# Design and Dynamic Simulation Verification of an On-Orbit Operation-Based Modular Space Robot
## 3 Dynamic Modeling of Spatial Modular Robot Based on R-W Method  
### Joint Motion Model  
The joint in the multi-body system is an abstraction of the constraints of the adjacent rigid bodies. The relative motion of the joint can be described by the relative motion relationship of the reference system fixed on the two parts. One of the two parts serves as a reference for relative motion, and the conjoined base on the part becomes the local base of the joint, which is recorded as \(\underline{e}^{h_{0}}\). Its base point is recorded as \(Q\) and the conjoined base of another component that moves relative to the base is called the moving base of the joint, denoted as \(\underline{e}^{h}\). Its base point is recorded as \(P\) and vector diameter is denoted as \(\boldsymbol{h}\), which describes the relative motion of the joint. The derivative of the vector \(\boldsymbol{h}\) in local basis \(\underline{e}^{h_{0}}\) with respect to time is \(\boldsymbol{v}_{r}\). The relative acceleration is \(\overset{\circ}{\boldsymbol{v}}_{r}\) and the direction cosine matrix of the dynamic basis relative to the bulk basis is recorded as \(\underline{A}^{h}\), which describes the posture of the dynamic base relative to the noumenal base. The relative angular velocity vector of the dynamic base relative to the body base is \(\omega_{r}\). The coordinate matrix of any vector such as \(\boldsymbol{a}\) in basis \(\underline{e}^{h_{0}}\) is called the local coordinate matrix or joint coordinate matrix of the joint, denoted as \(a^{\prime}\). The coordinate system is shown in Figure 13.  
For the space robot module developed in this work, its joint type is a rotating joint, which is a joint with only one relative rotational degree of freedom. Local basis \(\underline{e}^{h_{0}}\) and dynamic basis \(\underline{e}^{h}\) are established, respectively, at the midpoint of the rotation axis in which \(P\) coincides with \(Q\). The basis vector \(\underline{e}^{h_{0}}=\underline{e}^{h}=\boldsymbol{p}_{1}\), \(\boldsymbol{p}_{1}\) is the unit vector of the joint axis  
Figure 12: Module assembly experiment diagram.  
Figure 13: Joint motion model.  
and the rotation angle \(q_{1}\) is the generalized coordinate of the rotation joint. Assume that the initial rotation angle \(q_{10}=0\), then one has  
\[\underline{A}^{h}=\begin{pmatrix}1&0&0\\ 0&C_{1}&-S_{1}\\ 0&S_{1}&C_{1}\end{pmatrix}\]  
where \(S_{1}=\sin q_{1}\), \(C_{1}=\cos q_{1}\).  
The relative angular velocity vector of the rotating joint is \(\mathbf{\omega}_{r}=\mathbf{p}_{1}\hat{q}_{1}\). Since the vector \(\mathbf{p}_{1}\) is fixed to the joint foundation, the relative angular acceleration \(\overset{\circ}{\mathbf{\omega}_{r}}\) is \(\overset{\circ}{\mathbf{\omega}_{r}}=\mathbf{p}_{1}\hat{q}_{1}\). The local coordinate matrix of the unit vector is \({p^{\prime}}_{1}=\left(1\quad 0\quad 0\right)^{T}\), and the coordinate matrix of relative velocity and relative angular acceleration vector is \(\mathbf{\omega}^{\prime}_{r}={p^{\prime}}_{1}\hat{q}_{1}\), \(\overset{\circ}{\mathbf{\omega}^{\prime}_{r}}={p^{\prime}}_{1}\hat{q}_{1}\). Since the two base points \(Q\) and \(P\) always coincide, the relative displacement, relative velocity, and relative acceleration of the base point are all zero.  
The base in a free-floating state and 6 degrees of freedom is regarded as the case that is connected to the inertial reference frame through 6 degrees of freedom virtual joint.  
The generalized coordinate of relative motion of joints is \(\underline{q}=\left(\underline{q}^{TT}\quad\underline{q}^{\prime T}\right)^{T}\) and the vector diameter \(\mathbf{h}\) and generalized expressions of velocity, acceleration, angular velocity, and angular acceleration are \(\mathbf{h}=\underline{\mathbf{H}}^{\text{HT}}\underline{q}\), \(\overset{\circ}{v}_{r}=\overset{\circ}{\underline{H}}^{\text{HT}}\underline{ \tilde{q}}\), \(\overset{\circ}{\mathbf{\omega}_{r}}=\underline{\mathbf{H}}^{\text{MT}}\underline{ \tilde{q}}+\mathbf{\eta}\), respectively.