# Optimal Placement of Vibration Sensors for Industrial Robots Based on Bayesian Theory
## 2 Sensor Placement Method
### Simulation Analysis of Industrial Robot  
Sensor placement is an a priori problem in the case of only analyzing data; for example, they are obtained through finite element models [25]. Considering that before the optimization of the sensor location, there are many measuring points available for the actual industrial robot, to determine a more reasonable initial layout scheme, a modal simulation of the industrial robot is carried out to analyze its deformation under the influence of vibration. The parameters of the industrial robot are shown in Table 3. The modal simulation of the model is carried out using ANSYS Workbench software. The first six modal shapes of the model are shown in Figure 2.  
The simulation parameters are as follows: 254,880 divided nodes, 150,163 divided units, and the bottom of the base are set as the fixed support.  
The maximum natural frequencies and relative amplitudes of the first six orders of the whole machine are shown in Table 4.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Project**} & \multirow{2}{*}{**Type**} & **Number** & **Driving** & **Repeat Positioning** & **Range of** \\  & & **of Axes** & **Mode** & **Accuracy** & **Motion** \\ \hline Parameter & RB13 & 6 & AC servo & 0.07 mm & R499–R1404 mm \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameters of the industrial robot with the model.  
Figure 2: The first six modes of industrial robot: (**a**) 1 st order mode shape, (**b**) 2 nd order mode shape, (**c**) 3 rd order mode shape, (**d**) 4 th order mode shape, (**e**) 5 th order mode shape, and (**f**) 6 th order mode shape.  
The results of the modal analysis are as follows:  
First-order vibration: The natural frequency is 12.6 Hz, and the maximum amplitude is 0.11 M. The deformation mainly shifts along the \(x\)-axis and rotates around the \(z\)-axis. The closer to the end, the greater the amplitude.  
Second-order vibration: The natural frequency is 20.0 Hz, and the maximum amplitude is 0.09 M. The deformation mainly moves along the \(y\)-axis and rotates around the \(x\)-axis. The closer to the end, the greater the amplitude.  
Third-order vibration: The natural frequency is 30.0 Hz, and the maximum amplitude is 0.10 M. The main deformation is the torsion of the main arm around the \(y\)-axis, with a large amplitude at the elbow and end.  
Fourth-order vibration: The natural frequency is 70.3 Hz, and the maximum amplitude is 0.12 M. The main deformation is that the end of the main arm swings around the \(z\)-axis, and a larger amplitude is concentrated in the elbow and forearm.  
Fifth-order vibration: The natural frequency is 116.7 Hz, and the maximum amplitude is 0.15 M. The main deformation is the rotation of the arm and elbow around the \(z\)-axis, and a larger amplitude is concentrated in the elbow.  
Sixth order vibration: The natural frequency is 338.3 Hz, and the maximum amplitude is 0.14 M. The main deformation is that both ends of the jib swing around the \(y\)-axis, and a larger amplitude is concentrated at the two ends and the middle of the forearm.  
According to the simulation results of the modal analysis, there is a large amount of vibration deformation at the forearm, elbow, and related joints of the manipulator. When considering the initial location, the end of the forearm and elbow should be considered. Therefore, the initial placement of the sensor is selected as shown in Figure 3.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline Order & 1 & 2 & 3 & 4 & 5 & 6 \\ Natural frequency/Hz & 12.6 & 20.0 & 30.0 & 70.3 & 116.7 & 338.3 \\ Maximum relative amplitude/m & 0.11 & 0.09 & 0.10 & 0.12 & 0.15 & 0.14 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The first six natural frequencies and amplitudes.  
Figure 3: Selection of initial sensor placement.