Multi-spacecraft leader following formation control method based on neural adaptation

Abstract

The invention provides a multi-spacecraft leader following formation control method based on neural adaptation. The method comprises the following steps: firstly, establishing a multi-space aircraft system model with a leader-follower structure; then, designing a robust formation control scheme based on neural self-adaption; firstly, designing a self-adaptive distributed state observer based on local discontinuous interaction for a follower spacecraft; secondly, a distributed robust formation controller based on neural network fitting is designed for the follower spacecraft; and finally, designing an inter-aircraft communication mechanism based on a static event triggering mechanism for the follower spacecraft. According to the method, the problems of local knowing of leader information, limited communication, adverse influence of external disturbance on control performance and the like in the multi-spacecraft formation with a leader-following structure are considered, an effective formation control scheme is designed, and the robustness and reliability of a multi-spacecraft system are remarkably improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of multi-space aircraft formation control with a leader-follower structure, and in particular relates to a multi-space aircraft leader-follower formation control method based on neural adaptation. Background Art

[0002] In today's aerospace sector, multi-spacecraft formation flying technology is becoming a research hotspot, playing a key role in numerous important space missions, including space science exploration, satellite communications networking, and Earth observation. For example, when conducting scientific exploration of outer space beyond the Earth's atmosphere, multi-spacecraft formations can simultaneously acquire data from different angles and locations, providing a more comprehensive understanding of the cosmic environment. When building satellite communications networks, rational formations can achieve more efficient signal coverage and transmission.

[0003] However, multi-spacecraft formation control faces numerous challenges. First, in formation flight missions, leader information is often partially available; that is, not all follower vehicles can directly access all of the leader's state information. This makes it difficult for follower vehicles to accurately adjust to the leader's actions, thereby increasing the difficulty of formation control. Second, limited communication is another pressing issue. In complex space environments, communication between vehicles may be affected by factors such as distance and signal interference, resulting in information transmission delays or loss, further compromising the accuracy and stability of formation control. Furthermore, multi-spacecraft systems are inevitably subject to external disturbances during operation, such as impacts from space micrometeoroids and the effects of the solar wind. Furthermore, the system itself may also contain model uncertainties, such as slight changes in the vehicle's dynamic model parameters and errors in sensor accuracy. These external disturbances and model uncertainties can adversely affect the maintenance of formation formation and the smooth execution of the mission, and may even lead to the collapse of the formation structure.

[0004] In the existing technology, although some formation control methods have been proposed, most of them still have certain limitations when facing the above-mentioned complex situations. For example, some methods based on traditional control theory are weak in dealing with model uncertainty and have difficulty adapting to the complex space environment; while some methods that rely on frequent communication cannot effectively cope with situations where communication is limited. Therefore, how to effectively solve the formation control problem of multi-space aircraft systems when they are affected by external disturbances and model uncertainty in an environment where leader information is locally known and communication is limited, while ensuring the asymptotic stability of the system and avoiding the Zeno phenomenon that may occur in the event triggering mechanism (i.e., the system has an infinite number of event triggers in a finite time, resulting in frequent updates of control signals, increasing the system burden and even causing system instability) has become a key technical problem that needs to be solved in the current field of multi-space aircraft formation control technology. Summary of the Invention

[0005] The purpose of the present invention is to overcome the deficiencies of the prior art and provide a multi-spacecraft leader-follower formation control method based on neural adaptation. The present invention takes into account problems such as the partial availability of leader information, limited communication, and the adverse effects of external disturbances on control performance in a multi-spacecraft formation with a leader-follower structure, and designs an effective formation control scheme, significantly improving the robustness and reliability of the multi-spacecraft system.

