# Optimal Placement of Vibration Sensors for Industrial Robots Based on Bayesian Theory
## 2 Sensor Placement Method
### Numerical Method of Velocity  
To obtain the theoretical velocity value of the placement position, we first have to study the relevant knowledge of robotics. To calculate the velocity value of the precise position, we first need to calculate the velocity of the robot connecting rod, that is, the velocity transfer formula between connecting rods.  
\[{}^{i+1}v_{i+1}={}^{i+1}R_{i}({}^{i}v_{i}+{}^{i}\omega_{i}\times{}^{i}P_{i+1}) \tag{5}\]  
where \({}^{i+1}v_{i+1}\) is the velocity of the \(i+1\) connecting rod relative to the \(\{i+1\}\) coordinate system, \({}^{i}v_{i}\) is the velocity of the origin of the coordinate system \(\{i\}\) relative to the coordinate system \(\{i\}\), \({}^{i}w_{i}\) is the angular velocity of the connecting rod \(i\) relative to the coordinate system \(\{i\}\), \({}^{i+1}R_{i}\) is the rotation transformation matrix from the coordinate system \(\{i\}\) to the coordinate system \(\{i+1\}\), is the first three order matrix of \({}^{i+1}T_{i}\), and \({}^{i+1}P_{i}\) is the distance of the \(i+1\) connecting rod relative to the \(i\)th connecting rod.  
By introducing the parameters of each link into the expression, a theoretical solution for the speed of each link can be obtained. In the actual sensor layout, the speed of each measuring point is not the same, so it needs to be specific to each measuring point to calculate its corresponding speed. Therefore, based on the calculation formula for connecting rod speeds given by robotics, according to the distance between each point and adjacent joints, the speed calculation method of a specific measuring point is given. First, the velocity of each joint or connecting rod relative to its coordinate system \(\{i+1\}\) is calculated according to the formula. Then, the velocity of the point relative to the coordinate system \(\{i+1\}\) is calculated according to the distance from the point to the joint. The calculation formula of the velocity of the point relative to the coordinate system \(\{i+1\}\) is as follows:  
\[{}^{i+1}v_{p}={}^{i+1}v_{i+1}+{}^{i+1}w_{i+1}\times d \tag{6}\]  
where, \({}^{i+1}v_{p}\) is the velocity of the sensor point \(p\) relative to the coordinate system \(\{i+1\}\), \({}^{i+1}v_{i+1}\) is the velocity of the origin of the coordinate system \(\{i+1\}\) relative to the coordinate system \(\{i+1\}\), \({}^{i+1}w_{i+1}\) is the angular velocity of the connecting rod \(i+1\) relative to the coordinate system \(\{i+1\}\), and \(d\) is the distance of the sensor point relative to the origin of the coordinate system \(\{i+1\}\)  
Finally, the velocity in the base coordinate system is obtained through the corresponding transfer matrix. The coordinate transfer formula is as follows:  
\[{}^{0}v_{p}={}^{0}R_{i+1}{}^{i+1}v_{p} \tag{7}\]  
Among them, \({}^{0}v_{p}\) is the velocity of sensor distribution point \(p\) relative to the base coordinate system, \({}^{0}R_{i+1}\) is the rotation transformation matrix from the coordinate system \(\{i+1\}\) to the base coordinate system, and is the first three order square matrix of \({}^{0}T_{i+1}\).  
On the other hand, the flexible joint of the industrial robot and the vibration deformation of the manipulator material affect the actual speed of the placement. Therefore, a mechanical simulation analysis of the manipulator is needed.