## 2 Dynamic Modeling of 6R Robot System
### Graphical Description of the 6R Robot System  
The six rigid bodies that make up the 6R robot system are simply represented as homogeneous cylinders, denoted as \(B_{i}\) (\(i=1,2,\cdots,6\)), the length of \(B_{i}\) is denoted as \(l_{i}\) and the radius is denoted as \(R_{i}\), the center of mass of each rigid body is its geometric center, denoted as \(O_{ci}(i=1,2,\cdots,6)\). Establish the body-fixed base of each rigid body at the center of mass of each rigid body, denoted as \((O_{i},\mathbf{e}^{(i)})(i=1,2,\cdots,6)\) (The underscore indicates the matrix form), the three vectors \(\mathbf{e}_{1}^{(i)}\), \(\mathbf{e}_{2}^{(i)}\) and \(\mathbf{e}_{3}^{(i)}\) are the base vectors for each coordinate axis, the base vectors point to the direction of the central principal inertia axes. The body-fixed bases of the 6R robot system are shown in Figure 2, with the pose shown as the initial pose, the \(\mathbf{e}_{1}^{(i)}\) axis points from the inside to the outside along the length of the rigid body, the \(\mathbf{e}_{2}^{(i)}\) axis direction is perpendicular to the paper surface inwards and the \(\mathbf{e}_{3}^{(i)}\) axis direction is determined by the right-hand rule. Since \(B_{0}\) is firmly connected to the ground, in order to simplify the model and facilitate calculations, its volume and mass are both considered to be 0, its body-fixed base is \((O_{0},\mathbf{e}^{(0)})\), which is located at \(O_{1}\), \((O_{0},\mathbf{e}^{(0)})\) is used as a fixed reference base and \(O_{0}\) is used as a fixed reference point for this paper, the position of each center of mass is expressed relative to \(O_{0}\) and the posture of each rigid body is expressed relative to \((O_{0},\mathbf{e}^{(0)})\). The remaining rigid bodies are \(B_{1},B_{2},\cdots,B_{6}\) in the direction from \(B_{0}\) to the end-effector, the  
Figure 2: Schematic diagram of the 6R robot system body-fixed bases. If \(B_{j}\) is on the path from \(B_{0}\) to \(B_{i}\), then \(B_{j}\) is called the inside body of \(B_{i}\) and \(B_{i}\) is called the outside body of \(B_{i}\). In particular, if the inside rigid body \(B_{j}\) has incidence with \(B_{i}\), then \(B_{j}\) is called the inside connecting body of \(B_{i}\) and the \(O_{i}\) connecting \(B_{j}\) and \(B_{i}\) is called the inside joint of \(B_{i}\). For \(B_{j}\), \(B_{i}\) is called the outside connecting body of \(B_{j}\) and \(O_{i}\) is called the outside joint of \(B_{j}\). Based on the above basic concepts, we established the incidence matrix, the matrix of body-joint vector, the path matrix of force element and the vector matrix of force element of the 6R robot system by graph theory, and deduced the resultant force and the resultant moment of the force element received by the rigid body. First, the incidence matrix \(\underline{S}\) of the 6R robot system is established. Assuming that the robot system has \(n\) rigid bodies, define an element:  
\[S_{ij}=\left\{ \begin{array}{ll}1&O_{j}\text{ has incidence with }B_{i}\text{ and starts with }B_{i}\\ -1&O_{j}\text{ has incidence with }B_{i}\text{ and ends with }B_{i}\\ 0&O_{j}\text{ has no incidence with }B_{i}\end{array} \right.\qquad\qquad\qquad\qquad(i,j=1,2, \cdots,n). \tag{8}\]  
The \(n\times n\) square matrix composed of \(S_{ij}\) is called the incidence matrix of the system, denoted as \(\underline{S}\). \[\underline{S}=\left[ \begin{array}{ccc}S_{11}&\cdots&S_{1n}\\ \vdots&\ddots&\vdots\\ S_{n1}&\cdots&S_{mn}\end{array} \right]. \tag{9}\]  
For the 6R robot system,  
\[\underline{S}=\left[ \begin{array}{cccccc}-1&1&0&0&0&0\\ 0&-1&1&0&0&0\\ 0&0&-1&1&0&0\\ 0&0&0&-1&1&0\\ 0&0&0&0&-1&1\\ 0&0&0&0&0&-1\end{array} \right]. \tag{10}\]  
Second, the matrix of body-joint vector \(\underline{C}\) of the 6R robot system is established. The vector pointing from \(O_{ci}\) to any joint \(O_{j}\) that has incidence with \(B_{j}B_{i}\) is called the body-joint vector, denoted as \(\boldsymbol{c}_{ij}\). The multiplication of \(\boldsymbol{c}_{ij}\) and \(S_{ij}\) is called the weighted body-joint vector, denoted as \(\boldsymbol{c}_{ij}\).