Spacecraft relative attitude and orbit state feedback safety control method under DoS attack

Abstract

The invention discloses a spacecraft relative attitude and orbit state feedback safety control method under DoS attack, and belongs to the field of spacecraft attitude and orbit integrated control. According to the invention, the problem that the existing control method is still poor in security and stability when facing a system with DoS attacks is solved. According to the invention, dual quaternions are utilized to establish a single spacecraft attitude and orbit integrated kinematics and dynamics model, dual quaternion operation is utilized to establish a relative attitude and orbit kinematics and dynamics model between a controlled spacecraft and a target spacecraft, and a spacecraft attitude and orbit error system with actuator DoS attack is established. And designing a virtual controller and a control law by using a backstepping method. And a segmented stability model of the closed-loop system is given, a global stability model of the closed-loop system is established, parameter limiting conditions are given, and finally it is proved that under the DoS attack, the control method can achieve state feedback stabilization of the relative attitude and orbit system of the spacecraft. The method can be applied to spacecraft relative attitude and orbit integrated control under DoS attacks.

Description

Technical Field

[0001] The present invention belongs to the field of integrated attitude and orbit control of spacecraft, and particularly relates to a safety control method for relative attitude and orbit state feedback of spacecraft under DoS attacks. Background Technique

[0002] With the rapid development of the spacecraft attitude and orbit control system, spacecraft attitude control methods and spacecraft orbit control methods have been widely studied. By applying the two systems, separate control of spacecraft attitude and orbit parameters can be achieved. However, in practical applications, the attitude and orbit systems of spacecraft have complex coupling characteristics. At the same time, the normal operation of the spacecraft control system highly depends on the communication system, and the communication system may be subject to denial-of-service (DoS) attacks, which in turn affect the normal operation of spacecraft attitude and orbit control and even cause great losses. Therefore, it is of practical significance to study the state feedback safety control problem of the integrated spacecraft attitude and orbit system under DoS attacks.

[0003] The patent application with the publication number CN111596677A discloses an autonomous control method and system for spacecraft proximity operations based on online learning, which establishes a pose-integrated dynamics model of the spacecraft based on dual quaternions, gives motion constraint equations using dual quaternions, and globally models the spacecraft attitude and orbit coupling system, having a consistent and concise mathematical form, and being able to control the attitude and orbit parameters of the spacecraft simultaneously, but does not consider the spacecraft attitude and orbit safety control problem under DoS attacks. The patent application with the publication number CN114153222A discloses a pose-orbit integrated tracking control method under multiple constraint conditions, which establishes a spacecraft pose-orbit integrated dynamics model and can solve the control problem of the spacecraft system during pose-orbit coupling. However, it also does not consider the countermeasure against DoS attacks and cannot solve the spacecraft pose-orbit integrated safety control problem.

[0004] In summary, although the existing technologies have made certain progress in integrated attitude and orbit control, there are still problems of insufficient safety and stability when facing systems with DoS attacks. Therefore, developing a spacecraft pose-orbit integrated safety control system considering DoS attacks is an issue that needs to be solved in the current spacecraft attitude and orbit control field. Summary of the Invention

[0005] The purpose of the present invention is to address the problem that existing control methods still have poor safety and stability when facing systems with DoS attacks, and to propose a safety control method for relative attitude and orbit state feedback of spacecraft under DoS attacks.

[0006] The technical solution adopted by the present invention to solve the above technical problems is: a safety control method for relative attitude and orbit state feedback of spacecraft under DoS attacks, and the method specifically includes the following steps:

[0007] Step S1, using dual quaternions to establish an attitude-orbit integrated kinematics and dynamics model of the controlled spacecraft and an attitude-orbit integrated kinematics and dynamics model of the target spacecraft;

[0008] Step S2: Based on the attitude-orbit integrated kinematic and dynamic models of the controlled spacecraft and the target spacecraft established in step S1, a relative attitude-orbit kinematic and dynamic model between the controlled spacecraft and the target spacecraft is established using dual quaternion operations;

[0009] Step S3: Based on the relative attitude and orbit kinematics and dynamics model between the controlled spacecraft and the target spacecraft, a spacecraft attitude and orbit error system with actuator DoS attack is established;

[0010] Step S4: Based on the established spacecraft attitude and orbit error system, a virtual controller and control law are designed using the backstepping method.