[0006] The purpose of the present invention is achieved through the following technical solutions: A multi-spacecraft leader-follower formation control method based on neural adaptation, the steps are as follows:

[0007] S1. Establish a multi-spacecraft system with a leader-follower structure; the specific steps are as follows:

[0008] S11. Construct a six-degree-of-freedom dynamic model of the follower spacecraft: Use the modified Rodriguez parameters and the CW equations to describe the attitude change of the spacecraft in the geocentric inertial coordinate system and the relative motion in the orbital coordinate system respectively;

[0009] S12. Construct the dynamic model of the leader, and the leader dynamic model is described in the following form:

[0010]

[0011] Where respectively represent the state and the rate of change of the state of the leader l, represents the control input of the leader, which is unknown and bounded for all follower spacecraft, that is Let

[0012] S13. Determine the communication constraints between spacecraft: The communication relationship between follower spacecraft is described by an undirected and connected topology This topology The adjacency matrix of is Where a ij ≥0; a ij >0 means that follower spacecraft i and j can communicate with each other, a ij =0 means that there is no communication between follower spacecraft i and j; the degree matrix is Where Topology The Laplacian matrix of

[0013] Use a directed graph To describe the communication relationship between the leader and follower spacecraft, a directed graph has an adjacency matrix of where b i ≥0; b i >0 indicates that the follower spacecraft i can directly obtain the real-time information of the leader. Otherwise, b i =0; at least one of the follower spacecraft can obtain the information of the leader, and the communication topology among the multi-spacecraft systems is connected; define the matrix Considering the connectivity of the topology, the matrix is a positive definite matrix; use to represent the element in the i-th row and j-th column of the matrix respectively;

[0014] S14. Clearly define the formation control problem of the multi-spacecraft system. The formation control needs to satisfy the following conditions:

[0015]

[0016] where d li represents the expected state deviation vector of the follower spacecraft i relative to the leader;

[0017] S2. Design an adaptive state observer based on local interaction; design the following adaptive distributed observer for the follower spacecraft i to reconstruct the information of the leader:

[0018]

[0019] where β1, β2, γ1 > 0, representing gains; represent the estimated values of the information ξ l , of the leader by the follower spacecraft i respectively; represents the estimated value of the upper bound l of the control input f of the leader by the follower spacecraft i; the variable is designed in the following form:

[0020]

[0021] where represent the information broadcast by the spacecraft i to its neighbor spacecraft at the k-th time respectively;

[0022] The measurement error is expressed in the following form:

[0023]

[0024] S3. Design a distributed robust formation controller based on neural network fitting as follows:

[0025] S31. Use a neural network to fit the composite uncertainty. First, design the following auxiliary variables for the follower spacecraft i:

[0026]

[0027] where k p , k d > 0; Substitute the auxiliary variables into the dynamic equation of spacecraft i, and use a radial basis neural network to fit the uncertainty in the composite dynamic equation.

[0028] S32. Design a distributed robust formation controller based on neural network fitting. First, define the consensus formation error of spacecraft i:

[0029]

[0030] Considering the discontinuous communication between spacecraft, rewrite the above formula as follows:

[0031]

[0032] where represents the information broadcast by spacecraft j to its neighbor spacecraft for the k_j -th time, and define the measurement error It should be noted that Then write it in a compact form:

[0033]

[0034] where

[0035] Based on the above analysis, design the following controller for spacecraft i:

[0036]

[0037] where k s > 0;

[0038] S4. Construct an inter - spacecraft communication mechanism based on a static event - triggered mechanism as follows:

[0039] S41. Adopt an inter - spacecraft communication mechanism based on static event - triggering to adjust the communication frequency between spacecraft. For spacecraft i, the static event - triggered mechanism is designed in the following form:

[0040]

[0041] where the static event - trigger function is designed in the following form:

[0042]

[0043] where κ ki , η ki > 0;

[0044] S5. Design a neural - adaptive - based robust formation control scheme based on S2 - S4 to complete the formation control of multiple spacecraft: Deploy the neural - adaptive - based robust formation control scheme into a multiple - spacecraft system with a leader - follower structure to achieve the formation control of multiple spacecraft.

