Collision risk quantitative evaluation method applied to low-altitude airspace

Abstract

The invention relates to a collision risk quantitative evaluation method applied to a low-altitude airspace, belongs to the technical field of low-altitude risk evaluation, and solves the problem that a method for effectively evaluating collision risks for multiple types of aircrafts lacks in the prior art. If the invading aircraft belongs to the cooperative unmanned aerial vehicle, executing the step S1, if the invading aircraft belongs to the non-cooperative unmanned aerial vehicle, executing the step S2, and if the invading aircraft belongs to the manned unmanned aerial vehicle, executing the step S3; step S1, determining related conflict mitigation measures, generating and outputting the collision probability of the local unmanned aerial vehicle and the cooperative unmanned aerial vehicle, and executing step S4; s2, determining related conflict mitigation measures, generating and outputting the collision probability of the local unmanned aerial vehicle and the non-cooperative unmanned aerial vehicle, and executing a step S4; s3, determining related conflict mitigation measures, generating and outputting the collision probability of the local machine and the manned machine, and executing the step S4; and S4, according to the received collision probability, evaluating the safety situation of the local aircraft and the invading aircraft.

Description

Technical Field

[0001] The present invention relates to the technical field of low-altitude risk assessment, and in particular to a collision risk quantitative assessment method applied to low-altitude airspace. Background Art

[0002] Low-altitude airspace is a valuable resource and an important activity space for new types of vehicles such as drones and eVOTLs, which contains great economic and social value. The explosive growth in the number and scale of drone operations in recent years has brought unprecedented challenges to low-altitude safety. Obviously, how to objectively and effectively assess the collision risk between low-altitude aircraft is the primary issue facing the development of low-altitude transportation.

[0003] At present, the applicable objects of collision risk assessment methods are mostly concentrated between drones or between drones and civil aircraft, and there is a lack of a unified framework and method for multi-type aircraft collision risk assessment. In addition, the existing assessment methods assume that the risk is a process of continuous deterioration and do not take into account the various means of risk containment in modern air traffic management. Therefore, the assessment results obtained by such methods are usually biased and lack objectivity and practical significance.

[0004] In summary, this technical field urgently needs to establish a unified framework for low-altitude multi-type aircraft collision risk assessment, analyze the risk prevention and control measures currently implemented by the management units, and quantitatively evaluate the probability of failure of prevention and control measures at each stage according to the chronological order of the accidents, so as to obtain a scientific quantitative assessment method that is compatible with the above framework. Summary of the invention

[0005] In order to solve the above problems, the present invention provides a collision risk quantitative assessment method applied to low-altitude airspace.

[0006] According to an embodiment of the present invention, a collision risk quantitative assessment method applied to low-altitude airspace is provided, comprising:

[0007] Step S0, obtaining data information of low-altitude airspace, determining the type of intruding aircraft relative to the host, if the intruding aircraft is a cooperative UAV, executing step S1, if it is a non-cooperative UAV, executing step S2, if it is a manned aircraft, executing step S3, wherein the host is a UAV;

[0008] Step S1, determine the timeline for the cooperative UAV collision risk assessment and the conflict mitigation measures involved, build a conflict probability assessment model based on track uncertainty during the encounter process and an arrival time estimation model based on heading adjustment, and calculate the DAA endogenous failure rate, generate and output the collision probability between the aircraft and the cooperative UAV, and execute step S4;

[0009] Step S2, determining the timeline for the non-cooperative UAV collision risk assessment and the conflict mitigation measures involved, constructing a dynamic GAS model based on random positions and headings to characterize the probability of program conflict mitigation failure, generating and outputting the collision probability between the aircraft and the non-cooperative UAV, and executing step S4;

[0010] Step S3, determining the timeline for manned aircraft collision risk assessment and the conflict mitigation measures involved, establishing a GAS model based on the terminal area arrival and departure route structure and operation rules to quantify the probability of procedural conflict mitigation failure; combining the above-mentioned arrival time estimation model based on heading adjustment to quantify the probability of tactical conflict mitigation failure, generating and outputting the collision probability between the aircraft and the manned aircraft, and executing step S4;

[0011] Step S4, based on the received collision probability, evaluate the safety situation of the aircraft and the intruding aircraft to provide a basis for formulating a safety prevention and control strategy;

[0012] Wherein, the step S1 specifically includes the following steps:

[0013] Step S1.1, determining the timeline for the cooperative UAV collision risk assessment and the conflict mitigation measures involved, which include strategic conflict mitigation, tactical conflict mitigation and airborne collision avoidance of the aircraft relative to the cooperative UAV;

[0014] Step S1.2, construct a conflict probability assessment model during the encounter process based on track uncertainty to obtain the failure probability of strategic conflict mitigation;

[0015] Step S1.3, constructing a heading adjustment-based arrival time estimation model for cooperative UAVs to obtain the failure probability of tactical conflict mitigation;

[0016] Step S1.4, calculating the DAA endogenous failure rate based on historical data to obtain the failure probability of airborne collision avoidance;

[0017] Step S1.5, based on the failure probability of strategic conflict mitigation, tactical conflict mitigation and airborne collision avoidance for cooperative UAVs, obtain the collision probability between the aircraft and the cooperative UAV, output and execute step S4.

[0018] Optionally, the step S1.2 specifically includes the following steps:

[0019] Step S1.2.1 models the collision area between the local aircraft and the cooperative UAV, selects a cylinder as the collision area, expands the collision area based on the original size, and constructs the collision area between the local aircraft and the cooperative UAV as follows:

[0020]

[0021] Among them, L en is the radius of the collision area, H en is the half height of the collision area, R a is the actual radius of the machine, H a is the actual half-height of the machine, σ r is the radius expansion factor of the collision area, σ h is the height expansion factor of the collision area;

[0022] Step S1.2.2, modeling the uncertainty of the trajectory of the cooperative UAV, the error envelope of the cooperative UAV is an ellipse in the horizontal direction, and the two axes of the ellipse are respectively along the track direction and perpendicular to the track direction;

[0023] Step S1.2.3, establish a coordinate system for calculating the collision probability between the local aircraft and the cooperative UAV, wherein firstly, a probability distribution model of the position of the local aircraft or the cooperative UAV in the global coordinate system is constructed respectively:

[0024]

[0025]

