\) vectors, respectively, and the displacement of point \(P_{1}\) relative to point \(P_{2}\) is the vector \(\mathbf{\Delta}\). When the shortest distance \(d\) between \(\mathbf{L}_{1}\) and \(\mathbf{L}_{2}\) is zero, it means that vector \(\mathbf{L}_{1}\) and vector \(\mathbf{L}_{2}\) intersect in 3D space, where  
\[d=\frac{\mathbf{\Delta}\cdot(\mathbf{L}_{1}\times\mathbf{L}_{2})}{\|\mathbf{L}_{1}\times\mathbf{L} _{2}\|}. \tag{30}\]  
Consider an arbitrary curved track in space rotating around a fixed \(\mathbf{Z}\)-axis in space at constant angular velocity \(\omega\), and the \(<\mathbf{x}\mathbf{y}\mathbf{z}>_{\mathcal{O}}\) coordinate synchronized with the spin axis is selected as the reference coordinate system, as shown in Figure 5b. First, the perpendicular line from the \(i\)-th ball \(B_{i}\) to the spin \(\mathbf{Z}\)-axis (\(\mathbf{Z}=\mathbf{z}\)) is described as the spatial vector \(\mathbf{L}_{1i}\). The curvature radius of the curved track at \(B_{i}\) is \(R_{i}\), the curvature center is \(C_{i}\) and its unit tangent vector is \(\hat{\mathbf{n}}_{li}\). Define the vector pointing from \(C_{i}\) to \(B_{i}\) as \(\mathbf{\Delta}_{i}\), and the coordinate system \(<\mathbf{x}_{i}\mathbf{y}_{i}\mathbf{z}_{i}>_{C_{i}}\) with \(C_{i}\) as the origin. The \(\mathbf{x}_{i}\mathbf{y}_{i}\) is the plane formed by the  
Figure 4: Centripetal force and attitude of ball positioning. two vectors \(\hat{\mathbf{n}}_{ti}\) and \(\mathbf{\Delta}_{i}\), and the angular position of the vector \(\mathbf{\Delta}_{i}\) relative to the \(x_{i}\)-axis is denoted as \(\mathbf{\alpha}_{i}\). For the equilibrium position \(B_{i}\) in the space-curved track, the vector \(\hat{\mathbf{n}}_{ti}\) is perpendicular to \(\mathbf{\Delta}_{i}\). The \(z^{m}\)-axis is the central axis of curvature perpendicular to the \(\hat{\mathbf{n}}_{ti}\mathbf{\Delta}_{i}\) plane, the unit vector of \(z^{m}\)-axis is represented by the spatial vector \(\mathbf{L}_{2i}\). Next, use the Euler coordinate transformation as in Section 3.1. Taking the \(z\)-axis as the roll axis, \(\phi_{1}\) as the pitch angle, and \(\phi_{2}\) as the yaw angle. The forms of matrices \(\mathbf{T}_{1i}\) and \(\mathbf{T}_{2i}\) are shown in Equation (1). Denote the vectors \(\mathbf{\Delta}_{i}\) and \(\mathbf{L}_{2i}\) in the \(<x\mathbf{y}\mathbf{z}>_{O}\) coordinate system as, respectively:  
\[\mathbf{\Delta}_{i}=\mathbf{T}_{1i}{}^{-1}\mathbf{T}_{2i}{}^{-1} \begin{bmatrix}R_{i}\ cos \alpha_{i}\\ R_{i}\ sin\alpha_{i}\\ 0\end{bmatrix} \text{ and }\mathbf{L}_{2i}=\mathbf{T}_{1i}{}^{-1}\mathbf{T}_{2i}{}^{-1} \begin{bmatrix}0\\ 0\\ 1\end{bmatrix} . \tag{31}\]  
In the \(<x\mathbf{y}\mathbf{z}>_{O}\) coordinate system, the position vector of \(C_{i}\) is described as \(\mathbf{r}_{\text{OC}_{i}}= \begin{bmatrix}x_{C_{i}}&y_{C_{i}}&z_{C_{i}}\end{bmatrix}  ^{T}\) and the position vector of \(B_{i}\) is described as \(\mathbf{r}_{\text{OB}_{i}}=\mathbf{r}_{\text{OC}_{i}}+\mathbf{\Delta}_{i}\). \(\mathbf{L}_{1i}\) can be obtained from \(\mathbf{r}_{\text{OB}_{i}}\) and the unit vector \(\hat{\mathbf{k}}\) of the \(z\)-axis.