## 3 Error-Kinematic Model of Space Robot  
As shown in Figure 1, the proposed space robot mainly includes two 6-DOF PUMA560-type robotic arms, and the dual-arm is mounted on the satellite base. Arm-1 is the mission arm and Arm-2 is the balance arm.  
According to momentum conservation and the pose-feedback modeling idea, the pose-error kinematic relationship of end-effector and base [16] can be expressed as:  
\[\left\{\begin{array}{l}\dot{\mathbf{e}}_{1}=\mathbf{I}_{c}\mathbf{V}_{e}-\mathbf{I}_{ad}\bm {V}_{cd}=\mathbf{I}_{c}\big{(}\mathbf{J}_{0}V_{0}+\mathbf{I}_{n}^{1}\dot{\mathbf{\theta}}^{1} \big{)}-\mathbf{I}_{cd}\mathbf{V}_{ad}\\ \dot{\mathbf{e}}_{0}=\mathbf{J}_{b}\mathbf{V}_{0}-\mathbf{I}_{b}\mathbf{V}_{0d}=\mathbf{J}_{b}\mathbf{H}_{0 }^{-1}\big{(}\mathbf{C}-\mathbf{J}_{c}^{1}\dot{\mathbf{\theta}}^{1}-\mathbf{J}_{c}^{2}\dot{ \mathbf{\theta}}^{2}\big{)}-\mathbf{J}_{bd}\mathbf{V}_{0d}\end{array}\right. \tag{6}\]  
where \(\mathbf{e}_{1}=\big{[}\begin{array}{cc}\mathbf{e}_{e}&\mathbf{\sigma}_{e}\end{array} \big{]}^{T}\) and \(\mathbf{e}_{0}=\big{[}\begin{array}{cc}\mathbf{e}_{b}&\mathbf{\sigma}_{b}\end{array} \big{]}^{T}\), in which the subscripts \(e\) and \(\sigma\) denote the position and attitude error of FFSR, respectively; \(\mathbf{C}\) is the initial pose momentum of the system; \(\mathbf{J}_{0}\) and \(\mathbf{J}_{m}^{1}\) are the Jacobian matrixes of the base and Arm-1 of FFSR, respectively; \(\mathbf{I}_{c}^{1}\) and \(\mathbf{J}_{c}^{2}\) are the coupling inertia matrices of Arm-1 and Arm-2, respectively; \(\mathbf{V}_{e}=\big{[}\begin{array}{cc}\mathbf{v}_{e}&\mathbf{\omega}_{e}\end{array} \big{]}^{T}\) is the velocity vector of the end-effector; \(\mathbf{V}_{0}=\big{[}\begin{array}{cc}\mathbf{v}_{b}&\mathbf{\omega}_{b}\end{array} \big{]}^{T}\) is the velocity of base; \(\mathbf{V}_{cd}=\big{[}\begin{array}{cc}\mathbf{v}_{ed}&\mathbf{\omega}_{ed}\end{array} \big{]}^{T}\) is the desired velocity of end-effector; \(\mathbf{V}_{0d}\) is the desired velocity of base which is set to 0 here; \(\mathbf{\theta}^{1}\in\mathbb{R}^{6\times 1}\) and \(\mathbf{\theta}^{2}\in\mathbb{R}^{6\times 1}\) are the joint angular velocity vector of Arm-1 and Arm-2 (\(j=1,2\)), respectively; \(\mathbf{H}_{0}\) is the coupling inertia  
Figure 1: Model of a Dual-arm Space Robot.  
matrix of momentum conservation of (free-floating space robot) FFSR. \(J_{e,b}\) and \(J_{ed,bd}\) are with (7),  
\[\mathbf{J}_{e,b}=\left[\begin{array}{cc}\mathbf{E}_{3}&0\\ 0&\frac{1}{4}\mathbf{G}(\mathbf{\sigma}_{e,b})\end{array}\right],\ \ \mathbf{I}_{ed,bd}=\left[ \begin{array}{cc}\mathbf{E}_{3}&0\\ 0&\frac{1}{4}\mathbf{G}(\mathbf{\sigma}_{ed,bd})R_{ed,bd}\end{array}\right], \tag{7}\]  
in which \(\mathbf{E}_{3}\) is a 3-order identity matrix; \(R_{ed,bd}\in\mathbb{R}^{3\times 3}\) is the rotational matrix which transforms the reference frame to the frame fixed on the end-effector or base, \(\mathbf{G}(\mathbf{\sigma})=\big{(}1-\mathbf{\sigma}^{T}\mathbf{\sigma}\big{)}\mathbf{E}_{3}+2 \tilde{\mathbf{\sigma}}+2\sigma\mathbf{\sigma}^{T}\).  
**Remark 1**.: _The attitude error is in terms of Modified Rodrigues Parameter(MRP) [16], because the MRP can make attitude computation for robotic system small._