A collision avoidance control method for complex-shaped spacecraft based on algebraic conditions

Abstract

A method for collision avoidance control of complex-shaped spacecraft based on algebraic conditions belongs to the field of spacecraft formation control. The invention solves the problem of collision when spacecraft perform close-range posture maneuvers in space. The method of the invention comprises the following steps: S1, establishing kinematic and dynamic equations of a tracking satellite relative to a space tumbling target spacecraft based on Lie group SE (3); S2, introducing a configuration error potential function to obtain a configuration error vector; S3, designing a collision avoidance mechanism, including collision avoidance between a tracking satellite and a target spacecraft and collision avoidance between tracking satellites, wherein the tracking satellite and the target spacecraft both construct a minimum envelope ellipsoid; S4, designing a continuously differentiable artificial potential function for the collision avoidance constraint; S5, designing a collision avoidance feedback control law in combination with S1-S4.

Description

Technical Field

[0001] The invention relates to a spacecraft formation control method considering collision avoidance, and belongs to the field of spacecraft formation control. Background Art

[0002] With the advancement of space technology, on-orbit servicing of high-value, failed spacecraft has become an indispensable technology. A key step in on-orbit servicing is to approach the target closely, observing and monitoring the target during this process to extract characteristic information for subsequent capture and repair missions. For example, damage and fault characteristics on the target spacecraft's surface can be used to identify structural features that can be captured by a robotic arm. Compared to traditional one-on-one on-orbit servicing, utilizing mass-produced small spacecraft for close-range on-orbit operations reduces production costs while improving mission execution efficiency and success rates.

[0003] When considering spacecraft collision avoidance, there are two approaches to modeling collision avoidance constraints based on the tracker satellite's geometry. One approach involves treating the tracker satellite as a point mass and modeling the target spacecraft's dimensions as accurately as possible. For example, safe path constraints can be established to meet safe docking requirements. Alternatively, a spherical collision avoidance envelope can be constructed for the target spacecraft to achieve a collision-free rendezvous. Some studies have used superquadratic surfaces to construct the target spacecraft's geometry envelope. This method can produce a non-convex geometry that more closely approximates the spacecraft's geometry. However, in this approach, the tracker satellite's attitude maneuvers could cause its appendages to collide with the target spacecraft. The other approach involves using a simple spherical envelope to describe the tracker satellite's geometry. For example, a composite shape envelope combining a sphere and an ellipsoid can be constructed for the target spacecraft, followed by trajectory planning based on the Gaussian pseudospectral method for collision avoidance. Alternatively, a composite shape envelope combining a sphere and an ellipsoid can be constructed for the target spacecraft, followed by a rolling horizon and sequential convex programming approach to solve the collision avoidance problem during close approach. Some scholars regard individual tracking stars as atoms from a molecular perspective, and regard the collision avoidance force between tracking stars as the force between isotropic molecules. Based on the Lanner-Jones potential, they design a distributed collision avoidance mechanism for spacecraft formations.

[0004] For tracking satellites carrying large sails or accessories, using an ellipsoidal envelope to describe the tracking satellite's dimensions can further reduce the conservatism of constraint modeling. However, there is no direct algorithm for calculating the distance between two ellipsoids. Existing research first converts the tracking satellite's ellipsoidal envelope into a unit sphere and then uses an eigenvalue algorithm to calculate the distance between the two ellipsoids in space. However, this algorithm ignores the ellipsoid's attitude motion.

[0005] In summary, existing collision avoidance constraint modeling methods often simplify the tracking satellite as a point mass or sphere, which is overly conservative. When considering the geometry of both the tracking satellite and the target spacecraft, existing optimization-based spacecraft collision avoidance strategies require significant computational resources, making them inefficient for engineering practice. Furthermore, existing spacecraft collision avoidance methods fail to consider the coupling between the spacecraft's geometry and relative attitude at the 3D level, which can lead to collisions during close-range maneuvers in space.