[0026] Among them, P is the spatial distribution of the position of the local or cooperative UAV, θ′ is the heading angle of the local or cooperative UAV at the prediction time, is the climb angle of the local or cooperative UAV at the predicted time, is the predicted position of the local or cooperative UAV in the global coordinate system, is the prediction error of the local or cooperative UAV in the body coordinate system, Matrix A = diag(σ1, σ2, σ3), where diag represents a diagonal matrix, σ1, σ2, and σ3 represent the position errors of the local or cooperative UAV in the three directions of the coordinate axis, N3 represents a three-dimensional normal distribution, and R represents a coordinate transformation matrix;

[0027] Then the prediction error of the local or cooperative UAV in the global coordinate system is:

[0028] Q=RAR T ;

[0029] The local drone is taken as the reference drone, and the invading cooperative drone is taken as the random drone. The relative position AP and the combined error matrix M of the reference drone and the random drone are:

[0030]

[0031] Among them, P r is the reference position of the UAV, P s is the position of the random drone, Q ris the prediction error of the reference UAV in the global coordinate system, Q s is the prediction error of the random UAV in the global coordinate system;

[0032] Step S1.2.4, perform coordinate system transformation and calculate the analytical solution of collision probability in three-dimensional space.

[0033] The collision probability between the local drone and the cooperative drone in the horizontal direction is constructed as:

[0034]

[0035] Where Δx c represents the length of the integration interval in the x-axis direction, Δy c represents the length of the integration interval in the y-axis direction, Δy is the distance of the random UAV relative to the reference UAV in the y-axis direction, p(x) and p(y) are the probability density functions of the standard normal distribution decoupled to the x-axis and y-axis respectively, x represents the function variable of p(x), and y represents the function variable of p(y);

[0036] The probability of conflict between the local aircraft and the cooperative UAV in the vertical direction can be defined as the integral of the one-dimensional normal distribution in the vertical direction of the combined conflict area:

[0037]

[0038] in, is the half height of the collision area between the local drone and the cooperative drone combination, Δz is the distance of the random drone relative to the reference drone in the z-axis direction, p(z) is the one-dimensional normal distribution function, and z represents the function variable of p(z);

[0039] The probability of conflict between the host and the cooperative UAV during the encounter, that is, the probability of failure of strategic conflict mitigation between the host and the cooperative UAV is:

[0040] Optionally, the step S1.3 specifically includes the following steps:

[0041] Step S1.3.1, calculate the arrival time of the aircraft and the cooperative UAV, that is, the time from the failure of strategic conflict mitigation to the taking of maneuvering actions:

[0042]

[0043] Among them, D x represents the allowed flight distance, Δv is the relative speed between the aircraft and the cooperative UAV;

[0044] Step S1.3.2, establish the failure probability model of tactical conflict mitigation of the aircraft relative to the cooperative UAV:

[0045]

[0046] in, represents the probability of failure of tactical conflict mitigation of the aircraft relative to the cooperative UAV, Represents the normal distribution function.

[0047] Optionally: Step S1.4 includes calculating the DAA endogenous failure rate based on historical data to obtain the failure probability of airborne collision avoidance

[0048] The step S1.5 specifically includes obtaining the collision probability between the aircraft and the cooperative UAV based on the failure probability of strategic conflict mitigation, tactical conflict mitigation and airborne collision avoidance for the cooperative UAV:

[0049]

[0050] The obtained collision probability between the host and the cooperative UAV is output, and step S4 is executed.

[0051] Optionally, the step S2 specifically includes the following steps:

[0052] Step S2.1, determining a timeline for a collision risk assessment of a non-cooperative UAV and the conflict mitigation measures involved, the conflict mitigation measures including a procedural mitigation of the aircraft relative to the non-cooperative UAV;

[0053] Step S2.2, setting the area swept by the aircraft at a relative speed within a certain period of time as a rectangle as the combined collision area, and the non-cooperative UAVs overlapping with the combined collision area are intruders;

[0054] Step S2.3, calculate the volume of the area swept by the machine as:

[0055]

[0056] in, The radius of the combined collision zone between the own aircraft and the non-cooperative drone, is the half height of the collision area between the local UAV and the non-cooperative UAV, Δv is the relative speed between the local UAV and the non-cooperative UAV, t mission The time for task execution;

[0057] Step S2.4, determine whether there is a collision risk with the non-cooperative UAV:

[0058]

[0059] Among them, (x′ j , y′ j , z′ j) represents the spatial position of the jth non-cooperative UAV, represents the space area swept by the aircraft relative to the jth non-cooperative UAV;

[0060] Step S2.5, based on historical data evaluation, obtain the number of non-cooperative UAVs N that appear during the mission execution uav , the probability of program mitigation failure of non-cooperative UAV is:

[0061]

[0062] Step S2.6, the failure probability of the program mitigation of the own aircraft relative to the non-cooperative UAV is the collision probability of the own aircraft and the non-cooperative UAV:

[0063]

[0064] in, The probability of failure of conflict mitigation procedures for non-cooperative UAVs;

[0065] The obtained collision probability between the local aircraft and the non-cooperative UAV is output, and step S4 is executed.

[0066] Optionally, the step S3 specifically includes the following steps:

[0067] Step S3.1, determining the timeline for manned aircraft collision risk assessment and the conflict mitigation measures involved, the conflict mitigation measures including procedural mitigation and tactical conflict mitigation of the aircraft relative to the manned aircraft;

[0068] Step S3.2, determine the area scanned by the aircraft relative to the manned aircraft:

[0069]

[0070] in, is the radius of the collision area between the aircraft and the manned aircraft combination, is the half height of the collision area between the aircraft and the manned aircraft, Δv is the relative speed between the aircraft and the manned aircraft, t mission The time for task execution;

[0071] Step S3.3, calculate the probability that the take-off and landing flight of a manned aircraft falls into the area swept by the aircraft relative to the manned aircraft:

[0072] L sw / L tol

[0073] Among them, L sw L is the length of the manned segment in the area swept by the aircraft. tol is the length of the overall take-off and landing pattern of manned aircraft;

[0074] Step S3.4, for the entire airport terminal area, the probability of the procedure mitigation failure of the own aircraft relative to the manned aircraft is:

[0075]

[0076] in, is the length swept by the aircraft relative to the mth take-off and traffic route, is the total length of the mth take-off and landing route, M′ represents the total number of take-off and landing routes in the entire airport terminal area during the aircraft's mission;

[0077] Step S3.5, the probability of failure of tactical conflict mitigation of the aircraft relative to the manned aircraft is:

[0078]

[0079] in, represents the normal distribution function, t ap ′ represents the time between failure of procedure mitigation and taking maneuvering action;

[0080] Step S3.6, the collision probability between the aircraft and the manned aircraft is:

[0081]

[0082] The obtained collision probability between the aircraft and the manned aircraft is output, and step S4 is executed.

