## 2 Robot Kinematic Modeling  
The system of the robot has a clamping platform with eight degrees of freedom (DOF), including a robot and a platform, which allows expanding the machining space (Figure 1). The robot is designed according to the robot model of Asea Brown Boveri, which is a six DOF serial robot, consisting of six moveable links connected to a fixed base. The cutting tool is mounted on the end effector of the robot. The platform has two DOF, including two moveable links; the workpiece is fixed on the upper link. Names of the robot links and platform links of the system in turn are: L0, L1,..., L6, B and D, respectively. Link L0 is the base, link L6 is the end effector, link B connects the prismatic joint to the fixed base, and link D carries the workpiece. The robot and the clamping platform coordinate programmatically to perform the form-shaping motion of the machining surface. Normally, the machining object is clamped on a fixed platform, and a robot with 6 DOF is sufficient to obtain positions and orientations of the end effector for the machining performance. However, for the purpose of expanding the operational space of the robot, or choosing a convenient machining posture, a machining system with a two DOF clamping platform is utilized. As a result, the robot can machine large parts with many complex surfaces, reducing the number of clamping operations. Robots with a combination of two such component robots or more are often referred to Mechanism of Relative Manipulation robots--MRM [73, 74, 75, 76, 77, 78, 79, 80, 81]. For a simple presentation, the robot-platform system is hereinafter called the robot system or the robot. The article applies the Denavit-Hartenberg convention to define the link frames and derive kinematic equations of the robot kinematic chains, using generalized coordinates to represent and calculate the kinematic chain of the platform. The coordinate systems, kinematic parameters and corresponding homogeneous coordinate transformation matrices are shown in Table 1.  
In Table 1, O\({}_{1}\)x\({}_{1}\)y\({}_{1}\)z\({}_{1}\) (i = 1,..., 6) is the frame that is attached to the end of link i, and represents the position and orientation of frame O\({}_{1}\)x\({}_{1}\)y\({}_{1}\)z\({}_{1}\) with respect to frame O\({}_{1-1}\)x\({}_{1-1}\)y\({}_{1-1}\)z\({}_{1-1}\) by the Denavit-Hartenberg homogeneous coordinate transformation matrix \({}^{1-1}\)A\({}_{1}\), with respect to frame O\({}_{0}\)x\({}_{0}\)y\({}_{0}\)z\({}_{0}\) by matrix \({}^{0}\)A\({}_{1}\). O\({}_{6}\)x\({}_{6}\)y\({}_{6}\)z\({}_{6}\) is called the end-effector frame. O\({}_{0}\)x\({}_{0}\)y\({}_{0}\)z\({}_{0}\) is called the base frame for the entire robot-platform system, with matrix E being the \(4\times 4\) identity matrix that represents the position and orientation of the base frame with respect to itself. The parameters in the third column of Table 1 represent the relative positions between two consecutive frames, in which \(\theta_{1}\) (i = 1,..., 6), \(y_{\rm b}\), x\({}_{\rm d}\) are the joint coordinates, and the other parameters are constants. The vector of joint coordinate is denoted by q, represented by Equation (1). \[\rm{q}=\left[q_{1},\ldots,q_{8}\right]^{T}=\left[\theta_{1},\ldots,\theta_{6 },y_{\rm b},x_{\rm d}\right]^{T} \tag{1}\]  
O\({}_{0}\)x\({}_{0}\)y\({}_{0}\)z\({}_{0}\) is the fixed frame that has the origin at position O\({}_{0^{\prime}}\) and the axes parallel to the corresponding ones of the base frame, used to location the platform.