[0006] Therefore, in response to the above shortcomings, it is necessary to provide a collision avoidance technology that takes into account the geometric shapes of the tracking star and the target spacecraft at the same time, which can solve the spacecraft collision avoidance method that couples the relative posture and geometric shape in three-dimensional space, so as to meet the safety requirements of multiple spacecraft in close proximity when there are motion constraints. Summary of the Invention

[0007] In view of the above shortcomings, the present invention provides a collision avoidance control method for spacecraft with complex shapes based on algebraic conditions.

[0008] The present invention provides a method for collision avoidance control of a spacecraft with a complex shape based on algebraic conditions, the method comprising the following steps:

[0009] S1. Based on Lie group SE (3), the kinematic and dynamic equations of the tracking star relative to the space tumbling target spacecraft are established;

[0010] S2. Introduce the configuration error potential function to obtain the configuration error vector

[0011] S3. Designing a collision avoidance mechanism, including collision avoidance between the tracking satellite and the target spacecraft and collision avoidance between the tracking satellites, wherein the tracking satellite and the target spacecraft both construct a minimum envelope ellipsoid;

[0012] S4. Design a continuously differentiable artificial potential function φ for collision avoidance constraints pk ;

[0013] S5. Combine S1-S4 to design the collision avoidance feedback control law:

[0014]

[0015] in,

[0016] represents the control input of the kth tracking star, k=1,...,N, N is the number of tracking stars,

[0017] The potential function generates the control quantity for collision avoidance on the kth tracking star,

[0018]

[0019] Where: It means to find the gradient about (·), is the attitude error of the kth tracking satellite relative to the target spacecraft, ψ k represents the collision avoidance potential function, (·) ∨ Represents mapping, mapping (·) ∨ : Indicates that The matrix in is mapped to the isomorphic real space represents the position error of the kth tracking satellite relative to the target spacecraft, is the expected attitude of the kth tracking satellite relative to the target spacecraft, is the expected position of the kth tracking satellite relative to the target spacecraft,

[0020] is the basic feedback controller of the k-th tracking star that is not related to the potential function,

[0021]

[0022] Where: k f 、k l is the positive definite control gain matrix to be designed, and

[0023] Represents the coordinate system of the kth tracking star Relative to the target spacecraft body coordinate system The velocity vector, and Respectively Relative to The rotational speed and translational speed;

[0024] represents the gravitational effect on the k-th tracking star,

[0025] J k ,m k denote the moment of inertia and mass of the kth tracking star, respectively, and E3 denotes the third-order unit matrix;

[0026] Ad g is the adjoint operator of the element g on the Lie group SE(3), ad ξ is the adjoint operator of ξ

[0027]

[0028] express relatively speed, and Respectively Relative to The rotational speed and translational speed;

[0029] is the estimated value of the upper bound of interference.

[0030] Preferably, the adjoint operator Ad g 、ad ξ The matrix expression of is:

[0031] (·) × Indicates the antisymmetric matrix.

[0032] Any Any ξ=[ω v], where R∈SO(3) represents the rotation matrix, represents the rotation vector, represents the angular velocity of rotation, Indicates the translational velocity.

[0033] Preferably, the kinematic and dynamic equations of the tracking satellite in S1 relative to the space tumbling target spacecraft are the dynamic equations of the relative motion system, expressed as follows:

[0034]

[0035] Where:

[0036]

[0037] Where R tk for arrive The coordinate transformation matrix, Indicates the relative position vector of the kth tracking satellite relative to the target spacecraft. The coordinates under R It Represents the target spacecraft body coordinate system To the Earth-centered inertial coordinate system The coordinate transformation matrix, r t I and The target spacecraft and the kth tracking satellite are respectively in the target spacecraft body coordinate system The position vector under

[0038] The gravitational effect on the kth tracking star represents the gravity term, The direction of this vector is from the center of the Earth to the center of mass of the target spacecraft or the kth tracking satellite. μ represents the Earth's gravitational constant, μ = 398600.47 km 3 / s 2 , R Ik Represents the coordinate system of the kth tracking star To the Earth-centered inertial coordinate system The coordinate transformation matrix,

[0039] represents the external interference acting on the kth tracking satellite.

