Low-cost unmanned aerial vehicle cluster collaborative guidance method for air maneuvering target

Abstract

The invention discloses a low-cost unmanned aerial vehicle cluster cooperative guidance method for an air maneuvering target, and the method comprises the following steps: setting target detection equipment on a main unmanned aerial vehicle, and building a relative motion system model of the main unmanned aerial vehicle and the target; based on the relative motion model, establishing a sliding mode surface, and obtaining a guidance law of the main unmanned aerial vehicle along a sight line normal direction by adopting a sliding mode control method; setting the acceleration of the main unmanned aerial vehicle in the sight direction, and obtaining the guidance law of the main unmanned aerial vehicle in combination with the guidance law of the main unmanned aerial vehicle in the normal direction of the sight; and the slave unmanned aerial vehicles are formed according to the position of the master unmanned aerial vehicle, so that the unmanned aerial vehicle group flies in a spherical arrangement manner to cooperatively intercept the target. According to the method disclosed by the invention, the unmanned aerial vehicle cluster can simultaneously intercept the maneuvering incoming unmanned aerial vehicles in the three-dimensional space in the form of spherical blockade, so that the interception rate and the damage effect are effectively improved.

Description

Technical Field

[0001] The present invention relates to a low-cost unmanned aerial vehicle (UAV) cluster cooperative guidance method for aerial maneuvering targets, belonging to the technical field of aircraft control. Background Art

[0002] The method of using interceptors to counter UAVs is an important means of UAV countermeasure. Interceptors can search for targets in the air, are not easily interfered by ground protruding obstacles, and can quickly implement precise interception after detecting the targets, with high flexibility and mobility.

[0003] In practical applications, the maneuverability of the incoming UAVs may be comparable to or even better than that of the own interceptors. In this case, traditional guidance methods are difficult to ensure the guidance accuracy.

[0004] Multi-UAV cooperative interception is an effective way to solve the above problems. Multiple interceptors intercept the incoming UAVs simultaneously from multiple directions and different angles to form an encirclement situation, which can significantly improve the interception rate and damage effect.

[0005] Current cooperative guidance strategies are divided into two categories: open-loop and closed-loop according to whether there is real-time information exchange between participating aircraft. Open-loop cooperation means that before launching, each aircraft presets cooperative information such as the terminal position, terminal arrival time, or terminal angle to achieve cooperation. After the aircraft are launched, there is no communication means between UAVs. However, for open-loop cooperation, the UAV cluster cannot adjust the overall state according to the individual state during flight and is highly sensitive to external interference. Therefore, it is usually only applicable to the cooperative strike of fixed targets or slow-moving targets and is not applicable to the interception of highly maneuverable incoming UAVs.

[0006] Closed-loop cooperative guidance introduces information exchange between aircraft. The UAV cluster can adjust its own state in real time according to the information of each aircraft, and can achieve a more accurate guidance effect compared with open-loop cooperation. However, the existing closed-loop cooperative guidance requires each interceptor in the cluster to carry detection equipment, resulting in a high cost.

[0007] In addition, the existing closed-loop cooperative guidance usually needs to use the remaining flight time as a coordinated variable. However, due to the influence of target maneuvering, the remaining flight time is often difficult to accurately estimate. Due to the inaccurate estimation of the remaining flight time, the accuracy of the cooperative guidance law will be greatly reduced in practical applications.

[0008] Therefore, it is necessary to conduct a more in-depth study on the UAV cluster cooperative guidance method to solve the above problems. Summary of the Invention

[0009] To overcome the above problems, the present inventors have conducted in-depth research and proposed a low-cost UAV swarm cooperative guidance method for airborne maneuvering targets, including the following steps:

[0010] S1. Set up a target detection device on the master UAV and establish a relative motion system model between the master UAV and the target;

[0011] S2. Based on the relative motion model, establish a sliding surface and use the sliding mode control method to obtain the guidance law of the master UAV along the line-of-sight normal direction;

[0012] S3. Set the acceleration of the master UAV along the line-of-sight direction, and combine it with the guidance law of the master UAV along the line-of-sight normal direction to obtain the guidance law of the master UAV;

[0013] S4. The slave UAVs form a formation according to the position of the master UAV, so that the UAV swarm flies in a spherical arrangement and cooperatively intercepts the target.

[0014] In a preferred embodiment, no target detection device is provided on the slave UAVs.