[0011] Furthermore, the process of establishing the attitude-orbit integrated kinematic model of the controlled spacecraft is as follows:

[0012] Step 1. Establish the geocentric inertial coordinate system Ox o y o z o , controlled spacecraft body coordinate system O f x f y f z f and the target spacecraft body coordinate system O l x l y l z l ;

[0013] The geocentric inertial coordinate system Ox o y o z o Taking the center of mass of the earth as the origin, Ox o The axis points to the vernal equinox, Oz o The axis points to the North Pole, Oy o The axis is determined by the right-hand rule;

[0014] The controlled spacecraft body coordinate system O f x f y f z f The origin of is fixed to the controlled spacecraft, O f x f Axis points to the controlled spacecraft main axis, O f y f Within the controlled spacecraft cross section, O f z f Determined by the right-hand rule;

[0015] The target spacecraft body coordinate system O lx l y l z l The origin of which is fixedly connected to the target spacecraft, O l x l The axis points to the main axis of the target spacecraft, O l y l In the cross-section of the target spacecraft, O l z l Is determined by the right-hand rule;

[0016] Steps 1 and 2: Denote the dual quaternion representing the attitude and orbit parameters of the controlled spacecraft as q f Is the attitude quaternion of the controlled spacecraft; ε is the dual unit; Is the position coordinate of the controlled spacecraft relative to the earth's center in the body coordinate system of the controlled spacecraft; q′ f Represents the dual part in the dual quaternion ; Represents the conjugate of q f ; Represents quaternion multiplication;

[0017] Denote the dual spinor representing the velocity parameters of the controlled spacecraft as Is the angular velocity of the controlled spacecraft body in the body coordinate system of the controlled spacecraft; Is the linear velocity of the controlled spacecraft relative to the earth's center in the body coordinate system of the controlled spacecraft;

[0018] Then the attitude and orbit integrated kinematic model of the controlled spacecraft is:

[0019]

[0020] Wherein, Represents the first derivative of q f ; Represents The first derivative of; Represents The first derivative of;

[0021]

[0022] Furthermore, the establishment process of the attitude and orbit integrated dynamic model of the controlled spacecraft is:

[0023] Denote the dual matrix representing the mass parameters of the controlled spacecraft as m f Represents the mass of the controlled spacecraft; J fDenote the inertia matrix of the controlled spacecraft as \(I\); 3×3 Denote the identity matrix;

[0024] The dual force exerted on the controlled spacecraft by the Earth's gravity is

[0025]

[0026] where \(c\) e1 、\(c\) e2 and \(c\) e3 are all constants; \(\text{diag}(1, 1, 3)\) represents a diagonal matrix with diagonal elements \(1\), \(1\), and \(3\) respectively; \(\|\cdot\|\) represents the calculation of the 2-norm;

[0027] Denote the dual control torque of the controlled spacecraft as \(F\) u represents the control force of the controlled spacecraft in the body coordinate system of the controlled spacecraft, and \(F'\) u represents the control torque of the controlled spacecraft in the body coordinate system of the controlled spacecraft;

[0028] Then the integrated attitude and orbit dynamics model of the controlled spacecraft is:

[0029]

[0030] where represents the first derivative of.

[0031] Furthermore, the establishment process of the integrated attitude and orbit kinematic model of the target spacecraft is as follows:

[0032] Denote the dual quaternion representing the attitude and orbit parameters of the target spacecraft as \(q\) l is the attitude quaternion of the target spacecraft; is the position coordinate of the target spacecraft relative to the geocenter in the body coordinate system of the target spacecraft; \(q'\) l represents the dual part in the dual quaternion , represents the conjugate of \(q\) l ;

[0033] Denote the dual spinor representing the velocity parameters of the target spacecraft as is the angular velocity of the target spacecraft body in the body coordinate system of the target spacecraft; is the linear velocity of the target spacecraft relative to the geocenter in the body coordinate system of the target spacecraft;

[0034] Then the kinematic model of the integrated attitude and orbit of the target spacecraft is as follows:

[0035]

[0036] Wherein, denotes the first derivative of.

[0037] Furthermore, the establishment process of the integrated attitude and orbit dynamic model of the target spacecraft is as follows:

[0038] Denote the dual matrix representing the mass parameters of the target spacecraft as m l represents the mass of the target spacecraft, and J l represents the inertia matrix of the target spacecraft;

[0039] Denote the dual force exerted on the target spacecraft by the earth's gravity as

[0040]

[0041] Then the integrated attitude and orbit dynamic model of the target spacecraft is as follows:

[0042]

[0043] Wherein, denotes the first derivative of.