[0040] Preferably, the configuration error vector in S2 According to the error kinematic equation of the kth tracking satellite relative to the target spacecraft, it is obtained:

[0041]

[0042] In the equation:

[0043] Intermediate variables

[0044] Intermediate variables

[0045] (·) ∧ Indicates that The vectors in are mapped to the isomorphic matrix space

[0046] Preferably, the collision avoidance mechanism of S3 is:

[0047] A target spacecraft T and N tracking satellites each construct a minimum envelope ellipsoid, and the N+1 ellipsoids are defined as the ellipsoid system set Each ellipsoid is described by the matrix When i=1,...,N, g i is the position of the kth tracking satellite relative to the target spacecraft. When i = T, g i is the position of the target spacecraft, where is the parameter matrix related to the ellipsoid envelope, and a i 、b i 、c i They represent the semi-axis lengths of the xyz axes of the i-th ellipsoid envelope respectively;

[0048] Collision avoidance mechanism between tracker and target spacecraft:

[0049] when When it is established, it indicates that the kth tracking satellite has not collided with the target spacecraft, otherwise it is characterized as a collision;

[0050] △ kT is the collision constraint function between the kth tracking satellite and the target spacecraft, are the discriminants used to determine the contact between the kth tracking star ellipsoid and the target spacecraft ellipsoid on the xy, xz, and yz two-dimensional projection planes;

[0051] The collision avoidance mechanism between N tracking satellites is:

[0052]

[0053] When μ k When >0, it indicates that the kth tracking star has not collided with other tracking stars, otherwise it indicates a collision;

[0054] △ kj is the collision constraint function between the kth tracking star and the jth tracking star, are the discriminants used to determine the contact between the kth tracking star ellipsoid and the jth tracking star ellipsoid on the xy, xz, and yz two-dimensional projection planes respectively;

[0055] is the sum of the discriminants used to determine the collision between the kth tracking star and the jth tracking star in the xy, xz, yz two-dimensional projection planes,

[0056] is the constant to be designed and satisfies the relationship is the value of the discriminant corresponding to the lower bound of the safety radius between the kth tracking star and the jth tracking star.

[0057] Preferably, the artificial potential function φ in S4 pk :

[0058]

[0059] Where: l pk , κ pk , m pk is the normal number to be designed,

[0060] is the group error, is the configuration error between the kth tracking satellite and the target spacecraft,

[0061] Beneficial effects of the present invention: Compared with the existing collision avoidance strategies between spacecraft, the present invention can solve the collision avoidance problem when considering the relative posture and geometric shape of multiple spacecraft at the same time. It avoids the high conservatism caused by over-simplification of the spacecraft shape. The introduction of the posture error potential function facilitates the design of the posture integrated control law. At the same time, the collision avoidance control law is designed in combination with the artificial potential function method. Compared with the existing optimization method, it avoids the high dependence on computing resources when considering the geometric shapes of multiple tracking satellites and target spacecraft at the same time. This method takes into account the geometric shape and relative posture of the spacecraft at the 3D level, which will avoid collisions when the spacecraft performs posture maneuvers in three-dimensional space. BRIEF DESCRIPTION OF THE DRAWINGS

[0062] Figure 1 It is the three-dimensional transfer trajectory of the tracking star, one of which does not consider the geometric shape of the tracking star, and the other one does consider the geometric shape of the tracking star;

[0063] Figure 2 It is a structural diagram of the spacecraft simplified into an ellipsoid system, including the target spacecraft and the tracking satellite. DETAILED DESCRIPTION

[0064] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making any creative efforts shall fall within the scope of protection of the present invention.

[0065] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments may be combined with each other.

[0066] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, but they are not intended to limit the present invention.

