# Capturing a Space Target Using a Flexible Space Robot
## 5 Numerical Simulations
In controller design, the matrices \(K\) and \(D\) in Equation (15) are both \(26\times 26\) matrices, and all elements of \(K\) and \(D\) are chosen to be zeros except \(K_{25,25}=K_{26,26}=200\) and \(D_{25,25}=D_{26,26}=50\). \(K_{25,25}\), \(K_{26,26}\), \(D_{25,25}\), and \(D_{26,26}\) correspond to the control weightings of \(B_{6}\) and \(B_{7}\). The desired rotational velocities of \(B_{6}\) and \(B_{7}\) are both 0.1 rad/s. **Case 2:** the six joints of the manipulator and the two hands \(B_{6}\) and \(B_{7}\) are all controlled in capturing the target. For this case, active compliance controls are applied on the six joints of the manipulator to offset the capture of the robot. In controller design, all elements of \(K\) are chosen to be zeros except \(K_{25,25}=K_{26,26}=200\), and \(D\) is chosen as \(D=\text{diag}\ \big{[}\mathbf{0}_{1\times 6}\ 100\ 100\ \mathbf{0}_{1\times 6}\ 100\ 100\ 100\ 50\ 50\big{]}\). The first \(\mathbf{0}_{1\times 6}\) in \(D\) is the control weighting of the robot base, the second and third \(\mathbf{0}_{1\times 6}\) are the control weightings of modal coordinates of the two flexible links \(B_{2}\) and \(B_{3}\), respectively; and the six 100 are the control weightings of six joint coordinates of the manipulator. The desired rotational velocities of \(B_{6}\) and \(B_{7}\) are both 0.1 rad/s too. Figure 4 depicts the state of the space robot in the pre-capture and post-capture phases. Figures 5-8 present the simulation results for Cases 1 and 2, where Figures 5 and 7 are the time histories of the attitude of the robot base, and Figures 6 and 8 are those of the rotation angle of the hands \(B_{6}\) and \(B_{7}\); the solid line represents the results of Case 1 and the dotted line Case 2. From Figures 5 and 7, it can be observed that capturing operation only causes the plane motion of the robot system and does not produce rotational motions in the \(Y\) and \(Z\) directions. This is because the space robot is initially in the O-YZ plane, and the middle position of the target is grasped. The change in rotational angle of the robot base around the \(X\)-axis using an active compliance controller is smaller than that without using the controller, which demonstrates the effectiveness of the controller in reducing the capture impact on the robot. From Figures 6 and 8, we can observe that the target can finally be firmly grasped for the two cases. Figure 4: State of the space robot in the process of capturing: (**a**) pre-capture, and (**b**) post-capture. Figure 5: Time histories of angular displacement of the robot base \(B_{1}\) with and without active compliance control of the manipulator (0.05m/s): (**a**) the \(X\) axis, (**b**) the \(Y\) axis, and (**c**) the \(Z\) axis. Figure 6: Time histories of rotation angle of the two hands \(B_{6}\) and \(B_{7}\) with and without active compliance control of the manipulator (0.05 m/s): (**a**) \(B_{6}\), and (**b**) \(B_{7}\). the target cylinder \(B_{8}\) is (0, \(v\) m/s, 0) for the first and third impact cases (Figure 9a,c), and they are (0, 0, 0) and (0, \(-v\) m/s, 0) for the second and fourth impact cases (Figure 9b,d). In other words, the target cylinder \(B_{8}\) moves at a constant speed \(v\) m/s along the \(Y\) direction with no rotation towards or away from the robot. The six joints of the manipulator and the two hands \(B_{6}\) and \(B_{7}\) are all controlled in the capturing process. In controller design, \(\mathbf{K}\) and \(\mathbf{D}\) are chosen as the same as those in Case 2, namely all elements of \(\mathbf{K}\) are chosen to be zeros except \(K_{25,25}=K_{26,26}=200\), and \(\mathbf{D}=\text{diag}\) {\(\mathbf{0}_{1\times 6}\) 100 100 \(\mathbf{0}_{1\times 6}\) 100 \(\mathbf{0}_{1\times 6}\) 100 100 100 50 50]. The desired rotational velocities of \(B_{6}\) and \(B_{7}\) are both also 0.1 rad/s. The simulation results are given in Figures 10-19, where Figures 10 and 14 are the attitude change of the robot base \(B_{1}\) for \(v\) = 0.05 m/s and \(v\) = 0.1 m/s, respectively.