[0044] Furthermore, the specific process of step S2 is as follows:

[0045] Denote the relative dual quaternion representing the relative attitude and orbit parameters between the controlled spacecraft and the target spacecraft as Specifically:

[0046]

[0047] Wherein, q fl represents the relative attitude quaternion, denotes fl the conjugate of q; represents the relative position vector between the controlled spacecraft and the target spacecraft in the body coordinate system of the controlled spacecraft,

[0048] Denote the relative dual spinor representing the relative velocity parameters between the controlled spacecraft and the target spacecraft as Specifically:

[0049]

[0050] Among them, represents the relative body angular velocity between the controlled spacecraft and the target spacecraft in the body coordinate system of the controlled spacecraft; represents the relative linear velocity between the controlled spacecraft and the target spacecraft in the body coordinate system of the controlled spacecraft, denotes the first derivative of;

[0051] Then the relative kinematic model between the controlled spacecraft and the target spacecraft is:

[0052]

[0053] The relative dynamic model between the controlled spacecraft and the target spacecraft is:

[0054]

[0055] Among them, denotes the first derivative of; denotes the first derivative of; denotes the inverse of.

[0056] Furthermore, the actuator DoS attack is expressed as:

[0057]

[0058] Among them, h i represents the start time of the i-th attack, τ i represents the duration of the i-th attack, and θ(t) represents the DoS attack signal value at time t.

[0059] Furthermore, the specific process of step S3 is:

[0060]

[0061] Among them, the error system state (η fl , ξ fl ) = q fl , η fl represents the scalar part of the quaternion q fl , ξ fl represents the vector part of the quaternion q fl , η′ fl represents q′ flThe scalar part, ξ′ fl represents the vector part of q′ fl The superscript T represents transpose, represents the first derivative of x1, represents the first derivative of η fl The superscript ′ represents the first derivative of ξ fl The superscript ′ represents the first derivative of η′ fl The superscript ′ represents the first derivative of ξ′ fl The superscript ′Δ = -θBu, control law

[0062] f r and f d satisfy:

[0063]

[0064] Furthermore, the virtual controller and the control law are designed by using the backstepping method. The specific process is as follows: Step S41, construct the state transformation by using the backstepping method:

[0065]

[0066] where α1 ∈ R 6 represents the virtual control law to be designed;

[0067] Step S42, select the Lyapunov function V1 as:

[0068]

[0069] Derive the Lyapunov function V1:

[0070] where, represents the first derivative of V1,

[0071] then design the virtual control law α1 as:

[0072]

[0073] where c1 > 0 is the design parameter;

[0074] ​​Step S43. Rewrite the error system dynamics equation as:

[0075]

[0076] Wherein,

[0077]

[0078] Wherein, represents the vector part of the quaternion ;

[0079] Select the Lyapunov function V2 as:

[0080]

[0081] Derive the Lyapunov function V2 to obtain:

[0082]

[0083] Then the designed control law is:

[0084]

[0085] Wherein, c2>0 is a design parameter.

[0086] The beneficial effects of the present invention are:

[0087] The present invention uses dual quaternions to establish a kinematic and dynamic model for the integrated attitude and orbit of a single spacecraft, uses dual quaternion operations to establish a relative kinematic and dynamic model for the attitude and orbit between a controlled spacecraft and a target spacecraft, establishes a spacecraft attitude and orbit error system with actuator DoS attacks, and designs a virtual controller and a control law using the backstepping method. And a piecewise stability model of the closed-loop system is given, a global stability model of the closed-loop system is established and parameter limiting conditions are given. Finally, it is proved that under DoS attacks, the control method of the present invention can achieve state feedback stabilization of the spacecraft relative attitude and orbit system, indicating that the method of the present invention can effectively solve the state feedback safety control problem of the spacecraft relative attitude and orbit system under actuator DoS attacks and ensure the safety and stability of the system. Description of the Drawings

[0088] Figure 1 is a flowchart of a method for state feedback safety control of spacecraft relative attitude and orbit under DoS attacks of the present invention;

[0089] Figure 2 is a system operation diagram of a system for state feedback safety control of spacecraft relative attitude and orbit under DoS attacks of the present invention;

[0090] Figure 3 is a DoS attack signal curve diagram of the present invention;

[0091] Figure 4 This is the actual control force signal curve of the present invention;

[0092] In the figure, the vertical coordinate is the control force (control orbit) signal component corresponding to the control law, with the unit of N;

[0093] Figure 5 This is the actual control torque signal curve of the present invention;

[0094] In the figure, the vertical coordinate is the control torque (control attitude) signal component corresponding to the control law, with the unit of N·m;

[0095] Figure 6 This is the relative attitude quaternion error signal curve of the present invention;

[0096] In the figure, the vertical coordinate is the relative attitude quaternion error signal component, dimensionless;

[0097] Figure 7 This is the relative position error signal curve of the present invention;

[0098] In the figure, the vertical coordinate is the relative position error signal component, with the unit of m. Detailed implementation manners