[0067] Specific implementation method 1: Figure 1 and Figure 2 This embodiment describes a method for collision avoidance control of a spacecraft with a complex shape based on algebraic conditions. The method includes the following steps:

[0068] S1. Based on Lie group SE (3), the kinematic and dynamic equations of the tracking star relative to the space tumbling target spacecraft are established;

[0069] S2. Introduce the configuration error potential function to obtain the configuration error vector

[0070] S3. Designing a collision avoidance mechanism, including collision avoidance between the tracking satellite and the target spacecraft and collision avoidance between the tracking satellites, wherein the tracking satellite and the target spacecraft both construct a minimum envelope ellipsoid;

[0071] S4. Design a continuously differentiable artificial potential function φ for collision avoidance constraints pk ;

[0072] S5. Combine S1-S4 to design the collision avoidance feedback control law.

[0073] Regarding step S1, in order to describe the motion of the tracking star, first define the target spacecraft body coordinate system The origin is located at the target spacecraft mass center, the z-axis is along the normal of the orbital plane, the x-axis is along the direction of the target spacecraft orbital radius, and the y-axis is determined by the right-hand rule. Suppose there are N tracking satellites, and the body coordinate system of the kth tracking satellite is defined as The origin is located at the center of mass of the tracking star, and its coordinate axes are along the direction of the principal axis of inertia. In the present invention, the symbols (k, t, I) written in the upper right corner of the characters represent the expression of the variable in the corresponding coordinate system. Down.

[0074] In order to establish the integrated dynamic model of spacecraft attitude and orbit, it is necessary to unify the parameters and model forms of the two motions of attitude and orbit. First, the kinematic equation of the kth tracking satellite relative to the target is expressed in the following compact form:

[0075]

[0076] in, represents the position configuration of the tracking satellite relative to the target spacecraft, R tk for arrive The coordinate transformation matrix. Indicates the relative position vector of the kth tracking satellite relative to the target spacecraft. The coordinates under r t I and The target spacecraft and the kth tracking satellite are respectively in the target spacecraft body coordinate system The position vector under (the direction of the vector is from the center of the earth to the center of mass of the target spacecraft and the center of mass of the k-th tracking satellite). and Respectively Relative to The rotational speed and translational speed of . Mapping (·) ∧ : Indicates that The vectors in are mapped to the isomorphic matrix space in(·) × Indicates that the antisymmetric matrix is ​​obtained.

[0077] At the same time, for the elements of Lie group SE(3) ξ=[ω v], where R∈SO(3) represents the rotation matrix, represents the rotation vector, represents the angular velocity of rotation, represents the translational velocity, and defines two adjoint operators as follows

[0078]

[0079] And there

[0080] Secondly, when the relative velocity of the two spacecraft can be directly measured, the dynamic equation of the relative motion system can be expressed as follows:

[0081]

[0082] in, J k ,m k They represent the moment of inertia and mass of the kth tracking star respectively, and E3 represents the third-order unit matrix. represents the control input of the kth tracking star. represents the external interference acting on the kth tracking star. According to the aforementioned adjoint operator Ad g 、ad ξ The definition of and The expression:

[0083]

[0084] and Respectively and relatively speed. represents the gravitational effect on the k-th tracking star, represents the gravity term and is of the form

[0085]

[0086] Assume that for time-varying external disturbance There exists an unknown positive constant d k ,satisfy

[0087] The goal of this invention is to design a control strategy that enables each tracking satellite to achieve safe and continuous specific tracking of the target spacecraft. During the entire mission, all motion constraints should be complied with.

[0088] Regarding step S2, the expected configuration and configuration error function are introduced.

[0089] Before performing motion constraint analysis, the expected relative configuration between the kth tracking satellite and the target spacecraft is given as follows:

[0090]

[0091] in, is the expected attitude of the kth tracking satellite relative to the target spacecraft, is the expected position of the kth tracking satellite relative to the target spacecraft. The design must satisfy the requirements of motion constraints.

[0092] Then, the group error is selected as follows

[0093]

[0094] in, represents the attitude error of the kth tracking satellite relative to the target spacecraft, Represents the position error of the kth tracking satellite relative to the target spacecraft.