[0015] In a preferred embodiment, in S1, the relative motion system model is expressed as:

[0016]

[0017] where x1, x2, x3, x4 are system variables, and u1, u2 are system control variables; r represents the distance between the master UAV and the target, q θ represents the line-of-sight inclination angle of the master UAV, a Ty represents the acceleration of the target in the y direction in the line-of-sight coordinate system, a Tz represents the acceleration of the target in the z direction in the line-of-sight coordinate system, represents the line-of-sight deviation angle of the master UAV, represents the desired line-of-sight inclination angle of the master UAV, q θf represents the desired line-of-sight deviation angle of the master UAV, a uy represents the acceleration of the master UAV in the y direction in the line-of-sight coordinate system, a uz represents the acceleration of the master UAV in the z direction in the line-of-sight coordinate system.

[0018] In a preferred embodiment, in S2, there are two sliding surfaces s, namely s1 and s2, which are set as:

[0019]

[0020] where s1, s2 are the constructed sliding surfaces, α1, α2, β1, β2 are all constants greater than zero, and sgn() is the sign function.

[0021] In a preferred embodiment, in the sliding mode control, the sliding mode reaching law is set as:

[0022]

[0023] where ε1 and ε2 are constants greater than zero.

[0024] In a preferred embodiment, the guidance law of the master UAV along the line-of-sight normal direction is expressed as:

[0025]

[0026] In a preferred embodiment, in S3, the acceleration of the master UAV along the line-of-sight direction adopts PI control and is expressed as:

[0027] a ux =k p (v dx -v ux )+k i ∫(v dx -v ux )dt

[0028] where a ux is the acceleration of the master UAV along the line-of-sight direction, that is, the acceleration of the master UAV in the x direction in the line-of-sight coordinate system, v ux is the velocity of the master UAV along the line-of-sight direction, k p and k i are control parameters, and v dx is the desired velocity of the master UAV along the line-of-sight direction.

[0029] In a preferred embodiment, in S4, the center of the sphere is located on the line connecting the master UAV and the target.

[0030] In a preferred embodiment, the radius of the sphere is set as:

[0031]

[0032] where ρ represents the radius of the sphere, and c1 and c2 are adjustable parameters.

[0033] In a preferred embodiment, the cooperative guidance law of the slave UAVs is set as:

[0034]

[0035] where the subscripts i, j represent the serial numbers of the slave UAVs, all the slave UAVs form a directed graph, and a ij represents the element corresponding to the i-th and j -th slave UAV in the adjacency matrix of the directed graph, NDenote the total number of slave UAVs as \(u\). i Denote the guidance law of the \(i\)-th slave UAV as \(p_i\). i Denote the position of the \(i\)-th slave UAV in the inertial coordinate system as \(\mathbf{p}_i\). di Denote the desired position of the \(i\)-th slave UAV in the inertial coordinate system as \(\mathbf{v}_i\). i Denote the velocity of the \(i\)-th slave UAV as \(\mathbf{v}_i\), where \(\gamma\) is a constant greater than zero, and \(\mathbf{p}_1\) represents the position of the master UAV in the inertial coordinate system. Denote the rotation matrix from the inertial coordinate system to the line-of-sight coordinate system as \(\mathbf{R}\). Denote the desired position of the \(i\)-th slave UAV in the line-of-sight parallel coordinate system as \(\boldsymbol{\theta}_i\). di Denote the desired elevation angle of the \(i\)-th slave UAV in the spherical coordinate system as \(\varphi_i\). Denote the desired azimuth angle of the \(i\)-th slave UAV in the spherical coordinate system as \(\lambda_i\).

[0036] The beneficial effects of the present invention include:

[0037] (1) It enables the UAV swarm to simultaneously intercept maneuvering incoming UAVs in a spherical blockade formation in three-dimensional space, effectively improving the interception rate and damage effect. At the same time, during the guidance process, the azimuths of the UAVs in the spherical formation are fixed, avoiding the problem of UAVs colliding with each other;

[0038] (2) By reasonably designing the radius and the position of the center of the spherical blockade formation, the slave UAVs do not need to carry detection equipment and do not need to estimate the remaining flight time to achieve simultaneous strikes. Only the master UAV needs to carry expensive detection equipment, which can effectively reduce the interception cost. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] Figure 1 Shows a schematic flow chart of a low-cost UAV swarm cooperative guidance method for an air-mobile target according to a preferred embodiment of the present invention;