[0099] Detailed implementation manner 1: In combination with Figure 1 This detailed implementation manner is described. A relative attitude and orbit state feedback safety control method for a spacecraft under a DoS attack described in this detailed implementation manner specifically includes the following steps:

[0100] Step S1: Use dual quaternions to establish the kinematic and dynamic models of the integrated attitude and orbit of the controlled spacecraft and the kinematic and dynamic models of the integrated attitude and orbit of the target spacecraft;

[0101] Step S2: Based on the kinematic and dynamic models of the integrated attitude and orbit of the controlled spacecraft and the target spacecraft established in Step S1, use dual quaternion operations to establish the relative attitude and orbit kinematic and dynamic models between the controlled spacecraft and the target spacecraft;

[0102] Step S3: According to the relative attitude and orbit kinematic and dynamic models between the controlled spacecraft and the target spacecraft, establish a spacecraft attitude and orbit error system with actuator DoS attacks;

[0103] Step S4: Based on the established spacecraft attitude and orbit error system, use the backstepping method to design a virtual controller and a control law.

[0104] The control target can be achieved by the method of the present invention According to the properties of dual quaternions, at this time, the controlled spacecraft has the same attitude and orbit state as the target spacecraft.

[0105] Embodiment 2: The difference between this embodiment and Embodiment 1 is that the specific process of step S1 is as follows:

[0106] Step 11: Establish a geocentric inertial coordinate system Ox o y o z o , a controlled spacecraft body coordinate system O f x f y f z f and a target spacecraft body coordinate system O l x l y l z l ;

[0107] The geocentric inertial coordinate system Ox o y o z o has the center of the Earth as the origin, the Ox o axis points to the vernal equinox point, the Oz o axis points to the North Pole, and the Oy o axis is determined by the right-hand rule;

[0108] The origin of the controlled spacecraft body coordinate system O f x f y f z f is fixedly connected to the controlled spacecraft. The O f x f axis points to the main axis of the controlled spacecraft (by default, the longest axis is the main axis). The O f y f is in the cross-section of the controlled spacecraft, and the O f z f is determined by the right-hand rule, that is, the three axes are orthogonal;

[0109] The origin of the target spacecraft body coordinate system O l x l y l z l is fixedly connected to the target spacecraft. The O l x l axis points to the main axis of the target spacecraft (by default, the longest axis is the main axis). The O l y l is in the cross-section of the target spacecraft, and the O l z l is determined by the right-hand rule, that is, the three axes are orthogonal;

[0110] Step 12: Denote the dual quaternion representing the attitude and orbit parameters of the controlled spacecraft as q fis the attitude quaternion of the controlled spacecraft; ε is the dual unit; is the position coordinate of the controlled spacecraft relative to the geocenter in the body coordinate system of the controlled spacecraft; q′ f represents the dual part in the dual quaternion ; represents the conjugate of q f ; represents quaternion multiplication;

[0111] Denote the dual spinor representing the velocity parameter of the controlled spacecraft as is the angular velocity of the controlled spacecraft body in the body coordinate system of the controlled spacecraft; is the linear velocity of the controlled spacecraft relative to the geocenter in the body coordinate system of the controlled spacecraft;

[0112] Then the kinematic model of the integrated attitude and orbit of the controlled spacecraft is:

[0113]

[0114] Among them, represents the first derivative of q f ; represents 's first derivative; represents 's first derivative;

[0115]

[0116] Step 1-3: Denote the dual matrix representing the mass parameter of the controlled spacecraft as m f represents the mass of the controlled spacecraft; J f represents the inertia matrix of the controlled spacecraft; I 3×3 represents the identity matrix;

[0117] The dual force exerted on the controlled spacecraft by the earth's gravity is

[0118]

[0119] Among them, c e1 , c e2 and c e3 are all constants; c e1 = 398600.44×10 9 , c e2 = 1.08263×10 -3 , c e3= 6378140, diag(1, 1, 3) represents a diagonal matrix with diagonal elements 1, 1, and 3 respectively; ||·|| represents the calculation of the 2-norm;

[0120] Denote the dual control torque of the controlled spacecraft as F u represents the control force of the controlled spacecraft in the body coordinate system of the controlled spacecraft, and F′ u represents the control torque of the controlled spacecraft in the body coordinate system of the controlled spacecraft;

[0121] Then the integrated attitude and orbit dynamics model of the controlled spacecraft is:

[0122]

[0123] where, represents the first derivative of; represents the first derivative of;

