Simplified attitude distributed cooperative control method for directed communication leader-follower spacecraft formation

Abstract

The application provides a simplified attitude distribution cooperative control method for a directed communication leader-follower spacecraft formation, which comprises the following steps: firstly, describing the simplified attitude motion of the spacecraft in a unit celestial coordinate system, and converting the simplified attitude motion into a translation of the terminal point of a pointing vector on a unit celestial sphere; secondly, designing a distributed cooperative virtual control quantity under a directed communication topology according to the consistency theory, so as to ensure that the pointing vector of the follower spacecraft converges to a fixed pointing direction in the inertial space determined by the pointing vector of the leader spacecraft, and then calculating an ideal angular velocity vector of the follower spacecraft according to the virtual control quantity; thirdly, taking the difference between the angular velocity vector of the follower spacecraft and the ideal angular velocity vector as a sliding mode variable, and designing a sliding mode surface; and finally, designing a distributed cooperative control law for the follower spacecraft based on a super-twisting second-order sliding mode control method. The application can ensure that the pointing vector of the follower spacecraft in the directed communication formation converges to the pointing vector of the leader spacecraft fixed in the inertial space under a disturbance environment, and reduces the communication amount of the formation.

Description

Technical Field

[0001] This invention belongs to the field of spacecraft attitude control technology, specifically involving distributed cooperative control of simplified attitude of multiple spacecraft formations under directed communication. Background Technology

[0002] Spacecraft swarms can distribute the functions of a traditional large spacecraft across several smaller spacecraft; these smaller spacecraft can achieve the functions of a large spacecraft through mutual cooperation, and offer advantages such as low cost, high adaptability, and high reliability. Therefore, spacecraft swarms represent an important direction for future spacecraft development.

[0003] Attitude consistency of spacecraft formations is crucial for technologies such as distributed astronomical observations and distributed synthetic aperture radar. In some cases, it is only necessary to point the spacecraft's aiming vector in a specific direction in inertial space without requiring three-axis control. Rotational motion around the aiming vector does not affect the spacecraft's functionality. This motion, which only considers the aiming vector's direction in inertial space, is called simplified attitude motion, and the direction of the aiming vector represents the spacecraft's simplified attitude. Simplified attitude control stabilizes or tracks only two degrees of freedom of the spacecraft's attitude, saving fuel and avoiding attitude description singularities and attitude variable redundancy.

[0004] As spacecraft formations grow larger, communication between spacecraft consumes significant hardware resources and can even lead to communication congestion, causing formation failure. Directed communication, however, can drastically reduce the amount of communication between spacecraft. Under the same communication connection, the average communication throughput of directed communication is only 50% of that of undirected communication. Therefore, distributed control protocols based on directed communication are the foundation for realizing large-scale spacecraft formations in the future.

[0005] However, since the rotational motion around the aiming vector in simplified attitude control is not fully controlled, it is impossible to directly use the spacecraft aiming vector error to obtain a distributed cooperative control protocol for simplified attitude control of multiple spacecraft formations with directed communication topology based on traditional consistency theory. Summary of the Invention

[0006] To address the limitations of traditional consistency theory in the distributed cooperative control of simplified attitudes in multi-spacecraft formations under directed communication topologies, this invention proposes a distributed cooperative control method for simplified attitudes of guided and following spacecraft formations. The simplified attitude motion of the spacecraft is described in a unit celestial coordinate system. Combining consistency theory and the super-twisted second-order sliding mode control method, a distributed cooperative control protocol for simplified attitudes of guided and following multi-spacecraft formations under perturbation conditions is obtained. This protocol ensures that the aiming vector of the following spacecraft remains consistent with the fixed aiming vector of the guide spacecraft in inertial space under directed communication.

[0007] The technical solution of this invention is as follows:

[0008] A simplified attitude distribution cooperative control method for directional communication-guided navigation and following spacecraft formation includes the following steps:

[0009] Step 1: Describe the simplified attitude motion of the spacecraft in a unit celestial coordinate system, transforming the simplified attitude motion into translational motion of the target vector endpoint on a unit celestial surface; the spacecraft formation includes one lead spacecraft and n follower spacecraft, n>0; the communication topology between all formation spacecraft is shown in the figure. Description, and There exists a directed spanning tree;

[0010] Step 2: Based on the consistency theory, design a distributed cooperative virtual control variable to simplify the attitude of the spacecraft formation under the directed communication topology;

[0011] Step 3: Calculate the ideal angular velocity vector of the spacecraft based on the distributed cooperative virtual control variables designed in Step 2;

[0012] Step 4: Construct a sliding mode surface by using the difference between the angular velocity vector of each following spacecraft and the ideal angular velocity vector as the sliding mode variable;

[0013] Step 5: Based on the sliding surface constructed in Step 4, design the distributed cooperative control law for each following spacecraft using the super-twisted sliding mode control method.