[0040] Figure 2 Shows a schematic diagram of the line-of-sight parallel coordinate system;

[0041] Figure 3 Shows the trajectory simulation results of each UAV in Embodiment 1;

[0042] Figure 4 Shows the simulation results of the distance between each UAV and the target in Embodiment 1;

[0043] Figure 5 Shows the simulation results of the line-of-sight angle and line-of-sight angle rate curves of the master UAV in Embodiment 1;

[0044] Figure 6 Shows the simulation results of the three-axis acceleration curves of the master UAV in Embodiment 1;

[0045] Figure 7 Shows the velocity simulation results of the main UAV in Embodiment 1 along the line-of-sight direction;

[0046] Figure 8 Shows the simulation results of the three-axis acceleration curves of all UAVs in Embodiment 1;

[0047] Figure 9 Shows the simulation results of the distances between the UAVs in Embodiment 1. Detailed implementation manners

[0048] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Through these descriptions, the features and advantages of the present invention will become clearer and more definite.

[0049] The special term "exemplary" here means "serving as an example, embodiment or illustration". Any embodiment described as "exemplary" here does not have to be construed as superior to or better than other embodiments. Although various aspects of the embodiments are shown in the drawings, the drawings do not have to be drawn to scale unless otherwise specified.

[0050] A low-cost UAV swarm cooperative guidance method for an air-mobile target provided according to the present invention includes the following steps:

[0051] S1. Set up a target detection device on the main UAV and establish a relative motion system model between the main UAV and the target;

[0052] S2. Based on the relative motion model, establish a sliding mode surface and obtain the guidance law of the main UAV along the line-of-sight normal direction by using the sliding mode control method;

[0053] S3. Set the acceleration of the main UAV along the line-of-sight direction, and combine it with the guidance law of the main UAV along the line-of-sight normal direction to obtain the guidance law of the main UAV;

[0054] S4. The slave UAVs are formed into a formation according to the position of the main UAV, so that the UAV swarm flies in a spherical arrangement to perform cooperative interception of the target.

[0055] According to the present invention, no target detection device is set on the slave UAVs, and only the main UAV is relied on to detect the target. The slave UAVs construct the guidance law according to the guidance law of the main UAV to achieve cooperative interception of the target. Since only a detection device needs to be set on the main UAV, the overall cost of the UAV swarm can be greatly reduced.

[0056] In S1, the relative motion system model is expressed as:

[0057]

[0058] Among them, x1, x2, x3, x4 are system variables, and u1, u2 are system control variables; r represents the distance between the master UAV and the target, and q θ represents the line-of-sight inclination angle of the master UAV, and a Ty represents the acceleration of the target in the y direction in the line-of-sight coordinate system, and a Tz represents the acceleration of the target in the z direction in the line-of-sight coordinate system, represents the line-of-sight deviation angle of the master UAV, represents the desired line-of-sight inclination angle of the master UAV, q θf represents the desired line-of-sight deviation angle of the master UAV, a uy represents the acceleration of the master UAV in the y direction in the line-of-sight coordinate system, and a uz represents the acceleration of the master UAV in the z direction in the line-of-sight coordinate system.

[0059] In the relative motion system model, the acceleration of the target is measured by the detection equipment of the master UAV.

[0060] In S2, the sliding mode control is a commonly used control method, and its specific process is not elaborated in the present invention.

[0061] Furthermore, in the present invention, there are two sliding mode surfaces s, namely s1 and s2, which are set as:

[0062]

[0063] Among them, s1 and s2 are the constructed sliding mode surfaces, α1, α2, β1, and β2 are all constants greater than zero, and sgn() is the sign function.

[0064] Preferably, α1, α2 ∈ (0, 1).

[0065] In the sliding mode control, preferably, the sliding mode reaching law is set as:

[0066]

[0067] Among them, ε1 and ε2 are constants greater than zero.

[0068] The above reaching law can ensure that when the distance r between the master UAV and the target is large, the convergence rate to the sliding mode surface is small, thereby reducing unnecessary oscillations. When approaching the target, the rate of approaching the sliding mode surface continuously increases, so as to ensure that the system state can converge to the sliding mode surface before intercepting the target and enter the sliding state.