[0083] The present invention provides a collision risk quantitative assessment method applied to low-altitude airspace, including an air collision risk assessment framework for multiple types of aircraft, a collision risk quantitative assessment method for the aircraft and cooperative unmanned aerial vehicles, a collision risk quantitative assessment method for the aircraft and non-cooperative unmanned aerial vehicles, and a collision risk quantitative assessment method for the aircraft and manned aircraft. Among them, the collision risk quantitative assessment method for the aircraft and cooperative unmanned aerial vehicles involves three links: strategic conflict mitigation, tactical conflict mitigation, and airborne collision avoidance. A conflict probability assessment model based on track uncertainty during encounter, an arrival time estimation model based on heading adjustment, and a DAA system fault assessment method are proposed to calculate the failure probability of the three links, thereby obtaining the collision probability of the aircraft relative to the cooperative unmanned aerial vehicle. The collision risk quantitative assessment method for the aircraft and non-cooperative unmanned aerial vehicles involves a link of program conflict mitigation. A dynamic GAS model based on random position and heading is proposed to calculate the failure probability of this link, thereby obtaining the collision probability of the aircraft relative to the non-cooperative unmanned aerial vehicle. The quantitative assessment method for the collision risk between the aircraft and manned aircraft involves two links: procedural conflict mitigation and tactical conflict mitigation. The failure probabilities of the two links are calculated using the GAS model based on the approach and departure route structure and operating rules and the arrival time estimation model based on heading adjustment, thereby obtaining the collision probability of the aircraft relative to the manned aircraft.

[0084] Compared with the prior art, a collision risk quantitative assessment method applied to low-altitude airspace provided in accordance with an embodiment of the present invention has at least the following beneficial effects: a method for quantitative assessment of air collision risks for multiple types of aircraft is introduced, and the evolution of collision accidents is divided into several stages based on the risk prevention and control measures currently implemented by the management unit. At each stage, the failure risk of mitigation measures is modeled, and various environmental factors such as wind error, navigation error, flight technology error, communication delay, and response delay are considered to make the assessment results more objective and closer to reality. In addition, with reference to the risk assessment method for high-altitude civil aviation, the quantitative indicator of risk is defined as "number of collision accidents / hour" to connect the air collision risk and the ground loss cost, laying the foundation for the subsequent assessment of air-ground integration risks. BRIEF DESCRIPTION OF THE DRAWINGS

[0085] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments will be briefly introduced below. The features and advantages of the present invention can be more clearly understood by referring to the drawings. The drawings are schematic and should not be understood as limiting the present invention in any way. For ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

[0086] Figure 1 A framework diagram of a collision risk quantitative assessment method applied to low-altitude airspace provided according to an embodiment of the present invention.

[0087] Figure 2a to Figure 2c A relative geometric relationship diagram of two aircraft during an encounter process in an example of applying a collision risk quantitative assessment method for low-altitude airspace provided according to an embodiment of the present invention.

[0088] Figure 3a A two-dimensional plane schematic diagram after geometric coordinate conversion in an example of a collision risk quantitative assessment method applied to low-altitude airspace provided according to an embodiment of the present invention.

[0089] Figure 3b A schematic diagram of the Z-axis after geometric coordinate conversion in an example of a method for quantitatively assessing collision risk applied to low-altitude airspace provided according to an embodiment of the present invention. Figure 4 A schematic diagram of the minimum distance required to adjust the heading in an example of the method for quantitatively assessing collision risk applied to low-altitude airspace provided in accordance with an embodiment of the present invention.

[0090] Figure 5 A schematic diagram of approach time estimation in an example of a method for quantitatively assessing collision risk applied to low-altitude airspace provided according to an embodiment of the present invention.

[0091] Figure 6A schematic diagram of a dynamic GAS model based on random position and heading in an example of a method for quantitatively assessing collision risk applied to low-altitude airspace provided according to an embodiment of the present invention.

[0092] Figure 7 A schematic diagram of a GAS model based on an approach and departure route structure in an example of a method for quantitatively assessing collision risk applied to low-altitude airspace provided according to an embodiment of the present invention.

[0093] Figure 8 The present invention is a flowchart of a method for quantitatively assessing collision risk applied to low-altitude airspace according to an embodiment of the present invention. DETAILED DESCRIPTION

[0094] In order to more clearly understand the above-mentioned purpose, features and advantages of the present invention, the present invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments. It should be noted that the embodiments of the present invention and the features in the embodiments can be combined with each other without conflict.

[0095] In the following description, many specific details are set forth to facilitate a full understanding of the present invention, but the present invention may also be implemented in other ways different from those described herein. Therefore, the protection scope of the present invention is not limited to the specific embodiments disclosed below.

[0096] A method for quantitatively assessing collision risk applied to low-altitude airspace provided according to an embodiment of the present invention is described in detail below with reference to the accompanying drawings.

[0097] like Figures 1 to 8 As shown, a collision risk quantitative assessment method applied to low-altitude airspace provided in accordance with an embodiment of the present invention is a collision risk assessment framework for cooperative UAVs, non-cooperative UAVs and manned aircraft, comprising the following steps:

[0098] Step S0, obtain data information of low-altitude airspace, determine the type of intruding aircraft relative to the host, if the intruding aircraft is a cooperative UAV, execute step S1, if it is a non-cooperative UAV, execute step S2, if it is a manned aircraft, execute step S3. Among them, the host is a UAV.

[0099] Step S1, determine the timeline for the cooperative UAV collision risk assessment and the conflict mitigation measures involved, build a conflict probability assessment model based on track uncertainty during the encounter process to quantify the probability of strategic conflict mitigation failure; build an arrival time estimation model based on heading adjustment to quantify the probability of tactical conflict mitigation failure; use the DAA historical failure rate to quantify the probability of airborne collision avoidance failure. Generate and output the collision probability between the aircraft and the cooperative UAV, and execute step S4

[0100] Step S2, determine the timeline for the non-cooperative UAV collision risk assessment and the conflict mitigation measures involved. Construct a dynamic GAS model based on random positions and headings to characterize the probability of program conflict mitigation failure. Generate and output the collision probability between the aircraft and the non-cooperative UAV, and execute step S4.

