A three-dimensional master-slave multi-vehicle attack time-coordinated guidance method for unguided aircraft

Abstract

This invention designs a three-dimensional master-slave attack timing control cooperative guidance method for unguided aircraft, enabling all aircraft to simultaneously hit the target at the desired time, forming a fan-shaped blockade posture. The method first analyzes the mathematical model of the relative motion relationship between the master aircraft and the target in three-dimensional space, and the kinematic model of the slave aircraft, performing precise differential linearization. Then, it constructs an attack timing error variable and derives the attack timing control guidance law for the master aircraft. Furthermore, it proposes a spherical configuration with a variable radius, and based on this configuration, provides a method for calculating the desired position of the slave aircraft, and designs a cooperative guidance law for the slave aircraft. Even when only the master aircraft can detect target information and receive attack timing commands, and some slave aircraft cannot communicate directly with the master aircraft, the designed cooperative guidance law ensures that the master-slave aircraft group simultaneously hits the target in a fan-shaped blockade posture at the specified attack time.

Description

Technical Field

[0001] This invention relates to a guidance method for controllable attack time in three-dimensional space for some aircraft without seekers, belonging to the field of aircraft guidance and control. Specifically, it is an invention of a cooperative guidance method for attack time control of multiple aircraft under a single master-multiple slave architecture. Under the premise that only the master aircraft can obtain the attack time command and target information, the designed cooperative guidance law enables all aircraft to hit the target at the expected attack time and form a fan-shaped blockade of the target. Background Technology

[0002] Most existing cooperative guidance technologies require all aircraft to detect and locate the target. However, in environments with strong jamming denial, the target location devices of aircraft are sometimes suppressed and interfered with, rendering them inoperable. Furthermore, detection devices are expensive, and reducing the number of detection devices in multi-aircraft nodes of cooperative guidance can help reduce system costs. Therefore, it is necessary to study cooperative guidance methods when some aircraft are not equipped with target detection devices or when their detection devices fail. Simultaneously, in some special guided combat missions, there are often specific requirements for the timing of aircraft hitting the target. However, existing attack time control guidance methods require each aircraft in the missile swarm to have the ability to detect target information and receive attack time commands. Moreover, the information sharing capability between aircraft is poor. How to achieve simultaneous hits at the designated time by all aircraft on the target through cooperative guidance in three-dimensional flight space, under the premise that only some aircraft can detect target information and receive attack time commands, is of practical significance. In addition, some missions require multi-directional blockade strikes to achieve the requirement of highly efficient damage. Inspired by this, the present invention designs a three-dimensional master-slave attack time control cooperative guidance method for aircraft without seekers. When only the master aircraft can detect target information and receive attack time commands, the designed method enables all aircraft to hit the target simultaneously at a specified time and form a fan-shaped blockade of the target. Summary of the Invention

[0003] This invention considers the problem of three-dimensional space multi-vehicle cooperative guidance and blockade under attack time constraints, and designs a three-dimensional master-slave attack time control cooperative guidance method for a slave vehicle without a seeker head, so as to ensure that when only the master vehicle can detect target information and receive attack time commands, multiple vehicles hit the target simultaneously and form a fan-shaped blockade of the target.

[0004] The technical concept of this invention is as follows: First, a mathematical model of the relative motion relationship between the master aircraft and the target in three-dimensional space and a kinematic model of the slave aircraft in the inertial coordinate system are established, and the motion model of the slave aircraft is linearized by differentiation in order to design the guidance law; Second, based on the prediction of the remaining hit time of the master aircraft and the definition of the attack time error, the attack time control guidance law of the master aircraft is given; Then, through the design and calculation of the spherical formation configuration, the desired position of the slave aircraft under the spherical configuration with a variable radius is obtained; Finally, with the help of communication network information interaction, consistent cooperative variables are established, and a cooperative guidance law of the slave aircraft based on the desired position information is designed.

[0005] This invention relates to a three-dimensional master-slave multi-vehicle attack time-coordinated guidance method for unguided aircraft, comprising the following steps:

[0006] Step 1: Establish a motion model of the aircraft and linearize it with exact differentiation.

[0007] The multi-vehicle system consists of one master aircraft and n slave aircraft. The subscript i is used to represent the variable associated with the i-th aircraft, where i = 0, 1, ..., n; the master aircraft is labeled 0. The master aircraft can only send information to some of the slave aircraft that meet the communication requirements, but cannot receive information from the slave aircraft.