[0095] Because the expected configuration given in equation (4) Is a constant matrix, so, taking the derivative of formula (5) we can get

[0096]

[0097] Formula (6) shows that the group error and speed is consistent, so based on the group error Selected configuration error potential function and velocity For the convenience of controller design, the following positive definite Morse potential function is defined to evaluate the position error between the kth tracking satellite and the target spacecraft:

[0098]

[0099] and Represent the error functions of the posture and position parts respectively.

[0100]

[0101] in, Express Find the trace of a matrix.

[0102]

[0103] The time derivative of formula (7) can be obtained

[0104]

[0105] Mapping(·) ∨ : The matrix maps to the isomorphic real space

[0106] Formula (10) can be written as

[0107]

[0108] Among them, the configuration error function Gradient Also called the configuration error vector, its form is as follows

[0109]

[0110] and Represent the relative attitude error vector and relative position error vector respectively.

[0111] The time derivative of Equation (12) yields the error kinematic equation of the kth tracking satellite relative to the target spacecraft as follows:

[0112]

[0113] in,

[0114]

[0115]

[0116] By introducing the configuration error potential function, the configuration state error is given by Convert to This conversion makes the subsequent state error expression simpler and facilitates the design of the control law. Compared with the error function used to evaluate the attitude deviation The error function (8) used in the present invention is more reasonable. The error vector obtained by (8) is proportional to the rotation angle around the Euler axis between adjacent postures, which improves the posture tracking performance, especially when dealing with large-angle posture maneuvers with large initial posture errors.

[0117] Regarding step S3, collision avoidance mechanism.

[0118] To ensure the safety of inter-satellite relative motion, the present invention proposes a collision avoidance mechanism. This mechanism has two purposes: one is to avoid collisions between tracking satellites and target spacecraft, and the other is to avoid collisions between tracking satellites.

[0119] Common close-range collision avoidance studies rarely consider the physical form factors of both the tracker and target spacecraft. This step will analyze the tracker's orbital constraints, taking into account the physical form of both the tracker and target spacecraft, and design a corresponding obstacle function to meet collision avoidance requirements.

[0120] Since the target spacecraft usually carries related accessories, such as solar panels or antennas, its outer envelope is described by an ellipsoid to further reduce the conservatism brought by the spherical envelope. First, the definition of the ellipsoid envelope safety index is given as follows

[0121]

[0122] Among them, △ o is the safety index of the target spacecraft ellipsoid envelope, a t 、b t 、c t is the semi-axis length of the ellipsoid envelope, which is determined by the size of the spacecraft. t x t When the ellipsoid is rotated about the axis, the ellipsoid envelope parameters can be obtained by the following formula

[0123]

[0124] Among them, W t1 represents the maximum width of the target spacecraft appendage in the plane, L t1 Indicates that t1 The length of the corresponding accessory; L t2 represents the maximum length of the target spacecraft attachment in the plane, W t2 Indicates that L t2 The width of the corresponding attachment. Figure 2 The structural parameters of the target spacecraft can be obtained by reconnaissance satellites and then obtained using 3D reconstruction technology. Similarly, the minimum enveloping ellipsoid of the tracking star can be constructed.

[0125] In order to achieve collision avoidance, some existing documents regard the tracking star as a point mass. Substituting into formula (14), if △ o >0, it is considered that the tracking star and the target spacecraft have not collided. If the geometric shape of the tracking star is taken into consideration, o When it is close to 0, the tracking satellite and the target spacecraft may have collided, such as Figure 1As shown in Figure 2 , the present invention considers the geometric shapes of both the target spacecraft and the tracking satellite, treating them as ellipsoids with pose motion. Existing algorithms cannot directly calculate the shortest distance between ellipsoids with pose motion. Therefore, the present invention utilizes the following algebraic method for ellipsoid collision detection to evaluate the contact between the tracking satellite and the target spacecraft's ellipsoid envelope surface for collision warning.

[0126] Define an ellipsoid system set It includes a target spacecraft T and N tracking satellites (mission spacecraft), and there are N+1 ellipsoids in total. Each ellipsoid can be described as g i =g it The matrix A describing the ellipsoidal motion is the position of the k-th tracking satellite relative to the target spacecraft. i (g i ) is defined as in The parameter matrix associated with the ellipsoid envelope, and a i 、b i 、c i It represents the semi-axis length of the ellipsoid envelope of the i-th tracking star defined above. The calculation process refers to the semi-axis length of the ellipsoid envelope of the target spacecraft mentioned above.