[0101] Step S3, determine the timeline for manned aircraft collision risk assessment, establish a GAS model based on the terminal area arrival and departure route structure and operation rules to quantify the probability of procedural conflict mitigation failure; combine the above-mentioned arrival time estimation model based on heading adjustment to quantify the probability of tactical conflict mitigation failure. Generate and output the collision probability between the aircraft and the manned aircraft, and execute step S4. The manned aircraft can be, for example, a civil airliner, a transport aircraft or other manned aircraft.

[0102] Step S4, based on the received collision probability, evaluate the safety situation of the aircraft and the invading aircraft to provide a basis for formulating a safety prevention and control strategy.

[0103] like Figure 1 , Figure 2a to Figure 2c , Figure 3a to Figure 3b , Figure 4 and Figure 5 As shown, step S1 specifically includes the following steps, which are used to calculate the collision probability between the host and the cooperative UAV.

[0104] Step S1.1, determine the timeline for the cooperative UAV collision risk assessment and the conflict mitigation measures involved. The conflict mitigation measures include strategic conflict mitigation, tactical conflict mitigation and airborne collision avoidance of the aircraft relative to the cooperative UAV.

[0105] Step S1.2, construct a conflict probability assessment model during the encounter process based on trajectory uncertainty to quantify the probability of failure of strategic conflict mitigation, which specifically includes the following sub-steps.

[0106] Step S1.2.1 models the collision area between the local aircraft and the cooperative UAV. Referring to the definition of the risk envelope area (DAA WellClear) in the Detect-and-Avoid document released by NASA, this implementation method selects a cylinder as the collision area. Considering the non-horizontal maneuvers such as climbing and descending of the UAV during movement, as well as the interference of environmental factors such as wind and air pressure that may be encountered, the collision area is expanded accordingly on the basis of the original size. Combined with the concept of the minimum envelope circle, the collision area between the local aircraft and the cooperative UAV is constructed as:

[0107]

[0108] Among them, (L en , H en) are the radius and half height of the collision area, (R a , H a ) are the actual radius and half-height of the aircraft, (σ r ,σ h ) are the radius and height expansion factor of the collision area respectively.

[0109] Step S1.2.2, model the uncertainty of the trajectory of the cooperative UAV. The error envelope of the cooperative UAV is approximately elliptical in the horizontal direction, and the two axes of the ellipse are respectively along the track direction and perpendicular to the track direction. According to the provisions of the Performance-Based Navigation (PBN) Manual, the overall positioning error of the cooperative UAV is determined by the total system error (TSE), which is the sum of the navigation system error (NSE), the flight technical error (FTE) and the path definition error (PDE). The path definition error is the difference between the defined path and the required path, which can usually be ignored. The flight technical error usually increases along the track direction with the flight time, and the navigation system error is usually a constant. Therefore, the error envelope of the cooperative UAV is approximately elliptical in the horizontal direction, and the two axes of the ellipse are respectively along the track direction and perpendicular to the track direction.

[0110] Step S1.2.3, establish a coordinate system for calculating the collision probability between the local aircraft and the cooperative UAV. Assume that the position error of the aircraft (i.e., the local aircraft or the cooperative UAV) follows a Gaussian distribution, and assume that the error is independent in the three directions of the aircraft's body coordinate axis, namely σ1, σ2, and σ3. Construct a position distribution model of the local aircraft or the cooperative UAV in the global coordinate system, and after defining the prediction time span, the predicted position of the local aircraft or the cooperative UAV in the global coordinate system is The prediction error of the local or cooperative UAV in the body coordinate system is The coordinate transformation matrix is ​​R, and the probability distribution of the position of the local or cooperative UAV in the global coordinate system is obtained:

[0111]

[0112]

[0113] Where P is the spatial distribution of the position of the local or cooperative UAV, θ′ is the heading angle of the UAV (i.e., the local or cooperative UAV) at the prediction time, is the climb angle of the UAV (i.e., the own aircraft or cooperative UAV) at the prediction time, Matrix A=diag(σ1, σ2, σ3), diag represents a diagonal matrix, and N3 represents a three-dimensional normal distribution.

[0114] Therefore, the prediction error of the local or cooperative UAV in the global coordinate system can be obtained as: Q = RAR T .

[0115] The method for constructing the position probability distribution of the local drone and the cooperative drone in the global coordinate system is the same and can be obtained by the above method. When calculating the collision probability between the local drone and the cooperative drone, in this implementation, the local drone is used as the reference drone, the invading cooperative drone is used as the random drone, the prediction error of the reference drone is superimposed on the prediction error of the random drone, the collision area of ​​the random drone is superimposed on the collision area of ​​the reference drone, and the reference drone is used as the origin and the random drone is used as the mass point to construct the collision coordinate system, such as Figure 2a and Figure 2b As shown. The relative position and combined error matrix of the reference UAV and the random UAV are:

[0116]

[0117] Among them, P r is the reference position of the UAV, P s is the position of the random drone, Q r , Q s are the prediction errors of the reference UAV and the random UAV in the global coordinate system, ΔP is the relative position of the reference UAV and the random UAV, and M is the combined error matrix of the reference UAV and the random UAV.

[0118] Based on the above ideas, the instantaneous collision probability of the reference UAV and the random UAV can be calculated. The main purpose of this model is to calculate the total collision probability of the reference UAV and the random UAV during the entire encounter process (over a period of time). Therefore, it is assumed that the speed and error of the reference UAV and the random UAV are constant during the encounter. In this embodiment, in order to make the evaluation result more conservative, the maximum error is used for modeling. The combined collision area is extended in a direction parallel to the relative speed to form an extended collision area, the length of which depends on the time of the encounter, such as Figure 2c As shown in Figure 2, the collision probability between the reference drone and the random drone (the probability of failure of strategic conflict mitigation) is equal to the integral of the error probability density function in this extended region. Next, the 3-D space will be decoupled with the idea of ​​coordinate transformation and the analytical solution of this integral will be determined.

[0119] Step S1.2.4, perform coordinate system transformation and calculate the analytical solution of collision probability in three-dimensional space.

