# Capturing a Space Target Using a Flexible Space Robot
## 5 Numerical Simulations  
This section will carry out simulations to demonstrate the effectiveness of theoretical studies. As shown in Figure 1, the robot base is assumed to be a big rigid cylinder, of which the length is 24 m, and the radius is 1 m. The lengths of the four links (\(B_{2}\), \(B_{3}\), \(B_{4}\), and \(B_{5}\)) of the manipulator are 6.4 m, 7 m, 0.5 m, and 0.6 m, respectively. The first six modes of the flexible links \(B_{2}\) and \(B_{3}\) are considered in this paper, and the free-pinned beam's modal functions are both used. The first six natural frequencies of \(B_{2}\) are 5.9076 Hz, 25.8648 Hz, 368,781 Hz, 834,175 Hz, 1,026,270 Hz, and 172,7682 Hz, respectively, and those of \(B_{3}\) are 54,859 Hz, 240,087 Hz, 342,174 Hz, 773,491 Hz, 951,008 Hz, and 1,599,500 Hz. The space object is also assumed to be a rigid cylinder and its radius is \(R=0.05\) m. It is used to simulate the handle of a target spacecraft or other structures with a similar shape to be captured. The end effectors and the targets are steel, so [PERSON]'s modulus and Poisson's ratio in Equations (7) and (8) are \(E_{1}=E_{2}=2.06\times 10^{11}\) and \(v_{1}=v_{2}=0.3\). The other physical parameters of the robot system are given in Table 1. The space robot shown in Figure 1a is like the Shuttle Remote Manipulator System (SRMS) designed by the Canadian Space Agency (CSA). As shown in Figure 1, frame \(O\)-\(XYZ\) is an absolute reference frame; frame \(O_{1}\)-\(X_{1}Y_{1}Z_{1}\) paralleling to the frame \(O\)-\(XYZ\) fixes on the robot base \(B_{1}\) and the origin \(O_{1}\) is on the mass center of \(B_{1}\). Floating frames \(O_{2}\)-\(X_{2}Y_{2}Z_{2}\) and \(O_{3}\)-\(X_{3}Y_{3}Z_{3}\) fix on the two links \(B_{2}\) and \(B_{3}\), respectively, and the origins \(O_{2}\) and \(O_{3}\) are on the mass centers of the two links. Furthermore, frames \(O_{6}\)-\(X_{6}Y_{6}Z_{6}\) and \(O_{7}\)-\(X_{7}Y_{7}Z_{7}\) fixed on the body \(B_{5}\) are used to describe the relative rotational motion of the two hands of the end effector with respect to the body \(B_{5}\). In the simulations, the initial joint angles of \(B_{2}\)-\(B_{5}\) are chosen as (\({}^{\circ}\) 30\({}^{\circ}\)\(-\)57.2041\({}^{\circ}\) 27.2038\({}^{\circ}\) 0\({}^{\circ}\) 0\({}^{\circ}\)), where the axes of \(B_{4}\) and \(B_{5}\) are both parallel to the \(Y\) axis. The space robot is in the \(O\)-\(YZ\) plane before capturing. The initial and final positions of \(B_{6}\) and \(B_{7}\) are shown in Figure 3. The robot base is free-floating during capturing, while only the robot manipulator and the end-effector are controlled. The desired angular velocities of the manipulator (\(B_{2}\)-\(B_{5}\)) are all zero. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Body** & **Mass (kg)** & \(I_{xx}\) (kg\(\cdot\)**m\({}^{2}\))** & \(I_{yy}\) (kg\(\cdot\)**m\({}^{2}\))** & \(I_{zz}\) (kg\(\cdot\)**m\({}^{2}\))** & \(EI  
The effectiveness of the active controller designed in Section 4 is verified firstly. The relative position of mass center of the target cylinder \(B_{8}\) in the \(O_{1}\)-\(X_{1}Y_{1}Z_{1}\) frame is (0 m, 25.05 m, 1 m) and the cylinder axis parallels the \(X_{1}\) axis. The absolute angular velocity and the absolute translation velocity of the target cylinder \(B_{8}\) are (0, 0, 0) and (0, \(-\)0.05 m/s (\(-\)0.1 m/s), 0), respectively, namely \(B_{8}\) moves toward the space robot at 0.05 m/s and 0.1 m/s along the \(Y\) direction. Since the robot is at rest before impact, the absolute velocity of \(B_{8}\) is the relative one. The robot will grasp the middle position of \(B_{8}\), as shown in Figure 3c. The following two control cases are considered in the simulations:  
**Case 1:** the six joints of the manipulator are locked and the manipulator can be regarded as one body. The two hands \(B_{6}\) and \(B_{7}\) of the end effector are only controlled in the capture of the target.