[0045] The beneficial effects of the present invention are as follows: The present invention provides a neural - adaptive - based leader - follower formation control method for multiple spacecraft. First, a multiple - spacecraft system model with a leader - follower structure is established, and the potential impacts of external harsh environments and inaccurate modeling on the spacecraft dynamics model are fully considered. On this basis, a neural - adaptive - based robust formation control scheme is designed, and the design process of this scheme is divided into three key steps. First, considering that the leader information is only locally available and communication is limited, an adaptive distributed state observer based on local discontinuous interaction is designed for the follower spacecraft to reconstruct the leader information. Second, considering the adverse effects of external disturbances and model uncertainties on the system, a distributed robust formation controller based on neural network fitting is designed for the follower spacecraft. Then, an inter - vehicle communication mechanism based on a static event - triggering mechanism is designed for the follower spacecraft, thus reducing unnecessary information interaction between spacecraft to a certain extent. The present invention focuses on problems such as the locally available leader information, limited communication, and the adverse effects of external disturbances on the control performance in the formation of multiple spacecraft with a leader - follower structure, and designs an effective formation control scheme, significantly improving the robustness and reliability of the multiple - spacecraft system. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 is a flowchart of the neural - adaptive - based leader - follower formation control method for multiple spacecraft of the present invention;

[0047] Figure 2 is a schematic diagram of the communication topology relationship between the given follower spacecraft and the leader in this embodiment;

[0048] Figure 3 is the attitude observation error curve of the follower spacecraft with respect to the leader;

[0049] Figure 4 is the position observation error curve of the follower spacecraft with respect to the leader;

[0050] Figure 5 is the attitude tracking error curve of the follower space vehicle;

[0051] Figure 6 is the position tracking error curve of the follower space vehicle;

[0052] Figure 7 is the curve of the change in the attitude control input of the follower space vehicle;

[0053] Figure 8 is the curve of the change in the position control input of the follower space vehicle;

[0054] Figure 9 is the curve of the change in the F-norm of the neural network weight estimate of the follower space vehicle;

[0055] Figure 10 is the motion trajectory of the follower space vehicle in the orbital coordinate system;

[0056] Figure 11 is the communication time interval of all follower space vehicles under the inter-vehicle communication mechanism based on the event-triggered mechanism. Detailed implementation manner

[0057] To better describe the invention content, some symbols in the following text are explained here: I n represents the n-dimensional identity matrix, 1 n represents the n-dimensional column vector with all elements being 1. represents a column vector, represents a diagonal matrix. and respectively represent the minimum and maximum eigenvalues of matrix A. For the vector x = [x1, x2, x3] T , S(x) = [0, -x3, x2; x3, 0, -x1; -x2, x1, 0]. For the vector x ∈ R n , sgn(x) = x / ||x||.

[0058] The technical solution of the present invention is further described below with reference to the accompanying drawings.

[0059] As Figure 1 shown, a method for leader-follower formation control of multiple space vehicles based on neural adaptation of the present invention is as follows:

[0060] S1. Establish a multi-space vehicle system with a leader-follower structure; the specific steps are as follows:

[0061] S11. Construct the six-degree-of-freedom dynamic model of the follower spacecraft: Use the Modified Rodriguez Parameters (MRPs) and the CW (Clohessy-Wiltshire) equations to describe the attitude change of the spacecraft in the Earth-Centered Inertial (ECI) coordinate system and its relative motion in the Local Vertical Local Horizontal (LVLH) coordinate system respectively.

[0062] First, define σ i =[σ i1 ,σ i2 ,σ i3 T and ρ i =[ρ i1 ,ρ i2 ,ρ i3 T to represent the attitude of the follower spacecraft i in the ECI coordinate system and its position in the LVLH coordinate system respectively. The index i of the follower spacecraft belongs to where n represents the number of follower spacecraft. Then, the following dynamic equation can be obtained:

[0063]

[0064]

[0065] where ω i =[ω i1 ,ω i2 ,ω i3 T represents the projection of the angular velocity of the body coordinate system of the follower spacecraft i relative to the ECI coordinate system in the body coordinate system; J i ∈R 3×3 , m i ∈R represent the moment of inertia and mass of the follower spacecraft i respectively; τ i , τ di represent the projections of the control torque and disturbance torque acting on the follower spacecraft i in the body coordinate system respectively; F bi , f di represent the projection of the control force of the follower spacecraft i in the body coordinate system and the external disturbance it receives respectively. At the same time, G(σ i ), C ri , N ri , R lbi are expressed in the following forms respectively:

[0066]

[0067] where represents the angular velocity of the reference point rotating around the Earth in the LVLH coordinate system (the reference point is taken as the origin of the LVLH coordinate system); R​​​lbi , R le , R bei represent the rotation matrices from the body coordinate system of the follower spacecraft i to the LVLH system, from the ECI system to the LVLH system, and from the body system to the ECI system, respectively; R le can be calculated from the position and velocity of the reference point.