[0127] For any ellipsoid Its projection on the two-dimensional plane can be expressed as s represents a two-dimensional plane, where Represents the projection of the ellipsoid on the two-dimensional plane s. Figure 2 It can be seen that there are three states between the tracking star ellipsoid and the target spacecraft: separation, contact, and coincidence. and If external contact occurs, then the ellipse obtained by projecting the two ellipsoids on {xy,xz,yz} There must also be contact, i.e.

[0128]

[0129] and represent the inner areas of ellipsoids i and j respectively.

[0130] The equivalent condition of formula (16) can be written as

[0131]

[0132] Therefore, if and only if at least one s∈{xy,xz,yz}, When the ellipsoid and No collision occurred. This means that for at least one s∈{xy,xz,yz}, There must be one positive real root and two unequal negative roots, represents the characteristic polynomial, λ represents the eigenvalue variable, and det() represents the determinant of the matrix. express The discriminant of is used to determine the contact between the kth tracking star ellipsoid and the target spacecraft ellipsoid on the s two-dimensional projection plane. For at least one s∈{xy,xz,yz}, This must always be satisfied to ensure that the two plane ellipses do not touch. This means that there is a collision between the ellipsoids.

[0133] Define the smooth function σ(x), when x≤0, σ(x)=0, when x>0, When the ellipsoids collide on the two-dimensional projection plane, If and only if ellipsoid and No collision occurs. So far, the equivalent conditions for collision avoidance have been established. Since for all s∈{xy,xz,yz}, The discriminant of and The discriminant of Therefore, the present invention is a controllable spacecraft Define a continuously differentiable collision constraint function △ kj as follows

[0134]

[0135] Then, if the following inequality holds, it means that the k-th tracking satellite does not collide with the target spacecraft, and the collision constraint function between the k-th tracking satellite and the target spacecraft is:

[0136]

[0137] in, are the discriminants used to determine the contact between the kth tracking star ellipsoid and the target spacecraft ellipsoid on the xy, xz, and yz two-dimensional projection planes;

[0138] At the same time, since the present invention considers a multi-satellite system, in order to save inter-satellite computing resources, this step combines the collision avoidance mechanism with the tracking inter-satellite safety radius. Define the tracking inter-satellite safety radius △(g k ,g j ) is as follows

[0139]

[0140] where d con,k , d con,j They represent the safety radius of the kth tracking star and the jth tracking star respectively. Then there exists a positive constant satisfy Then, the intersatellite collision avoidance constraints are defined as follows:

[0141]

[0142] in, is the sum of the discriminants used to determine the collision between the kth tracking star and the jth tracking star in the xy, xz, yz two-dimensional projection planes, is the constant to be designed and satisfies the relationship is the discriminant value corresponding to the lower bound of the safety radius between the kth tracking star and the jth tracking star. k >0, it means that the kth tracking star has not collided with other tracking stars. When the distance between the kth tracking star and the jth tracking star exceeds the inter-satellite safety radius, it is not necessary to calculate the inter-satellite ellipsoid projection value.

[0143] Regarding step S4, collision avoidance potential function design

[0144] To ensure that the tracking satellite always meets these motion constraints during the mission, this step will propose a control method based on APF. First, the corresponding potential function is designed as follows.

[0145] In order to maintain the distance between the tracking star and the target spacecraft ellipsoid envelope surface and achieve the goal of the tracking star safely reaching the desired observation point, the following continuously differentiable repulsive potential function is defined for each spacecraft:

[0146]

[0147] Among them, l pk , κ pk , m pk is the normal number to be designed. From formula (22), it can be seen that when △ kT →0, φ pk →∞, the ellipsoids will collide with each other.

