## 1 Introduction  
Since the 1990s, space manipulators used to manipulate and capture payloads have received extensive attention [1; 2; 3; 4; 5; 6]. Space manipulator can not only be installed on a space station to assist astronauts in the maintenance of the space station [7; 8; 9], but can also be installed on an unmanned satellite to carry out in-orbit maintenance of the satellite and capture space debris [10; 11; 12; 13]. In 2007, the US Orbital Express project carried out capture and maintenance experiments of controlled target satellites using unmanned satellites equipped with a manipulator [14]. Unlike a manipulator with a fixed base on the ground, the base of a small space robot composed of a small unmanned satellite and manipulator is generally movable [15]. This kind of space robot with a movable base is a typical unrooted tree multibody dynamic system, which has the characteristics of many degrees of freedom (DOF), strong nonlinearity and strong coupling [16]. Its dynamic modeling and control are challenging and have attracted increasing attention from researchers in recent years [1; 15]. Compared with single-arm space robots, the establishment of the dynamic model of a dual-arm space robot is more complicated. So far, according to the different expressions of dynamic equations, the dynamic modeling methods of multibody systems such as robots are mainly divided into recursive methods based on the Newton-Euler equation and explicit methods based on the Lagrange equation. Recursive methods can obtain efficient recursive equations, but they are not suitable for the derivation of a high control law [17; 18]. Compared with recursive methods, the expression of the dynamic equation of explicit methods is more intuitive. The pseudo inertia matrix method [19; 20] and the generalized momentum method [21] are two common explicit dynamic modeling methods. However, these two explicit methods both have the problem that many intermediate variables need to be analyzed and calculated. Recently, an explicit multi-rigid-body dynamic modeling method was proposed, which can directly obtain the final canonical expression of a multibody system [22; 23]. Unlike ground robots with a fixed base, space robots have dynamic coupling between the arm and the movable base [24; 25]. If this is not controlled, it will lead to accidental movement of the base and affect the performance of the space robot [7; 26; 27]. For example, to ensure reliable communication with the ground or receive solar energy, the base of the satellite needs to accurately point in a fixed position, but the dynamic coupling between the arm and the base will interfere with the attitude of base [7]. [PERSON] and [PERSON] [28] divided space robots with movable bases into free-flying space robots and free-floating space robots according to the different control problems to be solved. Unlike free-flying space robots that need to control the position and attitude of the base at the same time, free-floating space robots only needs to consider the control of base attitude, which can save energy [29]. In order to simplify the kinematic analysis of free-floating space robots, [PERSON] and [PERSON] [30] proposed the concept of the virtual manipulator (VM). VM is a fixed base robot with the same kinematic equivalence as a free-floating space robot, and its first joint is a passive spherical joint, which represents the free-floating characteristics of the space robot [31]. [PERSON] [32] extended the concept of VM and proposed the dynamic equivalent manipulator (DEM), which can simultaneously simplify the kinematics and dynamics analysis of free-floating space robots. In the process of in-orbit service, a space robot will not only be subjected to unknown disturbances from the outside, but will also be constrained by the uncertainty of internal parameters of the system caused by the change in fuel quality and other factors [15]. Therefore, designing controllers for space robots is a challenge which has been studied by many researchers [33; 34; 35].