## 2 Robot Kinematic Modeling
O\({}_{\rm b}\)x\({}_{\rm b}\)y\({}_{\rm b}\)z\({}_{\rm b}\) is the frame that attached to link B; it is possible to translate in the direction y\({}_{\rm b}\)//y\({}_{0}\), corresponding to the joint coordinate y\({}_{\rm b}\), the coordinate transformation matrices as shown in Table 1. O\({}_{\rm d}\)x\({}_{\rm d}\)y\({}_{\rm d}\)z\({}_{\rm d}\) is the frame that attached to link D; it is possible to translate in the direction x\({}_{\rm d}\)//x\({}_{\rm b}\)//x\({}_{0}\), corresponding to the joint coordinate x\({}_{\rm d}\), the coordinate transformation matrices as shown in Table 1.  
Frame O\({}_{\rm d}\)x\({}_{\rm d}\)y\({}_{\rm d}\)z\({}_{\rm d}\), hereinafter called the platform frame or the jig frame, is used to locate the workpiece. The parameters of the workpiece's machined surface and toolpath are shown in the jig frame. In the machining process, the tool performs the relative motion with respect to the machined surface according to the law (trajectory) that is determined by the machining engineering. This trajectory is represented by parameters \({}^{\rm d}\)x\({}_{\rm E}\), \({}^{\rm d}\)y\({}_{\rm E}\), \({}^{\rm d}\)z\({}_{\rm E}\), \({}^{\rm d}\)\(\alpha\)e, \({}^{\rm d}\)g\({}_{\rm E}\), \({}^{\rm d}\)n\({}_{\rm UE}\) (Table 1) and their first and second  
Figure 1: The model of robot; clamping platform has eight degrees of freedom (DOF) in milling process. time derivatives with respect to time. Thereby, the vector of operational coordinates is called p, represented by (2). \[\text{p}=\text{{}^{d}}\text{p}_{\text{E}}=\text{{}^{d}}\text{p}_{1},\text{{}^{d}} \text{p}_{2},\ldots,\text{{}^{d}}\text{p}_{6}\text{{}^{I}}=\text{{}^{[d}}\text{ x}_{\text{E}},\text{{}^{d}}\text{y}_{\text{E}},\text{{}^{d}}\text{z}_{\text{E}}, \text{{}^{d}}\text{\alpha}_{\text{E}},\text{{}^{d}}\text{\beta}_{\text{E}}, \text{{}^{d}}\text{\eta}_{\text{E}}\text{]}\text{{}^{T}} \tag{2}\]  
Therefore, the frame O\({}_{\text{d}}\)x\({}_{\text{d}}\)y\({}_{\text{d}}\)z\({}_{\text{d}}\) is used as the operational frame. Frame O\({}_{\text{E}}\)x\({}_{\text{E}}\)y\({}_{\text{E}}\)z\({}_{\text{E}}\) is attached to the cutting edge of the cutter on the end effector, called the tool frame. According to the robot kinematic chain, matrix \({}^{0}\)A\({}_{\text{E}}\) represents the position and orientation of the tool frame O\({}_{\text{E}}\)x\({}_{\text{E}}\)y\({}_{\text{E}}\)z\({}_{\text{E}}\) with respect to the base frame O\({}_{0}\)x\({}_{0}\)y\({}_{0}\)z\({}_{0}\), determined by (3). \[{}^{0}\text{A}_{\text{E}}=\text{{}^{0}}\text{A}_{1}(\theta_{1})^{1}\text{A}_{ 2}(\theta_{2})\ldots\text{{}^{5}}\text{A}_{6}(\theta_{6})^{6}\text{A}_{\text{E}} \tag{3}\]  
According to the platform kinematic chain, \({}^{0}\)A\({}_{\text{E}}\) is determined by (4). \[{}^{0}\text{A}_{\text{E}}=\text{{}^{0}}\text{A}_{0}\text{{}^{0}}\text{{}^{A}} \text{A}_{\text{b}}\text{{}^{0}}\text{{}^{B}}\text{A}_{\text{d}}\text{(x}_{ \text{d}})\text{{}^{d}}\text{A}_{\text{E}}\text{(p)} \tag{4}\]  
From (3) and (4), we obtained (5). \[\text{{}^{d}}\text{A}_{\text{E}}\text{(p)}=\text{{}^{b}}\text{A}_{\text{d}}^{ -1}\text{(x}_{\text{d}})\text{{}^{0}}\text{{}^{A}}\text{A}_{\text{b}}\text{{} ^{-1}}\text{(y}_{\text{b}})\text{{}^{0}}\text{A}_{\text{1}}^{-1}\text{{}^{0} }\text{A}_{\text{1}}(\theta_{1})\ldots\text{{}^{5}}\text{A}_{6}(\theta_{6})^{6 }\text{A}_{\text{E}} \tag{5}\]  
The right hand side of (5) is the matrix representing the position and orientation of the cutting tool with respect to the platform frame O\({}_{\text{d}}\)x\({}_{\text{d}}\)y\({}_{\text{d}}\)z\({}_{\text{d}}\), whose elements are functions of the joint coordinates (1). Meanwhile, the elements of the matrix on the left hand side of (5) are functions of the operational coordinates (2). (5) is rewritten by (6).