## 2 Dynamic Contact Model of SNBSP
### Setting and Transformation of the Coordinate System
The transformation relation between \(CSOxyz\) and \(CSO^{\prime\prime}_{\varsigma}x^{\prime\prime}_{\varsigma}y^{\prime\prime}_{ \varsigma}z^{\prime\prime}_{\varsigma}\) can be derived from Equations (2) and (3). \[\{x,y,z,1\}^{T}=\mathbf{T}_{0-2\zeta}\{x^{\prime\prime}_{\varsigma},y^{\prime\prime }_{\varsigma},z^{\prime\prime}_{\varsigma},1\}^{T} \tag{5}\]  
where  
\[\mathbf{T}_{0-2\zeta}=\mathbf{T}_{0-1}\mathbf{T}_{1-2\zeta} \tag{6}\]  
Here, set the tiny position and the orientation of SNBSP in \(CSOxyz\) is \(\Delta P_{0}\), and it in \(CSO^{\prime\prime}_{\varsigma}x^{\prime\prime}_{\varsigma}y^{\prime\prime}_{ \varsigma}z^{\prime\prime}_{\varsigma}\) is \(\Delta P_{2\varsigma}\). Then, \(\Delta P_{0}\) and \(\Delta P_{2\zeta}\) can be express as follows [13],  
\[\Delta\mathbf{P}_{0}=\left[ \begin{array}{cccc}1&-\phi_{z}&\phi_{y}&dx\\ \phi_{z}&1&-\phi_{x}&dy\\ -\phi_{y}&\phi_{x}&1&dz\\ 0&0&0&1\end{array} \right] \tag{7}\]  
\[\Delta\mathbf{P}_{2\zeta}=\left[ \begin{array}{cccc}1&-\phi_{z_{2\zeta}}&\phi_{ y_{2\zeta}}&dx_{2\zeta}\\ \phi_{z_{2\zeta}}&1&-\phi_{x_{2\zeta}}&dy_{2\zeta}\\ -\phi_{y_{2\zeta}}&\phi_{x_{2\zeta}}&1&dz_{2\zeta}\\ 0&0&0&1\end{array} \right] \tag{8}\]  
The following formula can be obtained through differential transformation theory  
\[d\mathbf{T}=[\Delta\mathbf{P}_{0}-\mathbf{I}]\mathbf{T}_{0-2\zeta}=\mathbf{T}_{0-2\zeta}[\Delta \mathbf{P}_{2\zeta}-\mathbf{I}] \tag{9}\]  
Suppose the differential transform operators, respectively  
\[\Delta_{0}=\Delta\mathbf{P}_{0}-\mathbf{I} \tag{10}\]  
\[\Delta_{2\zeta}=\Delta\mathbf{P}_{2\zeta}-\mathbf{I} \tag{11}\]  
Put Equation (10) and Equation (11) into Equation (9), and transform it; \(\Delta_{2\zeta}\) can be represented by  
\[\Delta_{2\zeta}=\mathbf{T}_{0-2\zeta}^{-1}\Delta_{0}\mathbf{T}_{0-2\zeta} \tag{12}\]  
The right side of the Equation (12) is expanded and the result is simplified. It is clear through analyzing the characteristics of SNBSP that the tiny displacements in the six directions should be investigated in \(CSOxyz\), but only three tiny translation \(x_{2\zeta},y_{2\zeta}\) and \(z_{2\zeta}\) in \(CSO^{\prime\prime}_{\varsigma}x^{\prime\prime}_{\varsigma}y^{\prime\prime}_{ \varsigma}z^{\prime\prime}_{\varsigma}\). Therefore, Equation (12) can be represented as  
\[\{\mathrm{d}x_{25}\;\mathrm{d}y_{25}\;\mathrm{d}z_{25}\;\mathrm{d}x_{2N}\; \mathrm{d}y_{2N}\;\mathrm{d}z_{2N}\}^{\mathrm{T}}=\left[ \begin{array}{c}\bm {T}\mathbf{D}_{2S-0}\\ \mathbf{T}\mathbf{D}_{2N-0}\end{array} \right]\left(\mathrm{d}x\;\mathrm{d}y\;\mathrm{d }z\;\phi_{x}\;\phi_{y}\;\phi_{z}\right)^{\mathrm{T}} \tag{13}\]where \(\mathbf{TD}_{2z-0}\) denotes the homogeneous coordinate differential transformation matrix of \(CSO^{\prime\prime}_{z}x^{\prime\prime}_{z}y^{\prime\prime}_{z}z^{\prime\prime}_{z}\) and \(CSOxyz\)  
\[\mathbf{TD}_{2z-0}=\left[ \begin{array}{cccc}\sin\alpha_{z}\cos\theta+\cos\alpha_{ z}\sin\lambda\sin\theta&\sin\alpha_{z}\sin\theta-\cos\alpha_{z}\sin\lambda\cos \theta&\cos\alpha_{z}\cos\lambda&\zeta_{1}&\zeta_{2}&\zeta_{3}\\ -\cos\lambda\sin\theta&\cos\lambda\cos\theta&\sin\lambda&\zeta_{4}&\zeta_{5}& \zeta_{6}\\ -\cos\alpha_{z}\cos\theta+\sin\alpha_{z}\sin\lambda\sin\theta&-\cos\alpha_{z} \sin\theta-\sin\alpha_{z}\sin\lambda\cos\theta&\sin\alpha_{z}\cos\lambda& \zeta_{7}&\zeta_{8}&\zeta_{9}\end{array} \right] \tag{14}\]  
where  
\[