## 2 Grinding Robot Process Introduction and Kinematics Model
### Grinding Robot Kinematics Model  
This paper takes the KUKA KR series robot (grinding robot) as the research object, which is composed of a base frame, rotating column, link arm, in-line wrist, arm, and other parts. The main purpose is to install a grinding wheel at the end of the robot manipulator to grind the special steel bar with surface defects, the existing robot manipulator, and the steel bar grinding system (the work of this paper is to provide motion support for the realization of robot grinding steel bar) in the laboratory are shown in Figure 4.  
The essence of the kinematic analysis of the grinding robot is the process of matrix transformation between two adjacent coordinate systems, and then the transformed matrices are multiplied successively, and finally, the kinematic equation of the robot is obtained. The solution of the kinematics equation consists of two parts, forward kinematics solution, and inverse kinematics solution. The forward kinematics solution is based on the known angle of each joint variable to solve the position and orientation of the end-effector. The inverse kinematics is based on the known position and orientation of the end-effector to solve the angle of each joint variable [28]. The specific solution process is described in detail in the literature [8], which will not be described here. In order to obtain higher calculation accuracy and simplify the calculation process, a method based on a multi-layer forward propagation deep neural network was proposed to solve the kinematics equation of the robot. According to D-H model parameter method, the coordinate system of each link of grinding robot is established, as shown in Figure 5. The grinding robot is revolute joint, the end is connected with an actuator. No matter how the revolute joint 0 connecting the base frame and the rotating column rotates, its length has no influence on the movement of grinding robot, so the reference coordinate system 0 and coordinate system 1 are coaxial. According to the right-hand rule, the coordinate axes direction of remaining joints are determined [28,29]. Figure 4: Robot manipulator and grinding system. Figure 5: Schematic diagram of manipulator coordinate system. When the coordinate system of grinding robot manipulator is established, the corresponding kinematics analysis and calculation can be carried out according to the given parameters of each link of grinding robot. The variation range of D-H model parameters and related variables of each link is shown in Table 1.  
Where \(a_{i-1}\) is the length of the link and represents the common perpendicular of two adjacent joints axes connecting link \(i-1\), \(\alpha_{i-1}\) is the angle of the link and represents the angle between two adjacent joints, \(d_{i}\) is the offset of the link and represents the distance from the point where \(a_{i-1}\) intersects axis \(i\) to the point where \(a_{i}\) intersects axis \(i\), \(\theta_{i}\) is the angular displacement between adjacent joints and represents the angle between the extension of \(a_{i-1}\) and the extension of \(a_{i}\). The transformation matrix \({}_{i}^{i-1}T\) of coordinate system \(\{i\}\) with respect to coordinate system \(\{i-1\}\) is as follows. \[ \begin{array}{c}{}_{i}^{i-1}T=Rot(x,\alpha_{i-1})Trans(x,a_{i-1})Rot(z,\theta _{i})Trans(z,d_{i})\end{array}  \tag{1}\]  
\[ \begin{array}{c}{}_{i}^{i-1}T=Screw(x,a_{i-1},\alpha_{i-1})Screw(z,d_{i}, \theta_{i})\end{array}  \tag{2}\]  
where \(Screw(L,r,\varphi)\) is translating \(r\) along axis \(L\) and rotating \(\varphi\) around axis \(L\). The general expression of transformation matrix \({}_{i}^{i-1}T\) between links is as follows. \[ \begin{array}{c}{}_{i}^{i-1}T=\begin{bmatrix}\cos\theta_{i}&-\sin\theta_{i} &0&a_{i-1}\\ \sin\theta_{i}\cos\alpha_{i-1}&\cos\theta_{i}\cos\alpha_{i-1}&-\sin\alpha_{i-1 }&-d_{i}\sin\alpha_{i-1}\\ \sin\theta_{i}\sin\alpha_{i-1}&\cos\theta_{i}\sin\alpha_{i-1}&\cos\alpha_{i-1 }&d_{i}\cos\alpha_{i-1}\\ 0&0&0&1\end{bmatrix}\end{array}