## 3 Dynamics Modeling and Parameter Optimization of Double Motors Driving System
### Dynamics Modeling of Dual Drive Motors with Gear Clearance
According to Equation (13), Equation (11), Equation (6), Equation (7), and Equation (8), the dynamic model can be obtained as Equation (14). \[\left\{ \begin{array}{l}K_{e1}\theta_{c1}+I_{1}R_{1}+L_{1}\frac{dI_{1}}{dt}=U_{ 1}\\ K_{e2}\theta_{c2}+I_{2}R_{2}+L_{2}\frac{dI_{2}}{dt}=U_{2}\\ I_{1}\ddot{\theta}_{c1}+b_{1}\dot{\theta}_{c1}+K_{t1}\left(\theta_{c1}-i_{m} \theta_{m}\right)=K_{1}I_{1}\\ J_{2}\ddot{\theta}_{c2}+b_{2}\dot{\theta}_{c2}+K_{t2}\left(\theta_{c2}-i_{m} \theta_{m}\right)=K_{2}I_{2}\\ J_{m}\ddot{\theta}_{cm}+b_{m}\dot{\theta}_{cm}=i_{m}K_{t1}\left(\theta_{c1}-i_{ m}\theta_{m}\right)+i_{m}K_{t2}\left(\theta_{c2}-i_{m}\theta_{m}\right)\end{array} \right. \tag{14}\]  
Now consider the effect of gear clearance on the dynamic model. The clearance between drive gear and C-axis large gear can be expressed as \(\phi_{t}(t)\), and the difference between the two objects position can be expressed as \(\theta(t)\). Subsequently, the gear clearance model can be expressed as Equation (15). \[\phi_{1}(t)=\left\{ \begin{array}{ccc}\theta(t)-\Delta&\theta(t)\geq\Delta\\ 0&-\Delta<\theta(t)<\Delta\\ \theta(t)+\Delta&\theta(t)\leq-\Delta\end{array} \right. \tag{15}\]  
The non-linear functional expression describing the clearance between large gear and driving gear can be obtained from Equation (15), as shown in Equation (16). \[\phi_{1}(\theta_{c1}-i_{m}\theta_{m})=\left\{ \begin{array}{ccc}\theta_{c1}-i _{m}\theta_{m}-\Delta&\theta_{c1}-i_{m}\theta_{m}\geq\Delta\\ 0&-\Delta<\theta_{c1}-i_{m}\theta_{m}<\Delta\\ \theta_{c1}-i_{m}\theta_{m}+\Delta&\theta_{c1}-i_{m}\theta_{m}\leq-\Delta \end{array} \right. \tag{16}\]  
When considering the influence of tooth gap, Equation (16) is substituted into Equation (14) in order to obtain the dynamic model of double motor drive with tooth side clearance, as follows. \[\left\{ \begin{array}{l}K_{e1}\dot{\theta}_{c1}+I_{1}R_{1}+L_{1}\frac{dI_{1} }{dt}=U_{1}\\ J_{1}\ddot{\theta}_{c1}+b_{1}\dot{\theta}_{c1}+K_{t1}\phi_{1}(\theta_{c1}-i_{ m}\theta_{m})=K_{m1}I_{1}\\ K_{e2}\dot{\theta}_{c2}+I_{2}R_{2}+L_{2}\frac{dI_{2}}{dt}=U_{2}\\ J_{2}\ddot{\theta}_{c2}+b_{2}\dot{\theta}_{c2}+K_{t2}\dot{\theta}_{2}(\theta_{c2 }-i_{m}\theta_{m})=K_{m2}I_{2}\\ J_{m}\ddot{\theta}_{m}+b_{m}\dot{\theta}_{m}=i_{m}K_{t1}\phi_{1}(\theta_{c1}-i_{ m}\theta_{m})+i_{m}K_{t2}\phi_{2}(\theta_{c2}-i_{m}\theta_{m})\end{array} \right. \tag{17}\]  
where \(\phi_{1}(\theta_{c1}-i_{m}\theta_{m})=\phi_{1}(\theta_{c1}-i_{m}\theta_{m}- \beta_{1})\), \(\phi_{2}(\theta_{c2}-i_{m}\theta_{m})=\varphi_{2}(\theta_{c2}-i_{m}\theta_{m}- \beta_{2})\)  
\(\varphi_{1}\), \(\beta_{1}\), \(\varphi_{2}\), \(\beta_{2}\) are non-linear parameters of backlash. The specific expressions are shown in Equations (18)-(21). \[\varphi_{1}=\left\{ \begin{array}{ccc}1&|\theta_{c1}-i_{m}\theta_{m}|\geq \Delta\\ 0&|\theta_{c1}-i_{m}\theta_{m}|<\Delta\end{array} \right. \tag{18}\]  
\[\beta_{1}=\left\{ \begin{array}{ccc}\Delta&\theta_{1}-i_{m}\theta_{m}\geq \Delta\\ 0&|\theta_{1}-i_{m}\theta_{m}|<\Delta\\ \Delta&\theta_{1}-i_{m}\theta_{m}<-\Delta\end{array} \right. \tag{19}\]  
\[\varphi_{2}=\left\{ \begin{array}{ccc}1&|\theta_{c2}-i_{m}\theta_{m}|\geq \Delta\\ 0&|\theta_{c2}-i_{m}\theta_{m}|<\Delta\end{array} \right. \tag{20}\]  
\[\beta_{2}=\left\{ \begin{array}{ccc}\Delta&\theta_{2}-i_{m}\theta_{m}\geq \Delta\\ 0&|\theta_{2}-i_{m}\theta_{m}|<\Delta\\ \Delta&\theta_{2}-i_{m}\theta_{m}<-\Delta\end{array} \right. \tag{21}\]By substituting Equations (18)-(21) into Equation (17), the following can be obtained. \[\left\{