[0120] Let p and q represent the position of the aircraft (i.e., the reference drone or the random drone) in the original coordinate system and the transformed coordinate system, respectively. Assume q = Tp, where T is the change matrix to be determined, and W represents the inverse matrix of the change matrix T. Let W = T -1 , then p = Wq, and the combined error in the change matrix is Based on the Cholesky theorem, the combined error matrix M = LL T , where L is a lower triangular matrix, the combined error in the change matrix can be expressed as:

[0121]

[0122] Where H represents the orthogonal rotation matrix (HH T =I), I represents the unit matrix.

[0123] At this point, the combined error is transformed from the ellipsoid of the original coordinate system to the unit sphere of the new coordinate system (decoupled), and the combined collision area is transformed from the cylinder of the original coordinate system to the ellipsoid. Since the aircraft maintains horizontal flight during the encounter, it can be assumed that the horizontal and vertical motions are independent, and the above transformation can be decoupled into the collision probabilities in the horizontal and vertical directions by simple scaling, such as Figure 3a and Figure 3b Then, the present embodiment calculates the analytical solutions of the two directions respectively. Due to independence, the total probability of the two directions is the product of the two.

[0124] like Figure 3a As shown, for the horizontal probability, first, the orthogonal rotation matrix H mentioned above is used to rotate the transformed coordinate system so that the relative speed of the reference drone and the random drone is the positive direction of the x-axis. Assume that the relative speed between the two aircraft Δv = v r -v s , then the speed of the partial transformation in the transformed coordinate system is but

[0125]

[0126] Among them, v r is the speed of the reference drone, v s is the speed of the random drone, Δv x Represents the partial conversion speed matrix The component on the x-axis, Δv y Represents the partial conversion speed matrix The component on the y-axis.

[0127] Then determine the integral expression and the upper and lower limits of the integral for calculating the two-dimensional collision probability of two aircraft. Figure 3a As shown, the boundaries on the y-axis are the maximum and minimum values ​​of the ellipse boundary, and the integral area on the x-axis is the area swept by the relative velocity during the encounter time. c and Δq c Respectively represent the boundary distance of the y-axis in the original coordinate system and the transformed coordinate system, then:

[0128]

[0129] in, is the radius of the collision area between the aircraft and the cooperative drone, W = T -1 =LH -1 , if Δq c =[Δx c , Δy c ], Then square the above equation and find the discriminant, and then set the discriminant equal to zero to obtain the vector Δq c The element Δy in c , another element Δx c It is related to the encounter time as shown in the following formula:

[0130]

[0131] Where Δx c represents the length of the integration interval in the x-axis direction, Δy c represents the length of the integral interval in the y-axis direction, a, b, c represent the matrix W T The elements of W are known quantities. sr is the distance swept by the relative speed within the encounter period, Δv is the relative speed, t encounter For the duration of the encounter, is the radius of the collision area between the local aircraft and the cooperative UAV. In summary, the collision probability between the local aircraft and the cooperative UAV in the horizontal direction can be expressed as:

[0132]

[0133] Where Δy is the distance of the random UAV relative to the reference UAV in the y-axis direction, p(x) and p(y) are the probability density functions of the standard normal distribution decoupled to the x-axis and y-axis, respectively, x represents the function variable of p(x), and y represents the function variable of p(y).

[0134] The probability of conflict between the local aircraft and the cooperative UAV in the vertical direction can be defined as the integral of the one-dimensional normal distribution in the vertical direction of the combined conflict area:

[0135]

[0136] in, is the half-height of the collision area between the local drone and the cooperative drone combination, Δz is the distance of the random drone relative to the reference drone in the z-axis direction, p(z) is the one-dimensional normal distribution function, and z represents the function variable of p(z).

[0137] In summary, the probability of conflict during the encounter, that is, the probability of failure of strategic conflict mitigation between the aircraft and the cooperative UAV, is:

[0138] Step S1.3, construct a heading adjustment-based arrival time estimation model for cooperative UAVs to quantify the probability of tactical conflict mitigation failure, such as Figure 4 and Figure 5 As shown, it includes the following sub-steps:

[0139] Step S1.3.1, calculate the arrival time of the aircraft and the cooperative UAV, that is, the time from the failure of strategic conflict mitigation to the taking of maneuvering actions:

[0140]

[0141] Among them, D x It represents the allowed flight distance, and Δv is the relative speed between the host and the cooperative UAV.

[0142] The step S1.3.1 specifically includes the following steps.

[0143] Step S1.3.1.1, calculate the minimum maneuvering distance required for the aircraft to make heading adjustments.

[0144] During the heading adjustment process of the aircraft, it is assumed that the maximum turning angle that the aircraft can take is The maximum lateral turning overload n can be expressed as The minimum turning radius of this machine is:

[0145]

[0146] Among them, v re is the relative speed between the host and the cooperative UAV, g is the acceleration of gravity, R turn This is the minimum turning radius of the machine.

[0147] like Figure 4 As shown, when the machine is at the minimum radius R turn When turning, if the trajectory of the aircraft coincides with the combined collision area, the distance between the aircraft and the invading cooperative drone is L m is the minimum maneuvering distance. Assume that the heading angle of the relative speed Δv between the local aircraft and the cooperative UAV is θ, and the azimuth angle of the relative speed Δv between the local aircraft and the cooperative UAV is β. Taking the x-axis as the direction reference, the angle between the heading angle and the azimuth angle of the relative speed is β1=|β-θ|. Then, using the triangle theorem, the minimum maneuvering distance is calculated in the triangle with γ1 as the vertex angle. The value of γ1 depends on the heading angle θ. If θ>β, then γ1=π / 2+β1. If θ≤β, then γ1=π / 2-β1. In the above triangle, the minimum maneuvering distance L m It can be obtained by the following formula:

[0148]

[0149] Among them, γ1 represents the vertex angle of the solved triangle (such as Figure 4 ), The radius of the collision zone between this unit and the cooperative drone.

[0150] Step S1.3.1.2, the relative position relationship between the host and the invading cooperative UAV is as follows: Figure 5 As shown, the relative distance L between the two machines re and the allowed flight distance D x The relationship can also be described by the triangle theorem, and the angle γ2 of the triangle can be expressed as follows:

[0151]

[0152] Among them, γ2 represents the angle of the solved triangle (such as Figure 5 ), θ is the heading angle of the relative speed Δv between the local UAV and the cooperative UAV, β is the azimuth angle between the two UAVs, L re is the relative distance between the two aircraft, L m is the minimum maneuvering distance.