[0124] Step 14. Denote the dual quaternion representing the attitude and orbit parameters of the target spacecraft as q l is the attitude quaternion of the target spacecraft; is the position coordinate of the target spacecraft relative to the geocenter in the body coordinate system of the target spacecraft; q′ l represents the dual part in the dual quaternion , represents the conjugate of q l ;

[0125] Denote the dual spinor representing the velocity parameters of the target spacecraft as is the body angular velocity of the target spacecraft in the body coordinate system of the target spacecraft; is the linear velocity of the target spacecraft relative to the geocenter in the body coordinate system of the target spacecraft;

[0126] Then the integrated attitude and orbit kinematic model of the target spacecraft is:

[0127]

[0128] where, represents the first derivative of;

[0129] Step 15. Denote the dual matrix representing the mass parameter of the target spacecraft as ml Denote the mass of the target spacecraft as \(J\). l Denote the inertia matrix of the target spacecraft as

[0130] Denote the dual force exerted on the target spacecraft by the Earth's gravity as

[0131]

[0132] Then the integrated attitude and orbit dynamics model of the target spacecraft is as follows:

[0133]

[0134] Wherein, Denote as the first derivative of

[0135] The other steps and parameters are the same as those in the first specific implementation manner.

[0136] The third specific implementation manner: The difference between this implementation manner and the first or second specific implementation manner is that the specific process of step S2 is as follows:

[0137] Denote the relative dual quaternion representing the relative attitude and orbit parameters between the controlled spacecraft and the target spacecraft as Specifically:

[0138]

[0139] Wherein, Denote as the conjugate of Denote as the conjugate of ; \(q\) fl Denote the relative attitude quaternion as Denote fl as the conjugate of \(q\); Denote the relative position vector between the controlled spacecraft and the target spacecraft in the body coordinate system of the controlled spacecraft as

[0140] Denote the relative dual spinor representing the relative velocity parameters between the controlled spacecraft and the target spacecraft as Specifically:

[0141]

[0142] Wherein, Denote as the conjugate of (q′ fl )* denote \(q'\) fl as its conjugate; represents the relative body angular velocity between the controlled spacecraft and the target spacecraft in the body coordinate system of the controlled spacecraft; represents the relative linear velocity between the controlled spacecraft and the target spacecraft in the body coordinate system of the controlled spacecraft, denote as the first derivative of;

[0143] Then the relative kinematic model between the controlled spacecraft and the target spacecraft is:

[0144]

[0145] where, denote as the first derivative of;

[0146] The relative dynamic model between the controlled spacecraft and the target spacecraft is:

[0147]

[0148] where, denote as the first derivative of; denote as the first derivative of; denote as the first derivative of; denote as the first derivative of, denote as the inverse of,

[0149] Other steps and parameters are the same as those in the first or second specific implementation manners.

[0150] Specific implementation manner four: The difference between this implementation manner and any one of the first to third specific implementation manners is that the actuator DoS attack is expressed as:

[0151]

[0152] where, \(h\) i denotes the start time of the \(i\) -th attack, \(\tau\) i denotes the duration of the \(i\) -th attack, and \(\theta(t)\) represents the value of the DoS attack signal at time \(t\).

[0153] Other steps and parameters are the same as those in any one of the first to third specific implementation manners.

[0154] Assumption 1: The DoS attack frequency \(n(\tau,t)\) is limited, that is: for the interval \([\tau,t)\), there exists a constant with τ D ≥ 0 such that:

[0155]

[0156] Hypothesis 2: The DoS attack time |Ξ(τ, t)| is bounded, that is: for the interval [τ, t), there exist constants κ ≥ 0 and T ≥ 1 such that:

[0157]

[0158] Hypothesis 3: The function g(z1, z2, t) satisfies:

[0159]

[0160] where δ > 0 and χ > 0 are parameters.

[0161] Specific implementation manner five: Different from one of the specific implementation manners one to four, the specific process of step S3 is as follows:

[0162]

[0163] where the error system state (η fl , ξ fl ) = q fl , η fl represents the scalar part of the quaternion q fl , ξ fl represents the vector part of the quaternion q fl , η′ fl represents the scalar part of q′ fl , ξ′ fl represents the vector part of q′ fl , the superscript T represents the transpose, represents the first derivative of x1, represents the first derivative of x2, represents the first derivative of η fl , represents the first derivative of ξ fl , represents the first derivative of η′ fl , represents the first derivative of ξ′ fl , Δ = -θBu, control law fr and f d Satisfy:

[0164] Other steps and parameters are the same as those in any one of the first to fourth specific embodiments.