[0008] The equations of motion between the main aircraft and the fixed target in a three-dimensional line-of-sight coordinate system can be expressed as follows:

[0009]

[0010]

[0011]

[0012] Among them, a x a y and a z These represent the accelerations in the x, y, and z directions in the line-of-sight coordinate system, respectively. R represents the distance between the main aircraft and the target; q α ,q β These represent the line-of-sight tilt angle and line-of-sight deflection angle of the main aircraft relative to the target, respectively.

[0013] The simplified dynamic model of aircraft i in the ballistic coordinate system is as follows:

[0014]

[0015] Where g represents gravitational acceleration; V i ,θ i ,ψ Vi Represent the velocity, trajectory inclination angle, and trajectory deviation angle of aircraft i, respectively; nxi ,n yi ,n zi This represents the components of the overload of aircraft i on each coordinate axis of the ballistic coordinate system.

[0016] The kinematic model of aircraft i in the inertial coordinate system is as follows:

[0017]

[0018] Where, x i ,y i ,z i These represent the positions of aircraft i in the inertial coordinate system.

[0019] The state vector and control vector are selected as follows:

[0020]

[0021] The aircraft status feedback will be rewritten as follows:

[0022] v i =α(η) i )+β(η i )u i

[0023] Wherein, α(η) i ),β(η i ) is the transformation matrix

[0024]

[0025]

[0026] Therefore, the mathematical model of aircraft i is transformed into the following linear system:

[0027]

[0028] in, I n×n Let represent the n-dimensional identity matrix. This linear system transforms the formation control problem of a nonlinear aircraft system into a second-order multi-body consensus problem. Based on this linear system, an expression for the position tracking error is obtained, and the design of the control variables is made more convenient through a consensus algorithm.

[0029] Step 2: Construct attack time error variables

[0030] Define the attack time error variable as:

[0031]

[0032] Among them, T dThis indicates the desired attack time instruction, where t represents the current time. This represents an estimate of the remaining time.

[0033] Step 3: Determine the attack time control guidance law for the main aircraft.

[0034] The attack time control guidance law of the main aircraft is designed as follows:

[0035]

[0036]

[0037]

[0038] Where k1, k2, ..., k6 are the positive control gain coefficients, λ1, λ2 are the controller parameters to be designed, and a x Used to control the flight time of the main aircraft, a y and a z This is used to ensure the convergence of the line-of-sight tilt rate and the line-of-sight deflection rate, thereby enabling the main aircraft to hit the target at the specified attack time.

[0039] Step 4: Solve from the desired position of the aircraft.

[0040] The desired cooperative guidance configuration is defined on a sphere. The center of this sphere is defined on the line connecting the primary vehicle and the target, the distance between the primary vehicle and the target is denoted as R, and the radius of the sphere is defined as ρ. A variable scaling factor c is introduced, and ρ is given by the following calculation method:

[0041] ρ=cR

[0042]

[0043] Where c1 > c2 > 0 are the parameters to be designed.

[0044] With the center of the circle as the origin, the line connecting the main aircraft and the target is the x-axis, the y-axis is perpendicular to the x-axis and points upwards, and the z-axis is determined by the right-hand screw rule. Establish a line-of-sight parallel coordinate system Ox that is parallel to the main aircraft's line-of-sight coordinate system. d y d z d The desired position of aircraft i, expressed in spherical coordinates in a line-of-sight parallel coordinate system, is as follows: Where, θ pi , Let represent the desired elevation angle and azimuth angle of aircraft i in the spherical coordinate system, respectively.

[0045] From the aircraft i in the line-of-sight parallel coordinate system Ox d y d zd The expected position is:

[0046]

[0047] Transform it to the inertial coordinate system using coordinate transformation relationships. The position of the center of the circle in the inertial coordinate system is:

[0048]

[0049] Where (x0, y0, z0) represents the position of the main aircraft in the inertial coordinate system, and q α ,q β These represent the line-of-sight tilt angle and line-of-sight deflection angle of the main aircraft relative to the target, respectively.