[0069] According to the sliding mode surface and the sliding mode reaching law, the guidance law of the master UAV along the line-of-sight normal direction is obtained, which is expressed as:

[0070]

[0071] In S3, preferably, the acceleration of the master UAV along the line of sight is controlled by PI, expressed as:

[0072] a ux =k p (v dx -v ux )+k i ∫(v dx -v ux )dt

[0073] Where a ux is the acceleration of the master UAV along the line of sight, that is, the acceleration of the master UAV in the x - direction in the line - of - sight coordinate system, v ux is the velocity of the master UAV along the line of sight, k p and k i are control parameters, and v dx is the desired velocity of the master UAV along the line of sight.

[0074] Combining the guidance law of the master UAV along the normal direction of the line of sight, the guidance law of the master UAV is obtained, expressed as:

[0075]

[0076] In S4, the center of the sphere is located on the line connecting the master UAV and the target.

[0077] The radius of the sphere is set as:

[0078]

[0079] Where ρ represents the radius of the sphere, and c1, c2 are adjustable parameters.

[0080] According to the present invention, when r is large, the radius of the sphere will remain a constant value, avoiding the distance between UAVs being too far due to an overly large radius of the sphere; when r is small, as the master UAV approaches the target, ρ also gradually decreases until it converges to zero. At this time, the positions of the master UAV and the slave UAVs converge to the target point simultaneously, completing the multi - azimuth blockade and cooperative interception.

[0081] According to the present invention, the slave UAVs can adopt any cooperative guidance law as long as the UAV swarm can fly in a spherical arrangement. Preferably, the cooperative guidance law of the slave UAVs is set as:

[0082]

[0083] Where the subscripts i, j represent the serial numbers of the slave UAVs. All slave UAVs form a directed graph, and a ij represents the element corresponding to the i - th, j th slave UAV in the adjacency matrix of the directed graph. NDenote the total number of slave UAVs as \(u\). i Denote the guidance law of the \(i\)-th slave UAV as \(p\). i Denote the position of the \(i\)-th slave UAV in the inertial coordinate system as \(p\). di Denote the desired position of the \(i\)-th slave UAV in the inertial coordinate system as \(v\). i Denote the velocity of the \(i\)-th slave UAV. \(\gamma\) is a constant greater than zero, and \(p_1\) denotes the position of the master UAV in the inertial coordinate system. Denote the rotation matrix from the inertial coordinate system to the line-of-sight coordinate system. Denote the desired position of the \(i\)-th slave UAV in the line-of-sight parallel coordinate system as \(\theta\). di Denote the desired elevation angle of the \(i\)-th slave UAV in the spherical coordinate system. Denote the desired azimuth angle of the \(i\)-th slave UAV in the spherical coordinate system.

[0084] According to the present invention, the directed graph and the adjacency matrix are the general definitions in graph theory. The spherical coordinate system and the inertial coordinate system are the same as the general definitions in the art. The line-of-sight parallel coordinate system is as follows: The coordinate origin \(O\) p is the center of the desired spherical formation, located on the line connecting the master UAV and the target, and is at a distance \(\rho\) from the master UAV; The \(X\) p axis is the direction of the line connecting the master UAV and the target, with the direction pointing to the target being positive; The \(Z\) p axis is located in the vertical plane containing the \(X\) p axis, perpendicular to the \(X\) p axis, with the upward direction being positive; The \(Y\) p axis is determined according to the right-hand coordinate system, as shown in Figure 2 .

[0085] For the cooperative guidance law of the above-mentioned slave UAVs, the slave UAVs do not need to detect the target. They only need to communicate with each other to obtain the position information of other UAVs, and then the spherical formation control can be achieved. At the same time, during the flight, the azimuths of the UAVs in the spherical formation are fixed, avoiding the problem of collision between UAVs.

[0086] It should be understood that various forms of the processes shown above can be used, re-ordered, steps added or deleted. For example, the steps recorded in the disclosure of the present invention can be executed in parallel, sequentially, or in a different order, as long as the desired results of the technical solutions disclosed in the present invention can be achieved. This is not limited herein.

[0087] Embodiment

[0088] Embodiment 1

[0089] A simulation experiment is carried out. During the simulation process, 1 main UAV and 5 slave UAVs are used to cooperatively surround and capture the target. The initial position of the target UAV is (1200, 0, 50) m, the speed is (-10, -2, 0) m / s, and the acceleration in the first 30 s is (0, 0, 0) m / s 2 , and then it makes a maneuver of (0, 3sin(t), 3cos(t)) m / s 2 .