[0068] Define G i = G(σ i ), and then rewrite Equation (1) into the Euler - Lagrange form:

[0069]

[0070] where represent the first - order derivative and second - order derivative of σ i respectively. Then, considering the order - of - magnitude difference in the position and attitude of the follower spacecraft, the present invention normalizes the position of the spacecraft. Define the state variable of the follower spacecraft i as where ρ max represents the maximum distance between the follower spacecraft and the reference point, which is a quantity set artificially. Then, by combining Equation (2) and Equation (6), the six - degree - of - freedom dynamic equation of the follower spacecraft i can be obtained:

[0071]

[0072] where represents the state change rate of the follower spacecraft i,

[0073] Considering factors such as fuel consumption during the mission execution of the follower spacecraft, the mass m i and moment of inertia J i of the spacecraft i cannot be accurately obtained. At the same time, the present invention mainly considers the case where the external disturbance d i acting on the spacecraft in the space environment is unknown and bounded.

[0074] S12. Construct the dynamic model of the leader. The present invention is mainly applied to a multi - spacecraft system with a leader - follower structure. Therefore, it is necessary to introduce a leader for the follower spacecraft system. The leader dynamic model is described in the following form:

[0075]

[0076] where represent the state and state change rate of the leader l, respectively; Denote the control input of the leader \(l\), which is unknown and bounded for all follower spacecraft, i.e., Meanwhile, for the sake of convenient expression, let In the present invention, only the case of a single leader is considered, and the control input \(f\) of the leader l is bounded.

[0077] S13. Determine the communication constraints between spacecraft: The communication relationship between follower spacecraft is described by an undirected and connected topology This topology has an adjacency matrix where \(a\) ij ≥0; \(a\) ij >0 means that follower spacecraft \(i\) and \(j\) can communicate with each other, and \(a\) ij =0 means that there is no communication between follower spacecraft \(i\) and \(j\); meanwhile, the degree matrix is where The Laplacian matrix of the topology is

[0078] Use a directed graph to describe the communication relationship between the leader and follower spacecraft. The adjacency matrix of the directed graph is where \(b\) i ≥0; \(b\) i >0 indicates that follower spacecraft \(i\) can directly obtain the real-time information of the leader. Otherwise, \(b\) i =0; at least one of the follower spacecraft can obtain the information of the leader, and the communication topology between the multi-spacecraft systems is connected; define the matrix Considering the connectivity of the topology, the matrix is a positive definite matrix; use to represent the element in the \(i\)-th row and \(j\)-th column of the matrix respectively;

[0079] S14. Define the formation control problem of the multi-spacecraft system. When only some of the follower spacecraft can directly obtain the information of the leader, all follower spacecraft can complete the tracking task of the leader's trajectory with the desired attitude and formation configuration through local discontinuous interaction. That is, the formation control needs to satisfy the following conditions:

[0080]

[0081] where \(d\) liDenote the desired state deviation vector of the follower spacecraft \(i\) relative to the leader; in the present invention, it is considered as an internal parameter of the follower spacecraft \(i\), so there is no need to obtain it through communication.

[0082] S2. Design an adaptive state observer based on local interaction; specifically as follows:

[0083] S21. Considering that only some follower spacecraft can directly obtain the information of the leader, an adaptive distributed observer is designed for the follower spacecraft \(i\) to reconstruct the information of the leader as follows:

[0084]

[0085] where \(\beta_1,\beta_2,\gamma_1>0\) denote gains and are constants; respectively denote the estimated values of the information \(\xi\) of the leader by the follower spacecraft \(i\) l , ; denote the estimated value of the control input \(f\) of the leader by the follower spacecraft \(i\) l upper bound ; the variable is designed in the following form:

[0086]