[0050] The desired position of aircraft i in the inertial coordinate system is:

[0051] p i =p c +L1 -1 p di =[p xi p yi p zi ] T

[0052] Where, p xi ,p yi ,p zi These represent the projections of the desired position of aircraft i in the inertial coordinate system onto the three coordinate axes, respectively; L1 represents the transformation matrix from the line-of-sight coordinate system to the inertial coordinate system, specifically:

[0053]

[0054] Step 5: Define the position tracking error vector

[0055] The actual position of aircraft i in the inertial coordinate system is represented by p. xyzi To represent, p xyzi =[x i y i z i ] T The position tracking error vector of aircraft i is defined as follows:

[0056] e i =p xyzi -p i

[0057] Therefore, the systematic error can be expressed in the following form:

[0058]

[0059] in, This represents the second derivative of the desired position of aircraft i in the inertial coordinate system.

[0060] Step 6: Determine the cooperative guidance law for the aircraft.

[0061] The cooperative guidance law design for aircraft i is as follows:

[0062] u i =β -1 (η i (v) i -α(η i ))

[0063] Where, η i ,u i For the state vector and control vector; α(x) i ),β(x i ) is the transformation matrix; v i The control quantity given by the consensus algorithm in the linearized system is expressed as follows:

[0064]

[0065] Wherein, the function sig(x) α =sign(x)|x| α p j Let α represent the desired position of aircraft j in the inertial coordinate system, k1 and k2 be the control gain coefficients, and 0 < α < 1 be the controller parameters to be designed. ij Let $a$ represent the communication relationship between slave $j$ and slave $i$. If slave $i$ can receive information from slave $j$, then slave $j$ is said to be a neighbor of slave $i$ and $a$ is a neighbor of slave $i$. ij >0, otherwise a ij =0, This represents the set of neighbors excluding the master aircraft. It is assumed here that the directed communication graph consisting of the master and slave aircraft nodes is strongly connected, meaning there is always a path between any two nodes. (Use u) i Control the actual position p xyzi It eventually converges to the desired position p. i .

[0066] The beneficial effects of this invention are as follows: A three-dimensional master-slave attack timing control cooperative guidance method for aircraft without seekers is designed. This method is effective when only the master aircraft can detect target information and receive attack timing commands, and some slave aircraft cannot communicate directly with the master aircraft. The method utilizes the acceleration α of the master aircraft... x Control the attack time, use a y and a z Make From the aircraft via u i Control p xyzi Consistent with p i Within a finite time, the spherical configuration converges to the desired spherical configuration. As the distance between the main aircraft and the target decreases, the spherical configuration converges to the target point, thus ensuring that all aircraft hit the target simultaneously within the specified attack time. Attached Figure Description

[0067] Figure 1 This is the design process for a three-dimensional master-slave attack time control cooperative guidance method for aircraft without seekers.

[0068] Figure 2 This is a schematic diagram of the spherical configuration when the distance is relatively far.

[0069] Figure 3 This is a schematic diagram of the spherical configuration when the distance is close.

[0070] Figure 4 It is a coordinate system parallel to the line of sight.

[0071] Figure 5 It is the spatial motion trajectory curve of multiple aircraft.

[0072] Figure 6 It is the distance curve between the master and slave aircraft and the target.

[0073] Figure 7 It is derived from the aircraft consistency error curve. Detailed Implementation

[0074] To make the objectives, technical solutions, and advantages of this invention clearer, please refer to the appendix. Figure 1 —7. Further explanation of the present invention.

[0075] The design process of the invented method is as follows: Figure 1 As shown, the specific steps include:

[0076] Step 1: Establish a motion model of the aircraft and linearize it with exact differentiation.

[0077] The multi-vehicle system consists of one master aircraft and n slave aircraft. The subscript i is used to represent the variable associated with the i-th aircraft, where i = 0, 1, ..., n; the master aircraft is labeled 0. The master aircraft can only send information to some of the slave aircraft that meet the communication requirements, but cannot receive information from the slave aircraft.

[0078] The equations of motion between the main aircraft and the fixed target in a three-dimensional line-of-sight coordinate system can be expressed as follows:

[0079]

[0080]

[0081]

[0082] Among them, a x a y and a z These represent the accelerations in the x, y, and z directions in the line-of-sight coordinate system, respectively. R represents the distance between the main aircraft and the target; q α ,q β These represent the line-of-sight tilt angle and line-of-sight deflection angle of the main aircraft relative to the target, respectively.