[0014] Furthermore, the translational formula for the endpoint of the aiming vector obtained in step 1 is:

[0015]

[0016] Where p i Let η be the unit aiming vector of the i-th spacecraft. i The endpoint P i In the unit celestial coordinate system The position below, p i =[θ i ,φ i ] T θ i For longitude, φ i Latitude; E i The expression is

[0017]

[0018] Let P be the point i velocity vector in coordinate system The representation in:

[0019]

[0020] Where R FiI For coordinate system to inertial coordinate system The rotation matrix, Let the aiming vector of the i-th spacecraft be in the inertial coordinate system. The representation in the text, For volume coordinate system to inertial coordinate system The rotation matrix, The measured spacecraft relative to the inertial coordinate system The angular velocity vector in the body coordinate system The representation in;

[0021] The body coordinate system of the i-th spacecraft origin Located at the center of mass of the i-th spacecraft, The axis coincides with the principal axis of maximum inertia of the i-th spacecraft. The axis coincides with the minimum inertia principal axis of the i-th spacecraft. shaft and axis, The axes form a right-handed coordinate system;

[0022] Inertial coordinate system A right-handed coordinate system with its origin and coordinate axes fixed in inertial space;

[0023] Unit celestial coordinate system Origin and inertial coordinate system The origins coincide, unit celestial coordinate system The zero-latitude surface and the inertial coordinate system z I =0 coincides with the plane, when z I When the latitude is greater than 0, the zero longitude line is located in the inertial coordinate system. of y I =0,x I In the plane ≥0, along z I Longitude increases clockwise when viewed along the axis.

[0024] The coordinate system corresponding to the i-th spacecraft The origin is located at η i The endpoint P i , Axis and η i Parallel and pointing in the same direction, Axis and unit celestial coordinate system The meridians lie in the same plane and are tangent to each other, pointing in the direction of increasing latitude. shaft and axis, The axes form a right-handed coordinate system.

[0025] Furthermore, the distributed collaborative virtual control quantity designed in step 2 is... Make the aiming vector of the i-th spacecraft coincide with the aiming vector of the lead spacecraft, where:

[0026]

[0027] In the formula, k1>0 is a constant, p j In the body coordinate system of the j-th spacecraft The unit aiming vector η fixed to the spacecraft j The endpoint P j In the unit celestial coordinate system The position below; p0 is the body coordinate system of the pilot spacecraft. The endpoint P0 of the unit aiming vector η0, which is fixed to the spacecraft, lies in the unit celestial coordinate system. The position below; a ij Communication topology diagram between following spacecraft The adjacency matrix A = [a ij The element in ] is a when the i-th follower spacecraft can obtain information about the j-th follower spacecraft through communication. ij =1, otherwise a ij =0; a i0 The communication topology between the lead spacecraft and the follower spacecraft is represented by matrix H = diag([a 10 ,a 20 ,…,a n0 In the diagonal elements of ]), when the i-th following spacecraft can obtain information from the lead spacecraft, then a i0 =1, otherwise a i0 =0.

[0028] Furthermore, in step 3, the ideal angular velocity vector of the i-th following spacecraft is:

[0029]

[0030] in for volume coordinate system Relative to coordinate system The angular velocity vector in the coordinate system The following represents β i For any real number; coordinate system Relative to the inertial coordinate system Ideal angular velocity vector From the formula

[0031]

[0032] express, For coordinate system to inertial coordinate system rotation matrix; According to the formula

[0033]

[0034] This indicates that the function of Ξ(·) in the formula is defined as

[0035]

[0036] matrix for

[0037]

[0038] Furthermore, in step 4, the sliding surface is defined as... J i Let ω be the inertia matrix of the i-th following spacecraft. i Let be the angular velocity vector of the i-th following spacecraft.