[0087] where respectively denote the information broadcast by the spacecraft \(i\) to its neighbor spacecraft at the \(k_i\)-th time;

[0088] Meanwhile, considering the time delay caused by discontinuous communication, the measurement error is expressed in the following form:

[0089]

[0090] Furthermore, combining Equation (11) and Equation (12) can obtain the following equation:

[0091]

[0092] It should be noted that the \(\varphi\) i in Equation (13) represents two consensus observation errors of the spacecraft \(i\), which are mainly used for the stability analysis of the system, while the in Equation (11) is mainly used for observer design. Meanwhile, the compact form of the variable \(\varphi\) i can be expressed as:

[0093]

[0094] where

[0095] S22. After the design of the adaptive distributed observer is completed in step S21, the following Lyapunov function can be constructed for the stability analysis of the control scheme designed in the present invention below:

[0096]

[0097] where represents the estimation error of the follower spacecraft i with respect to ; since the matrix is positive definite, the Lyapunov function V1 is also positive definite.

[0098] According to equations (10) and (14), the derivative of V1 with respect to time is:

[0099]

[0100] where Furthermore, substituting equation (13) into the above equation and using the Young's inequality, equation (16) can be simplified to:

[0101]

[0102] where ε1 > 0 is a constant, Then, according to the boundedness of f l , substituting the update rate (10) of the adaptive parameter into equation (17) gives:

[0103]

[0104] Before further simplifying , the following important inequality is given first:

[0105]

[0106] Similarly, it can be obtained that:

[0107]

[0108] Thus, substituting inequalities (19) and (20) into equation (18) gives:

[0109]

[0110] where the parameters are: By reasonably designing the parameters, θ1 > 0 can be made.

[0111] S3. Design a distributed robust formation controller based on neural network fitting as follows:

[0112] S31. Use a neural network to fit the composite uncertainty. First, design the following auxiliary variables for spacecraft i:

[0113]

[0114] where \(k p , k d > 0\) is a constant. Substitute the auxiliary variables into the dynamic equation (7) of spacecraft i to obtain the following new dynamic equation:

[0115]

[0116] Let \(D i = k d B i . Considering that the dynamic model of the spacecraft is difficult to be accurately modeled and the external disturbances received by the spacecraft, the present invention uses radial basis function neural networks (RBFNNs) to fit the composite uncertainty \(r i \) to obtain the following equation:

[0117] r i = W i T h(s i ) + δ i , (25)

[0118] where \(W i \in R p×6 \) represents the ideal neural network weight, \(\delta i \in R 6×1 \) represents the neural network fitting residual, and there is an unknown upper bound \(\delta mi \); the radial basis function \(h(s i )\in R p×1 \) is designed in the form of a Gaussian function:

[0119]

[0120] where \(\mu k \in R 6×1 , c k \in R, k\in\{1, 2,..., p\}\).

[0121] Then, the estimated value of the composite uncertainty is obtained:

[0122]

[0123] where represents the estimated value of the ideal weight of the neural network, Denote the upper bound δ of the neural network fitting residual mi as the estimated value; Denote an adaptive factor, satisfying Meanwhile, their update rates are designed in the following form:

[0124]

[0125] where γ2, γ3, γ4 > 0 are constants.

[0126] S32. Design a distributed robust formation controller based on neural network fitting; First, define the consensus formation error of spacecraft i:

[0127]

[0128] Considering the discontinuous communication between spacecraft, rewrite the above formula in the following form:

[0129]

[0130] where denotes the information broadcast by spacecraft j to its neighbor spacecraft for the k_j-th time, and define the measurement error It should be noted that Then write Equation (30) in a compact form:

[0131]

[0132] where

[0133] Based on the above analysis, design the following controller for spacecraft i:

[0134]

[0135] where k s > 0;

[0136] S33. Considering the distributed robust formation controller based on neural network estimation designed in steps S31 and S32, the following Lyapunov function can be constructed for the stability analysis of the control scheme designed in the present invention in step S52:

[0137]

[0138] where denotes the neural network weight estimation error, denotes the neural network fitting residual upper bound estimation error, denotes the tracking error, p i > 0 ∈ R; Obviously, the Lyapunov function V2 is positive definite.