[0165] Specific Embodiment Six: The difference between this embodiment and any one of the first to fifth specific embodiments is that when designing the virtual controller and control law using the backstepping method, the specific process is as follows:

[0166] Step S41: Construct a state transformation using the backstepping method:

[0167]

[0168] where α1 ∈ R 6 represents the virtual control law to be designed;

[0169] Step S42: Select the Lyapunov function V1 as:

[0170]

[0171] Take the derivative of the Lyapunov function V1. According to the definition of quaternion operations, it can be known that:

[0172]

[0173] where, represents the first derivative of V1, represents the first derivative of z1,

[0174] Then, design the virtual control law α1 as:

[0175]

[0176] where c1 > 0 is a design parameter;

[0177]

[0178] Step S43: Rewrite the error system dynamic equation as:

[0179]

[0180] where,

[0181]

[0182] where, represents the quaternion vector part of;

[0183] Select the Lyapunov function V2 as follows:

[0184]

[0185] Derive the Lyapunov function V2 to obtain:

[0186]

[0187] Then the designed control law is:

[0188]

[0189] where c2 > 0 is a design parameter.

[0190] The other steps and parameters are the same as those in any one of the specific embodiments one to five.

[0191] The controller designed by the present invention can achieve the control objective, and at this time, the controlled spacecraft has the same attitude and orbit state as the target spacecraft.

[0192] The following gives the piecewise stability model of the closed-loop system, establishes the global stability model of the closed-loop system based on the piecewise stability model, and gives the parameter constraint conditions. The specific process is as follows:

[0193]

[0194] In the interval t ∈ [h i , h i + τ i ) with DoS attack, Δ = -Bu. According to Assumption 3 and the Young's inequality, we have:

[0195]

[0196] where:

[0197]

[0198] Substitute (26) and (27) into (25) to obtain:

[0199]

[0200] where The solution of the above formula (27) is:

[0201]

[0202] where,

[0203] In the interval without DoS attack Δ = 06 = [0, 0, 0, 0, 0, 0]T , then we have:

[0204]

[0205] where λ Λ = min{c1, 2c2}, and M Λ = 4c1.

[0206] The solution of the above equation (30) is:

[0207] [[ID=U18]]

[0208] where M Λ = 4c1.

[0209] Considering cases (29) and (31), we get:

[0210]

[0211] where Υ i = -λ Λ |Λ(h i + τ i , t)| + λ Ξ |Ξ(h i , t)|;

[0212] According to Assumption 2, we have:

[0213]

[0214] Let we have:

[0215]

[0216] where, and

[0217]

[0218] According to Assumption 1, we have

[0219]

[0220] Combining (35), (36) and (37), we can obtain:

[0221]

[0222] where the design parameters c1 and c2 satisfy and λ Λ = min{c1, 2c2}, then the relative dual - quaternion error system is ultimately uniformly bounded.

[0223] Experimental part

[0224] As shown Figure 2 in the overall schematic diagram composed of a control component and a physical system, a virtual control law is generated using the relative attitude and orbit system error, and the relative attitude and orbit system state and the target system state are used, and combined with the virtual control law to generate an actual control law to control the relative attitude and orbit system of a spacecraft with actuator DoS attacks, and a relative attitude and orbit error signal is generated using the system state.

[0225] To verify the effectiveness of the control method proposed by the present invention, the system state is set

[0226] The mass parameter is set to m f = 1, m l = 3 and The DoS attack signal is set as Figure 3 shown. θ = 1 indicates the existence of actuator DoS attacks, and θ = 0 indicates the DoS attack is dormant.

[0227] Select the parameters δ = 1, T = 5, c1 = 1.35, c2 = 12, then there is λ Λ = min{c1, 2c2} = 1.35 and That is Figure 4 is the actual control force signal curve; u f = [u f1 , u f2 , u f3 ; Figure 5 is the actual control torque signal curve; u t = [u t1 , u t2 , u t3 , [u f1 , u f2 , u f3 , u t1 , u t2 , u t3 T = (1 - θ)u; According to Figure 4 and Figure 5 it can be seen that when the DoS attack occurs, the control signal remains the zero vector. Figure 6 is the relative attitude quaternion error signal curve; q e = q fl - (1, 03) = (q e0 , [q e1 , q e2 , q e3 T ); Figure 7 is the relative position error signal curve;​​ According to Figure 6 and Figure 7 it can be known that the proposed safety control method of the present invention can stabilize the relative attitude and orbit system of the spacecraft. Thus, it can be seen that a state feedback safety control method and system for the relative attitude and orbit of a spacecraft proposed by the present invention can achieve the state feedback stabilization of the relative attitude and orbit system of the spacecraft under DoS attacks.