[0039] Furthermore, the distributed cooperative control law for each accompanying spacecraft is as follows:

[0040]

[0041]

[0042] Where τ i Let be the torque acting on the i-th following spacecraft; λ > 0, α > 0 are design parameters, and ⊙ represents the element-wise multiplication of two vectors; s i Write it in fractional form as s i =[s i,1 ,s i,2 ,s i,3 ] T , and sgn(s i The corresponding expressions are as follows: sgn(s i )=[sgn(s i,1 ),sgn(s i,2 ),sgn(s i,3 )] T .

[0043] Beneficial effects

[0044] This invention provides a distributed cooperative control method for simplifying the attitude of a navigator-follower spacecraft formation under directed communication. In a disturbed environment, when the directed communication topology of the navigator-follower spacecraft formation has a directed spanning tree, the distributed cooperative control method provided by this invention can achieve convergence of the aiming vector of the follower spacecraft to the aiming vector of the navigator spacecraft, which is fixed in inertial space, while reducing the amount of communication compared to undirected communication formations.

[0045] Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0046] The above and / or additional aspects and advantages of the present invention will become apparent and readily understood from the description of the embodiments taken in conjunction with the following drawings, in which:

[0047] Figure 1 Design Steps for a Simplified Distributed Cooperative Attitude Control Method for Directed Communication-Based Navigation and Spacecraft Formation Following

[0048] Figure 2 Inertial coordinate system Unit celestial coordinate system and coordinate system Schematic diagram

[0049] Figure 3 Communication topology between spacecraft in an example of the present invention

[0050] Figure 4 The initial state and aiming vector of the spacecraft in the examples of this invention

[0051] Figure 5 In an example of this invention, the trajectory of the target vector endpoint following the spacecraft on a unit celestial sphere is...

[0052] Figure 6 In an example of this invention, the trajectory of the target vector endpoint following the spacecraft lies in the longitude-latitude plane.

[0053] Figure 7 In an example of this invention, the 2-norm variation curve of the target vector endpoint position error between the follower spacecraft and the lead spacecraft is shown.

[0054] Figure 8 In an example of this invention, the control input variation curve follows that of the spacecraft. Detailed Implementation

[0055] The embodiments of the present invention are described in detail below. These embodiments are exemplary and intended to explain the present invention, and should not be construed as limiting the present invention.

[0056] This embodiment proposes a directional communication-based guide-follower spacecraft formation simplified attitude distribution cooperative control method, which can achieve convergence of the aiming vector of the follower spacecraft to the aiming vector of the guide spacecraft fixed in inertial space under disturbed environments, while reducing the amount of communication compared to undirected communication formation.

[0057] First, let's explain the relevant explanations regarding leading and following spacecraft formations:

[0058] The formation consists of n+1 spacecraft, with the lead spacecraft numbered 0 and the follower spacecraft numbered 1, 2, ..., n (n > 0). The communication topology between the follower spacecraft is shown in the diagram. The description is that its adjacency matrix is ​​A = [a ij When the i-th following spacecraft can obtain information about the j-th following spacecraft through communication, the j-th following spacecraft is called the neighbor of the i-th following spacecraft, and a ij =1; otherwise a ij =0, and the diagonal element a ii =0. The lead spacecraft has no neighbors, and the communication topology between the lead spacecraft and the follower spacecraft is represented by matrix H = diag([a 10 ,a 20 ,…,a n0 The description states that if the i-th following spacecraft can obtain information from the lead spacecraft, a i0 =1; otherwise a i0 =0. The communication topology between all formation spacecraft is shown in the diagram. Description, and There exists a directed spanning tree.

[0059] Let the body coordinate system of the i-th spacecraft be ? Its origin Located at the center of mass of the i-th spacecraft, The axis coincides with the principal axis of maximum inertia of the i-th spacecraft. The axis coincides with the minimum inertia principal axis of the i-th spacecraft. shaft and axis, The axes form a right-handed coordinate system. Let the inertial coordinate system be... Its origin and coordinate axes are fixed in inertial space, forming a right-handed coordinate system. (Unit celestial coordinate system) Origin and inertial coordinate system The origins coincide, unit celestial coordinate system The zero-latitude surface and the inertial coordinate system z I =0 coincides with the plane, when z I When the latitude is greater than 0, the zero longitude line is located in the inertial coordinate system. of y I=0,x I In the plane ≥0, along z I The longitude increases clockwise when viewed along the axis.