[0153] Then use the law of cosines to calculate the allowable flight distance D x , that is, the distance the aircraft would fly along its original trajectory without performing an avoidance maneuver:

[0154]

[0155] Step S1.3.1.3, calculate the arrival time of the host aircraft and the cooperative UAV, that is, the time from the failure of strategic conflict mitigation to the taking of maneuvering actions. The arrival time between the host aircraft and the cooperative UAV is t ap The reach distance is D, which is the allowed flight distance. x And the relative speed is:

[0156]

[0157] Among them, Δv is the relative speed between the local aircraft and the cooperative UAV.

[0158] Step S1.3.2, establish a model to calculate the probability of failure of tactical conflict mitigation of the own aircraft relative to the cooperative UAV.

[0159] The reason for the failure of tactical conflict mitigation is that the delay causes the time for tactical conflict mitigation measures to take effect to be longer than the time of separation loss between the two aircraft, that is, the aircraft has reached the separation loss state before completing the evasive action. Based on the above idea, assuming that the delay time follows a normal distribution, its distribution function is expressed as The arrival time of the two aircraft, that is, the time from the failure of strategic conflict mitigation to the taking of maneuvering actions, is defined as t ap , if tactical conflict mitigation fails, that is, the delay time is greater than t ap , otherwise, tactical conflict mitigation is effective and is considered to have no collision risk. Therefore, the failure probability of tactical conflict mitigation can be expressed as:

[0160]

[0161] in, Represents the probability of failure of tactical conflict mitigation of the own aircraft relative to the cooperative UAV.

[0162] Step S1.4: Calculate the DAA endogenous failure rate based on historical data to obtain the failure probability of the aircraft's airborne collision avoidance relative to the cooperative UAV.

[0163] Step S1.5, multiply the probability of failure of the mitigation measures in each stage to obtain the collision probability between the aircraft and the cooperative UAV:

[0164]

[0165] in, They are respectively the failure probabilities of strategic conflict mitigation, tactical conflict mitigation and airborne collision avoidance for the UAV and the cooperative UAV. The obtained failure probabilities are output and step S4 is executed.

[0166] like Figure 1 , Figure 6 and Figure 8 As shown, step S2 is to evaluate the collision probability between the local aircraft and the non-cooperative UAV. For non-cooperative UAVs, they may appear unexpectedly during the local aircraft's mission. Since their flight plans and trajectories are not easy to determine, unlike the above numerical calculation method, a dynamic GAS model based on random positions and headings is constructed to calculate the collision probability of non-cooperative UAVs. Specifically, the following steps are included:

[0167] Step S2.1, determine the timeline for the non-cooperative UAV collision risk assessment and the conflict mitigation measures involved. The conflict mitigation measures include the procedure mitigation of the aircraft relative to the non-cooperative UAV.

[0168] In step S2.2, it is assumed that the non-cooperative UAV is similar to gas molecules randomly distributed in the mission airspace, with random horizontal speeds and no tendency to climb or descend. The area swept by the UAV at a relative speed within a certain period of time is regarded as a rectangle as the combined collision area. The non-cooperative UAV overlapping with the combined collision area is the intruder that may conflict.

[0169] Step S2.3, calculate the area swept by the machine. Assume is the radius and half-height of the combined collision area of ​​the local UAV and the non-cooperative UAV, Δv is the relative speed between the local UAV and the non-cooperative UAV, t mission is the time for task execution. Due to the influence of relative speed, the area swept by the machine relative to different intruders will be different, so this model is a dynamic GAS model, and the volume of the swept area is:

[0170]

[0171] Step S2.4, determine whether a non-cooperative UAV has a collision risk. Determine whether there is a conflict between the local aircraft and the non-cooperative UAV. When the collision zones between the local aircraft and the non-cooperative UAV overlap during the mission, that is, the non-cooperative UAV is located in the area swept by the local aircraft, it is considered that there is a risk of conflict between the two aircraft. Otherwise, let The logic for determining whether there is a conflict between the aircraft and the non-cooperative drone is as follows:

[0172]

[0173] Among them, (x′ j , y′ j , z′ j ) represents the spatial position of the jth non-cooperative UAV, Represents the spatial area swept by the aircraft relative to the jth non-cooperative UAV.

[0174] Step S2.5, calculate the collision risk for non-cooperative UAVs in the airspace, that is, the probability of program mitigation failure. Based on historical data, the number of non-cooperative UAVs that may appear during the mission execution is evaluated, and N uav The probability of program mitigation failure of non-cooperative UAVs can be expressed as:

[0175]

[0176] Step S2.6, the failure probability of the program mitigation of the aircraft relative to the non-cooperative UAV is the collision probability between the aircraft and the non-cooperative UAV:

[0177]

[0178] in, The probability of failure of conflict mitigation procedures for non-cooperative UAVs.

[0179] Step S3 is to evaluate the collision probability between the aircraft and the manned aircraft, which specifically includes a dynamic GAS model based on random positions and headings, such as Figure 1, Figure 7 and Figure 8 As shown, the following steps are included:

[0180] Step S3.1, determine the timeline for manned aircraft collision risk assessment and the conflict mitigation measures involved. The conflict mitigation measures include procedural mitigation and tactical conflict mitigation of the aircraft relative to the manned aircraft.

[0181] Step S3.2, determine the area swept by the aircraft (the aircraft is a drone) relative to the manned aircraft. is the radius and half-height of the collision area between the aircraft and the manned aircraft combination, t mission is the time for mission execution, and Δv is the relative speed between the aircraft and the manned aircraft. For a certain runway at an airport, the take-off and landing heading is fixed, but the difference in the speed of the manned aircraft leads to different relative speeds, so the swept area is different. The corresponding calculation formula is:

[0182]

[0183] Step S3.3, calculate the probability that the take-off and landing flight falls into the area swept by the aircraft relative to the manned aircraft.

[0184] Assume that the positions of manned aircraft take-off and landing routes are evenly distributed, and the length of the flight segment in the swept area is L sw , the length of the overall take-off and landing route is L tol , then the probability that a manned aircraft happens to be in the swept area is L sw / L tol .

