## 2 Dynamics Modeling for the SISP
The harmonic drive enlarges \(T_{wg}(t)\) by \(N\) times and, at the same time, it reduces the angle by \(N\) times [33]. The relationship is as follows,  
\[N\theta_{m}(t)=\theta_{m}^{{}^{\prime}}(t) \tag{5}\]  
\[NT_{wg}(t)=T_{fs}(t) \tag{6}\]  
where \(T_{fs}(t)\) and \(\theta_{m}^{{}^{\prime}}(t)\) are the output torque and the output angle of the flex spline in the harmonic drive, respectively. The angular deformation in the harmonic drive \(\Delta\theta(t)\)[34] is expressed as follows,  
\[\Delta\theta(t)=\theta_{m}^{{}^{\prime}}(t)-\theta_{l}(t) \tag{7}\]  
where \(\theta_{l}(t)\) is the load inertial angle. The transmission compliance acting between \(\Delta\theta(t)\) and \(T_{fs}(t)\) is [35; 36] expressed as follows,  
\[u_{a}(t)-e_{a}(t)=L_{a}i_{a}(t)+R_{a}i_{a}(t) \tag{1}\]0  
where \(f(\Delta\theta(t))\) is a nonlinear function. The stiffness in the harmonic drive exhibits significant compliance when it is externally loaded [37]. The torque hysteresis nonlinear characteristic is approximated by a linear function of the torsion angle as a spring-damper element with stiffness and internal damping as described by \(K_{s}\) and \(B_{s}\). The nonlinear function (8) is presented as follows,  
\[u_{a}(t)-e_{a}(t)=L_{a}i_{a}(t)+R_{a}i_{a}(t) \tag{1}\]1  
The load rotation equation is as follows,  
\[u_{a}(t)-e_{a}(t)=L_{a}i_{a}(t)+R_{a}i_{a}(t) \tag{1}\]2where \(J_{l}\) and \(B_{l}\) are the load equivalent moment of inertia and the equivalent damping coefficient, respectively. When the motor drives the rotor and load to rotate, the reaction torque will also drive the base to rotate in reverse. The base rotation equation is expressed as follows,  
\[u_{a}(t)-e_{a}(t)=L_{a}i_{a}(t)+R_{a}i_{a}(t) \tag{1}\]3  
where \(T_{z}(t)\) is the external torque on the base, \(J_{b}\) is the base equivalent moment of inertia, \(B_{b}\) is the base equivalent damping coefficient, and \(\theta_{b}(t)\) is the base inertial angle. In the inertial space, the relative angle between the base and the load is measured by the encoder \(\theta_{e}(t)\):  
\[u_{a}(t)-e_{a}(t)=L_{a}i_{a}(t)+R_{a}i_{a}(t) \tag{1}\]4  
Therefore, by sorting Equations (3)-(12), the dynamics model differential equations can be expressed as follows,  
\[\mathbf{I} \begin{cases}\dfrac{J_{b}}{K_{a}}\ddot{\theta}_{b}(t)+\dfrac{B_{b}} {K_{a}}\dot{\theta}_{b}(t)=\dfrac{T_{z}(t)}{K_{a}}+u(t)-\dfrac{F_{m}(t)}{K_{ a}}\\ \dfrac{J_{m}}{NK_{a}}(\ddot{\theta}_{e}(t)+N\ddot{\theta}_{b}(t))+\dfrac{B_{m} }{NK_{a}}(\dot{\theta}_{e}(t)+N\dot{\theta}_{b}(t))+\dfrac{J_{l}}{NK_{a}}\ddot {\theta}_{l}(t)+\dfrac{B_{l}}{NK_{a}}\dot{\theta}_{l}(t)=u(t)-\dfrac{F_{m}(t) }{K_{a}}\\ J_{l}\ddot{\theta}_{l}(t)+(B_{l}+B_{s})\dot{\theta}_{l}(t)-B_{s}(\dot{\theta}_{ e}(t)+\dot{\theta}_{b}(t))-K_{s}(\theta_{e}(t)+\theta_{b}(t)-\theta_{l}(t))=0\end{cases}  \tag{13}\]  
Further, it can be written more generally as follows,  
\[\mathbf{J}\ddot{\Theta}(t)+\mathbf{B}\dot{\Theta}(t)+\mathbf{K}\Theta(t)= \mathbf{p}(t)+\mathbf{f}(t)+\mathbf{u}(t) \tag{14}\]  
where \(\Theta(t)=[\theta_{b}(t)\;\theta_{e}(t)\;\theta_{l}(t)]^{T}\) is the angular vector. The constant matrices are defined below,  
\[\mathbf{J}= \begin{bmatrix}J_{b}&0&0\\ NJ_{m}&J_{m}&J_{l}\\ 0&0&J_{l}\end{bmatrix}  \tag{15}\]  
\[\mathbf{B}= \begin{bmatrix}B_{b}&0&0\\ NB_{m}&B_{m}&B_{l}\\ -B_{s}&-B_{s}&B_{l}+B_{s}\end{bmatrix}  \tag{16}\]  
\[\mathbf{K}= \begin{bmatrix}0&0&0\\ 0&0&0\\ -K_{s}&-K_{s}&K_{s}\end{bmatrix}  \tag{17}\]  
with the time-variant vectors  
\[\mathbf{p}(t)= \begin{bmatrix}T_{z}(t)&0&0\end{bmatrix} ^{T} \tag{18}\]  
\[\mathbf{f}(t)= \begin{bmatrix}-F_{m}(t)&-NF_{m}(t)&0\end{bmatrix} ^{T} \tag{19}\]  
\[\mathbf{u}(t)= \begin{bmatrix}K_{a}u(t)&NK_{a}u(t)&0\end{bmatrix} ^{T} \tag{20}\]  
The traditional dynamics model simply considers the system as an ideal electromechanical system, which is equivalent to a second order differential equation. However, for the SISP system with a light base support, the traditional model has made some approximations and several problems cannot be clearly explained.