# Collision Risk Assessment and Operation Assistant Strategy for Teleoperation System
## 4 Collision Risk Assessment
### Collision Risk Assessment Model  
Fuzzy set theory was proposed by [PERSON] in 1965 and is extensively applied to model uncertain systems when there is no complete and precise information [31; 32]. As shown in Figure 8, A typical fuzzy system consists of three function layers: an input layer, a fuzzy inference layer, and an output layer. According to traditional set theory, an element either belongs to a particular set or not. However, in fuzzy set theory, the association of an element with a particular set lies between 0 and 1, which is called the membership degree. As with distance and collision risk, the degree of danger cannot be expressed precisely, but can be described in fuzzy terms, such as \"extremely dangerous\" or \"somewhat dangerous\". Additionally, the fuzzy inference layer combines expert experience, which can help decision makers in extracting and keeping expert knowledge. Therefore, the fuzzy theory was introduced in this study. 1. Fuzzy set and fuzzification  
The purpose of fuzzification is to transform the exact input variables into fuzzy variables with linguistic meaning. Before the performance of fuzzification, a scale transformation of the input values was executed to convert them into their corresponding domain ranges. The output domain was [0; 1; 2; 3; 4; 5], and the transformed input domains were \(T\) = [0; 1; 2; 3; 4; 5], \(V\) = [0; 5], and \(D\) = [1, 2, 3, 4, 5, 6; 7, 8, 9, 10] respectively. The transformation of the shortest  
Figure 8: Collision risk assessment model based on fuzzy inference. distance is nonlinear, which is set according to the discrete radius of the envelope and experience. The transformation formula is:  
\[\overline{D}=\left\{ \begin{array}{rl}1&(0\;\text{cm}<d\leq 1\;\text{cm})\\ 2&(1\;\text{cm}<d\leq 2\;\text{cm})\\ 3&(2\;\text{cm}<d\leq 3\;\text{cm})\\ 4&(3\;\text{cm}<d\leq 5\;\text{cm})\\ 5&(5\;\text{cm}<d\leq 7\;\text{cm})\\ 6&(7\;\text{cm}<d\leq 9\;\text{cm})\\ 7&(9\;\text{cm}<d\leq 12\;\text{cm})\\ 8&(12\;\text{cm}<d\leq 15\;\text{cm})\\ 9&(15\;\text{cm}<d\leq 20\;\text{cm})\\ 10&(20\;\text{cm}<d\leq 30\;\text{cm})\end{array} \right. \tag{6}\]  
where \(\overline{D}\) is the transformed distance, and \(d\) is the true distance. In fuzzification, common membership functions include the Triangular membership function, the Trapezoidal membership function, the Gaussian membership function, the Sigmoid membership function, etc. [31]. The speed membership function was used here as an example to illustrate the setting up process. As the division of different speed fuzzy sets was symmetrical and uniformly distributed, the linear triangular membership function and linear trapezoidal membership function were utilized. The speed within the region was roughly divided into five parts, and the slope of the trapezoidal and triangular membership functions were established according to the requirement that the membership function of each speed point belonging to the fuzzy subset was greater than 0.5. The final established membership function for a uniformly symmetric distribution is shown in Figures 9-11. The fuzzy sets of shortest distance in the form of linguistic rating variables were described as 'Very close (NB)' Relatively close (NS)', 'Moderate (M)', 'Relatively far (PS)', and 'Far (PB)'. Similarly, as is shown in Figure 12, a set of fuzzy affiliation functions were created for the model output, which were represented as 'Very dangerous (NB)', 'Relatively dangerous (NS)', 'Dangerous (M)', 'Relatively safe (PS)', and 'Safe (PB)', respectively. Figure 10: Membership function of the time delay \(T\). Variable \(T\): Very small delay (NB), Small delay (NS), Large delay (PS), and Very large delay (PB). Figure 9: Membership function of the shortest distance \(D\). Variable \(D\): Very close (NB), Relatively close (NS), Moderate (M), Relatively far (PS), and Far (PB).