# 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
### Position Motion of Each Rigid Body in the System  
For a multi-rigid body system connected by rotating joints such as rotary joints and universal joints, it will become a multi-rigid body system with rotating joints. Its characteristic is that the joint points on the adjacent objects of all joints coincide, and there is only relative rotation without position motion. The robot system developed in this work is exactly such a system. The position, angular velocity, and acceleration analysis of each rigid body's center of mass in the system will designate the distribution of the joint points on the rigid body. Thus, in describing the basic parameters of a multi-rigid body system, in addition to the topological configuration parameters of the system, it is also necessary to give the distribution of the joints on each object. Thus, before discussing the position, velocity, and acceleration of the center of mass of each rigid body in the system, we will first introduce the body joint vector and pathway vectors. For a multi-rigid body system composed of \(N\) objects, the center of mass of the rigid body \(B_{i}(i=1,\cdots,N)\) is recorded as \(C_{i}\). The vector pointing from \(C_{i}\) to a joint point on the rigid body is defined as the body joint vector of the joint \(H_{j}\), which is recorded as \(c_{ij}\), where \(i\) represents the fixed rigid body and \(j\) represents the joint pointed by this vector. If the joint \(H_{j}\) is not related to \(B_{i}\), \(c_{ij}=0\) for a root object whose motion is known, the starting point \(C_{0}\) of the body joint vector is generally taken at the joint \(H_{1}\). Since each joint involves two body joint vectors, the non-zero body joint vector for the tree system is \(2N-1\). Their coordinate matrix in the conjoined bases is one of the basic parameters describing the system. For a tree system, using the correlation matrix elements, the weighted body joint vectors are  
\[C_{ij}=S_{ij}c_{ij}\quad(i=1,\cdots,N;j=1,\cdots,N) \tag{11}\]  
There are still \(2N-1\) non-zero weighted body joint vectors. \(S_{ij}\) negates \(c_{ij}\), pointing to the inscribed joint and \(C_{ij}\) is in the same direction as the joint \(H_{j}\), deviating from the direction of \(B_{0}\). The weighted joint vector \(C_{ij}\) is used to form an \(N\)-order vector square matrix to become the body joint vector matrix \(\underline{\zeta}\). This vector matrix describes the distribution of joints of each object. The body joint vector matrix of the robot system is \[\underline{C}= \begin{pmatrix}-c_{11}&c_{12}&c_{13}&0&c_{15}&c_{16}\\ 0&-c_{22}&0&0&0&0\\ 0&0&-c_{33}&c_{34}&0&0\\ 0&0&0&-c_{44}&0&0\\ 0&0&0&0&-c_{55}&0\\ 0&0&0&0&0&-c_{66}\end{pmatrix} \]  
Each object has only one internal joint in the tree systems and the internal joint of the rigid body \(B_{i}\) is \(H_{i}\). The connected vector \(\mathbf{d}_{ik}\) from \(H_{i}\) in \(B_{i}\) can be defined as a pathway vector. When \(k\
eq i\), this vector points to another joint on \(B_{i}\), which is located on the path from \(B_{i}\) to \(B_{k}\). When \(k=i\), this vector points to the center of mass of the object. The \(N\)-order vector square matrix \(\underline{d}\) containing \(\mathbf{d}_{ik}\) is called the path vector matrix, which describes the joint distribution of each object. The path vector matrix \(\underline{d}\) of the tree system is  
\[\underline{d}= \begin{pmatrix}d_{11}&d_{12}&d_{13}&d_{14}&d_{15}&d_{16}\\ 0&d_{22}&0&0&0&0\\ 0&0&d_{33}&d_{34}&0&0\\ 0&0&0&d_{44}&0&0\\ 0&0&0&d_{55}&0\\ 0&0&0&0&d_{66}\end{pmatrix} = \begin{pmatrix}d_{11}&d_{12}&d_{13}&d_{13}&d_{15 }&d_{16}\\ 0&d_{22}&0&0&0&0\\ 0&0&d_{33}&d_{34}&0&0\\ 0&0&0&d_{44}&0&0\\ 0&0&0&0&d_{55}&0\\ 0&0&0&0&0&d_{66}\end{pmatrix} \]  
Both the body joint vector and the path vector describe the position of the joint point on the rigid body, but the starting point of the vector is different and their mathematical relationship can be described as \(\underline{d}=-\underline{C}T\).