State estimation method and device for autonomous navigation of spacecraft and storage medium

Abstract

The embodiment of the invention provides a state estimation method and device for autonomous navigation of a spacecraft and a storage medium, and the method comprises the steps: constructing an uncertainty nonlinear system for autonomous navigation of the spacecraft during the operation of the spacecraft, designing the gain of a state estimation error through employing a robust extended Kalman filtering algorithm, and guaranteeing the upper bound of the state estimation error; a quasi-consistency robust extended Kalman filtering algorithm is adopted to ensure that the actual covariance matrix of the state estimation error is equal to or smaller than the covariance matrix of the state estimation error generated by a filter, and an uncertainty nonlinear system with quasi-consistency is obtained; and solving the uncertainty nonlinear system with quasi-consistency to obtain state parameters of the spacecraft. The state estimation error upper bound is ensured, and the estimation precision of the filter can be improved; the uncertainty and the robustness stability of process noise and measurement noise can be ensured, and the state of the spacecraft can be accurately obtained.

Description

Technical Field

[0001] The present invention relates to the field of spacecraft autonomous navigation, and in particular to a state estimation method, device and storage medium for spacecraft autonomous navigation. Background Art

[0002] In the fields of communications and aerospace, especially for autonomous navigation of spacecraft, the existing technology uses the Kalman filter (KF) algorithm proposed by the famous mathematician Kalman in 1960. This algorithm takes random signals as the object and is based on the two assumptions of linear system and Gaussian distribution. However, most actual spacecraft autonomous navigation system models have a certain degree of nonlinearity, and the system noise does not necessarily meet the Gaussian distribution conditions, which will cause the KF algorithm to be unable to be directly applied. Compared with linear systems, the filtering problem of nonlinear systems is more complex and difficult. The reason is that even if the noise meets the distribution conditions in the initial state, after the nonlinear transformation, the original Gaussian distribution is destroyed, and the posterior probability density function cannot be represented by only first-order matrices and second-order matrices. For example Figure 8 At this point, the time update and measurement update equations of the KF algorithm are no longer applicable.

[0003] On the one hand, in order to solve the filtering problem of nonlinear systems corresponding to autonomous navigation of spacecraft, the most intuitive approach is to approximate the nonlinear function. This type of method is called nonlinear filtering method based on nonlinear function approximation. According to the approximation method, there are first-order extended Kalman filter (first-order EKF or EKF), second-order extended Kalman filter (second-order EKF) and polynomial filter, etc. Figure 9 As shown. However, for this type of nonlinear filtering method, a large number of studies on deterministic models have shown that the errors introduced during the linearization process may lead to inconsistent estimates. Consistency is a basic criterion for evaluating filter performance. In practical applications, since the true state of the system is unknown, if a filter is inconsistent, then the error of the filter is unknown, which will lead to unreliable estimation results. In addition, multi-order EKF will also increase the computational burden. On the other hand, uncertainty is a problem that most nonlinear systems have to face and solve. If the robustness of the filter is not considered, the filtering performance will be significantly reduced. Therefore, when estimating the state of spacecraft autonomous navigation, a state estimation method that maintains the consistency and robustness of the nonlinear system is required. Summary of the Invention

[0004] Embodiments of the present invention provide a state estimation method, device, and storage medium for autonomous navigation of a spacecraft. The state estimation method can maintain the consistency and robustness of a nonlinear system when estimating the state of the autonomous navigation of a spacecraft.

[0005] To achieve the above objectives, in a first aspect, an embodiment of the present invention provides a state estimation method for autonomous navigation of a spacecraft, comprising:

[0006] Step 1: When the spacecraft is in operation, an uncertainty nonlinear system for autonomous navigation of the spacecraft is constructed, where the uncertainty nonlinear system is used to estimate the state parameters of the spacecraft, where the state parameters include position, velocity, or acceleration.

[0007] Step 2: For the uncertain nonlinear system, a robust extended Kalman filter algorithm is used to design a gain of the state estimation error to ensure an upper bound of the state estimation error;

[0008] Step 3: Using the quasi-consistent robust extended Kalman filter algorithm to ensure that the actual covariance matrix of the state estimation error is equal to or less than the covariance matrix of the state estimation error generated by the filter, thereby obtaining an uncertain nonlinear system with quasi-consistency;

[0009] Step 4: Obtain the state of the spacecraft by solving the uncertain nonlinear system with quasi-consistency.

[0010] In a second aspect, an embodiment of the present invention provides a state estimation device for autonomous navigation of a spacecraft, comprising:

[0011] A state estimation system construction unit is used to construct an uncertainty nonlinear system for autonomous navigation of the spacecraft when the spacecraft is in operation, wherein the uncertainty nonlinear system is used to estimate the state parameters of the spacecraft, wherein the state parameters include position, velocity or acceleration;

[0012] A gain design unit is used to design a gain of a state estimation error using a robust extended Kalman filter algorithm for the uncertain nonlinear system, so as to ensure an upper bound of the state estimation error;

[0013] A system correction unit is used to use a quasi-consistent robust extended Kalman filter algorithm to ensure that the actual covariance matrix of the state estimation error is equal to or less than the covariance matrix of the state estimation error generated by the filter, thereby obtaining an uncertain nonlinear system with quasi-consistency;

[0014] The solving unit is used to obtain the state parameters of the spacecraft by solving the uncertain nonlinear system with quasi-consistency.

[0015] In a third aspect, an embodiment of the present invention provides a computer-readable storage medium, which stores one or more programs. When the one or more programs are executed by a computer device, the computer device executes the aforementioned state estimation method for autonomous navigation of the spacecraft.

[0016] In a fourth aspect, an embodiment of the present invention provides a computer device, including:

[0017] A processor; and a memory arranged to store computer-executable instructions, which, when executed, cause the processor to perform the aforementioned state estimation method for autonomous navigation of a spacecraft.

[0018] The above technical solution has the following beneficial effects: the robust extended Kalman filter algorithm is used to design the gain of the state estimation error, ensure the upper bound of the state estimation error, and improve the estimation accuracy of the filter; the quasi-consistent robust extended Kalman filter algorithm is used to ensure the robustness and stability of the uncertainty, process noise and measurement noise of the uncertain nonlinear system, and the state of the spacecraft can be accurately obtained. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0020] Figure 1 A state estimation method for autonomous navigation of a spacecraft according to an embodiment of the present invention;

[0021] Figure 2 A state estimation device for autonomous navigation of a spacecraft according to an embodiment of the present invention;

[0022] Figure 3 are Brownian noise and Gaussian white noise according to an embodiment of the present invention;

[0023] Figure 4 It is a centralized robust filtering algorithm;

[0024] Figure 5 is the error caused by the linear approximation of the embodiment of the present invention;

[0025] Figure 6 are the state estimates and actual state values ​​of EKF, REKF, QCEKF and QCREKF;

[0026] Figure 7 is the root mean square error of the four algorithms at each moment;

[0027] Figure 8 It is a nonlinear transformation of Gaussian distribution in the prior art;

[0028] Figure 9 It is a classification of the existing technology based on linearization nonlinear filtering methods. DETAILED DESCRIPTION

[0029] 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 creative efforts are within the scope of protection of the present invention.

[0030] like Figure 1 As shown, in combination with an embodiment of the present invention, a state estimation method for autonomous navigation of a spacecraft is provided, comprising:

[0031] Step 1: When the spacecraft is in operation, an uncertainty nonlinear system for autonomous navigation of the spacecraft is constructed, where the uncertainty nonlinear system is used to estimate the state parameters of the spacecraft, where the state parameters include position, velocity, or acceleration.