[0083] The simplified dynamic model of aircraft i in the ballistic coordinate system is as follows:

[0084]

[0085] Where g represents gravitational acceleration; V i ,θ i ,ψ Vi Represent the velocity, trajectory inclination angle, and trajectory deviation angle of aircraft i, respectively; n xi ,n yi ,n zi This represents the components of the overload of aircraft i on each coordinate axis of the ballistic coordinate system.

[0086] The kinematic model of aircraft i in the inertial coordinate system is as follows:

[0087]

[0088] Where, x i ,y i ,z i These represent the positions of aircraft i in the inertial coordinate system.

[0089] The state vector and control vector are selected as follows:

[0090]

[0091] The aircraft status feedback will be rewritten as follows:

[0092] v i =α(η) i )+β(η i )u i

[0093] Wherein, α(η) i ),β(η i ) is the transformation matrix

[0094]

[0095]

[0096] Therefore, the mathematical model of aircraft i is transformed into the following linear system:

[0097]

[0098] in, I n×n Let represent the n-dimensional identity matrix. This linear system transforms the formation control problem of a nonlinear aircraft system into a second-order multi-body consensus problem. Based on this linear system, an expression for the position tracking error is obtained, and the design of the control variables is made more convenient through a consensus algorithm.

[0099] Step 2: Construct attack time error variables

[0100] Define the attack time error variable as:

[0101]

[0102] Among them, T d This indicates the desired attack time instruction, where t represents the current time. This represents an estimate of the remaining time.

[0103] Step 3: Determine the attack time control guidance law for the main aircraft.

[0104] The attack time control guidance law of the main aircraft is designed as follows:

[0105]

[0106]

[0107]

[0108] Where k1, k2, ..., k6 are the positive control gain coefficients, λ1, λ2 are the controller parameters to be designed, and a x Used to control the flight time of the main aircraft, a y and a z This is used to ensure the convergence of the line-of-sight tilt rate and the line-of-sight deflection rate, thereby enabling the main aircraft to hit the target at the specified attack time.

[0109] Step 4: Solve from the desired position of the aircraft.

[0110] The desired cooperative guidance configuration is defined on a sphere. The center of this sphere is defined on the line connecting the primary vehicle and the target, the distance between the primary vehicle and the target is denoted as R, and the radius of the sphere is defined as ρ. Figure 2 Figure 3As shown, the variable scaling factors c and ρ are given by the following calculation method:

[0111] ρ=cR

[0112]

[0113] Where c1 > c2 > 0 are the parameters to be designed.

[0114] With the center of the circle as the origin, the line connecting the main aircraft and the target is the x-axis, the y-axis is perpendicular to the x-axis and points upwards, and the z-axis is determined by the right-hand screw rule. Establish a line-of-sight parallel coordinate system Ox that is parallel to the main aircraft's line-of-sight coordinate system. d y d z d like Figure 4 The desired position of aircraft i, expressed in spherical coordinates in a line-of-sight parallel coordinate system, is as follows: Where, θ pi Let represent the desired elevation angle and azimuth angle of aircraft i in the spherical coordinate system, respectively.

[0115] From the aircraft i in the line-of-sight parallel coordinate system Ox d y d z d The expected position is:

[0116]

[0117] Transform it to the inertial coordinate system using coordinate transformation relationships. The position of the center of the circle in the inertial coordinate system is:

[0118]

[0119] Where (x0, y0, z0) represents the position of the main aircraft in the inertial coordinate system, and q α ,q β These represent the line-of-sight tilt angle and line-of-sight deflection angle of the main aircraft relative to the target, respectively.

[0120] The desired position of aircraft i in the inertial coordinate system is:

[0121]

[0122] Where, p xi ,p yi ,p zi These represent the projections of the desired position of aircraft i in the inertial coordinate system onto the three coordinate axes, respectively; L1 represents the transformation matrix from the line-of-sight coordinate system to the inertial coordinate system, specifically:

[0123]

[0124] Step 5: Define the position tracking error vector

[0125] The actual position of aircraft i in the inertial coordinate system is represented by p. xyzi To represent, p xyzi =[x i y i z i ] T The position tracking error vector of aircraft i is defined as follows:

[0126] e i =p xyzi -p i

[0127] Therefore, the systematic error can be expressed in the following form:

[0128]

[0129] in, This represents the second derivative of the desired position of aircraft i in the inertial coordinate system.