[0090] The process of surrounding and capturing the target includes the following steps:

[0091] S1. Set up a target detection device on the main UAV and establish a relative motion system model between the main UAV and the target;

[0092] S2. Based on the relative motion model, establish a sliding surface and obtain the guidance law of the main UAV along the line-of-sight normal direction by using the sliding mode control method;

[0093] S3. Set the acceleration of the main UAV along the line-of-sight direction, and combine it with the guidance law of the main UAV along the line-of-sight normal direction to obtain the guidance law of the main UAV;

[0094] S4. The slave UAVs form a formation according to the position of the main UAV, so that the UAV swarm flies in a spherical arrangement and cooperatively intercepts the target.

[0095] In S1, the relative motion system model is expressed as:

[0096]

[0097] In S2, there are two sliding surfaces s, namely s1 and s2, which are set as:

[0098]

[0099] In the sliding mode control, the sliding mode reaching law is set as:

[0100]

[0101] The guidance law of the main UAV along the line-of-sight normal direction is expressed as:

[0102]

[0103] In S3, the acceleration of the main UAV along the line-of-sight direction is controlled by PI and is expressed as:

[0104] a ux =k p (v dx -v ux )+k i ∫(v dx -v ux )dt

[0105] In S4, the center of the sphere is located on the line connecting the master drone and the target.

[0106] The radius of the sphere is set to:

[0107]

[0108] The cooperative guidance law of the slave UAV is set as:

[0109]

[0110] The initial positions of the six UAVs in the simulation and the expected inclination and deflection angles in the line-of-sight parallel coordinate system are shown in Table 1. The upper limit of the acceleration of each UAV is 5 m / s, and the spherical formation radius parameters are c1=100 and c2=2.

[0111] Table 1 UAV initial position and expected inclination and deflection

[0112]

[0113]

[0114] In the simulation, the expected velocity along the line of sight is v dx =15m / s, the expected terminal sight inclination angle and sight deflection angle are q θf =15°, γ is taken as 3, and the main UAV guidance law control parameters are shown in Table 2.

[0115] Table 2 Main UAV guidance law control parameters

[0116] <![CDATA[k p > <![CDATA[k i > <![CDATA[α1]]> <![CDATA[β1]]> <![CDATA[ε1]]> <![CDATA[α2]]> <![CDATA[β2]]> <![CDATA[ε1]]> 2 0.1 0,9 0.4 0.4 0.9 0.2 0.2

[0117] The simulation results are as follows Figures 3 - 9 shown.

[0118] in, Figure 3 The simulation results of each UAV trajectory are shown. Figure 4 The simulation results of the distance between each UAV and the target are shown. Figure 3 , Figure 4 It can be seen that after taking off from different points, each slave drone can quickly adjust its position, form a spherical formation around the main drone, and hit the target at the same time at different angles in a siege posture to achieve interception.

[0119] Figure 5 The simulation results of the line of sight angle and line of sight angular rate curve of the main UAV are shown. It can be seen from the figure that the line of sight angle of the main UAV converges to the specified value in about 25s, and the corresponding line of sight rotation angular rate converges to zero.

[0120] Figure 6 The three-axis acceleration curve of the main UAV is shown, Figure 7 The velocity of the main UAV along the line of sight is shown, Figure 8 The three-axis acceleration curves of all UAVs are shown. It can be seen from Figures 6 - 8 this that the acceleration of the main UAV does not reach saturation for most of the time. Before the target starts to maneuver, the acceleration of the main UAV gradually tends to be near zero; once the target starts to maneuver, the main UAV makes corresponding compensating maneuvers to intercept the target. The acceleration change trends of each slave UAV are roughly the same as that of the main UAV and do not reach saturation for most of the time. Before the target makes a maneuver, the accelerations of each UAV converge to near zero. After the target makes a maneuver, each UAV quickly follows the main UAV to make corresponding acceleration compensations; and the component velocity of the UAV in the line-of-sight direction reaches the specified value at about 5 s, with a small overshoot and no obvious oscillation, indicating that the flight stability of the UAVs is high during the enclosure process.

[0121] Figure 9 The distances between the UAVs are shown. It can be seen from the figure that the UAVs maintain a certain distance throughout the process except at the terminal moment, and there is no phenomenon of collision between the UAVs.

[0122] The present invention has been described in combination with preferred embodiments, but these embodiments are only exemplary and only serve an illustrative purpose. On this basis, various substitutions and improvements can be made to the present invention, and all of these fall within the protection scope of the present invention.

