## 5 Simulation Analysis
### Details and Results of the First Simulation Condition
The space robot end is controlled to move from its initial position to the directly above the insertion position in 0-3 s. In 3-8 s, the space robot end adjusts its attitude and move to the insertion hole position. Then, in 8-13 s, the insertion operation is executed. Figure 4: 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 5: Joint trajectory tracking curve of on-orbit insertion operation. (**a**) The joint \(O_{1}\), (**b**) The joint \(O_{2}\), (**c**) The joint \(O_{3}\). Figure 5 shows the joint trajectory tracking curve. Figure 6 shows the two-dimensional trajectory in basic coordinate system \(X^{\prime}O^{\prime}Y^{\prime}\). The initial position of the space robot end and the insertion position are marked in the image. The control condition illustrated in Figure 7 is in accordance with the description in the beginning of this subsection. The space robot end impedance control is switched off in 0\(\sim\)3 s and switched on in 3\(\sim\)13 s. In 3-8 s, the output forces of space robot end are increased to 20 N gradually and linearly. Then, in 8-13 s, the output forces remain the expected value and the space robot begin to perform the operation. The joint tracking error curves of PPC method are given in Figure 8a, and which of SMC method are given in Figure 8b. For the convergence time, the error under the control of PPC method is stable convergence at 1.413 s and which is stable convergence under the control of SMC method at 3.554 s. For the error range, it is enveloped by the prescribed performance function under the control of PPC method while it is beyond the boundary of the same function under the control of SMC method. For the control accuracy, the PPC method is better than the SMC method. Analysis of the results shows that the PPC method can ensure that the tracking error is always within the predetermined boundaries. Therefore, the transient and steady state performance of space robot system has been improved. Figure 6: Space robot end X&Y-direction trajectory tracking curve of on-orbit insertion operation. Figure 7: Output force of space robot end tracking curve of on-orbit insertion operation. Figures 4-8 show that both the PPC method and SMC method can complete the on-orbit insertion operation. Figure 9 shows the output force of space robot end tracking error which is defined as \(e_{f}=F_{\text{output}}-F_{\text{desired}}=F_{\text{output}}-(F_{\text{d}}+2)\). It is obvious that the error of PPC method is more smaller fluctuating than that of SMC method in 3-4 s, which means the insertion process of PPC method is more flexible. From the perspective of engineering practice, it is not only conducive to improve the success rate of insertion operation, but also conducive to the protection of space robot and other hardware equipment. Figure 8: The joint tracking error curve of space robot end on-orbit insertion operation. (**a**) PPC. (**b**) SMC. Figure 9: Force tracking error curve of space robot end on-orbit insertion operation. Figure 10a-d shows the unknown system dynamic estimator errors of the carrier and joints. These figures illustrated that the estimator can precisely compensate the dynamic uncertainties which consists of \(\mathbf{H}(\mathbf{q},\dot{\mathbf{q}})\dot{\mathbf{q}}\) and \(\tau_{d}\). It also indicates that the robustness of space robot system is enhanced.