Abstract
With the development of aircraft flow under the existing airspace capacity now, the shortage of airspace resources and flight delays have become significantly severe. Therefore, building a safe and efficient mathematical model from quantitative analysis and improving the scientificity of route planning and management are essential propositions for future research on new navigation systems. Based on the traditional event collision risk assessment model, the collision module is upgraded to an ellipsoid according to the performance of aircraft, integrating the boundary curvature optimization characteristics of the TSRRT (task-space rapidly-exploring random trees) algorithm. An aircraft event lateral conflict resolution model based on the TSRRT algorithm is proposed. Taking the A320 aircraft and 737-800 aircraft as experimental subjects, the corresponding collision coefficients are imported into software tools for the simulation and the Kalman filtering is combined to verify the smoothness of the front and rear boundary curvatures. The result proves that the event side collision risk of aircraft based on TSRRT is 15% of the traditional event model and the smooth curvature error is reduced by 80% due to the improvement. Therefore, the improved event model is practical and valuable, providing a theoretical basis for future track-based operations.
1. Introduction
With the rapid increase of aircraft flow, the existing air route network has become congested with limited airspace resources, which not only limits the increase in flight traffic but also causes much insecurity to flight [1]. Therefore, it is necessary to evaluate the collision risk of navigation from the scientific quantification perspective.
Domestic and foreign scholars have made many efforts to research the airway collision risk model. In 1966, Reich proposed the earliest collision risk model when evaluating the safety of separation standards for the North Atlantic route [2]. In 1984, Brooker applied the Reich model to the North Atlantic lateral collision risk study [3]. Brewer-Dougherty et al. [4] established a collision risk model of PBN (performance-based navigation) based on the Reich model in 2016. However, Kim and Hwang [5] conducted a practical comparison and analysis of the event model and Reich model in 2018, finding out that the event models are more inclusive and the models have apparent advantages in applying multiple integrations and superimposed factors. The event model (also known as the post-Reich model) was proposed by professor Brooker [6] of the United Kingdom in 2003, and a traditional cuboid collision template was established. In 2008, Xiaohao et al. [7] proposed using a cylindrical collision template instead of the traditional one to realize the function of risk assessment and calculating lateral intervals. In 2010, Wenjun et al. [8] used a spherical collision template to replace the traditional template to assess the safety of cross-route separation. In 2011, Yuling et al. [9] established a collision risk model based on a given arrival time interval for the flight at the same altitude and the same direction with the help of the probability theory. In 2013, Zongping et al. [10] analyzed the approach risk of paired aircraft based on the accident tree analysis method and obtained the variation curve of the aircraft’s longitudinal collision risk with relevant parameters through simulation calculation. It was assessed by Xingwu and Zhaoning [11] in 2015. In 2017, Jian [12] fully considered the speed error, navigation error, initial safe distance, and other factors of the two aircraft during the paired approach of the short-distance parallel runway to establish a paired approach collision risk assessment model; in 2018, Xinsheng and Zhi [13] studied the flight interval and conflict relief threshold under the influence of multiple random factors. In 2019, Shuo [14] used a cylinder-rectangular collision temlate to study the collision probability of UAVs. In 2021, Xie et al. [15] comprehensively considered the influence of aircraft yaw and established a paired approach longitudinal collision risk assessment model for the longitudinal separation calculation of aircraft at each stage. However, none of the abovementioned event collision models can reflect the speed vector changes in the aircraft space and the aircraft sideslip curvature cannot be described. More importantly, due to the aircraft’s pitch angle limitation, the aircraft will generally change the height to prevent a collision when the two aircraft are approaching, so the space formed by the aircraft’s deflection around the lateral axis of the wing is not a cuboid or sphere, but a cylinder close to it. Therefore, a skid-slip event lateral conflict resolution model is proposed to analyze and study the lateral risk of the air routes.
2. Event Lateral Model Construction and Improvement
2.1. Event Model Construction
The event model evaluates the collision risk of adjacent-tracking aircraft (labeled AIR-A and AIR-B) with a lateral separation of . AIR-A is used as the center point to simulate a cylinder with a radius of ( and representing the horizontal and vertical lengths of the aircraft, respectively. It is assumed that the wingspan and the length of the aircraft are equal, that is, , both expressed in terms of aircraft length ) and a height of (the height of the aircraft is represented by , taking the center point of the aircraft as the center of the cylinder, and the height of the cylinder is ) [16]. AIR-B is used as the origin to simulate a plane with no thickness on the –-axes. When the spacing piece of AIR-B is in contact with the cylinder of AIR-A, that is, AIR-A and AIR-B are in an absolute collision, the schematic diagram is described in Figure 1 as follows.