[0060] If we consider the spacecraft as a rigid body, then the attitude of the i-th spacecraft can be determined by the body coordinate system fixed to the spacecraft. to inertial coordinate system rotation matrix R i In other words, matrix R i Belongs to the real special orthogonal group The rotational kinematics equation of the i-th spacecraft can be expressed as follows:

[0061]

[0062] In the formula: Let the i-th spacecraft be relative to the inertial coordinate system. The angular velocity vector in the coordinate system In the representation, the function (·) × Defined as

[0063]

[0064] The attitude dynamics equation of the i-th spacecraft is:

[0065]

[0066] In the formula: J i Let τ be the inertia matrix of the i-th spacecraft; i d represents the torque acting on the i-th spacecraft; i Let be the disturbance torque acting on the i-th spacecraft.

[0067] Based on the above explanation of the guidance and following of spacecraft formations, the simplified attitude distribution cooperative control method in this embodiment will be described in detail below:

[0068] Step 1: Describe the simplified attitude motion of the spacecraft in the unit celestial coordinate system, and transform the simplified attitude motion into translation of the endpoint of the aiming vector on the unit celestial surface.

[0069] Using η i Represents the body coordinate system of the i-th spacecraft The unit vector fixed to the spacecraft needs to be η when performing simplified attitude control. i A specific direction pointing into inertial space is called η. i Let η be the unit aiming vector for the i-th spacecraft. i In the inertial coordinate system Below is

[0070]

[0071] Differentiating the above equation, we obtain the simplified kinematic equations for the attitude.

[0072]

[0073] To facilitate the design of distributed collaborative virtual control variables, a coordinate system is defined. The origin is located at η i The End Pi ; Axis and η i Parallel and pointing in the same direction; Axis and unit celestial coordinate system The meridians lie in the same plane and are tangent to each other, pointing in the direction of increasing latitude; shaft and axis, The axes form a right-handed coordinate system. Let η... i In the inertial coordinate system Below, then η i The endpoint P i In the unit celestial coordinate system The position below is represented as p i =[θ i ,φ i ] T θ i For longitude, φ i For latitude. For p i Find the point P by derivation i The kinematic equations are

[0074]

[0075] In the formula: Let P be the point i velocity vector in coordinate system The representation in, i.e.

[0076]

[0077] in For coordinate system To standard system The rotation matrix, Let P be the point i velocity vector in coordinate system The representation in the text, Let the i-th spacecraft be relative to the coordinate system. The angular velocity vector in the coordinate system The representation in the text, Let the unit aiming vector of the i-th spacecraft be in the coordinate system. The representation in the text, For coordinate system To standard system The rotation matrix, The measured spacecraft relative to the coordinate system The angular velocity vector in the coordinate system In the representation, E i The expression is

[0078]

[0079] Step 2: Based on the consistency theory, design a distributed cooperative virtual control quantity for simplifying the attitude of the spacecraft formation under the directed communication topology.

[0080] Based on the consistency theory, design Virtual control quantity Make the aiming vector of the i-th spacecraft coincide with the aiming vector of the lead spacecraft:

[0081]

[0082] In the formula: k1>0 is a constant.

[0083] The closed-loop system under the action of the aforementioned virtual control variable is asymptotically convergent, as proven below:

[0084] Substitute the virtual control quantity into point P i The kinematic equations are obtained.

[0085]

[0086] Typically, the lead spacecraft points the aiming vector in a specific direction in inertial space for observation. In this case, p0 is constant, therefore the above equation is equivalent to...

[0087]

[0088] In the formula: Indicates the Kronecker product. It is the identity matrix. To follow the spacecraft communication topology diagram The Laplace matrix is ​​defined as follows:

[0089]

[0090] Due to the communication topology diagram There exists a directed spanning tree, therefore The eigenvalues ​​of have positive real parts. Clearly,

[0091] Step 3: Calculate the ideal angular velocity vector of the spacecraft based on the distributed cooperative virtual control quantity designed in Step 2.

[0092] because Therefore, a coordinate system can be obtained. Relative to the inertial coordinate system The ideal angular velocity vector is

[0093]

[0094] In the formula: For coordinate system to inertial coordinate system The rotation matrix.

