## 5 Simulation Analysis
### Details and Results of the First Simulation Condition  
Suppose that the friction in insertion only exists along its direction and remain unchanged. Then, an ideal physical process of on-orbit insertion operation can be described as:  
The end replaceable instrument is tightly held by the manipulator end that can be seen as a rigid body overall (hereinafter referred to as space robot end). Switching off the impedance controller, the space robot end is controlled to move directly above the insertion position, and its attitude is adjusted to make it perpendicular to the hole plane. The next, switching on the impedance controller, the output force of the space robot end are gradually increased to expected value before the impact happens. Then, the space robot begins to execute the on-orbit insertion operation. The settings of the condition are: the insertion position on front side of carrier in the base-coordinate system \(X^{\prime}O^{\prime}Y^{\prime}\) are \([\ \ 2.604\ \mathrm{m},\ \ 0.1\ \mathrm{m}\ \ ]^{\mathrm{T}}\); the initial attitude of the carrier, the initial position and attitude of space robot end, respectively, are \([\ \ 0\ \mathrm{rad},\ \ 3.350\ \mathrm{m},\ \ 1.126\ \mathrm{m},\ \ \frac{5}{6}\pi \mathrm{rad}\ ]^{\mathrm{T}}\); the friction are \([\ \ 0\ \mathrm{N}\cdot\mathrm{m},\ \ 0\ \mathrm{N},\ \ 18\ \mathrm{N},\ \ 0\ \mathrm{N}\ \cdot \mathrm{m}]^{\mathrm{T}}\); the simulation step is set to \(0.001\ \mathrm{s}\) and the total time is \(13\ \mathrm{s}\). According to the above description, the expected trajectory and output force are, respectively, designed as:  
\[\mathbf{X}_{d}=\left\{ \begin{array}{llll}[\ \ 0\ \mathrm{rad},\ \ 2.604\ \mathrm{m},\ \ 0.1\ \mathrm{m},\ \ \ \mathrm{\tau}\mathrm{rad}\ ]^{\mathrm{T}}&0\leq t<3\\ [\ \ 0\ \mathrm{rad},\ \ 2.604\ \mathrm{m},\ \ (0.16-0.02t)\ \mathrm{m},\ \ \ \mathrm{\tau}\mathrm{rad}\ ]^{\mathrm{T}}&3\leq t<13\end{array} \right., \tag{38}\]  
\[\mathbf{F}_{\mathrm{td}}=\left\{ \begin{array}{llll}[\ \ 0\ \mathrm{N}\cdot\mathrm{m},\ \ 0\ \mathrm{N},\ \ 0\ \mathrm{N},\ \ 0\ \mathrm{N}\cdot\mathrm{m}\ ]^{\mathrm{T}}&0\leq t<3\\ [\ \ 0\ \mathrm{N}\cdot\mathrm{m},\ \ 0\ \mathrm{N},\ \ (5(t-6))\ \mathrm{N},\ \ 0\ \mathrm{N}\cdot\mathrm{m}\ ]^{\mathrm{T}}&3\leq t<8\\ [\ \ 0\ \mathrm{N}\cdot\mathrm{m},\ \ 0\ \mathrm{N},\ \ 20\ \mathrm{N},\ \ 0\ \mathrm{N}\ \cdot\mathrm{m}]^{\mathrm{T}}&8\leq t<13\end{array} \right., \tag{39}\]  
The default performance function is \(\rho(t)=(2-0.01)e^{-5t}+0.01\); the upper and lower bounds are set to \(\overline{\mu}=[1,1,1,1]^{\mathrm{T}}\) and \(\underline{\mu}=[1,1,1,1]^{\mathrm{T}}\); the control gains are \(\mathbf{K}=diag\{2\}_{4\times 4}\), \(\mathbf{\Gamma}=diag\{5\}_{4\times 4}\); the filter coefficient of the dynamic estimator of the unknown system is \(k=0.001\). What's more, a control group using sliding mode control (SMC) is set up to verify the effectiveness of the proposed prescribed performance control algorithm (PPC). The simulation programs are executed on MATLAB R2020b and the simulation results are shown in the following figures. Figure 4a-d, respectively, are the carrier attitude, the space robot end position and attitude trajectory tracking curve of on-orbit insertion operation. Figure 4a,d correspond to the description in Section 4.1. The trajectory where the three curves coincide in Figure 4b correspond to the vertical trajectory in Figure 5 which represents the X-direction position of the insertion hole. From these results, it can be seen that the space robot end is controlled to move to the insertion hole with less than \(0.0001\ \mathrm{rad}\) control accuracy of carrier and manipulator end attitude and less than \(0.0001\ \mathrm{m}\) control accuracy of the X-direction position of manipulator end. Figure 4c indicates the change process of an ideal insertion in the direction of inserting.