[0032] Step 2: For the uncertain nonlinear system, a robust extended Kalman filter algorithm is used to design a gain of the state estimation error to ensure an upper bound of the state estimation error;

[0033] Step 3: Using the quasi-consistent robust extended Kalman filter algorithm to ensure that the actual covariance matrix of the state estimation error is equal to or less than the covariance matrix of the state estimation error generated by the filter, thereby obtaining an uncertain nonlinear system with quasi-consistency;

[0034] Step 4: Obtain the state parameters of the spacecraft by solving the quasi-consistent uncertain nonlinear system.

[0035] Preferably, step one includes:

[0036] For spacecraft autonomous navigation, an uncertain nonlinear system for spacecraft autonomous navigation is constructed:

[0037] x k =f(x k-1 )+B k η k φ(x k-1 )+w k (1)

[0038] y k =h(x k )+D k ξ k ψ(x k )+v k (2)

[0039] Among them, x k ∈R l 、y k ∈R mdenote the state vector and measurement vector of a parameter of an uncertain nonlinear system at time k, respectively. f(·) and h(·) are continuously differentiable nonlinear functions.

[0040] B k η k φ(x k-1 ) and D k ξ k ψ(x k ) represents the random uncertainty of the uncertain nonlinear system, and the matrix B k and D k It is obtained by using the prior information of the system composed of sensors carried by the spacecraft, and all of them are uncertain errors; B k ∈R l×l and D k ∈R m×l are all known time-varying matrices, φ(·) and ψ(·) are known nonlinear functions satisfying:

[0041] E[φ(x k )φ T (x k )]≤Ξ k E[ψ(x k )ψ T (x k )]≤Θ k (3)

[0042] Among them, E represents the mathematical expectation, Ξ k ∈R l×l 、Θ k ∈R l×l are all known matrices, η k ∈R,ξ k ∈R,w k ∈R l ,v k ∈R m are all uncorrelated noises, η k ∈R,ξ k ∈R,w k ∈R l ,v k ∈R m It has the following statistical properties:

[0043] E(η k )=0E(ξ k )=0E(w k )=0E(v k )=0(4)

[0044]

[0045] q k ≤1rk ≤1(6)

[0046] Q k ∈R l×l ,R k ∈R m×m are the covariance matrices of noise w, v, δ kj represents the Kronecker delta function.

[0047] Preferably, step 2 includes:

[0048] For the equations (1) and (2) corresponding to the uncertain nonlinear system, the robust extended Kalman filter algorithm is used to set the gain K of the state estimation error k , and get the corresponding state prediction and update model:

[0049]

[0050] in, and Represents the state estimation at time k and time k-1 respectively. The state is, for example, speed. K k is the gain of the undetermined state estimation error at time k, K k is a matrix, represents the predicted value of the state at time k, in k∣k-1 Indicates using the estimated value at time k-1 to predict the predicted value at time k, y k Represents the measurement value at time k.

[0051] Using equations (1) and (9), the state prediction error It is expressed as formula (11):

[0052]

[0053] In formula (1), f(x k-1 ) in state estimation Taylor expansion at:

[0054]

[0055] in, represents the state estimation error at time k-1, All represent the higher-order terms of Taylor series expansion; Transformed into a tractable equivalent:

[0056]

[0057] Among them, A k ∈R l×lis the scaling matrix, β k ∈R l×l is an unknown time-varying matrix introduced to consider the linearization error of state prediction; assuming β k is bounded:

[0058]

[0059] Introduce the matrix L∈R l×l To provide additional degrees of freedom to tune the robust extended Kalman filter algorithm;

[0060] In formula (2), h(x k )exist Taylor expansion at:

[0061]

[0062] in, C k ∈R m×l is the scale matrix, α k ∈R l×l is an unknown time-varying matrix, α k ∈R l×l Used to describe the linearization error of formula (2) corresponding to the measurement model, α k ∈R l×l Satisfy the assumptions:

[0063]

[0064] Substituting equations (12) and (13) into equation (11), we obtain:

[0065]

[0066] Then, the covariance matrix of the state prediction error can be written as:

[0067]

[0068] The state estimation error can be written as:

[0069]

[0070] Therefore, according to formula (19), the covariance matrix of the state estimation error can be written as:

[0071]

[0072] By setting the filter gain K k By minimizing the covariance of the final state estimation error, the upper bound of the state estimation error can be guaranteed.

[0073] Preferably, step 2 further includes:

[0074] The filter gain K is calculated as follows: k :

[0075] For the uncertain nonlinear system described by Equations (1) and (2), assuming that Equations (14) and (16) are satisfied, if there exists a positive constant γ, the following Riccati difference equation is obtained:

[0076] P0=Σ0(21)

[0077]

[0078] in,

[0079]

[0080] There is a positive solution P k and

[0081] γ 2 I-LP k L T >0 γ 2 I-LP k∣k-1 L T >0 (25)

[0082] Then the gain of the state estimation error using REKF shown in equations (9) and (10) is expressed as:

[0083]

[0084] In equations (21) and (26), P0 represents the covariance matrix of the state estimation error at the initial moment, Σ0 represents the covariance matrix of the state estimation error at the newly defined initial moment, and P k∣k-1 Represents the covariance matrix prediction value of the state prediction error at time k, P k∣k-1 The k|k-1 in the equation represents the covariance matrix of the state estimation error at time k-1 to predict the covariance matrix of the state prediction error at time k, P k-1 Represents the covariance matrix estimate of the state estimation error at time k-1, P k Represents the estimated value of the covariance matrix of the state estimation error at time k.

[0085] Preferably, step three includes:

[0086] The quasi-consistency of the quasi-consistent robust extended Kalman filter algorithm is defined as follows: a filter is quasi-consistent, then the actual covariance matrix of the state estimation error estimated by it is equal to or less than the covariance matrix of the state estimation error generated by the filter;

[0087] According to the quasi-consistency definition of the quasi-consistent robust extended Kalman filter algorithm, based on equations (1) and (2) corresponding to the uncertain nonlinear system, assuming that it satisfies equations (9) and (10), if there exists a positive constant: γ, satisfying

[0088] γ 2 I-LP k L T >0 γ 2 I-LP k∣k-1 L T >0 (37)

[0089] Then, the equations (9), (39), (26), (10), (42), (43), and (44) corresponding to the quasi-consistent robust extended Kalman filter algorithm are quasi-consistent:

[0090]

[0091] P k∣k-1 =F k P k-1 F k T +ΔQ k (39)

[0092]

[0093] P k =(IK k H k )P k∣k-1 (IK k H k ) T +ΔR k (42)

[0094] in:

[0095]

[0096] Σ0=P0(21)

[0097] Among them, E represents the mathematical expectation, represents the state estimate at time k-1, ΔQ k , ΔR k Indicates uncertainty about the parameters that need to be adjusted.

[0098] Preferably, step three further includes:

[0099] For the uncertain nonlinear system corresponding to equations (1) and (2), the quasi-consistent robust extended Kalman filter algorithm is determined by equations (9), (39), (26), (10), (42), (43), (44), and (21), and satisfies equations (37), (14), and (16), and the following assumptions hold for k = 0, 1, …, n:

[0100] det[IK k (H k +C k α k L)]≠0 det(F k +A k β k L)≠0 det(L)≠0 (56)

[0101] Then there is

[0102]

[0103] Among them, det represents the absolute value, represents the state estimation error at time n, represents the state estimation error at the initial moment, symbol Defined as the weighted (weight P) L2 norm square mean of x:

[0104] Equation (57) shows that the quasi-consistent robust extended Kalman filter algorithm can ensure the robustness and stability of uncertainty, process noise and measurement noise of uncertain nonlinear systems.