[0095] According to the coordinate system and inertial coordinate system The relationship between them can be obtained

[0096]

[0097] It can be represented as

[0098]

[0099] The function of Ξ(·) in the formula is defined as

[0100]

[0101] matrix for

[0102]

[0103] therefore,

[0104]

[0105] Because around η i The angular velocity of rotation will not affect η. i The direction of the coordinates has an impact, so the volume coordinate system Relative to the inertial coordinate system The ideal angular velocity vector is

[0106]

[0107] In the formula: for volume coordinate system Relative to coordinate system The angular velocity vector in the coordinate system The following indicates, It can be freely selected without affecting the direction of the aiming vector. Thus, it is controlled by virtual variables. The ideal angular velocity vector following the spacecraft was obtained.

[0108] Step 4: Use the difference between the angular velocity vector of each following spacecraft and the ideal angular velocity vector as the sliding mode variable to construct the sliding mode surface.

[0109] When the angular velocity vectors of all following spacecraft are equal to their respective ideal angular velocity vectors Therefore, the angular velocity vector ω that follows the spacecraft can be... i With the corresponding ideal angular velocity vector The difference is used as a sliding mode variable to design the sliding surface. The sliding surface is defined as...

[0110]

[0111] Step 5: Based on the sliding surface constructed in step 4, design the distributed cooperative control protocol, i.e., the control law, for each following spacecraft using the super-twisted sliding mode control method.

[0112] For s i Differentiate, and we get

[0113]

[0114] Based on the super-twisted second-order sliding mode control theory, the reaching law is designed as follows:

[0115]

[0116]

[0117] In the formula: λ>0, α>0 are design parameters, ⊙ represents the element-wise multiplication of two vectors; s i Write it in fractional form as s i =[s i,1 ,s i,2 ,s i,3 ] T ,but and sgn(s i The corresponding expressions are as follows: sgn(s i )=[sgn(s i,1 ),sgn(s i,2 ),sgn(s i,3 )] T Therefore, the distributed cooperative control law following the spacecraft can be obtained as follows:

[0118]

[0119]

[0120] The effectiveness of the above method is verified through numerical simulation.

[0121] Consider a simplified attitude distributed cooperative control problem involving a formation of five spacecraft as the lead and follower. The communication topology between the spacecraft is as follows: Figure 3 As shown, this communication topology is a directed graph with a directed spanning tree. (Communication topology diagram of the spacecraft) The adjacency matrix is

[0122]

[0123] The communication topology matrix H between the follower spacecraft and the lead spacecraft is

[0124]

[0125] The inertia matrix of the spacecraft is

[0126]

[0127] The initial attitude conditions and aiming vectors of each follower spacecraft are shown in [link to documentation]. Figure 4 .

[0128] The aiming vector of the navigator spacecraft in the inertial coordinate system Below is The control parameters are set as k1 = 0.2, α = 0.05, and λ = 1. External disturbance set

[0129]

[0130] The simulation results of this example are as follows: Figures 5-8 As shown. From Figure 5 and Figure 6 As can be seen, even when only follower spacecraft 3 can obtain information from the lead spacecraft, the aiming vectors of all follower spacecraft eventually converge to the same direction as the aiming vector of the lead spacecraft. Figure 7 The 2-norm variation curves of the aiming vector endpoint position errors of all follower and lead spacecraft are presented, and the final convergence error is less than 2 × 10⁻⁶. -6 This verifies the high precision of the present invention. Figure 8 The curve of the control torque variation following the spacecraft is presented. The control torque variation is smooth and there is no chattering phenomenon.

[0131] Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those skilled in the art can make changes, modifications, substitutions and variations to the above embodiments within the scope of the present invention without departing from the principles and spirit of the present invention.

