# Capturing a Space Target Using a Flexible Space Robot
## 2 Dynamic Model of Space Robot  
This paper uses a simplified flexible space robot and a target (shown in Figure 1) to study capturing operation. The space robot is composed of a spacecraft base (the body \(B_{1}\)), a manipulator (the bodies \(B_{2}\)-\(B_{5}\)), and an end effector (the bodies \(B_{6}\) and \(B_{7}\)). The manipulator consists of two flexible links (the bodies \(B_{2}\) and \(B_{3}\)) and two rigid links (the bodies \(B_{4}\) and \(B_{5}\)) and has six degrees of freedom (Axes 1-6). The body \(B_{2}\) can rotate around Axes 1 and 2, \(B_{3}\) around Axis 3, \(B_{4}\) around Axis 4, and the body \(B_{5}\) around Axes 5 and 6. The end effector is composed of three same cylinder-shaped hands, two of which form \(B_{6}\), while the other one forms \(B_{7}\). The body \(B_{8}\) is the space target to capture, denoted by a rigid cylinder that simulates the handle of a captured spacecraft. The space robot shown in Figure 1a is like the Shuttle Remote Manipulator System (SRMS) designed by the Canadian Space Agency (CSA). Based on the Jourdain's velocity variation principle, the dynamic equation of the robot system can be expressed as [34]\[\sum_{i=1}^{n}\Delta\mathbf{v}_{i}^{\rm T}(-\mathbf{M}_{i}\dot{\mathbf{v}}_{i}-\mathbf{f}_{i}^{ \omega}-\mathbf{f}_{i}^{\omega}+\mathbf{f}_{i}^{\omega})+\Delta P=0 \tag{1}\]  
where \(\mathbf{v}_{i}\) is the vector of configuration velocity of the \(i\)-th body, \(\mathbf{M}_{i}\dot{\mathbf{v}}_{i}\) denotes the acceleration-dependent inertia force acting on the \(i\)-th body, \(\mathbf{f}_{i}^{\omega}\) is the velocity-dependent inertia force acting on the \(i\)-th body, \(\mathbf{f}_{i}^{\omega}\) is the external force of the system acting on the \(i\)-th body, \(\mathbf{f}_{i}^{\omega}\) is the elastic force acting on the \(i\)-th body (if the \(i\)-th body is a rigid body, \(\mathbf{f}_{i}^{\omega}\) will be a zero vector), and \(\Delta P\) is the sum of virtual power of the inner forces of the system. \(n=7\) in Equation (1) since the robot system shown in Figure 1 is composed of seven bodies. By the assumed mode method, the elastic force \(\mathbf{f}_{i}^{\omega}\) can be expressed as [34]. \[\mathbf{f}_{i}^{\omega}=\left[0^{\rm T}\ \ 0^{\rm T}\ \ \left(\mathbf{C}_{a_{i}}\dot{\mathbf{a }}_{i}+\mathbf{K}_{a_{i}}\mathbf{a}_{i}\right)^{\rm T}\right]^{\rm T}\in\Re^{(6+s)\times 1} \tag{2}\]  
where \(\mathbf{a}_{i}\) is the modal coordinate vector of the \(i\)-th body, \(\mathbf{C}_{a_{i}}\in\Re^{s\times s}\) and \(\mathbf{K}_{a_{i}}\in\Re^{s\times s}\) are the modal damping and stiffness matrices of the \(i\)-th body, \(s\) denotes the highest order of the mode chosen. For the robot system given in Figure 1, the bodies \(B_{2}\) and \(B_{3}\) are flexible, so only these two bodies have elastic force. In the simulations, the first six modes of \(B_{2}\) and \(B_{3}\) are both chosen, so \(s=6\) in this paper. In matrix form, Equation (1) can be written as  
\[\Delta\mathbf{v}^{\rm T}(-\mathbf{M}\dot{\mathbf{v}}-\mathbf{f}^{\omega}-\mathbf{f}^{\omega}+\bm {f}^{\prime})+\Delta P=0 \tag{3}\]  
where \(\mathbf{v}=\left[\mathbf{v}_{1}^{\rm T},\ \cdots,\ \mathbf{v}_{n}^{\rm T}\right]^{\rm T}\) is the configuration velocity of the system, \(\mathbf{M}=diag[\mathbf{M}_{1},\ \cdots,\ \mathbf{M}_{n}]\) is the mass matrix, \(\mathbf{f}^{\omega}=\left[\mathbf{f}_{1}^{\rm T},\ \cdots,\ \mathbf{f}_{n}^{\rm T}\right]^{\rm T}\), \(\mathbf{f}^{\omega}=\left[\mathbf{f}_{1}^{\rm T},\ \cdots,\ \mathbf{f}_{n}^{\rm T}\right]^{\rm T}\), and \(\mathbf{f}^{\omega}=\left[\mathbf{f}_{1}^{\rm T},\ \cdots,\ \mathbf{f}_{n}^{\rm T}\right]^{\rm T}\) are the elastic force, the velocity-dependent inertia force, and the external force, respectively. Figure 1: Space robot and the object to be captured: (**a**) the total diagram of the robot system, and (**b**) enlarged drawing of the end effector.