[0228] A state feedback safety control method and system for the relative attitude and orbit of a spacecraft under DoS attacks provided by the present invention uses dual quaternions to establish the kinematics and dynamics models of the attitude and orbit of a single spacecraft, uses dual quaternion operations to establish the relative attitude and orbit kinematics and dynamics models between the controlled spacecraft and the target spacecraft, establishes an attitude and orbit error system of the spacecraft with actuator DoS attacks, designs a virtual controller and a control law using the backstepping method and gives a piecewise stability model of the closed-loop system, establishes a global stability model of the closed-loop system and gives parameter limitation conditions, and realizes the state feedback stabilization of the relative attitude and orbit system of the spacecraft under DoS attacks.

[0229] The above-mentioned numerical examples of the present invention are only to illustrate in detail the calculation models and calculation processes of the present invention, rather than to limit the implementation manners of the present invention. For those of ordinary skill in the art, other different forms of changes or modifications can be made based on the above description. It is impossible to list all the implementation manners here. Any obvious changes or modifications derived from the technical solutions of the present invention still fall within the protection scope of the present invention.

Claims

1. A safety control method for the relative attitude and orbit state feedback of a spacecraft under DoS attacks, characterized in that The method specifically includes the following steps: Step S1: Establish the kinematic and dynamic models of the integrated attitude and orbit of the controlled spacecraft and the kinematic and dynamic models of the integrated attitude and orbit of the target spacecraft by using dual quaternions; Step S2: Based on the kinematic and dynamic models of the integrated attitude and orbit of the controlled spacecraft and the target spacecraft established in Step S1, establish the relative kinematic and dynamic models of the attitude and orbit between the controlled spacecraft and the target spacecraft by using dual quaternion operations; Step S3: Establish a spacecraft attitude and orbit error system with actuator DoS attacks according to the relative kinematic and dynamic models of the attitude and orbit between the controlled spacecraft and the target spacecraft; Step S4: Based on the established spacecraft attitude and orbit error system, design a virtual controller and a control law by using the backstepping method.

2. The safety control method for the relative attitude and orbit state feedback of a spacecraft under a DoS attack according to claim 1, characterized in that The establishment process of the kinematic model of the integrated attitude and orbit of the controlled spacecraft is as follows: Step 1: Establish the geocentric inertial coordinate system Ox o y o z o , the body coordinate system O of the controlled spacecraft f x f y f z f and the body coordinate system O of the target spacecraft l x l y l z l ; The geocentric inertial coordinate system Ox o y o z o With the Earth's mass center as the origin, the Ox o axis points to the vernal equinox point, the Oz o axis points to the North Pole, and the Oy o axis is determined by the right-hand rule; The origin of the controlled spacecraft body coordinate system O f x f y f z f is fixedly connected to the controlled spacecraft. The O f x f axis points to the main axis of the controlled spacecraft. The O f y f is in the cross-section of the controlled spacecraft. The O f z f is determined by the right-hand rule; The origin of the body coordinate system O of the target spacecraft l x l y l z l is fixedly connected to the target spacecraft. The O l x l axis points to the main axis of the target spacecraft. The O l y l is in the cross-section of the target spacecraft. The O l z l is determined by the right-hand rule; Steps 1 and 2: Denote the dual quaternion representing the attitude and orbit parameters of the controlled spacecraft as q f is the attitude quaternion of the controlled spacecraft; ε is the dual unit; is the position coordinate of the controlled spacecraft relative to the geocenter in the body coordinate system of the controlled spacecraft; q′ f represents the dual part in the dual quaternion ; represents q f conjugate; represents quaternion multiplication; Denote the dual spinor representing the velocity parameters of the controlled spacecraft as is the angular velocity of the controlled spacecraft's body in the body coordinate system of the controlled spacecraft; is the linear velocity of the controlled spacecraft relative to the earth's center in the body coordinate system of the controlled spacecraft; Then the kinematic model of the integrated attitude and orbit of the controlled spacecraft is: Among them, represents the first derivative of q f ; represents the first derivative of represents the first derivative of; 3. A method for spacecraft relative attitude and orbit state feedback safety control under DoS attack according to claim 2, characterized in that The establishment process of the dynamic model of the integrated attitude and orbit of the controlled spacecraft is as follows: Denote the dual matrix representing the mass parameters of the controlled spacecraft as m f representing the mass of the controlled spacecraft; J f representing the inertia matrix of the controlled spacecraft; I 3×3 representing the identity matrix; The dual force exerted on the controlled spacecraft by the Earth's gravity is where c e1 , c e2 and c e3 are all constants; diag(1, 1, 3) represents a diagonal matrix with diagonal elements 1, 1, and 3 respectively; ||·|| represents the calculation of the 2-norm; Denote the dual control torque of the controlled spacecraft as F u which represents the control force of the controlled spacecraft in the body coordinate system of the controlled spacecraft, and F u ′ represents the control torque of the controlled spacecraft in the body coordinate system of the controlled spacecraft; Then the dynamic model of the integrated attitude and orbit of the controlled spacecraft is: Among them, denotes the first derivative of.