Claims

1. A simplified attitude distribution cooperative control method for directional communication-guided navigation and following spacecraft formations, characterized in that: The method comprises the following steps: Step 1: describe the simplified attitude motion of a spacecraft in a unit celestial coordinate system, and convert the simplified attitude motion into a translation of the terminal point of a pointing vector on a unit celestial sphere; the spacecraft formation includes a lead spacecraft and n follower spacecrafts, n>0; the communication between all the spacecrafts in the formation is described by a topological graph is described, and there is a directed spanning tree; The translational formula of the aiming vector end point is: in For the first Unit aiming vector of a spacecraft The End In the unit celestial coordinate system The lower position, , Longitude Latitude; The expression is for a point the velocity vector of a point in a coordinate system is represented by wherein is the rotation matrix from the body coordinate system to the inertial coordinate system is the rotation matrix from the body coordinate system to the inertial coordinate system is the representation of the pointing vector of the i-th spacecraft in the inertial coordinate system is the rotation matrix from the body coordinate system to the inertial coordinate system is the rotation matrix from the body coordinate system to the inertial coordinate system is the representation of the measured angular velocity vector of the i-th spacecraft in the body coordinate system with respect to the inertial coordinate system is the rotation matrix from the body coordinate system to the inertial coordinate system wherein the body coordinate system of the first spacecraft origin is located at the center of mass of the first spacecraft, the z-axis coincides with the maximum moment of inertia principal axis of the first spacecraft, the y-axis coincides with the minimum moment of inertia principal axis of the first spacecraft, the x-axis is perpendicular to the xy-plane, the x-axis, the y-axis, and the z-axis form a right-handed coordinate system; Inertial coordinate system is a right-handed coordinate system with its origin at the center of mass of the vehicle and its axes fixed in inertial space; Unit celestial coordinate system Origin and inertial coordinate system The origins coincide, unit celestial coordinate system The zero-latitude surface and the inertial coordinate system of When the planes coincide, Latitude is greater than 0; the zero longitude line is located in the inertial coordinate system. of In the plane, along Longitude increases clockwise when viewed along the axis. The first coordinate system corresponds to the first spacecraft The second coordinate system corresponds to the second spacecraft The origin of the coordinate system is located at The terminal point of the coordinate system , The axis is parallel to the axis and points in the same direction The axis is parallel to the axis and points in the same direction The axis is parallel to the axis and points in the same direction The axis is parallel to the axis and points in the same direction The axis is parallel to the axis and points in the same direction The axis is parallel to the axis and points in the same direction The axis is parallel to the axis and points in the same direction Step 2: According to the consistency theory, a distributed cooperative virtual control quantity for simplifying the formation attitude of the leader and follower spacecrafts in a directed communication topology is designed; Step 3: The ideal angular velocity vector of the follower spacecraft is calculated based on the distributed cooperative virtual control quantity designed in step 2; Step 4: The difference between the angular velocity vector of each follower spacecraft and the ideal angular velocity vector is taken as a sliding mode variable, and a sliding mode surface is constructed; Step 5: Based on the sliding mode surface constructed in step 4, a distributed cooperative control law of each follower spacecraft is designed by using a super-twisting sliding mode control method.

2. The simplified attitude distributed cooperative control method for a formation of leader-follower spacecrafts with directional communication according to claim 1, wherein: The distributed cooperative virtual control variable designed in Step 2 is such that the pointing vector of the th spacecraft coincides with the pointing vector of the lead spacecraft, wherein: In the formula, It is a constant. In the first The spacecraft's body coordinate system Unit aiming vector fixed to the spacecraft The End In the unit celestial coordinate system The lower position; In the body coordinate system of the pilot spacecraft Unit aiming vector fixed to the spacecraft The End In the unit celestial coordinate system The lower position; Communication topology diagram between following spacecraft adjacency matrix The element in, when the first The following spacecraft can obtain the information via communication. When following information from a spacecraft ,otherwise ; A matrix is ​​used for the communication topology between the lead spacecraft and the follower spacecraft. The diagonal element in the middle, when the first A follower spacecraft can obtain information from the lead spacecraft, then ,otherwise .

3. The simplified attitude distributed cooperative control method for a formation of leader-follower spacecrafts with directional communication according to claim 2, characterized in that: In step 3, the ideal angular velocity vector of the second following spacecraft is: ​ where is the body coordinate system is the coordinate system is the angular velocity vector of the coordinate system in the coordinate system is an arbitrary real number; the coordinate system is the inertial coordinate system is the ideal angular velocity vector of the coordinate system is given by the formula denotes, is the rotation matrix from the body coordinate system to the inertial coordinate system ; according to the formula represents, in which is defined as matrix for 。 4. The simplified attitude distributed cooperative control method for a formation of leader-follower spacecrafts with directional communication according to claim 3, wherein: The sliding surface is defined in step 4 as where is the inertia matrix of the th follower spacecraft, is the angular velocity vector of the th follower spacecraft.

5. The simplified attitude distributed cooperative control method for a formation of leader-follower spacecrafts with directional communication according to claim 4, wherein: The distributed cooperative control law of each follower spacecraft is: wherein is a torque acting on the first spacecraft; , is a design parameter, denotes the multiplication of the corresponding elements of two vectors; written in component form as , and are respectively denoted as , .

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