## 7 Conclusions  
Based on a new dynamic modeling theory, a novel dynamic control method is proposed for the free-floating dual-arm space robot. This method can not only realize the tracking of the desired trajectory by the two arms of the space robot, but also realize the stable control of the space robot base. The in-orbit service mission of space debris capture was designed and simulated. The main contributions of this paper are as follows:  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Pseudo-Inertia** & **Generalized** & **Explicit** \\ \multicolumn{1}{c}{**Numbers**} & \multicolumn{2}{c}{**Matrix**} & \multicolumn{1}{c}{**Momentum**} & \multicolumn{1}{c}{**Canonical Method**} \\ \cline{3-5}  & & **Method** & **Method** & **Canonical Method** \\ \hline Inertia & Multiplication & \(21n^{3}+63n^{2}+66n\) & \(7n^{3}+12n^{2}+3n\) & \(3.5n^{3}+16n^{2}+12.5n\) \\ \cline{2-5} matrix & Addition & \(18n^{3}+49.5n^{2}+43.5n\) & \(5n^{3}+10n^{2}+2n\) & \(3.5n^{3}+10n^{2}+6.5n\) \\ \hline Bias force & Multiplication & \(91n^{2}+157n\) & - & \(16.5n^{2}+43.5n\) \\ \cline{2-5} vector & Addition & \(67n^{2}+105n\) & - & \(7.5n^{2}+27.5n\) \\ \hline \hline \end{tabular} (3) Due to the floating characteristics of the base of the free-floating space robot, it is difficult to combine the traditional explicit dynamic modeling method with DEM, because it requires tedious and complex analysis processes, such as the definition and operation of complex intermediate variables. When combined with DEM, our explicit canonical dynamic modeling method can avoid the tedious and complex analysis of intermediate variables. (3) Due to the inconvenient modeling method, most of the literature in the study of space robot dynamics and control only simulates the plane situation of the two-DOF manipulator, which makes it difficult to reflect the real working state of the complex multi-DOF space robot. When using the proposed explicit dynamic modeling method, it is easy to establish a dynamic model for complex multi-DOF space robot for control research. (4) Compared with other traditional control methods, research on the NMPC method applied to space robots is not extensive enough. Recently, [PERSON] [11; 44] explored the possibility of using the NMPC method to control manipulators mounted on free-floating space robots. However, [PERSON] [11; 44] only considered the trajectory tracking control of the end manipulator of the free-floating space robot, and did not consider the stable control of the attitude of the base of the free-floating space robot. In addition, [PERSON] [11; 44] only carried out dynamic modeling and simulation verification of the NMPC method for a two-DOF planar manipulator. This paper not only considers the trajectory tracking control of the end manipulator of the free-floating space robot, but also considers the stable control of the base attitude of the free-floating space robot. In addition, the dynamic modeling and control simulation of the free-floating dual-arm space robot with nine DOF are carried out in this paper. (2) The free-floating space robot has high requirements on the accuracy of trajectory tracking and the stability of the base attitude. The traditional NMPC method has difficulty meeting these requirements. This paper innovatively introduces the idea of sliding mode variable structure to improve the traditional NMPC method. The proposed improved NMPC method has better robustness and can better meet the requirements of trajectory tracking accuracy and base attitude stability for free-floating space robots serving in orbit. (3) Inspired by the literature [15], the control system in this paper consists of a combined controller for the base and the arms, rather than two separate controllers, which can reduce the risk of failure. (4)