4. The safety control method for the relative attitude and orbit state feedback of a spacecraft under a DoS attack according to claim 3, characterized in that The establishment process of the kinematic model of the integrated attitude and orbit of the target spacecraft is as follows: Denote the dual quaternion representing the attitude and orbit parameters of the target spacecraft as q l which is the attitude quaternion of the target spacecraft; and is the position coordinate of the target spacecraft relative to the geocenter in the body coordinate system of the target spacecraft; q l ′ represents the dual part in the dual quaternion and represents the conjugate of q l . Denote the dual spinor representing the velocity parameters of the target spacecraft as is the angular velocity of the target spacecraft's body in the body coordinate system of the target spacecraft; is the linear velocity of the target spacecraft relative to the earth's center in the body coordinate system of the target spacecraft; Then the kinematic model of the integrated attitude and orbit of the target spacecraft is: Among them, denotes the first derivative of.

5. A method for spacecraft relative attitude and orbit state feedback safety control under DoS attack according to claim 4, characterized in that The establishment process of the dynamic model of the integrated attitude and orbit of the target spacecraft is as follows: Denote the dual matrix representing the mass parameters of the target spacecraft as m l represents the mass of the target spacecraft, and J l represents the inertia matrix of the target spacecraft; Denote the dual force exerted on the target spacecraft by the Earth's gravity as Then the dynamic model of the integrated attitude and orbit of the target spacecraft is: Among them, denotes the first derivative of.

6. The spacecraft relative attitude and orbit state feedback safety control method under DoS attack according to claim 5, characterized in that The specific process of Step S2 is as follows: The relative dual quaternion representing the relative attitude and orbit parameters between the controlled spacecraft and the target spacecraft is denoted as Specifically: where q fl represents the relative attitude quaternion, represents the conjugate of q fl ; represents the relative position vector between the controlled spacecraft and the target spacecraft in the body coordinate system of the controlled spacecraft, Denote the relative dual spinor representing the relative velocity parameter between the controlled spacecraft and the target spacecraft as Specifically: Among them, represents the relative body angular velocity between the controlled spacecraft and the target spacecraft in the body coordinate system of the controlled spacecraft; represents the relative linear velocity between the controlled spacecraft and the target spacecraft in the body coordinate system of the controlled spacecraft, denotes the first derivative of; Then the relative kinematic model between the controlled spacecraft and the target spacecraft is: The relative dynamic model between the controlled spacecraft and the target spacecraft is: Among them, denotes the first derivative of; denotes the first derivative of; denotes the inverse of.

7. A method for spacecraft relative attitude and orbit state feedback safety control under DoS attack according to claim 6, characterized in that The actuator DoS attack is expressed as: Among them, h i represents the start time of the i-th attack, and τ i represents the duration of the i-th attack, and θ(t) represents the value of the DoS attack signal at time t.

8. A method for safe control of the relative attitude and orbit state feedback of a spacecraft under a DoS attack according to claim 7, characterized in that, The specific process of Step S3 is as follows: where the error system state (η fl ,ξ fl ) = q fl , η fl represents the scalar part of the quaternion q fl , ξ fl represents the vector part of the quaternion q fl , ′η fl represents the scalar part of q′ fl , ξ′ fl represents the vector part of q′ fl , The superscript T represents transpose, represents the first derivative of x1, represents the first derivative of x2, represents the first derivative of η fl , represents the first derivative of ξ fl , represents the first derivative of η′ fl , represents the first derivative of ξ′ fl , Δ = -θBu, control law f r and f d satisfy:

9. The safety control method for the relative attitude and orbit state feedback of a spacecraft under a DoS attack according to claim 8, characterized in that The specific process of designing a virtual controller and a control law by using the backstepping method is as follows: Step S41: Construct a state transformation by using the backstepping method: where α1 ∈ R 6 represents the virtual control law to be designed; Step S42: Select the Lyapunov function V1 as: Take the derivative of the Lyapunov function V1: Among them, represents the first derivative of V1, Then design the virtual control law α1 as: where c1 > 0 is a design parameter; Step S43: Rewrite the dynamic equation of the error system as: Among them, Among them, represents the vector part of the quaternion ; select the Lyapunov function V2 as: Take the derivative of the Lyapunov function V2 to get: Then design the control law as: where c2 > 0 is a design parameter.

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