[0148] When considering the intersatellite collision avoidance constraint, the expression of the collision avoidance potential function is as follows

[0149]

[0150] Among them, κ 1_kj , γ 1_kj and κ 1_kjare all positive constants to be designed. When the distance between the kth tracking star and the jth tracking star exceeds the inter-satellite safety radius, When kj →0, ψ k →∞, external contact will occur between tracking stars.

[0151] From equations (22) and (23), we can see that when the constraints (19) and (21) are satisfied at the initial moment, and φ pk and ψ k are all bounded, which means that the kth tracking star has not collided with the target spacecraft, nor with the jth tracking star. When all tracking stars have reached the desired configuration, φ pk +ψ k reaches the global minimum.

[0152] Regarding step S5, the collision avoidance feedback control law

[0153] In this step, a posture tracking controller will be designed for the tracking star to control each tracking star to reach its desired state so as to carry out the corresponding observation task. At the same time, in the process of reaching the desired state, the tracking star needs to comply with the motion constraints (19) and (21). The estimated error of the upper bound of the interference on the kth tracking star is defined as is the estimated value of the upper bound of the disturbance. Through the design of the adaptive law, it can be proved that the estimation of the upper bound of the disturbance can properly compensate for the disturbance of the system. The relative motion system under the control law is almost globally asymptotically stable. The control law is designed as follows

[0154]

[0155]

[0156]

[0157] in, is a basic feedback controller that is not related to the potential function; The potential function generates the control quantity for collision avoidance on the kth tracking star. is the positive definite control gain matrix to be designed. Ask for information about The gradient, Ask for information about The estimated value of the upper bound of the interference is updated by the following adaptive law

[0158]

[0159] in, is the positive definite estimated gain matrix to be designed.

[0160] According to the definition of adjoint operator, we can get

[0161] Although the present invention is described herein with reference to specific embodiments, it should be understood that these embodiments are merely illustrative of the principles and applications of the present invention. It should be understood that many modifications may be made to the illustrative embodiments, and that other arrangements may be devised, without departing from the spirit and scope of the present invention as defined by the appended claims. It should be understood that features described in various dependent claims and in this invention may be combined in ways other than those described in the original claims. It should also be understood that features described in conjunction with individual embodiments may be used in conjunction with other described embodiments.

Claims

1. A collision avoidance control method for complex-shaped spacecraft based on algebraic conditions, characterized in that: The method comprises the following steps: S1. Based on Lie group SE (3), the kinematic and dynamic equations of the tracking star relative to the space tumbling target spacecraft are established; S2. Introduce the configuration error potential function to obtain the configuration error vector S3. Designing a collision avoidance mechanism, including collision avoidance between the tracking satellite and the target spacecraft and collision avoidance between the tracking satellites, wherein the tracking satellite and the target spacecraft both construct a minimum envelope ellipsoid; S4. Design a continuously differentiable artificial potential function φ for collision avoidance constraints pk ; S5. Combine S1-S4 to design the collision avoidance feedback control law: in, represents the control input of the kth tracking star, k=1,...,N, N is the number of tracking stars, The potential function generates the control amount for collision avoidance on the kth tracking star, Where: It means to find the gradient about (·), is the attitude error of the kth tracking satellite relative to the target spacecraft, ψ k represents the collision avoidance potential function, (·) ∨ Represents mapping, mapping Indicates that The matrix in is mapped to the isomorphic real space represents the position error of the kth tracking satellite relative to the target spacecraft, is the expected attitude of the kth tracking satellite relative to the target spacecraft, is the expected position of the kth tracking satellite relative to the target spacecraft, is the basic feedback controller of the kth tracking star that is not related to the potential function, Where: k f 、k l is the positive definite control gain matrix to be designed, and Represents the coordinate system of the kth tracking star Relative to the target spacecraft body coordinate system The velocity vector, and Respectively Relative to The rotational speed and translational speed; represents the gravitational effect on the k-th tracking star, J k ,m k denote the moment of inertia and mass of the kth tracking star, respectively, and E3 denotes the third-order unit matrix; Ad g is the adjoint operator of the element g on the Lie group SE(3), ad ξ is the adjoint operator of ξ; express relatively speed, and Respectively Relative to The rotational speed and translational speed; is the estimated value of the upper bound of interference.

