# Optimal Placement of Vibration Sensors for Industrial Robots Based on Bayesian Theory
## 2 Sensor Placement Method
### Optimal Sensor Placement Based on Bayesian
#### 2.4.1 Bayesian Estimation of Motion Joint Position  
Whether the running state of the industrial robot can be better expressed is an important index for measuring the sensor placement of the industrial robot, and the joint motion state of the industrial robot is the main component of the running state of the robot. First, according to the operational characteristics of the industrial robot, an appropriate event is established, and the corresponding probability distribution is given. Then, assuming that the running joint of the industrial robot in the current state is \(r\) and the running joint is the event \(A\), the uncertainty of event \(A\) is quantified by the probability distribution, which is updated according to the data measured by the sensors arranged on the industrial robot. If the sensor arranged on the industrial robot can detect the motion of the position due to the motion of joint \(r\), it can detect the number of the moving joint, and then it can correctly judge event \(A\). Therefore, the optimal sensor placement problem can also be understood. The sensor placement we determined can make the best estimation of event \(A\) (the number of joints in motion). Since the model of the industrial robot has been established in the previous chapter, and the theoretical calculation formula of industrial robot speed has been given, now suppose that the distance from the manipulator to the origin of the base is taken as the coordinates of sensor placement, and a reasonable initial sensor layout is determined according to the simulation results in Section 2.2. The corresponding acceleration sensor is arranged at each position to obtain the acceleration of the point, and then the acceleration is integrated to obtain the corresponding velocity. \(V^{\prime}(r;s)\) is the predicted value of the velocity measured at the point \(s\), which is obtained by calculating the theoretical velocity of \(s\) when \(r\) joint moves? Moreover, assuming that the prior distribution of event \(A\) exists and is known, let the prior probability distribution be \(p(r)\). Then, when the measured value \(V_{t}\) of the sensor is known, the posterior distribution \(p(r|V_{t},s)\) of event \(A\) can be determined. According to the Bayesian principle, the posterior distribution \(p(r|V_{t},s)\) is proportional to the product of its prior distribution \(p(r)\) and likelihood \(p(V_{t}|r,s)\), namely \(p(r|V_{t},s)\propto p(r)\cdot p(V_{t}|r,s)\). The likelihood equation represents the probability that the measured value \(y\) comes from the real moving joint \(r\) after a given sensor placement \(s\). Since there is an error between the real measured value and the theoretically calculated value, assuming that the error is \(\varepsilon(s)\), the relationship among them is as follows. \[V_{t}=V^{\prime}(r;s)+\varepsilon(s) \tag{8}\]  
The principle of maximum entropy is a criterion for selecting the distribution of random variables whose statistical characteristics are most consistent with objective conditions. It is an effective criterion for selecting the distribution of random variables with maximum entropy. In the discrete case, the entropy of the equiprobability model is the maximum, but the detection of the joint motion state of the industrial robot is not an equiprobability model. Therefore, the discrete model does not meet the requirements. Multivariate Gaussian distribution is the most natural expression of random variables in ignorance. When the mean and covariance are constant, the random variable with a normal distribution has maximum entropy. Therefore, it can be assumed that the error of the above formula conforms to the definition and that \(\varepsilon(s)\) obeys a multivariate Gaussian distribution with amean value of 0 and a specific covariance matrix.