[0130] Step 6: Determine the cooperative guidance law for the aircraft.

[0131] The cooperative guidance law design for aircraft i is as follows:

[0132] u i =β -1 (η i (v) i -α(η i ))

[0133] Where, η i ,u i For the state vector and control vector; α(x) i ),β(x i ) is the transformation matrix; v i The control quantity given by the consensus algorithm in the linearized system is expressed as follows:

[0134]

[0135] Wherein, the function sig(x) α =sign(x)|x| α p j Let α represent the desired position of aircraft j in the inertial coordinate system, k1 and k2 be the control gain coefficients, and 0 < α < 1 be the controller parameters to be designed. ij Let $a$ represent the communication relationship between slave $j$ and slave $i$. If slave $i$ can receive information from slave $j$, then slave $j$ is said to be a neighbor of slave $i$ and $a$ is a neighbor of slave $i$.ij >0, otherwise a ij =0, This represents the set of neighbors excluding the master aircraft. It is assumed here that the directed communication graph consisting of the master and slave aircraft nodes is strongly connected, meaning there is always a path between any two nodes. (Use u) i Control the actual position p xyzi It eventually converges to the desired position p. i .

[0136] The effectiveness of the designed master-slave attack time-controlled cooperative guidance method was verified using the Matlab simulation platform. One master aircraft and three slave aircraft were selected. The initial positions, trajectory inclinations, and trajectory deflections are shown in the table below:

[0137]

[0138] The initial velocities of both the master and slave aircraft are 330 m / s; the control parameters of the master aircraft are: k1 = k2 = k3 = k4 = k5 = k6 = 10, λ1 = 0.5, λ2 = 2, T d =45s; the spherical cooperative control configuration design parameters are c1=5, c2=1; the controller parameters of the aircraft are: k1=3.5, k2=5, α=0.8.

[0139] Simulation results are as follows Figures 5-7 As shown, Figure 5 This represents the ballistic curves of the primary and secondary aircraft during their flight. Figure 6 This represents the curve showing the change in the distance between the primary and secondary aircraft and the target over time. Figure 7 The curve represents the change in the consistency and coordination error of the three aircraft over time. Figure 5 Figure 6 It can be seen that all four aircraft were able to hit the target, and the final hit time was consistent with the expected attack time of 45 seconds. Figure 7 The consistent positional coordination error can be stably converged to 0, and the simulation results verify the effectiveness of the design method.

Claims

1. A three-dimensional master-slave multi-vehicle attack time-coordinated guidance method for unguided aircraft, characterized in that, Includes the following steps: Step 1: Establish a motion model of the aircraft and linearize it using exact differentiation; Multi-vehicle system consists of one main aircraft and n Each component consists of an aircraft, using subscripts. i To indicate the first i Variables related to aircraft i =0,1,…, n The master aircraft is designated as 0. The master aircraft only sends information to some of the slave aircraft that meet the communication requirements, but cannot receive information from the slave aircraft. The equations of motion between the main aircraft and the fixed target in the three-dimensional line-of-sight coordinate system are expressed as follows: ； in, , and In the line-of-sight coordinate system x , y and z Acceleration in the direction of; R Indicates the distance between the main aircraft and the target; , These represent the tilt angle and deflection angle of the main aircraft relative to the target, respectively. Step 2: Construct the attack time error variable; ； in, Indicates the desired attack timing command. t Indicates the current time; Indicates an estimate of the remaining time; Step 3: Design the attack time control guidance law for the main aircraft; ； in, It is a positive control gain coefficient. For the parameters of the controller to be designed, Used to control the flight time of the main aircraft. and Used to ensure the convergence of the line-of-sight tilt rate and the line-of-sight deflection rate, thereby enabling the main aircraft to hit the target at the specified attack time; Step 4: Solve from the desired position of the aircraft; The desired cooperative guidance configuration is defined as lying on a sphere; the center of this sphere is defined on the line connecting the main vehicle and the target, and the distance between the main vehicle and the target is denoted as... R The radius of a sphere is defined as Introducing a variable scaling factor , The following calculation method is given: ； in, The parameters to be designed; Step 5: Define the position tracking error vector; From the aircraft The actual position in the inertial coordinate system is used To indicate, , They represent from the aircraft i Position in the inertial coordinate system, from the aircraft The position tracking error vector is defined as: ； in, From the aircraft The desired position in the inertial coordinate system; Therefore, the error can be expressed in the following form: ； in, Represents from aircraft The second derivative of the desired position in the inertial coordinate system; ; Step 6: Determine the cooperative guidance law from the aircraft; From the aircraft The cooperative guidance law is designed as follows: ； in, η i , u i These are the state vector and the control vector; The transformation matrix; This is the control quantity given by the consensus algorithm in the linearized system.