[0185] Step S3.4, for the entire airport terminal area, calculate the probability of collision between the aircraft and the manned aircraft during the mission execution, that is, the probability of procedure mitigation failure. Assuming that there are M′ take-off and landing routes in the entire airport terminal area during the mission of the aircraft, the probability of procedure mitigation failure of the aircraft relative to the manned aircraft is:

[0186]

[0187] in, is the length swept by the aircraft relative to the mth take-off and traffic route, is the total length of the mth take-off and landing route.

[0188] Step S3.5, the probability of failure of tactical conflict mitigation of the aircraft relative to the manned aircraft The arrival time estimation model based on heading adjustment can be obtained by referring to the above step S1.3:

[0189]

[0190] in, represents the normal distribution function, tap ' represents the time between the failure of the procedure mitigation and the taking of maneuvering actions. It will not be described in detail here.

[0191] Step S3.6, the collision probability of the manned aircraft is the product of the program mitigation failure probability of the own aircraft relative to the manned aircraft and the tactical conflict mitigation failure probability. The collision probability of the own aircraft and the manned aircraft is:

[0192]

[0193] in, They are respectively the failure probabilities of program conflict mitigation and tactical conflict mitigation for manned aircraft. The obtained collision probability between the aircraft and the manned aircraft is output, and step S4 is executed.

[0194] Based on the same technical solution, the present invention also discloses a software system of the above method, a collision risk assessment system applied to low-altitude airspace, including: a collision risk assessment module for cooperative UAVs, which writes program code based on the above model, and calculates the probability of strategic conflict mitigation failure, the probability of tactical conflict mitigation failure, and the probability of airborne conflict avoidance failure respectively, and the product of the three is the overall collision risk. A collision risk assessment module for non-cooperative UAVs, which writes program code based on the above model, calculates the probability of program mitigation failure, and regards this as the overall collision risk. A collision risk assessment module for manned aircraft, which writes program code based on the above model, calculates the probability of program mitigation failure and the probability of tactical conflict mitigation failure, and regards this as the overall collision risk.

[0195] In the above system, the data processing flow of each module is consistent with the corresponding steps of the method, and will not be repeated here.

[0196] All the above optional technical solutions can be arbitrarily combined to form optional embodiments of the present application, which will not be described one by one here.

[0197] It should be understood that the order of execution of the steps in the above embodiment does not necessarily mean the order of execution. The execution order of each process should be determined by its function and internal logic, and should not constitute any limitation on the implementation process of the embodiment of the present invention.

[0198] The above description is only a preferred specific implementation manner of the present invention, but the protection scope of the present invention is not limited thereto. Any changes or substitutions that can be easily conceived by any technician familiar with the technical field within the technical scope disclosed by the present invention should be covered within the protection scope of the present invention.

Claims

1. A method for quantitatively assessing collision risk in low-altitude airspace, characterized in that: include: Step S0, obtaining data information of low-altitude airspace, determining the type of intruding aircraft relative to the host, if the intruding aircraft is a cooperative UAV, executing step S1, if it is a non-cooperative UAV, executing step S2, if it is a manned aircraft, executing step S3, wherein the host is a UAV; Step S1, determine the timeline for the cooperative UAV collision risk assessment and the conflict mitigation measures involved, build a conflict probability assessment model based on track uncertainty during the encounter process and an arrival time estimation model based on heading adjustment, and calculate the DAA endogenous failure rate, generate and output the collision probability between the aircraft and the cooperative UAV, and execute step S4; Step S2, determining the timeline for the non-cooperative UAV collision risk assessment and the conflict mitigation measures involved, constructing a dynamic GAS model based on random positions and headings to characterize the probability of program conflict mitigation failure, generating and outputting the collision probability between the aircraft and the non-cooperative UAV, and executing step S4; Step S3, determining the timeline for manned aircraft collision risk assessment and the conflict mitigation measures involved, and establishing a GAS model based on the terminal area arrival and departure route structure and operation rules to quantify the probability of failure of procedural conflict mitigation; Combine the above-mentioned arrival time estimation model based on heading adjustment to quantify the failure probability of tactical conflict mitigation, generate and output the collision probability between the aircraft and the manned aircraft, and execute step S4; Step S4, based on the received collision probability, evaluate the safety situation of the aircraft and the intruding aircraft to provide a basis for formulating a safety prevention and control strategy; Wherein, the step S1 specifically includes the following steps: Step S1.1, determining the timeline for the cooperative UAV collision risk assessment and the conflict mitigation measures involved, which include strategic conflict mitigation, tactical conflict mitigation and airborne collision avoidance of the aircraft relative to the cooperative UAV; Step S1.2, construct a conflict probability assessment model during the encounter process based on track uncertainty to obtain the failure probability of strategic conflict mitigation; Step S1.3, constructing a heading adjustment-based arrival time estimation model for cooperative UAVs to obtain the failure probability of tactical conflict mitigation; Step S1.4, calculating the DAA endogenous failure rate based on historical data to obtain the failure probability of airborne collision avoidance; Step S1.5, based on the failure probability of strategic conflict mitigation, tactical conflict mitigation and airborne collision avoidance for cooperative UAVs, obtain the collision probability between the aircraft and the cooperative UAV, output and execute step S4.