[0105] like Figure 2 As shown, in combination with an embodiment of the present invention, a state estimation device for autonomous navigation of a spacecraft is provided, comprising:

[0106] A state estimation system construction unit 21 is used to construct an uncertainty nonlinear system for autonomous navigation of the spacecraft when the spacecraft is in operation, wherein the uncertainty nonlinear system is used to estimate the state parameters of the spacecraft, wherein the state parameters include position, velocity or acceleration;

[0107] A gain design unit 22 is configured to design a gain of a state estimation error using a robust extended Kalman filter algorithm for the uncertain nonlinear system to ensure an upper bound on the state estimation error;

[0108] The system correction unit 23 is used to use a quasi-consistent robust extended Kalman filter algorithm to ensure that the actual covariance matrix of the state estimation error is equal to or less than the covariance matrix of the state estimation error generated by the filter, thereby obtaining an uncertain nonlinear system with quasi-consistency;

[0109] The solving unit 24 is used to obtain the state of the autonomous navigation of the spacecraft by solving the uncertain nonlinear system with quasi-consistency.

[0110] Preferably, the state estimation system construction unit 21 is specifically configured to:

[0111] For spacecraft autonomous navigation, an uncertain nonlinear system for spacecraft autonomous navigation is constructed:

[0112] xk=f(xk-1)+Bkηkφ(xk-1)+wk (1)

[0113] y k =h(x k )+D k ξ k ψ(x k )+v k (2)

[0114] Among them, x k ∈R l 、y k ∈R m denote the state vector and measurement vector of a parameter of an uncertain nonlinear system at time k, respectively. f(·) and h(·) are continuously differentiable nonlinear functions.

[0115] B k η k φ(x k-1 ) and D k ξ k ψ(x k ) represents the random uncertainty of the uncertain nonlinear system, and the matrix B k and D k It is obtained by using the prior information of the system composed of sensors carried by the spacecraft, and all of them are uncertain errors; B k ∈R l×l and D k ∈R m×l are all known time-varying matrices, φ(·) and ψ(·) are known nonlinear functions satisfying:

[0116] E[φ(x k )φ T (x k )]≤Ξ k E[ψ(x k )ψ T (x k )]≤Θ k (3)

[0117] Among them, E represents the mathematical expectation, Ξ k ∈Rl×l 、Θ k ∈R l×l are all known matrices, η k ∈R,ξ k ∈R,w k ∈R l ,v k ∈R m are all uncorrelated noises, η k ∈R,ξ k ∈R,w k ∈R l ,v k ∈R m It has the following statistical properties:

[0118] E(η k )=0E(ξ k )=0E(w k )=0E(v k )=0(4)

[0119]

[0120] q k ≤1r k ≤1(6)

[0121] Q k ∈R l×l ,R k ∈R m×m are the covariance matrices of noise w, v, δ kj represents the Kronecker delta function.

[0122] Preferably, the gain design unit 22 is specifically configured to:

[0123] For the equations (1) and (2) corresponding to the uncertain nonlinear system, the robust extended Kalman filter algorithm is used to set the gain K of the state estimation error k , and get the corresponding state prediction and update model:

[0124]

[0125] in, and Represents the state estimation at time k and time k-1 respectively. The state is, for example, speed. K k is the gain of the undetermined state estimation error at time k, K k is a matrix, represents the predicted value of the state at time k, in k∣k-1 Indicates using the estimated value at time k-1 to predict the predicted value at time k, y kRepresents the measurement value at time k.

[0126] Using equations (1) and (9), the state prediction error It is expressed as formula (11):

[0127]

[0128] In formula (1), f(x k-1 ) in state estimation Taylor expansion at:

[0129]

[0130] in, represents the state estimation error at time k-1, All represent the higher-order terms of Taylor series expansion; the higher-order terms Transformed into a tractable equivalent:

[0131]

[0132] Among them, A k ∈R l×l is the scaling matrix, β k ∈R l×l is an unknown time-varying matrix introduced to consider the linearization error of state prediction; assuming β k is bounded:

[0133]

[0134] Introduce the matrix L∈R l×l To provide additional degrees of freedom to tune the robust extended Kalman filter algorithm;

[0135] In formula (2), h(x k )exist Taylor expansion at:

[0136]

[0137] in, C k ∈R m×l is the scale matrix, α k ∈R l×l is an unknown time-varying matrix, α k ∈R l×l Used to describe the linearization error of formula (2) corresponding to the measurement model, α k ∈R l×l Satisfy the assumptions:

[0138]

[0139] Substituting equations (12) and (13) into equation (11), we obtain:

[0140]

[0141] Then, the covariance matrix of the state prediction error can be written as:

[0142]

[0143] The state estimation error can be written as:

[0144]

[0145] Therefore, according to formula (19), the covariance matrix of the state estimation error can be written as:

[0146]

[0147] By setting the filter gain K k By minimizing the covariance of the final state estimation error, the upper bound of the state estimation error can be guaranteed.

[0148] Preferably, the gain design unit 22 is specifically configured to:

[0149] The filter gain K is calculated as follows: k :

[0150] For the uncertain nonlinear system described by Equations (1) and (2), assuming that Equations (14) and (16) are satisfied, if there exists a positive constant γ, the following Riccati difference equation is obtained:

[0151] P0=Σ0(21)

[0152]

[0153] in,

[0154]

[0155] There is a positive solution P k and

[0156] γ 2 I-LP k L T >0γ 2 I-LP k∣k-1 L T >0(25)

[0157] Then the gain of the state estimation error using REKF shown in equations (9) and (10) is expressed as:

[0158]

[0159] In equations (21) and (26), P0 represents the covariance matrix of the state estimation error at the initial moment, Σ0 represents the covariance matrix of the state estimation error at the newly defined initial moment, and P k∣k-1 Represents the covariance matrix prediction value of the state prediction error at time k, P k∣k-1 in k∣k-1 Indicates that the covariance matrix of the state estimation error at time k is used to predict the covariance matrix of the state prediction error at time k, P k-1 Represents the covariance matrix estimate of the state estimation error at time k-1, P k Represents the estimated value of the covariance matrix of the state estimation error at time k.

[0160] Preferably, the system correction unit 23 is specifically configured to:

[0161] The quasi-consistency of the quasi-consistent robust extended Kalman filter algorithm is defined as follows: a filter is quasi-consistent, then the actual covariance matrix of the state estimation error estimated by it is equal to or less than the covariance matrix of the state estimation error generated by the filter;

[0162] According to the quasi-consistency definition of the quasi-consistent robust extended Kalman filter algorithm, based on equations (1) and (2) corresponding to the uncertain nonlinear system, assuming that it satisfies equations (9) and (10), if there exists a positive constant: γ, γ>0, satisfying

[0163] γ 2 I-LP k L T >0 γ 2 I-LP k∣k-1 L T >0 (37)

[0164] Then, the equations (9), (39), (26), (10), (42), (43), and (44) corresponding to the quasi-consistent robust extended Kalman filter algorithm are quasi-consistent:

[0165]

[0166] P k∣k-1 =F k P k-1 F k T +ΔQ k (39)

[0167]

[0168] P k =(IK k H k )P k∣k-1 (IK k H k ) T +ΔR k (42)

[0169] in:

[0170]

[0171] Σ0=P0(21)

[0172] Among them, E represents the mathematical expectation, represents the state estimate at time k-1, ΔQ k , ΔR k Indicates uncertainty about the parameters that need to be adjusted.

