# Optimal Placement of Vibration Sensors for Industrial Robots Based on Bayesian Theory
## 2 Sensor Placement Method
### Derivation of Forward Kinematics of Industrial Robots  
The mechanical arm of the six-axis industrial robot used in the simulation in this paper is assembled by a series of connecting rods, so a corresponding coordinate system should be constructed to express the robot. At present, the two commonly used link coordinate system construction methods in robotics are the standard type and the improved D-H coordinate system. Among them, the improved D-H refers to adding a new parameter on the basis of the standard four parameters, through which the singularity that occurs when the adjacent connecting rods are in a parallel relationship can be solved. Since the general six-axis industrial robot does not have parallel links, an improved D-H coordinate system that uses more parameters is not used. On the other hand, in the existing standard D-H coordinate system, there are also two different establishment methods. The first is that the origin of the coordinate system \(o_{i-1}\) is unified with the joint \(i\); the corresponding second is that the coordinate system \(o_{i-1}\) is unified with the joint \(i-1\). Due to the problem of the tree structure, the first coordinate system will be ambiguous when dealing with it. Considering the diversity of industrial robots, the second method of establishing a system was chosen after a comprehensive comparison. The first method of establishing a system is described in detail below, and the MATLAB model effects corresponding to the two coordinate systems are given. The establishment method of the connecting rod coordinate system \(o_{i}x_{i}y_{i}z_{i}\) is shown in Table 1. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Origin \(o_{i}\)** & \(z_{i}\)-**Axis** & \(x_{i}\)-**Axis** & \(y_{i}\)-**Axis** \\ \hline When the axis of joint \(i\) intersects with the axis of joint \(i+1\), the intersection point is taken & If the joint axis \(i\) intersects \(i+1\), it is & \\   of joint \(i+1\), the intersection point is taken & perpendicular to the plane where the joint axis & \\ \hline When the axis of joint \(i\) is out of plane with the axis of joint \(i+1\), take the intersection of the common perpendicular of the two axes and the axis of joint \(i\) & Coincides with the axis of joint \(i\) & On the common perpendicular of the links \(i\) & Determined by the \\  & & axis of joint \(i\) & and \(i+1\), its direction is from \(i\) to \(i+1\) & right-hand rule \\ \hline When the axis of joint \(i\) is parallel to the axis of joint \(i+1\), take the intersection of the common perpendicular of the axis of joint \(i+1\) with the axis of joint \(i\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Method for establishing a D-H coordinate system. The coordinate system fixed on the robot base (link 0) is coordinate system {0}. This coordinate system has the property of being invariant and can be used as a reference. The reference coordinate system {0} itself is not specially set, but considering the subsequent calculation, the reference coordinate system {0} is set to coincide with the coordinate system {1}. For the revolute joint n, \(\theta_{n}\) is set to 0. According to the coordinate system establishment criteria described in this paper, the connecting rod coordinate system of the six-axis industrial robot is solved, as shown in Figure 1.  
The link coordinate system of the industrial robot contains 4 parameters. They are defined as follows:  
1. Link length \(a_{i}\): along the \(x_{i}\) axis, the distance from \(z_{i-1}\) to \(z_{i}\);
2. The torsion angle \(\alpha_{i}\) of the connecting rod: the angle from \(z_{i-1}\) to \(z_{i}\) around the \(x_{i}\) axis;
3. Link offset \(d_{i}\): along the \(z_{i}\) axis, the distance from \(x_{i-1}\) to \(x_{i}\);
4. Joint angle \(\theta_{i}\): the angle of rotation from \(x_{i-1}\) to \(x_{i}\) around the \(z_{i}\) axis.