[0139] Taking the derivative with respect to V2 and substituting Eqs. (23), (27), (28) and (32) into it, we can obtain:

[0140]

[0141] For the Euler-Lagrange system, we have Meanwhile, substituting Eqs. (10), (22), (25) and (31) into Eq. (34), we can obtain:

[0142]

[0143] where ∈1, ∈2 > 0, By reasonably designing the parameters, we can make the parameters θ3, θ5 > 0.

[0144] S4. Construct an inter-vehicle communication mechanism based on a static event-triggering mechanism; specifically as follows:

[0145] S41. In order to reduce the communication times between adjacent space vehicles, the present invention adopts an inter-vehicle communication mechanism based on static event-triggering to adjust the communication frequency between follower space vehicles; for space vehicle i, the static event-triggering mechanism is designed in the following form:

[0146]

[0147] The static event-triggering function therein is designed in the following form:

[0148]

[0149] where κ ki , η ki > 0; meanwhile, according to the previous design, reasonably designing the parameters can make θ0 - θ6 > 0.

[0150] S5. Design a neural adaptive robust formation control scheme based on S2 - S4 to complete the formation control of multiple space vehicles; the specific content is as follows:

[0151] S51. Design a neural adaptive robust formation control scheme based on S2 - S4. The neural adaptive robust formation control scheme designed by the present invention mainly consists of three parts, which are respectively: 1. An adaptive distributed state observer based on local interaction (10); 2. A distributed robust formation controller based on neural network fitting (32); 3. An inter-vehicle communication mechanism based on a static event-triggering mechanism (36).

[0152] S52. Based on the designed neural adaptive robust formation control scheme, the formation control of multiple space vehicles can be realized.

[0153] Consider the Lyapunov function \(V = V_1+V_2\) of the entire closed-loop system. Combining equations (21) and (35), the derivative of \(V\) with respect to time is:

[0154]

[0155] According to the event-triggering condition, for the space vehicle \(i\) in the to this time period, there is \(f\) ki ≥ 0. Then equation (38) can be simplified to:

[0156]

[0157] Then integrating both sides of equation (39) gives:

[0158]

[0159] It can be seen from equation (40) that the Lyapunov function \(V\) is bounded. Then, combining equations (15) and (33), we know that \(\varphi\), s i , are all bounded. Combining equations (10) and (14) and the boundedness of \(\varphi\), , we can obtain Furthermore, we can get Also considering the boundedness, we can get \(\xi\) i \(\in L\) ∞ . Combining with equation (22), we can get Furthermore, we get Considering the boundedness, we can obtain Then, combining equations (24) and (32), we can get \(u\) i , \(r\) i \(\in L\) ∞ . Further, combining with equation (23), we can get Summarizing the above results, we can obtain

[0160] Then, it can be intuitively seen from equation (40) that By using Babarlet’s Lemma, when \(t\rightarrow\infty\), we have \(\varphi\), s i , Combining with equation (14), since so we have when \(t\rightarrow\infty\). Combining with equation (22), since so we have When \(t\rightarrow\infty\). Based on the above analysis, we can obtain Therefore, the designed scheme can achieve the formation control of multiple space vehicles.

[0161] S53. The designed inter - vehicle communication mechanism based on the static event - trigger mechanism will not exhibit the Zeno phenomenon.

[0162] First, for Taking the derivative with respect to time, we can obtain the following equation:

[0163]

[0164] Because the stability proof of the closed - loop system has been completed in step S52, we can obtain s ei ,

[0165] are all bounded. Therefore also has an upper bound, and thus we have:

[0166]

[0167] where \(Z\) ki > 0. In the time domain Integrating both sides of equation (42) gives:

[0168]

[0169] Because at time, we have Then, combining with equation (43), we can obtain that at time, the following inequality holds:

[0170]

[0171] Furthermore, combining the designed event - trigger mechanism (36) and event - trigger function (37), at time, we have:

[0172]

[0173] Then, by combining equation (44) and equation (45), we can obtain:

[0174]

[0175] From equation (46), it can be seen that the value of \(T\) is always positive. Therefore, the designed static event - trigger mechanism will not exhibit the Zeno phenomenon.