[0173] Preferably, the system correction unit 23 is specifically configured to:

[0174] For the uncertain nonlinear system corresponding to equations (1) and (2), the quasi-consistent robust extended Kalman filter algorithm is determined by equations (9), (39), (26), (10), (42), (43), (44), and (21), and satisfies equations (37), (14), and (16), and the following assumptions hold for k = 0, 1, …, n:

[0175] det[IK k (H k +C k α k L)]≠0det(F k +A k β k L)≠0det(L)≠0(56)

[0176] Then there is

[0177]

[0178] Among them, det represents the absolute value, represents the state estimation error at time n, represents the state estimation error at the initial moment, symbol Defined as the weighted (weight P) L2 norm square mean of x:

[0179] Equation (57) shows that the quasi-consistent robust extended Kalman filter algorithm can ensure the robustness and stability of the uncertainty, process noise and measurement noise of the uncertain nonlinear system, and can accurately obtain the state of the spacecraft.

[0180] In combination with an embodiment of the present invention, a computer-readable storage medium is provided, which stores one or more programs. When the one or more programs are executed by a computer device, the computer device executes the aforementioned state estimation method for autonomous navigation of a spacecraft.

[0181] In accordance with an embodiment of the present invention, a computer device is provided, including:

[0182] A processor; and a memory arranged to store computer-executable instructions, which, when executed, cause the processor to perform the aforementioned state estimation method for autonomous navigation of a spacecraft.

[0183] The above technical solutions of the embodiments of the present invention are described in detail below with reference to specific application examples. For technical details not introduced during the implementation process, please refer to the relevant description above.

[0184] 1. Problem Statement

[0185] For spacecraft autonomous navigation, an uncertain nonlinear system for spacecraft autonomous navigation is constructed:

[0186] x k =f(x k-1 )+B k η k φ(x k-1 )+w k (1)

[0187] y k =h(x k )+D k ξ k ψ(x k )+v k (2)

[0188] Among them, x k ∈R l 、y k ∈R m They represent the state vector and measurement vector of a state parameter (state includes one or more of position, velocity or acceleration) of the uncertain nonlinear system at time k. For example, the state parameter is velocity, then x k is the speed at time k, y k is the measurement vector of the velocity at time k, f(·) and h(·) are continuously differentiable nonlinear functions; x on the right side of formula (1)k-1 Is the estimated value of the parameter obtained at the previous moment (k-1 moment), and the predicted value of the parameter at the kth moment is obtained by formula (1) (belonging to the state vector), but the predicted value is not accurate, so the measured value (belonging to the measurement vector) is obtained by formula (2). Using formula (1) and formula (2) to obtain the predicted value and measurement respectively, and using the algorithm weighting, the state estimate at the kth moment is obtained

[0189] B k η k φ(x k-1 ) and D k ξ k ψ(x k ) represents the random uncertainty of the uncertain nonlinear system, and the matrix B k and D k It is obtained by using the prior information of the actual system (the system composed of sensors carried by the spacecraft), and all of them are uncertain errors; B k ∈R l×l and D k ∈R m×l are all known time-varying matrices, φ(·) and ψ(·) are known nonlinear functions satisfying:

[0190] E[φ(x k )φ T (x k )]≤Ξ k E[ψ(x k )ψ T (x k )]≤Θ k (3)

[0191] Among them, E represents the mathematical expectation, Ξ k ∈R l×l 、Θ k ∈R l×l are all known matrices, η k ∈R,ξ k ∈R,w k ∈R l ,v k ∈R m is uncorrelated noise, η k ∈R,ξ k ∈R,w k ∈R l ,v k ∈R m It has the following statistical properties:

[0192] E(η k )=0E(ξ k )=0E(w k )=0E(vk )=0(4)

[0193]

[0194] q k ≤1r k ≤1(6)

[0195] Q k ∈R l×l ,R k ∈R m×m are the covariance matrices of noise w, v, δ kj represents the Kronecker delta function.

[0196] Compared with the deterministic linear system, the uncertain nonlinear system corresponding to Equation (1) and Equation (2) will bring the following two problems.

[0197] (1) The mathematical model abstracted from the working principle of the spacecraft as a whole has uncertainty, such as: B in Equation (1) and Equation (2) k η k φ(x k-1 ), D k ξ k ψ(x k ), as well as the uncertainty of noise, such as flicker noise, impulse noise, Brownian noise, etc., will cause the KF algorithm (Kalman filter algorithm) and the first-order and second-order EKF filtering accuracy to decrease or even diverge. Figure 3 The figure below is a comparison of Rand noise and Gaussian white noise. In recent years, some achievements have been made in the design of robust nonlinearity, and different robust Kalman filters have been developed for different forms of noise. For example Figure 4 The robust filtering algorithms shown in Figure 1 use Huber updating and maximum entropy updating. However, these robust Kalman filters only guarantee an upper bound on the estimation error for the dynamic model uncertainty. In reality, simply guaranteeing an upper bound on the error often fails to accurately linearize the state function and, consequently, to produce an accurate estimation result. Therefore, while guaranteeing an upper bound on the state estimation error, it is important to ensure that the actual mean squared error is less than or equal to the mean squared error calculated by the filter.

[0198] (2) When B k =D k = 0, Equations (1) and (2) are deterministic nonlinear systems. When the state of a deterministic nonlinear system is estimated using a first-order or second-order extended Kalman filter (EKF), the nonlinear function f(·) is linearized at the current state estimate, i.e.:

[0199]

[0200] in, is the state estimation error at time k, represents the state estimation error at time k-1, represents the state estimate at time k.

[0201] EKF does not consider high-order terms during the linearization process That is to say, directly:

[0202]

[0203] Figure 5 This linearization process is described in detail, which determines the possible errors associated with the linear approximation of nonlinear functions. Figure 5 As shown, the previous nonlinear state estimation is somewhere on the function surface, but far from the true state x k The projection of the nonlinear state estimate onto the approximate linear state estimate yields the current state estimate (The red dot is the ideal state estimate of the current nonlinear state function) but it is limited by the truncation error. Obviously, Distance x k The farther away, the larger the truncation error introduced by the approximation. More importantly, this approximate linear state estimate may not satisfy the original nonlinear state estimate specified by the system. Therefore, it is desirable to reduce the truncation error introduced by this approximation by including higher-order terms while maintaining the problem's computational tractability.

[0204] To address these issues, this embodiment of the present invention first briefly introduces the robust extended Kalman filter (REKF) algorithm for uncertain nonlinear systems. Secondly, while introducing higher-order terms in the linearization process, the quasi-consistency of the new algorithm, QCREKF, is ensured through appropriate parameters. Finally, the performance of QCREKF is analyzed and verified through simulation examples.

[0205] 2. Robust Design

[0206] Filtering is a signal processing technique that aims to remove unwanted components from a signal while retaining useful ones. The uncertainty of mathematical models severely degrades filtering performance. The robust extended Kalman filter (REKF) appropriately designs filter gains to reduce the state estimation's reliance on uncertainty information while minimizing the upper bound of the estimation error. For uncertain nonlinear systems, the following state prediction and update model is shown after setting gains for equations (1) and (2) using the robust extended Kalman filter (REKF).

[0207] The robust extended Kalman filter REKF adopts the state prediction and update model of EKF:

[0208]

[0209] in, and Represents the state estimation at time k and time k-1 respectively. The state is, for example, speed. K k is the undetermined gain at time k, K k is a matrix, represents the predicted value of the state at time k, in k∣k-1 Indicates using the estimated value at time k-1 to predict the predicted value at time k, y k Represents the measurement value at time k.