2. The three-dimensional master-slave multi-vehicle attack time-coordinated guidance method for a seekerless aircraft as described in claim 1, characterized in that: In step 1, from the aircraft i The simplified dynamic model in the ballistic coordinate system is as follows: ； in, Represents gravitational acceleration; They represent from the aircraft i Speed, trajectory inclination angle, trajectory deviation angle; Indicates from the aircraft i The overload components on each coordinate axis of the ballistic coordinate system.

3. A three-dimensional master-slave multi-vehicle attack time-coordinated guidance method for a seekerless aircraft as described in claim 1 or 2, characterized in that: From the aircraft i The kinematic model in the inertial coordinate system is as follows: ； in, They represent from the aircraft i Position in the inertial coordinate system They represent from the aircraft i Speed, trajectory inclination angle, trajectory deviation angle.

4. The three-dimensional master-slave multi-vehicle attack time-coordinated guidance method for a seekerless aircraft as described in claim 3, characterized in that: The state vector and control vector are selected as follows: ； The aircraft status feedback will be rewritten as follows: ； in, Transformation matrix ； 。 5. A three-dimensional master-slave multi-vehicle attack time-coordinated guidance method for a seekerless aircraft as described in claim 4, characterized in that: From the aircraft i The mathematical model is transformed into the following linear system: ； in, ; ; express n 3D identity matrix.

6. The three-dimensional master-slave multi-vehicle attack time-coordinated guidance method for unguided aircraft as described in claim 1, characterized in that: In step 4, with the center of the circle as the origin, the line connecting the main aircraft and the target is... x d axis, y d Axis perpendicular to x d The axis is upward. z d The axis is determined by the right-hand screw rule, establishing a line-of-sight parallel coordinate system parallel to the main aircraft's line-of-sight coordinate system. From the aircraft The desired position, expressed in spherical coordinates in a coordinate system parallel to the line of sight, is as follows: ,in, They represent from the aircraft The desired elevation and azimuth angles in spherical coordinates.

7. A three-dimensional master-slave multi-vehicle attack time-coordinated guidance method for a seekerless aircraft as described in claim 6, characterized in that: From the aircraft In the coordinate system parallel to the line of sight The expected position is: ； Transform it to the inertial coordinate system using coordinate transformation relationships. The position of the center of the circle in the inertial coordinate system is: ； in,( x 0, y 0, z 0 This indicates the position of the main aircraft in the inertial coordinate system. These represent the line-of-sight tilt angle and line-of-sight deflection angle of the main aircraft relative to the target, respectively.

8. A three-dimensional master-slave multi-vehicle attack time-coordinated guidance method for a seekerless aircraft as described in claim 7, characterized in that: From the aircraft The desired position in the inertial coordinate system is: ； in, p xi ,p yi ,p zi They represent from the aircraft The projection of the desired position onto the three coordinate axes in the inertial coordinate system. This represents the transformation matrix from the line-of-sight coordinate system to the inertial coordinate system.

9. A three-dimensional master-slave multi-vehicle attack time-coordinated guidance method for a seekerless aircraft as described in claim 8, characterized in that: The transformation matrix from the line-of-sight coordinate system to the inertial coordinate system is as follows: 。 10. A three-dimensional master-slave multi-vehicle attack time-coordinated guidance method for a seekerless aircraft according to claim 1, characterized in that: In step 6, ； Among them, the function , p j Indicates from the aircraft j The desired position in the inertial coordinate system. To control the gain coefficient, The parameters of the controller to be designed; a ij Indicates from the aircraft j With the aircraft i The communication relationship between them, if the first i The first one received from the aircraft j Information from an aircraft is called information from the aircraft. j From the aircraft i Neighbors and ,on the contrary , This represents the set of neighbors excluding the master spacecraft; the directed communication graph consisting of the master spacecraft nodes and slave spacecraft nodes is strongly connected, meaning there is always a path between any two nodes; through Control the actual position It eventually converges to the desired position. .

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