For AIR-A and AIR-B, is used to represent the voyage time and the number of collisions per unit time of voyage can be expressed as follows: where is the number of aircraft logs per hour and is the probability that AIR-A and AIR-B overlap horizontally. is the probability that AIR-A and AIR-B overlap vertically when the horizontal overlap occurs, which generally takes a fixed value of . is the average vertical relative speed. is the average horizontal relative speed. When studying the lateral risk model of adjacent trajectories, the voyage report period is set to hours, the number of planes is set to when calculating the lateral interval, and it is assumed that the two planes fly at the cruise level. When the lateral interval is unknown, it can be replaced by and the number of collisions per hour on the same track can be expressed as follows:
is the communication and controller intervention buffer value based on the CPDLC data link (controller and pilot data link communication); the horizontal overlap probability is based on the vertical overlap probability and lateral overlap at . So, the probability can be expressed as follows: where is the probability of the longitudinal overlap and is the probability of the lateral overlap of 2 aircraft on adjacent levels of the track.
2.2. Collision Ratio
2.2.1. Conavigation Collision Relationship Ratio
Figure 2 shows the movement of the crash box laterally and longitudinally across the spacer in the case of codirectional flight. If a collision occurs, the aircraft point must be within the line segment . When the aircraft point is outside the line segment on the line segment , that is, on the line segment and the line segment , there will be no collision. In the event model, since the collision box is the cuboid shown in the figure, during the entire process of crossing the spacer, as long as point is on the line segment , a collision may occur. When flying on the same heading, the collision box laterally traverses the spacer and moves in the lateral and longitudinal directions. According to the geometric principles, point collides in the line segment . Otherwise, it does not crash. According to the characteristics of the event model, it can be described that when point is on the line segment , AIR-A and AIR-B will hit each other.
Assuming that is relatively stationary, considering the collision probability relationship ratio from a geometric point of view, the longitudinal movement distance of AIR-A is , which can be expressed as follows:
, , and are the relative velocities of A-AIR through the AIR-B spacer in the longitudinal, lateral, and vertical directions when sailing in the same direction [17]. At the same time, they can float in the same direction according to the geometry and the event model motion principle. The collision probability relationship ratio is as follows:
2.2.2. Reverse Navigation Collision Relationship Ratio
The longitudinal relative speed of AIR-A and AIR-B when cruising in the reverse direction is . It is assumed that the relative speed change value in the lateral and vertical movement is 0, that is, the time for the reverse sailing to pass through the interval is equal to the time in the same direction. The schematic diagram of the collision cylinder passing through the spacing piece when traveling in the reverse direction is shown in Figure 3.
According to Figure 3, considering the geometric ratio relationship, the length of and can be expressed as follows:
Since is much larger than , the approximate length of the line segment is and the collision probability relationship between AIR-A and AIR-B when sailing in the reverse direction is as follows:
2.3. Integration of Skid-Slip Event Model
Since the TSRRT algorithm will integrate kinematic model constraints into the route planning, the curvature of the overall planning can be optimized so that the path curvature can meet the need for parameter continuity [18]. Also, it is easy to solve the length of the line segment and rough value of in the same direction and reverse navigation collision geometric model. Because the TSRRT algorithm adopts the Dubins curve, it can effectively reduce the convergence time of risk assessment when the sailing demands are met.
The motion model of the Dubins curve only allows the moving body to move forward and not to return, and the line can connect the shortest path in a two-dimension surface under the circumstances that the curve is satisfied with path constraints. This article uses RSL to represent the shortest path connecting the two control points of and , where the radius in the curve circle is the radius of the bottom surface of the cylinder. That is, and are equal and the corresponding RSL diagram is shown in Figure 4 as follows.
The TSRRT firstly samples and marks the nodes in the areas prone to lateral overlap, integrating the heuristic function to calculate the value of the central point of the lateral adjacent overlap area. Let be the combination of adjacent nodes with the smallest total value of the current risk.
is the substitute value of adjacent overlapping risk areas. Let be the total risk calculation cost from the initial risk overlapping point to the current period, and is the estimated value at risk from the current period to the target point.
is the approximate length of the adjacent point and the Bezier curve, and is the Bezier order, which is the side collision surface of A-AIR and B-AIR running in the same or reverse direction.
3. Kalman Filter Error Model Based on RNP
The fusion of the TSRRT algorithm and the event model could reduce the convergence time and increase the accuracy of under geometric analysis. However, it cannot meet the requirements of free navigation in the ICAO.DOC9750 global navigation plan [19]. The yaw probability-based Kalman filtering is introduced to evaluate the multifactor superimposition effect on the collision model so that the practicability, accuracy, and reproducibility of the event lateral conflict model can be improved in future navigation systems [20, 21]. The total yaw error TSE (total system error) under lateral confliction error is generally composed of NSE (navigation system error), PDE (path definition error), and FTE (flight technical error). The RNP (required navigation performance) program can be selected as a benchmark when measuring the error probability under lateral conflict. RNPN can be described as the aircraft’s actual position being within ±95% of the total flight time. In other words, it is safe in the range of [22, 23] and there will be no risk of collision, as shown in Figure 5.
