## 2 Description and Design of the Woodworking Manipulator and Worktable Under Study
### Kinematic Analysis and Modeling  
After simplifying the structure of the woodworking manipulator and worktable, the forward kinematics modeling is carried out by the D-H parametric method [27]. Due to the advantages of few parameters, a simple form, and easy mathematical modeling, the D-H parametric method is widely used in the kinematic analysis of manipulators, which is generally divided into the standard D-H parametric method (SDH) and the modified D-H parametric method (MDH), and the main difference between the two is the way of placing the coordinate system, and its definition of a transformation matrix is different. The standard D-H method focuses on the common plumb line between the axes, which is suitable for most manipulator models with simple joint arrangements, but it can lead to cumbersome modeling of complex mechanisms. The improved D-H method simplifies the coordinate system arrangement, reduces redundant calculations, and is suitable for modeling complex robotic systems. In this paper, the modified D-H parameter method is used to model the manipulator and worktable more easily. After homogeneous transformation by using the modified D-H parameter method, the transformation matrix of the adjacent linkage coordinate system can be obtained. Tables 1 and 2 show the D-H parameter tables of the manipulator and worktable. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(i\) & \(\theta_{i}\)/(rad) & \(d_{i}\)/(mm) & \(\alpha_{i-1}\)/(\({}^{\circ}\)) & \(a_{i-1}\)/(mm) & **Joint Range** \\ \hline
1 & \(-\pi/2\) & \(d_{1}\) & 0 & 0 & 0–1500 mm \\
2 & \(\pi/2\) & \(d_{2}\) & \(-\pi/2\) & 0 & 0–1500 mm \\
3 & \(\theta_{3}\) & \(d_{3}\) & \(-\pi/2\) & 0 & \(-\pi\)-\(\pi\) \\
4 & \(\theta_{4}\) & \(d_{4}\) & \(\pi/2\) & 0 & \(-\pi\)-\(\pi\) \\ \hline \hline \end{tabular} where \(\theta_{3}\) and \(\theta_{4}\) represent the joint angle variables corresponding to revolute joints 3 and 4; the domain of the \(\theta_{3}\) and \(\theta_{4}\) variables spans the interval [\(-\)180\({}^{\circ}\), 180\({}^{\circ}\)]; \(d_{1}\) and \(d_{2}\) represent the joint displacement variables corresponding to joints 1 and 2; the domain of the \(d_{1}\) and \(d_{2}\) variables span the interval [0, 1500]; \(d_{3}\) = 672 mm; and \(d_{4}\) = 286.5 mm.  
\end{table} 
Table 1: Modified D-H parameters of woodworking manipulator. Figure 1: Overall structure of woodworking manipulator and worktable. Figure 2: Simplified woodworking manipulator and worktable. The parameters in the above table are inserted into the coordinate system transformation matrices of the joints of the manipulator and worktable to realize the transformation from the base coordinate system to the tool coordinate system, and then the chi-square transformation matrices \(T_{1}\) and \(T_{2}\) of the corresponding end-effector are obtained. In the Equations (1)-(4), \({}^{i-1}A_{i}\) is the transformation matrix of the neighboring linkage coordinate system of the \(i\)-th joint, \(C\theta\) denotes \(\cos\theta\), \(S\theta\) denotes \(\sin\theta\), \(T_{1}\) and \(T_{2}\) are the tool coordinate system transformation matrices of the manipulator and worktable, respectively; \(T_{01}\) and \(T_{02}\) are the base coordinate system transformation matrices of the manipulator and worktable, respectively; and \(Tool_{1}\) and \(Tool_{2}\) are the tool coordinate system transformation matrices, respectively. \[ \begin{split} T_{1}&=T_{01}\cdot{}^{0}A_{1}\cdot{} ^{1}A_{2}\cdot{}^{2}A_{3}\cdot{}^{3}A_{4}\cdot Tool_{1}\\ &=\begin{bmatrix}r_{11}&r_{12}&r_{13}&P_{x}\\ r_{21}&r_{22}&r_{23}&P_{y}\\ r_{31}&r_{32}&r_{33}&P_{z}\\ 0&0&0&1\end{bmatrix}\\ &=\begin{bmatrix}-C\theta_{4}S\theta_{3}&S\theta_{3}S\theta_{4}&C\theta_{3}&d _{2}+d_{4}C\theta_{3}+yS\theta_{3}S\theta_{4}\\ \delta\theta_{3}&C\theta_{4}&0&d_{3}+yC\theta_{4}\\ -C\theta_{3}C\theta_{4}&C\theta_{3}S\theta_{4}&-S\theta_{3}&d_{1}+z-d_{4}S \theta_{3}+yC\theta_{3}S\theta_{4}\\ 0&0&0&1\end{bmatrix}\end{split}