2. The method for collision avoidance control of a complex-shaped spacecraft based on algebraic conditions according to claim 1, characterized in that: Adjoint operator Ad g 、ad ξ The matrix expression of is: (·) × Indicates the antisymmetric matrix to be obtained. Any Any ξ=[ω v], where R∈SO(3) represents the rotation matrix, represents the rotation vector, represents the angular velocity of rotation, Indicates the translational velocity.

3. The method for collision avoidance control of a complex-shaped spacecraft based on algebraic conditions according to claim 2, characterized in that: The kinematic and dynamic equations of the tracking satellite in S1 relative to the space tumbling target spacecraft are the dynamic equations of the relative motion system, expressed as: Where: Where R tk for arrive The coordinate transformation matrix, Indicates the relative position vector of the kth tracking satellite relative to the target spacecraft. The coordinates under R It Represents the target spacecraft body coordinate system To the Earth-centered inertial coordinate system The coordinate transformation matrix, r t I and The target spacecraft and the kth tracking satellite are respectively in the target spacecraft body coordinate system The position vector under The gravitational effect on the kth tracking star represents the gravity term, The direction of this vector is from the center of the Earth to the center of mass of the target spacecraft or the kth tracking satellite. μ represents the Earth's gravitational constant, μ = 398600.47 km 3 / s 2 , R Ik Represents the coordinate system of the kth tracking star To the Earth-centered inertial coordinate system The coordinate transformation matrix, represents the external interference acting on the kth tracking satellite.

4. The method for collision avoidance control of a complex-shaped spacecraft based on algebraic conditions according to claim 2, characterized in that: Configuration error vector in S2 According to the error kinematic equation of the kth tracking satellite relative to the target spacecraft, it is obtained: In the equation: Intermediate variables Intermediate variables (·) ∧ Indicates that The vectors in are mapped to the isomorphic matrix space 5. The method for collision avoidance control of a complex-shaped spacecraft based on algebraic conditions according to claim 2, characterized in that: The collision avoidance mechanism of S3 is: A target spacecraft T and N tracking satellites each construct a minimum envelope ellipsoid, and the N+1 ellipsoids are defined as the ellipsoid system set Each ellipsoid is described by the matrix When i=1,...,N, g i is the position of the kth tracking satellite relative to the target spacecraft. When i = T, g i is the position of the target spacecraft, where is the parameter matrix related to the ellipsoid envelope, and a i 、b i 、c i They represent the semi-axis lengths of the xyz axes of the i-th ellipsoid envelope respectively; Collision avoidance mechanism between tracker and target spacecraft: when When it is established, it indicates that the kth tracking satellite has not collided with the target spacecraft, otherwise it is characterized as a collision; Δ kT is the collision constraint function between the kth tracking satellite and the target spacecraft, are the discriminants used to determine the contact between the kth tracking star ellipsoid and the target spacecraft ellipsoid on the xy, xz, and yz two-dimensional projection planes; The collision avoidance mechanism between N tracking satellites is: When μ k When >0, it indicates that the kth tracking star has not collided with other tracking stars, otherwise it indicates a collision; Δ kj is the collision constraint function between the kth tracking star and the jth tracking star, are the discriminants used to determine the contact between the kth tracking star ellipsoid and the jth tracking star ellipsoid on the xy, xz, and yz two-dimensional projection planes respectively; is the sum of the discriminants used to determine the collision between the kth tracking star and the jth tracking star in the xy, xz, yz two-dimensional projection planes, is the constant to be designed and satisfies the relationship is the lower bound of the discriminant corresponding to the safety radius between the kth tracking star and the jth tracking star.

6. The method for collision avoidance control of a complex-shaped spacecraft based on algebraic conditions according to claim 5, characterized in that: Artificial potential function φ in S4 pk : Where: l pk , κ pk , m pk is the normal number to be designed, is the group error, is the configuration error between the kth tracking satellite and the target spacecraft,

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