The Kalman filter is based on the theory of the state equation and motion vector equation of the moving aircraft, and the least mean square error is used as the reference to estimate the system state [24]. Therefore, the minimum mean square error threshold can be selected to correspond to the RNP regulations. The filter error is used to measure the event’s lateral direction, the applicability, and the rationality of the conflict model. The Kalman system equation is as follows: where is the state vector, which represents the value of the motion vector at time . is the state transition matrix used to describe the motion of the target object. is the interference transition matrix. represents the system noise of the motion model. represents the observation vector, describing the observation value at time . is the observation matrix; is the motion observation noise generated during the estimation process, and and are independent of each other. Its statistical characteristics are as follows:
At the same time, the Kalman equation of motion can be divided into the time update equation and the observation update equation, here named where is the symmetric nonnegative definite azimuth matrix of the system noise . is the symmetric positive definite variance matrix of the observation noise . is the error variance matrix. is the filter gain matrix [25].
According to the definition of the abovementioned system equations and the physical relationship of the parameters, the state transition matrix and the observation matrix can be obtained.
4. Simulation Calculations
To test the effectiveness and efficiency of the aircraft event lateral conflict resolution model based on TSRRT, an Intel Core i7 processor is selected as the hardware operating environment with a 16 GB memory capacity.
The wingspan, fuselage, and fuselage heights of the A320 aircraft are 34.1 m, 37.6 m, and 11.8 m, respectively, (34.1, 37.6) (unit: m). The wingspan, fuselage, and fuselage height of the B737-800 aircraft are 35.79 m, 39.47 m, and 12.57 m, respectively, (35.79 m, 39.47 m). The ADS-B-based position and time information of the A320 and B737-800 aircraft are selected as the primary simulation data in the experiment. Some of the position and time data are shown in Tables 1 and 2.
According to the requirements of 《introduction to safety assessment of flight intervals》 [26], considering the maximum risk of lateral collision, the lateral relative velocity is set to 6.44 m/s and the longitudinal relative velocity is set to 514 m/s, vertical relative velocity , , , , and . According to the literature [27–29], the lateral overlap probability is , the longitudinal overlap probability , the vertical overlap probability , the longitudinal proximity rate is 0.01, , the longitudinal interval of the aircraft is 10 km, the target offset rate , the maximum curvature , the maximum sideslip steering angle , and the sampling step length , as shown in Table 3.
The event parameters set in Table 3 are brought into the lateral conflict model, and the collision risk values of the original cylindrical event lateral conflict model and the improved skid-slip event lateral conflict model are obtained; the obtained calculation result (retaining two decimal places) is shown in Table 4.
As the safety target level specified in ICAO Annex 6 11th Edition [30] is , the collision probabilities of the traditional event model and the improved skid-slip event model are and , respectively, when choosing the 1st combination (A320), which meets the target level specified by ICAO, and the optimized event collision probability becomes 15% (). Similarly, the collision probabilities of the traditional event model and the improved skid-slip event model are and when using the B737-800 combination. The collision probability of the B737-800 combination model before and after the improvement is less than , and the optimized event collision probability becomes 15% () of the traditional event model.
5. Main Factor Analysis and Improved Model Test
5.1. Analysis of Main Factors
It can be seen in Tables 3 and 4 that the collision risk probability is mainly affected by the flight time and the speed error. The following mainly analyzes the collision risk from the two aspects of the flight time change and the speed error parameter. Let the airplane speed , the value of airplane speed error parameter , and the time interval be 5 min. The safety target-level risk time specified in ICAO Annex 6 is times/flight hour [30], and the logarithm based on 10 is , which is different for different RNP specifications. The changing trend of collision risk is shown in Figure 6.
It can be seen in Figure 6 that the risk of collision and the collision specifications of different navigation sources also differ significantly as the flight time increases. Taking RNP4 as an example, the collision risk changes from 10−18 to 10−9 overtime. The collision risk change interval of RNP10 is [10−11, 11−9] because the navigation error parameter value of RNP10 is 5.1, which is 2.5 in the case of RNP4. The difference in navigation specifications makes the flight error significantly increase. When the flight time is 90 minutes, the collision risk of RNP10 is (which is more than the ), which cannot meet the given safety target level of ICAO. The requirements can be met only by shortening the flight time or increasing the time interval. As shown in Figure 7, the number of collisions under different navigation specifications increases when the speed error parameter value changes. When , the number of collisions of RNP4, RNP5, and RNP10 are −27.1, −22.0, and −11.6, respectively. Under the same speed error , , and as the speed error parameter value increases, the collision risk changes decrease.