[0176] S54. Deploy the robust formation control scheme based on neural adaptation proposed in S51 to a multi-spacecraft system with a leader-follower structure to achieve formation control of multi-spacecraft.

[0177] In this embodiment, four follower spacecraft are considered. The control objective is that, based on the direct or indirect estimation of the leader's information, the follower spacecraft can form a tetrahedral formation through discontinuous local interactions and track the leader's trajectory, as Figure 2 shown. Moreover, the attitudes of all follower spacecraft must be consistent with that of the leader. However, only the first and second spacecraft can directly obtain the leader's information, and the remaining follower spacecraft can only communicate and interact with their neighbors. At the same time, the parameters of each follower spacecraft are m = 50 kg, J = diag(27, 25, 40) kg·m 2 .

[0178] The initial position of the leader is at the reference point. Assume that the orbital eccentricity of the reference point is 0, the right ascension of the ascending node is 45 degrees, and the orbital radius is R c = 6678140 m. Define ω l = 0.002 rad / s. The state of the leader is:

[0179] ζ l = [0.2cos(ω l t), 0.2cos(ω l t), 0.2sin(ω l t), 50cos(ω l t), 50sin(ω l t), 50sin(ω l t)] T

[0180] The initial values of the position and velocity of the follower spacecraft are:

[0181] ρ1(0) = [-20, -25, -25] T , ρ2(0) = [-20, -30, 95] T , ρ3(0) = [20, -35, 20] T

[0182] ρ4(0) = [-40, -35, 20] T (m), v i = [0, 0, 0] T (m / s)

[0183] The initial values of the attitude and angular velocity of the follower spacecraft are:

[0184] σ1(0) = [0.25, 0.01, 0.08] T , σ2(0) = [0.05, 0.1, 0.1] T , σ3(0) = [0.03, 0.36, 0.04] T

[0185] σ1(0) = [0.3, 0.01, 0.0015] T , ω i = [0, 0, 0] T (rad / s)

[0186] The deviation vector of the follower spacecraft from the desired state of the virtual leader is:

[0187] d l1 = [0, 0, 0, 0, -75, 0] T , d 2l = [0, 0, 0, 25, 0, 35] T

[0188] d l3 = [0, 0, 0, 25, 0, -35] T , d l4 = [0, 0, 0, -50, 0, 0] T

[0189] Considering the control input saturation, the maximum control force of all spacecraft is 50 N, and the maximum control torque is 0.05 Nm.

[0190] Reasonably design the parameters so that the designed multi - spacecraft leader - follower formation control method based on neural adaptation can complete the robust formation control task of multi - spacecraft. The parameters are shown in Table 1:

[0191] Table 1 Numerical simulation parameters

[0192]

[0193]

[0194] For the neural network, the variance of the Gaussian function is c k = 1, select p = 25 neural network nodes, and the centers of the basis functions are uniformly selected in μ k = [-1, 1]. At the same time, consider the external interference received by the spacecraft as:

[0195] f di = 10 -3 [3cos(0.5t) + 10sin(0.1t) + 15sin(0.05t)]·13(N)

[0196] τ di = 10 -4 [sin(0.1t) + cos(0.01t) + sin(0.05t)]·13 (Nm)

[0197] The example results are given in Figures 3 to 11 the following. Figure 3 and Figure 4 respectively show the attitude observation error curve and position observation error curve of the follower spacecraft with respect to the leader. Figure 5 and Figure 6 respectively show the attitude tracking error curve and position tracking error curve of the follower spacecraft. Figure 7 and Figure 8 respectively show the attitude control input variation curve and position control input variation curve of the follower spacecraft. Figure 9 and Figure 10 respectively show the variation curve of the F - norm of the neural network weight estimation value of the follower spacecraft and the motion trajectory of the follower spacecraft in the orbital coordinate system. Figure 11 shows the communication time intervals of all follower spacecraft under the inter - vehicle communication mechanism based on the event - triggered mechanism. It can be seen that in most time periods, the communication intervals between spacecraft are greater than 0.2 s, and the inter - vehicle communication intervals are greatly reduced compared with the time - triggered communication mechanism (communication period is 0.05 s).

