## 5 Simulation Analysis
### Details and Results of the Second Simulation Condition  
Suppose that there is a sudden change of the friction happened when the insertion operation proceeding to four fifths. Other conditions of the second simulation are same as the first. The friction of second condition can be expressed as:  
\[\mathbf{F}_{e}=\left\{ \begin{array}{llll}\left[\begin{array}{cccc}0&\mathrm{N }\cdot\mathrm{m},&0&\mathrm{N},&18&\mathrm{N},&0&\mathrm{N}\end{array}\cdot \mathrm{m}\right]^{\mathrm{T}}&0\leq t<12\\ \left[\begin{array}{cccc}0&\mathrm{N}\cdot\mathrm{m},&0&\mathrm{N},&30& \mathrm{N},&0&\mathrm{N}\end{array}\cdot\mathrm{m}\right]^{\mathrm{T}}&12\leq t <19\end{array} \right., \tag{40}\]  
The control strategy is to stop the insertion and gradually increase the output force until it exceeds the friction after the sudden change happened. The expected trajectory and output force are, respectively, designed as:  
\[\mathbf{X}_{d}=\left\{ \begin{array}{llll}\left[\begin{array}{cccc}0&\mathrm{ rad},&2.604&\mathrm{m},&0.1\,\mathrm{m},&\mathrm{\pi rad}\end{array}\right]^{ \mathrm{T}}&0\leq t<3\\ \left[\begin{array}{cccc}0&\mathrm{rad},&2.604&\mathrm{m},&(0.16-0.02t)& \mathrm{m},&\mathrm{\pi rad}\end{array}\right]^{\mathrm{T}}&3\leq t<12\\ \left[\begin{array}{cccc}0&\mathrm{rad},&2.604&\mathrm{m},&-0.08\,\mathrm{m},& \mathrm{\pi rad}\end{array}\right]^{\mathrm{T}}&12\leq t<15\\ \left[\begin{array}{cccc}0&\mathrm{rad},&2.604&\mathrm{m},&(-0.005-0.005t)& \mathrm{m},&\mathrm{\pi rad}\end{array}\right]^{\mathrm{T}}&15\leq t<19\end{array}  \right., \tag{41}\]  
\[F_{\mathrm{td}}=\left\{ \begin{array}{llll}\left[\begin{array}{cccc}0& \mathrm{N}\cdot\mathrm{m},&0&\mathrm{N},&0&\mathrm{N}\cdot\mathrm{m}\end{array} \right]^{\mathrm{T}}&0\leq t<3\\ \left[\begin{array}{cccc}0&\mathrm{N}\cdot\mathrm{m},&0&\mathrm{N},&(5(t-6))& \mathrm{N},&0&\mathrm{N}\cdot\mathrm{m}\end{array}\right]^{\mathrm{T}}&3 \leq t<8\\ \left[\begin{array}{cccc}0&\mathrm{N}\cdot\mathrm{m},&0&\mathrm{N},&20& \mathrm{N}&0&\mathrm{N}\cdot\mathrm{m}\end{array}\right]^{\mathrm{T}}&8\leq t <12\\ \left[\begin{array}{cccc}0&\mathrm{N}\cdot\mathrm{m},&0&\mathrm{N},&(4(t-7))& \mathrm{N},&0&\mathrm{N}\cdot\mathrm{m}\end{array}\right]^{\mathrm{T}}&12\leq t <15\\ \left[\begin{array}{cccc}0&\mathrm{N}\cdot\mathrm{m},&0&\mathrm{N},&32& \mathrm{N}&0&\mathrm{N}\cdot\mathrm{m}\end{array}\right]^{\mathrm{T}}&15\leq t <19\end{array} \right., \tag{42}\]  
Figure 10: The tracking error of unknown system dynamic estimator. (**a**) The carrier \(O_{0}\). (**b**) The joint \(O_{1}\). (**c**) The joint \(O_{2}\). (**d**) The joint \(O_{3}\). The second simulation control parameters are the same as the first and the total simulation time is 19 s. Its simulation results are shown in the following figures. Figure 11c shows that the sudden change of friction happens at 12 s and the space robot end stop the insertion during 12-15 s. Then, the operation restart at a slower speed until the space robot end force output by impedance controller is greater the friction after sudden change. It also can be found that what expressed in Figures 13 and 14 is consistent with the above content. Figures 12a-c and 13 are extended descriptions of the second simulation condition. That is the space robot joint trajectories and end trajectory of XY two dimensional. Figure 12a-c show that the convergence rate is about 2 s faster than that of SMC method. In addition, the control accuracy of PPC method is better at \(10^{-4}\). This is conducive to the rapid response of the space manipulator when executing the operation of inserting and extracting. Figure 11: Space robot end trajectory tracking curve of on-orbit insertion operation. (**a**) The carrier attitude. (**b**) The X-direction position. (**c**) The Y-direction position. (**d**) The space robot end attitude. Figure 14 is the force tracking curve of space robot end.