5.2. Improved Model Checking
When the Kalman filter is used to test the yaw probability model, the sampling period is controlled within the interval of and the observed noise value , , , the system noise parameter is B(n|n-1) =0.01 × I6×1, with equations (12)–(15), and the corresponding motion filter is shown in Figures 8 and 9,
According to the requirements of ICAO.DOC4444《air traffic management》 [31], the width of the route is 20 km, with 10 km on both sides of the centerline. When conditions restrict a specific route section, the width can be reduced but it should not be less than 8 km. The maximum distance of ADS-B pulse yaw in Figure 8 is 10 km, and the minimum is 4 km, so it can well meet the navigation requirements of ICAO. At the same time, the curve clusters are analyzed. The comparison of of the central mainline to the right benchmark mainly fluctuates in the range of [−10,0], and the comparison benchmarks of a small part of the edge curves are mainly in [−30, −25] (unit: m) range. It can also meet the requirements for CNS in ICAO Annex 6 [28]. At the same time, the aircraft lateral slip analysis is performed on the filtering of Figure 9. The minimum error distance mainly fluctuates in the interval of [200,300] (unit: m). Because the general route width is ±20 km, the relative route slip ratio during the sailing time is 0.01. The RNP is selected as the flight navigation specification in the air route. The required coefficient of the sideslip time is ≤0.05, and the minimum sideslip coefficient of the RNP is 0.05. The filter sideslip rate of Figure 9 is much less than 0.05. The test shows that the event lateral conflict model based on TSRRT is effective and reasonable. It can meet the needs of TBO (trajectory-based operation). where is the mean function; is the covariance function; is the Dirac function; when , ; otherwise, it is 0.
At the same time, because the Kalman filter is based on the state equation and motion vector equation of the moving aircraft, the minimum mean square error is used as the basis to estimate the system state [24]. Gaussian process regression (GPR) includes the noise variance test and Gaussian process prior. The maximum likelihood function in the Gaussian process regression is selected as the test standard. The lateral offset margin can be estimated while predicting the accurate point of the route to provide higher safety and reliability for future free navigation. Putting the parameters in Table 3 into formula (17) (), the predicted values and intervals of its traditional event model and skid-slip event model can be obtained, as shown in Figures 10 and 11.
As shown in Figure 10, in the traditional event model, the upper limit of the interval is 122° and the lower limit of the interval is 115°; that is, the prediction interval is [115°,122°], the maximum difference of the prediction interval is 7°, and the interval boundary is volatile. It is not convenient for maneuvering in the main and auxiliary sections of the RNP route. In the improved event model of Figure 11, the upper limit of the interval is 119° and the lower limit of the interval is 116°; that is, the prediction interval is [116°,119°] and the maximum difference is 3°, compared with before the improvement, the sideslip angle is reduced by 4°. In addition, the interval boundary line is relatively smooth, which is more in line with RNP route specifications and operation well.
6. Conclusion
This paper selects the event lateral conflict resolution model based on TSRRT and takes the combination of A320 aircraft and B737-800 aircraft as the experimental subject. The calculation shows that the side collision probability of the improved event model is 15% of the collision probability of the traditional event model and the dynamic sideslip curvature error per unit time is reduced by 80%, indicating that the improved event model is effective and reasonable.
Through the analysis of the flight time change parameters and speed error parameter values in the event model, the collision risk increases with the parameter value. The collision risk decreases with the RNP error change within a given time, and it will be free to sail in the future. The performance of the CNS system can be improved to shorten the flight time or appropriately increase the safety margin.
Because the Kalman filter can analyze errors and improve the prediction accuracy, this article uses ICAO.DOC4444 as a guide to calculate the data obtained by the ADS-B query response pulse using the Kalman filter to verify that the improved event model is efficient.
The research object of this paper is only studying the lateral conflict model to satisfy the 4D trajectory operation in ASBU (aviation system block upgrade). It can be studied from the longitudinal, vertical, and mixed directions in the future to obtain a more practical conflict resolution model.
Data Availability
The numerical simulation data used to support the findings of this study are included in the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was sponsored in part by the “Visual Flight Programming and Approval Guidelines Project of RNAV Procedures” (no. 14002600100015J013), “Special Guidance Found of Building World-Class Universities (Disciplines) and Characteristic Development” (no. D202103), and “Innovation and Entrepreneurship Training Program for College Students of Sichuan Province” (no. 202210624005).We are grateful for the support of these funds.