[0210] State prediction error It can be expressed as formula (11) using formula (1) and formula (9):

[0211]

[0212] In formula (1), f(x k-1 ) in state estimation Taylor expansion at:

[0213]

[0214] in, represents the state estimation error at time k-1, Both represent the higher-order terms of Taylor series expansion. In order to facilitate the subsequent derivation, the higher-order terms Transformed into a tractable equivalent:

[0215]

[0216] Among them, A k ∈R l×l is the scaling matrix, β k ∈R l×l is an unknown time-varying matrix introduced to consider the linearization error of state prediction. Assume that β k is bounded, that is,

[0217]

[0218] Introduce the matrix L∈R l×l To provide an extra degree of freedom to tune the robust extended Kalman filter REKF. In general, it can be set as the identity matrix I.

[0219] Similarly, h(x k )exist Taylor expansion at:

[0220]

[0221] in, C k ∈R m×l is the scale matrix, α k ∈R l×l Is an unknown time-varying matrix used to describe the linearization error of equation (2) corresponding to the measurement model. Satisfy the assumption:

[0222]

[0223] Substituting equations (12) and (13) into equation (11), we obtain:

[0224]

[0225] Then, the covariance matrix of the state prediction error can be written as:

[0226]

[0227] Similarly, the state estimation error can be written as:

[0228]

[0229] Therefore, according to formula (19), the covariance matrix of the state estimation error can be written as:

[0230]

[0231] Error generally refers to the difference between the measured value, estimated value or predicted value and the true value; state estimation error refers to the difference between the estimated value and the true state when estimating the current or past state through observation data and filtering algorithm; state prediction error refers to the difference between the predicted value and the actual value when predicting the future state.

[0232] In summary, the key point of the REKF algorithm is to design an appropriate gain: K k is the undetermined gain matrix at time k. The gains are used to dynamically adjust the key parameters of the predicted values ​​and observation weights to minimize the covariance of the final state estimation error and ensure an upper bound on the state estimation error. Theorem 1 is used to calculate the gains.

[0233] Theorem 1: For the uncertain nonlinear system described by Equations (1) and (2), assume that Equations (14) and (16) are satisfied. If there exists a positive constant γ such that the following Riccati difference equation is:

[0234] P0=Σ0(21)

[0235]

[0236] in,

[0237]

[0238] There is a positive solution P k and

[0239] γ 2 I-LP k L T >0 γ 2 I-LP k∣k-1 L T >0 (25)

[0240] Then the gain of the state estimation error using REKF shown in equations (9) and (10) is:

[0241]

[0242] In equations (21) and (26), P0 represents the covariance matrix of the state estimation error at the initial moment, Σ0 represents the covariance matrix of the state estimation error at the newly defined initial moment, and P k∣k-1 Represents the covariance matrix prediction value of the state prediction error at time k, P k∣k-1 in k∣k-1 Indicates that the covariance matrix of the state estimation error at time k is used to predict the covariance matrix of the state prediction error at time k, P k-1 Represents the covariance matrix estimate of the state estimation error at time k-1, P k Represents the estimated value of the covariance matrix of the state estimation error at time k.

[0243] In summary, the gain calculation formula (26) is obtained according to the Riccati difference equation, and the robust extended Kalman filter REKF is summarized as follows:

[0244] State prediction: Using Equations (9) and (22), we can obtain the state prediction and its corresponding state prediction error covariance matrix (18).

[0245] Update model: The covariance matrix (20) of the state estimation and the corresponding state estimation error is calculated using equations (10) and (23), respectively. Before state estimation, the state prediction is performed, and the gain (weight) is used to balance the state prediction and state estimation.

[0246] 3. Consistency Improvement

[0247] When linearizing uncertain nonlinear systems, the introduction of truncation error (the third term to the right of the equal sign in Equation (12)) can cause inconsistent estimates in the robust extended Kalman filter (REKF). Consistency is defined as the filter being unbiased. To address this issue, we need to consider how to ensure the consistency of the robust extended Kalman filter (REKF) while maintaining robustness. The analysis of the robust extended Kalman filter (REKF) consistency is based on Theorem 2.

[0248] Theorem 2: The robust extended Kalman filter is called consistent if the state estimation error of the robust extended Kalman filter satisfies:

[0249]

[0250] Lemma 1: Given matrices A, B, C, D with compatible dimensions, such that CC T ≤I. Let U be a symmetric positive definite matrix, and a be an arbitrary positive constant such that a -1 I-DUD T > 0, then the following matrix inequality holds:

[0251] (A+BCD)U(A+BCD) T ≤A(U -1 -aD T D) -1 A T +a -1 BB T (61)

[0252] According to Theorem 2, the consistency of the robust extended Kalman filter REKF is analyzed below. Using equations (3), (6) and Lemma 1, we have the following inequality:

[0253]

[0254] in,

[0255]

[0256] If ΔΣ k > 0, the actual mean square error is not equal to the calculated covariance Σ k Therefore, for general uncertain nonlinear systems, REKF cannot guarantee the consistency of the filtering results. The actual state estimation error is unknown and the calculated covariance Σ cannot be used. k To make the estimate, we can make the error covariance Σ k This characteristic can be illustrated by the following example.

[0257] Example 1

[0258] Consider the system

[0259]

[0260] in, is the state transition matrix, X k =[x 1,k ,x 2,k ] T is a two-dimensional vector, y k is a one-dimensional measurement. k ,v k is zero-mean uncorrelated Gaussian white noise with covariances of R k =1,k=0,1,…. Initial value x 2,0 ∈[-0.5,0.5], initial state Independent of v0, and the covariance of the initial estimate is known to be:

[0261]

[0262] Assumptions D1 = [0.2 0], and η k =0.01,ξ k =0.02, k=0,1,…, we can get H k =[1 0], let C k =0,k=0,1,….

[0263] Initial value When, according to REKF, and Pick So So according to (18) and (22) we have

[0264]

[0265] Depend on Available gains

[0266]

[0267] According to formula (23), we can get

[0268]

[0269] It can be seen that P1 is still a diagonal matrix, and obviously there is

[0270]

[0271] And P1 and With {y k}independent.

[0272] Similarly, since B k ,D k As a time-varying matrix, let D k =[0.2k 0]. At this time, So

[0273]

[0274] Obviously P k is divergent, so Σ k It is also divergent.

[0275] 4. Quasi-consistent robust extended Kalman filter algorithm

[0276] The quasi-consistent robust extended Kalman filter algorithm QCREKF is used for discrete-time nonlinear systems with uncertainties, and its performance is analyzed.

[0277] 4.1 Quasi-consistent robust extended Kalman filter algorithm design process

[0278] 4.1.1 Quasi-consistent robust extended Kalman filter algorithm steps

[0279] The quasi-consistency of the quasi-consistent robust extended Kalman filter algorithm QCREKF is based on the following definition: a filter is quasi-consistent if the actual covariance matrix (i.e., mean square error) of the state estimation error estimated by it is equal to or less than the covariance matrix of the state estimation error generated by the filter.

[0280] According to the definition of quasi-consistency, the main result is given by Theorem 3.