Claims

1. A low-cost UAV swarm cooperative guidance method for air-mobile targets, characterized in that It includes the following steps: S1. Set up a target detection device on the master UAV and establish a relative motion system model between the master UAV and the target; S2. Based on the relative motion model, establish a sliding surface and obtain the guidance law of the master UAV along the line-of-sight normal direction by using the sliding mode control method; S3. Set the acceleration of the master UAV along the line-of-sight direction, and combine it with the guidance law of the master UAV along the line-of-sight normal direction to obtain the guidance law of the master UAV; S4. The slave UAVs form a formation according to the position of the master UAV, so that the UAV swarm flies in a spherical arrangement and cooperatively intercepts the target.

2. The low-cost UAV swarm cooperative guidance method for air-mobile targets according to claim 1, characterized in that No target detection device is set on the slave UAVs.

3. The low-cost UAV swarm cooperative guidance method for air-mobile targets according to claim 1, characterized in that In S1, the relative motion system model is expressed as: Among them, x1, x2, x3, x4 are system variables, and u1, u2 are system control variables; r represents the distance between the master UAV and the target, and q θ represents the line-of-sight inclination angle of the master UAV, and a Ty represents the acceleration of the target in the y direction in the line-of-sight coordinate system, and a Tz represents the acceleration of the target in the z direction in the line-of-sight coordinate system, represents the line-of-sight deviation angle of the master UAV, represents the desired line-of-sight inclination angle of the master UAV, and q θf represents the desired line-of-sight deviation angle of the master UAV, and a uy represents the acceleration of the master UAV in the y direction in the line-of-sight coordinate system, and a uz represents the acceleration of the master UAV in the z direction in the line-of-sight coordinate system.

4. The low-cost UAV swarm cooperative guidance method for air-mobile targets according to claim 3, characterized in that In S2, there are two sliding surfaces s, namely s1 and s2, which are set as: where s1 and s2 are the constructed sliding surfaces, α1, α2, β1, and β2 are all constants greater than zero, and sgn() is the sign function.

5. The low-cost UAV swarm cooperative guidance method for air-mobile targets according to claim 4, characterized in that In the sliding mode control, the sliding mode reaching law is set as: where ε1 and ε2 are constants greater than zero.

6. The low-cost UAV swarm cooperative guidance method for air-mobile targets according to claim 5, characterized in that The guidance law of the master UAV along the line-of-sight normal direction is expressed as:

7. The low-cost UAV swarm cooperative guidance method for air-mobile targets according to claim 1, characterized in that In S3, the acceleration of the master UAV along the line-of-sight direction is controlled by PI and is expressed as: a ux = k p (v dx - v ux ) + k i ∫(v dx - v ux )dt Among them, a ux is the acceleration of the master UAV along the line of sight direction, that is, the acceleration of the master UAV in the x direction in the line-of-sight coordinate system, v ux is the velocity of the master UAV along the line of sight direction, k p and k i are control parameters, v dx is the desired velocity of the master UAV along the line of sight direction.

8. The low-cost UAV swarm cooperative guidance method for air-mobile targets according to claim 1, characterized in that In S4, the center of the sphere is located on the line connecting the master UAV and the target.

9. The low-cost UAV swarm cooperative guidance method for air-mobile targets according to claim 1, characterized in that The radius of the sphere is set as: where ρ represents the radius of the sphere, and c1 and c2 are adjustable parameters.

10. The low-cost UAV swarm cooperative guidance method for air-mobile targets according to claim 1, characterized in that The cooperative guidance law of the slave UAVs is set as: Among them, the subscripts i and j represent the serial numbers of the drones. All the slave drones form a directed graph. a ij represents the element corresponding to the i-th and j-th slave drones in the adjacency matrix of the directed graph. N represents the total number of slave drones. u i represents the guidance law of the i-th slave drone. p i represents the position of the i-th slave drone in the inertial coordinate system. p di represents the desired position of the i-th slave drone in the inertial coordinate system. v i represents the velocity of the i-th slave drone. γ is a constant greater than zero. p1 represents the position of the master drone in the inertial coordinate system. represents the rotation matrix from the inertial coordinate system to the line-of-sight coordinate system. represents the desired position of the i-th slave drone in the line-of-sight parallel coordinate system. θ di represents the desired elevation angle of the i-th slave drone in the spherical coordinate system. represents the desired azimuth angle of the i-th slave drone in the spherical coordinate system.

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