[0198] Those of ordinary skill in the art will realize that the embodiments described herein are for helping the reader understand the principles of the present invention. It should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. Those of ordinary skill in the art can make various other specific deformations and combinations without departing from the essence of the present invention based on the technical revelations disclosed in the present invention, and these deformations and combinations are still within the protection scope of the present invention.

Claims

1. A multi-spacecraft leader-follower formation control method based on neural adaptation, characterized in that, The steps are as follows: S1. Establish a multi-spacecraft system with a leader-follower structure. The specific steps are as follows: S11. Construct a six-degree-of-freedom dynamic model of the follower spacecraft: Use the modified Rodriguez parameters and the CW equations to describe the attitude change of the spacecraft in the geocentric inertial coordinate system and the relative motion in the orbital coordinate system respectively. S12. Construct the dynamic model of the leader. The leader dynamic model is described in the following form: where ξ l , represent the state and the rate of change of state of the leader l, respectively, denotes the control input of the leader, which is unknown and bounded for all follower spacecraft, i.e., Let ; S13. Determine the communication constraints between space vehicles: The communication relationship between follower space vehicles is described by an undirected and connected topology which is The adjacency matrix of this topology is where a ij ≥ 0; a ij > 0 means that follower space vehicles i and j can communicate with each other, and a ij = 0 means that there is no communication between follower space vehicles i and j; the degree matrix is where The topology The Laplacian matrix of Using a directed graph to describe the communication relationship between the leader and follower spacecraft, the directed graph has an adjacency matrix of where b i ≥ 0; b i > 0 indicates that follower spacecraft i can directly obtain real-time information of the leader. Otherwise, b i = 0; at least one of the follower spacecraft can obtain information of the leader, and the communication topology among the multi-spacecraft systems is connected; define the matrix Considering the connectivity of the topology, the matrix is a positive definite matrix; Usage respectively represent the element in the i-th row and j-th column of the matrix ; S14. Define the formation control problem of the multi-spacecraft system. The formation control needs to satisfy the following conditions: where d li represents the desired state deviation vector of the follower spacecraft i relative to the leader; S2. Design an adaptive state observer based on local interaction. Design the following adaptive distributed observer for the follower spacecraft i to reconstruct the information of the leader: where β1, β2, γ1 > 0, representing gains; respectively represent the estimated value of the information ξ of the leader by the follower space vehicle i l , ; represents the estimated value of the upper bound of the control input f of the leader by the follower space vehicle i l ; the variable is designed in the following form: ​ Among them respectively represent the information broadcast by the space vehicle i to its neighboring space vehicles at the kith time; Measurement error It is expressed as follows: S3. Design a distributed robust formation controller based on neural network approximation. Specifically as follows: S31. Use neural network approximation to fit the composite uncertainty. First, design the following auxiliary variable for the follower spacecraft i: where k p ,k d >0; Substitute the auxiliary variables into the dynamic equation of aircraft i, and use radial basis neural network to fit the uncertain terms in the composite dynamic equation; S32. Design a distributed robust formation controller based on neural network approximation. First, define the consensus formation error of spacecraft i: Considering the discontinuous communication between spacecraft, rewrite the above formula in the following form: wherein represents the information broadcast by the j-th space vehicle to its neighboring space vehicles for the k_j-th time, and defines the measurement error It should be noted that Then it is written in a compact form as: Among them Based on the above analysis, design the following controller for spacecraft i: where k s > 0; S4. Construct an inter-spacecraft communication mechanism based on the static event-triggering mechanism. Specifically as follows: S41. Adopt an inter-spacecraft communication mechanism based on static event-triggering to adjust the communication frequency between spacecraft. For spacecraft i, the static event-triggering mechanism is designed in the following form: The static event-triggering function is designed in the following form: where κ ki , η ki > 0; S5. Based on S2 - S4, design a neural adaptive robust formation control scheme to complete the formation control of multi-spacecraft: Deploy the neural adaptive robust formation control scheme to the multi-spacecraft system with a leader-follower structure to achieve the formation control of multi-spacecraft.

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