[0281] Theorem 3: Based on the equations (1) and (2) corresponding to the uncertain nonlinear system, it is assumed that it satisfies equations (9) and (10). If there exists a positive constant: γ, satisfying

[0282] γ 2 I-LP k L T >0γ 2 I-LP k∣k-1 L T >0(37)

[0283] Then, the equations (9), (39), (26), (10), (42), (43) and (44) corresponding to the quasi-consistent robust extended Kalman filter algorithm QCREKF are quasi-consistent, that is,

[0284]

[0285] P k∣k-1 =F k P k-1 F k T +ΔQ k (39)

[0286]

[0287] P k =(IK k H k )P k∣k-1 (IK k H k ) T +ΔR k (42)

[0288] in:

[0289]

[0290]

[0291] Σ0=P0(21)

[0292] E represents the mathematical expectation, represents the state estimate at time k-1, ΔQ k , ΔR k Indicates uncertainty about the parameters that need to be adjusted.

[0293] The detailed proof of Theorem 3 is as follows.

[0294] To prove Theorem 3, we need to use the following lemma.

[0295] Lemma 1: Given matrices A, B, C, D with compatible dimensions, such that CC T ≤I. Let U be a symmetric positive definite matrix, and a be an arbitrary positive constant such that a -1 I-DUD T > 0, then the following matrix inequality holds:

[0296] (A+BCD)U(A+BCD) T ≤A(U -1 -aD T D) -1 A T +a -1 BB T (61)

[0297] Lemma 2 (Matrix Inversion Lemma): Given matrices A, B, and C of compatible dimensions, where A is a non-singular matrix, the following holds:

[0298] (A+BC) -1 =A -1 -A -1 B(I+CA -1 B) -1 CA -1 (62)

[0299] From equations (18) and (20), we can see that if the covariance matrix Σ k-1 ≤P k-1 , then the following inequality holds:

[0300] Σ k∣k-1 (Σ k-1 )≤Σ k∣k-1 (P k-1 )(63)

[0301] Σ k (Σ k∣k-1 (Σ k-1 ))≤Σ k (Σ k∣k-1 (P k-1 ))(64)

[0302] Using mathematical induction, we know that Σ0=P0, and assume that at time k-1, Linearize the process equation, when ΔQ k When equation (43) is satisfied, applying Lemma 1 and Lemma 2, the prediction error at time k is:

[0303]

[0304] in,

[0305] Similarly, linearizing the measurement equation and applying Lemma 1 and Lemma 2 yields

[0306]

[0307] in, ΔR k Satisfies inequality (44).

[0308] The calculation process of the QCREKF algorithm in one time step is given in Table 1.

[0309] Table 1. Calculation process of QCREKF in one time step

[0310]

[0311] Note 1: When ΔQk The corresponding formula (43) and ΔR k When the corresponding equation (44) takes the equal sign, QCREKF degenerates into REKF. If the influence of linearization error and model uncertainty can be ignored, then A k ,B k ,C k and D k Should be set to zero. In addition, if robust performance is not required, L can also be set to zero, in which case QCREKF degenerates into EKF.

[0312] Note 2: There are many free design parameters in the QCREKF algorithm. Introducing the matrix A k ,B k ,C k ,D k ,ΔQ k and ΔR k The linearization error of the dynamic model, the uncertainty of the dynamic model, the linearization error of the measurement model, the uncertainty of the measurement model, the variance boundary of the nonlinear state function truncation error and the variance boundary of the nonlinear measurement function truncation error are scaled respectively. This makes the filter gain K k It becomes possible to consider different types of interference in the design of the filter, thereby reducing the filter's dependence on the dynamic or observation model with uncertainty.

[0313] 4.1.2 Quasi-Consistent Applications

[0314] The quasi-consistency of the QCREKF algorithm has been proved in Appendix A. The following example 1 illustrates the quasi-consistency of the QCREKF algorithm.

[0315] Example 2

[0316] In Example 1, from formula (36), P k The divergence mainly comes from P k (2,2). Let’s analyze how to choose ΔQ in QCREKF to avoid diverging results. k and ΔR k The value of .

[0317] Substituting the initial values ​​in Example 1 into QCREKF equations (22) and (43) yields

[0318]

[0319] and

[0320]

[0321] It can be seen that P k|k-1 The value of (2,2) depends on ΔQk The size of (2,2).

[0322] According to formula (33),

[0323]

[0324] Substituting equation (48) into equations (42) and (44) in QCREKF, we can obtain

[0325]

[0326] Visible P k The value of (2,2) also depends on ΔQ k (2,2). This is consistent with the result of formula (36).

[0327] Take the following

[0328]

[0329]

[0330] Substituting equations (51) and (52) into equations (46) and (49) respectively, we can obtain

[0331]

[0332] Easy to obtain Since QCREKF is quasi-consistent, we have

[0333]

[0334] Therefore the mean square error is bounded.

[0335] In this example, ΔQ can be found k and ΔR k The value of is very important. Wrong value will lead to P k Diverge.

[0336] 4.2 Robustness Performance Analysis

[0337] In this section, Theorem 4 proves a key property of the quasi-consistent robust extended Kalman filter (QCREKF) algorithm: robustness. Robustness is crucial for ensuring system stability and reliability. It describes a system's ability to maintain stable functionality despite changes in its internal structure or external environment. Theorem 4 illustrates the robustness of the quasi-consistent robust extended Kalman filter (QCREKF).

[0338] Theorem 4: For the uncertain nonlinear system corresponding to Equations (1) and (2), the quasi-consistent robust extended Kalman filter algorithm is determined by Equations (9), (39), (26), (10), (42), (43), (44), and (21), and satisfies Equations (37), (14), and (16), and the following assumptions hold for k = 0, 1, …, n:

[0339] det[IK k (H k +C k α k L)]≠0 det(F k +A k β k L)≠0 det(L)≠0 (56)

[0340] Then there is

[0341]

[0342] Among them, det represents the absolute value, represents the state estimation error at time n, represents the state estimation error at the initial moment, symbol It is defined as the weighted (weight is P) L2 norm square mean of x, that is,

[0343] Equation (57) shows that the quasi-consistent robust extended Kalman filter algorithm QCREKF can ensure the robustness of system model uncertainty, as well as process noise and measurement noise.

[0344] The proof of Theorem 4 will be as follows.

[0345] ΔQ k and ΔR k Substituting into equation (39) and equation (42) respectively, we can obtain:

[0346]

[0347] Substituting formula (17) into formula (19), we have

[0348]

[0349] then

[0350]

[0351] The following inequality can be obtained by using equations (16), (37), (42) and Lemma 1:

[0352]

[0353] Based on formula (71), the second formula in formula (70) can be written as:

[0354]

[0355] Substitute equation (71) into equation (72) and apply the matrix inversion lemma to obtain:

[0356]

[0357] Using formula (26), the third formula in formula (70) can be written as:

[0358]

[0359] And there is

[0360]

[0361] Therefore, the following inequality holds:

[0362]

[0363] According to equation (71) and the first equation (70), it can be expressed as:

[0364]

[0365] From Lemma 1 and Lemma 2, we can get the following form of the right side of Equation (77):

[0366]

[0367] Substituting formula (67) into the first formula in formula (78) yields:

[0368]

[0369] Therefore, formula (77) can be written as:

[0370]

[0371] Therefore, Equation (70) can be expressed using Equations (73), (76) and (80):

[0372]

[0373] Summing equation (81) from k = 0, 1, ..., n, we can obtain:

[0374]

[0375] The robust performance of QCREKF is then expressed as:

[0376]

[0377] The proof is complete.

[0378] 5. Simulation

[0379] Two numerical examples are used to illustrate the importance of parameter selection in QCREKF. Comparative analysis with EKF, REKF and QCEKF verifies that QCREKF has better robustness and higher accuracy.