2. The method for quantitatively assessing collision risk in low-altitude airspace according to claim 1 is characterized in that: The step S1.2 specifically includes the following steps: Step S1.2.1 models the collision area between the local aircraft and the cooperative UAV, selects a cylinder as the collision area, expands the collision area based on the original size, and constructs the collision area between the local aircraft and the cooperative UAV as follows: Among them, L en is the radius of the collision area, H en is the half height of the collision area, R a is the actual radius of the machine, H a is the actual half-height of the machine, σ r is the radius expansion factor of the collision area, σ h is the height expansion factor of the collision area; Step S1.2.2, modeling the uncertainty of the trajectory of the cooperative UAV, the error envelope of the cooperative UAV is an ellipse in the horizontal direction, and the two axes of the ellipse are respectively along the track direction and perpendicular to the track direction; Step S1.2.3, establish a coordinate system for calculating the collision probability between the local aircraft and the cooperative UAV, wherein firstly, a probability distribution model of the position of the local aircraft or the cooperative UAV in the global coordinate system is constructed respectively: Among them, P is the spatial distribution of the position of the local or cooperative UAV, θ′ is the heading angle of the local or cooperative UAV at the prediction time, is the climb angle of the local or cooperative UAV at the predicted time, is the predicted position of the local or cooperative UAV in the global coordinate system, is the prediction error of the local or cooperative UAV in the body coordinate system, Matrix A = diag(σ1,σ2,σ3), where diag represents a diagonal matrix, σ1, σ2, and σ3 represent the position errors of the local or cooperative UAV in the three directions of the coordinate axis, N3 represents a three-dimensional normal distribution, and R represents a coordinate transformation matrix; Then the prediction error of the local or cooperative UAV in the global coordinate system is: Q=RAR T ; The local drone is taken as the reference drone, and the invading cooperative drone is taken as the random drone. The relative position ΔP and the combined error matrix M of the reference drone and the random drone are: Among them, P r is the reference position of the UAV, P s is the position of the random drone, Q r is the prediction error of the reference UAV in the global coordinate system, Q s is the prediction error of the random UAV in the global coordinate system; Step S1.2.4, perform coordinate system transformation and calculate the analytical solution of collision probability in three-dimensional space. The collision probability between the local drone and the cooperative drone in the horizontal direction is constructed as: Where Δx c represents the length of the integration interval in the x-axis direction, Δy c represents the length of the integration interval in the y-axis direction, Δy is the distance of the random UAV relative to the reference UAV in the y-axis direction, p(x) and p(y) are the probability density functions of the standard normal distribution decoupled to the x-axis and y-axis respectively, x represents the function variable of p(x), and y represents the function variable of p(y); The probability of conflict between the local aircraft and the cooperative UAV in the vertical direction can be defined as the integral of the one-dimensional normal distribution in the vertical direction of the combined conflict area: in, is the half height of the collision area between the local drone and the cooperative drone combination, Δz is the distance of the random drone relative to the reference drone in the z-axis direction, p(z) is the one-dimensional normal distribution function, and z represents the function variable of p(z); The probability of conflict between the host and the cooperative UAV during the encounter, that is, the probability of failure of strategic conflict mitigation between the host and the cooperative UAV is:

3. The method for quantitatively assessing collision risk in low-altitude airspace according to claim 2 is characterized in that: The step S1.3 specifically includes the following steps: Step S1.3.1, calculate the arrival time of the aircraft and the cooperative UAV, that is, the time from the failure of strategic conflict mitigation to the taking of maneuvering actions: Among them, D x represents the allowed flight distance, Δv is the relative speed between the aircraft and the cooperative UAV; Step S1.3.2, establish the failure probability model of tactical conflict mitigation of the aircraft relative to the cooperative UAV: in, represents the probability of failure of tactical conflict mitigation of the aircraft relative to the cooperative UAV, Represents the normal distribution function.

4. The method for quantitatively assessing collision risk in low-altitude airspace according to claim 3 is characterized in that: The step S1.4 includes calculating the DAA endogenous failure rate based on historical data to obtain the failure probability of airborne collision avoidance The step S1.5 specifically includes obtaining the collision probability between the aircraft and the cooperative UAV based on the failure probability of strategic conflict mitigation, tactical conflict mitigation and airborne collision avoidance for the cooperative UAV: The obtained collision probability between the host and the cooperative UAV is output, and step S4 is executed.

5. The method for quantitatively assessing collision risk in low-altitude airspace according to claim 1, characterized in that: The step S2 specifically includes the following steps: Step S2.1, determining a timeline for a collision risk assessment of a non-cooperative UAV and the conflict mitigation measures involved, the conflict mitigation measures including a procedural mitigation of the aircraft relative to the non-cooperative UAV; Step S2.2, setting the area swept by the aircraft at a relative speed within a certain period of time as a rectangle as the combined collision area, and the non-cooperative UAVs overlapping with the combined collision area are intruders; Step S2.3, calculate the volume of the area swept by the machine as: in, The radius of the combined collision zone between the own aircraft and the non-cooperative drone, is the half height of the collision area between the local UAV and the non-cooperative UAV, Δv is the relative speed between the local UAV and the non-cooperative UAV, t mission The time for task execution; Step S2.4, determine whether there is a collision risk with the non-cooperative UAV: Among them, (x j ′ ,y j ′ ,z j ′ ) represents the spatial position of the jth non-cooperative UAV, represents the space area swept by the aircraft relative to the jth non-cooperative UAV; Step S2.5, based on historical data evaluation, obtain the number of non-cooperative UAVs N that appear during the mission execution uav , the probability of program mitigation failure of non-cooperative UAV is: Step S2.6, the failure probability of the program mitigation of the own aircraft relative to the non-cooperative UAV is the collision probability of the own aircraft and the non-cooperative UAV: in, The probability of failure of conflict mitigation procedures for non-cooperative UAVs; The obtained collision probability between the local aircraft and the non-cooperative UAV is output, and step S4 is executed.

6. The method for quantitatively assessing collision risk in low-altitude airspace according to claim 1, characterized in that: The step S3 specifically comprises the following steps: Step S3.1, determining the timeline for manned aircraft collision risk assessment and the conflict mitigation measures involved, the conflict mitigation measures including procedural mitigation and tactical conflict mitigation of the aircraft relative to the manned aircraft; Step S3.2, determine the area scanned by the aircraft relative to the manned aircraft: in, is the radius of the collision area between the aircraft and the manned aircraft combination, is the half height of the collision area between the aircraft and the manned aircraft, Δv is the relative speed between the aircraft and the manned aircraft, t mission The time for task execution; Step S3.3, calculate the probability that the take-off and landing flight of a manned aircraft falls into the area swept by the aircraft relative to the manned aircraft: L sw / L tol Among them, L sw L is the length of the manned segment in the area swept by the aircraft. tol is the length of the overall take-off and landing pattern of manned aircraft; Step S3.4, for the entire airport terminal area, the probability of the procedure mitigation failure of the own aircraft relative to the manned aircraft is: in, is the length swept by the aircraft relative to the mth take-off and traffic route, is the total length of the mth take-off and landing route, M′ represents the total number of take-off and landing routes in the entire airport terminal area during the aircraft's mission; Step S3.5, the probability of failure of tactical conflict mitigation of the aircraft relative to the manned aircraft is: in, represents the normal distribution function, t ap ′ represents the time between failure of procedure mitigation and taking maneuvering action; Step S3.6, the collision probability between the aircraft and the manned aircraft is: The obtained collision probability between the aircraft and the manned aircraft is output, and step S4 is executed.

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