[0380] Example 3

[0381] In order to compare the nonlinear approximation accuracy of EKF, REKF, QCEKF and QCREKF, the nonlinear random scalar system shown in Equation (58) is selected:

[0382]

[0383] Where {w k} and {v k} is a random noise sequence with variance 1.

[0384] In the simulation, assuming the initial state x0 = 10, the initial state estimate The initial error covariance P0=1, the number of simulation steps is 50, and the EKF, REKF, QCEKF and QCREKF algorithms are used to simulate x k The state of is estimated, and the simulation results are shown in the figure below.

[0385] from Figure 6 and Figure 7 It can be seen that the estimation robustness of QCREKF is good and the accuracy is the highest. Compared with EKF, REKF and QCEKF, the accuracy is improved by 29.59%, 74.5% and 27.13%, respectively. Table 2 shows the comparison of the root mean square error (RMSE) of the four algorithms under this simulation, where the RMSE calculation formula is (59).

[0386]

[0387] Where n is the number of simulation steps.

[0388] Table 2 RMSE comparison

[0389] EKF REKF QCEKF QCREKF RMSE 1.2794 3.5327 1.2361 0.9008

[0390] Finally, we select the two-dimensional random nonlinear system shown in Equation (60) and test the accuracy of the four algorithms again.

[0391]

[0392] Among them, Xk =[x 1,k ,x 2,k ] T is the two-dimensional state vector, Y k =[y 1,k ,y 2,k ] T is the two-dimensional measurement vector, w k =[w 1,k ,w 2,k ] T ,v k =[v 1,k ,v 2,k ] T . Noise {w k} and {v k} is a random noise sequence with variance 1.

[0393] In the simulation, assume that the initial state X0 = [5,10] T , initial state estimate Initial error covariance P0 = I 2×2 , the number of simulation steps is 50, and EKF, REKF, QCEKF and QCREKF algorithms are used to simulate X k The state of is estimated, and the simulation results are as follows Figure 8 As shown, the root mean square error at each moment is calculated according to formula (59): Figure 9 .

[0394] Depend on Figure 8 and Figure 9 It can be seen that the robust performance of QCREKF is still very good and maintains the lowest error among the four algorithms. According to Table 3, compared with EKF, QCEKF and REKF, the accuracy of QCREKF is improved by 35.67%, 92.98% and 65.32% respectively.

[0395] Table 3 RMSE comparison

[0396] EKF QCEKF REKF QCREKF RMSE [0.0370,0.2652] [0.5145,1.9323] [0.4167,0.3025] [0.0204,0.1950]

[0397] Since QCEKF is designed for nonlinear systems without unknown inputs, it does not perform very well in this example. However, the QCEKF algorithm is still of great significance in ensuring that the state error covariance results do not diverge.

[0398] The beneficial technical effects achieved by the embodiments of the present invention are as follows:

[0399] For discrete-time nonlinear systems with random uncertainties, the quasi-consistent robust extended Kalman filter (QCREKF) algorithm is used to minimize the upper bound of the estimation error, improving the filter's estimation accuracy while ensuring that the actual mean square error is less than or equal to the mean square error calculated by the filter. This ensures that the actual mean square error is within the minimized upper bound of the estimation error, while also enabling real-time estimation of the error boundary.

[0400] During the linearization process, high-order terms that are discarded in the EKF are added to improve the calculation accuracy, and the consistency of the proposed algorithm is guaranteed by introducing appropriate parameters.

[0401] Compared with the quasi-consistent extended Kalman filter (QCEKF) algorithm, the uncertainty factor is taken into account in the model, making the system closer to reality and thus improving the estimation accuracy of the filter.

[0402] Robust to process noise, measurement noise, and model uncertainty.

[0403] It should be understood that the specific order or hierarchy of steps in the disclosed processes is an example of an exemplary method. Based on design preferences, it should be understood that the specific order or hierarchy of steps in the process can be rearranged without departing from the scope of the present disclosure. The accompanying method claims present elements of the various steps in an exemplary order and are not intended to be limited to the specific order or hierarchy described.

[0404] In the foregoing detailed description, various features are grouped together in a single embodiment to simplify the disclosure. This method of disclosure should not be interpreted as reflecting an intention that embodiments of the claimed subject matter require more features than are expressly recited in each claim. On the contrary, as reflected in the appended claims, the invention comprises less than all the features of any individual disclosed embodiment. The appended claims are hereby expressly incorporated into the detailed description, with each claim standing on its own as a separate preferred embodiment of the invention.

[0405] The specific implementation methods described above further illustrate the objectives, technical solutions and beneficial effects of the present invention in detail. It should be understood that the above description is only a specific implementation method of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A state estimation method for autonomous navigation of a spacecraft, characterized in that: include: Step 1: When the spacecraft is in operation, an uncertainty nonlinear system for autonomous navigation of the spacecraft is constructed, where the uncertainty nonlinear system is used to estimate the state parameters of the spacecraft, where the state parameters include position, velocity, or acceleration. Step 2: For the uncertain nonlinear system, a robust extended Kalman filter algorithm is used to design a gain of the state estimation error to ensure an upper bound of the state estimation error; Step 3: Using the quasi-consistent robust extended Kalman filter algorithm to ensure that the actual covariance matrix of the state estimation error is equal to or less than the covariance matrix of the state estimation error generated by the filter, thereby obtaining an uncertain nonlinear system with quasi-consistency; Step 4: Obtain the state parameters of the spacecraft by solving the quasi-consistent uncertain nonlinear system.

2. The state estimation method for spacecraft autonomous navigation according to claim 1, characterized in that: The step one comprises: For spacecraft autonomous navigation, an uncertain nonlinear system for spacecraft autonomous navigation is constructed: x k =f(x k-1 )+B k or k φ(x k-1 )+w k (1) y k =h(x k )+D k x k ψ(x k )+v k (2) Among them, x k ∈R l 、y k ∈R m denote the state vector and measurement vector of a parameter of an uncertain nonlinear system at time k, respectively. f(·) and h(·) are continuously differentiable nonlinear functions. B k η k φ(x k-1 ) and D k ξ k ψ(x k ) represents the random uncertainty of the uncertain nonlinear system, and the matrix B k and D k It is obtained by using the prior information of the system composed of sensors carried by the spacecraft, and all of them are uncertain errors; B k ∈R l×l and D k ∈R m ×l are all known time-varying matrices, φ(·) and ψ(·) are known nonlinear functions satisfying: E[φ(x k )φ T (x k )]≤Ξ k E[ψ(x k )ψ T (x k )]≤Θ k (3) Among them, E represents the mathematical expectation, Ξ k ∈R l×l 、Θ k ∈R l×l are all known matrices, η k ∈R,ξ k ∈R,w k ∈R l ,v k ∈R m are all uncorrelated noises, η k ∈R,ξ k ∈R,w k ∈R l ,v k ∈R m It has the following statistical properties: E(η k )0 E(ξ k )0 E(w k )0 E(v k )0 (4) q k ≤1 r k ≤1 (6) Q k ∈R l×l ,R k ∈R m×m are the covariance matrices of noise w, v, δ kj represents the Kronecker delta function.

3. The state estimation method for spacecraft autonomous navigation according to claim 2, characterized in that: The second step includes: For the equations (1) and (2) corresponding to the uncertain nonlinear system, the robust extended Kalman filter algorithm is used to set the gain K of the state estimation error k , and get the corresponding state prediction and update model: in, and Represents the state estimation at time k and time k-1 respectively. The state is, for example, speed. K k is the gain of the undetermined state estimation error at time k, K k is a matrix, represents the predicted value of the state at time k, in k∣k-1 Indicates that the estimated value at time k is used to predict the prediction at time k, y k Represents the measurement value at time k. Using equations (1) and (9), the state prediction error It is expressed as formula (11): In formula (1), f(x k-1 ) in state estimation Taylor expansion at: in, represents the state estimation error at time k-1, All represent the higher-order terms of Taylor series expansion; Transformed into a tractable equivalent: Among them, A k ∈R l×l is the scaling matrix, β k ∈R l×l is an unknown time-varying matrix introduced to consider the linearization error of state prediction; assuming β k is bounded: Introduce the matrix L∈R l×l To provide additional degrees of freedom to tune the robust extended Kalman filter algorithm; In formula (2), h(x k )exist Taylor expansion at: in, C k ∈R m×l is the scale matrix, α k ∈R l×l is an unknown time-varying matrix, α k ∈R l×l Used to describe the linearization error of formula (2) corresponding to the measurement model, α k ∈R l×l Satisfy the assumptions: Substituting equations (12) and (13) into equation (11), we obtain: Then, the covariance matrix of the state prediction error can be written as: The state estimation error can be written as: Therefore, according to formula (19), the covariance matrix of the state estimation error can be written as: By setting the filter gain K k By minimizing the covariance of the final state estimation error, the upper bound of the state estimation error can be guaranteed.

4. The state estimation method for spacecraft autonomous navigation according to claim 3, characterized in that: The second step also includes: The filter gain K is calculated as follows: k : For the uncertain nonlinear system described by Equations (1) and (2), assuming that Equations (14) and (16) are satisfied, if there exists a positive constant γ, the following Riccati difference equation is obtained: P0=Σ0(21) in, There is a positive solution P k and γ 2 The LP k L T >0 γ 2 The LP k∣k-1 L T >0 (25) Then the gain of the state estimation error using REKF shown in equations (9) and (10) is expressed as: In equations (21) and (26), P0 represents the covariance matrix of the state estimation error at the initial moment, Σ0 represents the covariance matrix of the state estimation error at the newly defined initial moment, and P k∣k-1 Represents the covariance matrix prediction value of the state prediction error at time k, P k∣k-1 The k|k-1 in the equation represents the covariance matrix of the state estimation error at time k-1 to predict the covariance matrix of the state prediction error at time k, P k-1 Represents the covariance matrix estimate of the state estimation error at time k-1, P k Represents the estimated value of the covariance matrix of the state estimation error at time k.

5. The state estimation method for autonomous navigation of a spacecraft according to claim 4, characterized in that: The step three includes: The quasi-consistency of the quasi-consistent robust extended Kalman filter algorithm is defined as follows: a filter is quasi-consistent, then the actual covariance matrix of the state estimation error estimated by it is equal to or less than the covariance matrix of the state estimation error generated by the filter; According to the quasi-consistency definition of the quasi-consistent robust extended Kalman filter algorithm, based on equations (1) and (2) corresponding to the uncertain nonlinear system, assuming that it satisfies equations (9) and (10), if there exists a positive constant: γ, γ>0, satisfying γ 2 The LP k L T >0 γ 2 The LP k∣k-1 L T >0 (37) Then, the equations (9), (39), (26), (10), (42), (43), and (44) corresponding to the quasi-consistent robust extended Kalman filter algorithm are quasi-consistent: P k∣k-1 =F k P k-1 F k T +ΔQ k (39) P k =(IK k H k )P k∣k-1 (IC k H k ) T +ΔR k (42) in: Σ0=P0 (21) Among them, E represents the mathematical expectation, represents the state estimate at time k-1, ΔQ k , ΔR k Indicates uncertainty about the parameters that need to be adjusted.

6. The state estimation method for spacecraft autonomous navigation according to claim 5, characterized in that: The step three also includes: For the uncertain nonlinear system corresponding to equations (1) and (2), the quasi-consistent robust extended Kalman filter algorithm is determined by equations (9), (39), (26), (10), (42), (43), (44), and (21), and satisfies equations (37), (14), and (16), and the following assumptions hold for k = 0, 1, …, n: det[IK k (H k +C k α k L)]≠0 det(F k +A k β k L)≠0 det(L)≠0 (56) Then there is Among them, det represents the absolute value, represents the state estimation error at time n, represents the state estimation error at the initial moment, symbol Defined as the weighted (weight P) L2 norm square mean of x: Equation (57) shows that the quasi-consistent robust extended Kalman filter algorithm can ensure the robustness of the uncertainty, process noise and measurement noise of the uncertain nonlinear system, stabilize the system model uncertainty, and the robustness of process noise and measurement noise.

7. A state estimation device for autonomous navigation of a spacecraft, characterized in that: include: A state estimation system construction unit is used to construct an uncertainty nonlinear system for autonomous navigation of the spacecraft when the spacecraft is in operation, wherein the uncertainty nonlinear system is used to estimate the state parameters of the spacecraft, wherein the state parameters include position, velocity or acceleration; A gain design unit is used to design a gain of a state estimation error by using a robust extended Kalman filter algorithm for the uncertain nonlinear system, so as to ensure an upper bound of the state estimation error; A system correction unit is used to use a quasi-consistent robust extended Kalman filter algorithm to ensure that the actual covariance matrix of the state estimation error is equal to or less than the covariance matrix of the state estimation error generated by the filter, thereby obtaining an uncertain nonlinear system with quasi-consistency; The solving unit is used to obtain the state parameters of the spacecraft by solving the uncertain nonlinear system with quasi-consistency.

8. The state estimation device for autonomous navigation of a spacecraft according to claim 7, characterized in that: State estimation system building blocks, specifically for: For spacecraft autonomous navigation, an uncertain nonlinear system for spacecraft autonomous navigation is constructed: x k =f(x k-1 )+B k or k φ(x k-1 )+w k (1) y k =h(x k )+D k x k ψ(x k )+v k (2) Among them, x k ∈R l 、y k ∈R m denote the state vector and measurement vector of a parameter of an uncertain nonlinear system at time k, respectively. f(·) and h(·) are continuously differentiable nonlinear functions. B k η k φ(x k-1 ) and D k ξ k ψ(x k ) represents the random uncertainty of the uncertain nonlinear system, and the matrix B k and D k It is obtained by using the prior information of the system composed of sensors carried by the spacecraft, and all of them are uncertain errors; B k ∈R l×l and D k ∈R m ×l are all known time-varying matrices, φ(·) and ψ(·) are known nonlinear functions satisfying: E[φ(x k )φ T (x k )]≤Ξ k E[ψ(x k )ψ T (x k )]≤Θ k (3) Among them, E represents the mathematical expectation, Ξ k ∈R l×l 、Θ k ∈R l×l are all known matrices, η k ∈R,ξ k ∈R,w k ∈R l ,v k ∈R m are all uncorrelated noises, η k ∈R,ξ k ∈R,w k ∈R l ,v k ∈R m It has the following statistical properties: E(η k )0 E(ξ k )0 E(w k )0 E(v k )0 (4) q k ≤1 r k ≤1 (6) Q k ∈R l×l ,R k ∈R m×m are the covariance matrices of noise w, v, δ kj represents the Kronecker delta function.

9. A computer-readable storage medium, characterized in that The computer-readable storage medium stores one or more programs, which, when executed by a computer device, enable the computer device to execute the state estimation method for autonomous navigation of a spacecraft according to any one of claims 1 to 6.

10. A computer device, characterized in that: include: processor; and a memory arranged to store computer-executable instructions, wherein when the instructions are executed, the processor executes the state estimation method for autonomous navigation of a spacecraft according to any one of claims 1 to 6.

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