Cubature Kalman Filtering Method Suitable for High-dimensional GNSS/INS Deep Coupling

Abstract

The present invention discloses a cubature Kalman filtering method suitable for high-dimensional GNSS/INS deep coupling, including: S1, constructing a high-dimensional GNSS/INS deep coupling filter model; S2, generating an initialization cubature point for the constructed filter model by using standard cubature rules; S3, performing CKF filtering by using novel cubature point update rules. The present invention is suitable for high-dimensional GNSS/INS deep coupling filtering with high precision and high stability.

Description

TECHNICAL FIELD

The present invention relates to filtering techniques, and particularly to a cubature Kalman filtering method suitable for high-dimensional GNSS/INS deep coupling.

BACKGROUND

Nonlinear filtering is a common information fusion technique in the fields of integrated navigation and target tracking. Different from the significant nonlinearity in the target tracking model, the nonlinear error in the integrated navigation is generally weak. However, due to factors such as filter structure, sensor sampling rate, measurement quality and filter update rate, etc., the nonlinearity in the integrated navigation shows stronger time-varying characteristics. Especially when the system has a large initial error and large measurement noise, and the system carrier operates with high maneuver, the nonlinear state estimation based on Kalman filter (KF) is often difficult to work stably. Extended Kalman filter (EKF) is widely used in engineering. It solves the nonlinear problem of the system based on the Jacobian matrix. The state estimation accuracy can reach the first-order level of Taylor series expansion, and good results are obtained when the carrier is stationary or low dynamic, but the estimation accuracy of strong nonlinear system is poor. The sampling point approximation filtering method represented by unscented Kalman filter (UKF) is rapidly applied as soon as it is proposed. It replaces the Jacobian matrix of a single-point in EKF with the expected value of the Jacobian matrix after multipoint sampling, and has a better smoothing effect on the uncertainty caused by nonlinearity. Cubature Kalman filter (CKF) has better accuracy and stability than UKF in the state estimation of high-dimensional systems. Although its mean accuracy can reach the third-order of Taylor series in the nonlinear Gaussian noise model, the estimation accuracy of its variance is only at the first-order of the Taylor series. The UKF adopts the improved UT sampling method to deal with the non-local sampling problem of sampling points in high-dimensional filtering, which improves the accuracy of variance estimation. Because CKF adopts a symmetric sampling strategy, it cannot use the center point weight to capture the high-order information of the variance.

The accuracy problem and consistency problem of state error variance estimation are more prominent in high-dimensional integrated navigation systems. For example, in engineering, the GNSS/INS loose integration generally adopts 15 dimensions, and the tight integration generally adopts 17 or higher dimensions. However, in deep coupling, due to the need to ensure the auxiliary effect of the INS output on the GNSS tracking loop, more inertial device state errors are required. Meanwhile, in order to eliminate the correlation error between the auxiliary information and the measured value, the state error variable of the code tracking loop and carrier tracking loop are required to be increased into the state variable of the integrated filter. Taking the conventional four visible satellites as an example, the state variable of the increased tracking loop is 12 dimensions, so the existing nonlinear filtering method is difficult to apply to the state estimation problem of high-dimensional GNSS/INS deep coupling.

SUMMARY

Objective of the present invention: the present invention provides a cubature Kalman filtering method suitable for high-dimensional GNSS/INS deep coupling in view of the problems existing in the prior art.

Technical solution: the cubature Kalman filtering method suitable for high-dimensional GNSS/INS deep coupling as described in the present invention includes:

S1, constructing a high-dimensional GNSS/INS deep coupling filter model;

S2, generating an initialization cubature point for the constructed filter model by using standard cubature rules;

S3, performing CKF filtering by using novel cubature point update rules.

Further, the step S1 includes:

S11, setting the INS state variable of the subsystem of the INS as x_I and the GNSS state variable of the subsystem of the GNSS as x_G, respectively, wherein,

x _I =[δP δV ψ k _a b _a k _ω b ₁₀₇ ],

x _G =[b _c d _c δp_dll δφ_pll δf_pll],

and wherein in the formula, [δP δV ψ k_a b_a k_ω b₁₀₇ ] are respectively and successively a 3D position error, a velocity error, an attitude error, an accelerometer coefficient error, an accelerometer coefficient null bias, a gyroscope coefficient error and a constant value drift; [b_c d_c δp_dll δφ_pll δf_pll ] are respectively and successively a receiver clock bias, a clock drift, a phase-discrimination pseudo-range error of the code tracking loop, a phase error of the carrier tracking loop, and a frequency error of the carrier tracking loop;

S12, establishing a system model of GNSS/INS deep coupling according to the state variables x_I and x_G as:

$\left[\begin{array}{c}{\stackrel{.}{x}}_I\\ {\stackrel{.}{x}}_G\end{array}\right]=\left[\begin{array}{cccc}{F}_I\left(t\right)& O_{21\times 2}& O_{21\times 4}& O_{21\times 8}\\ O_{2\times 21}& F_G\left(t\right)& O_{2\times 4}& O_{2\times 8}\\ F_{\mathrm{ID}}\left(t\right)& O_{4\times 2}& F_D\left(t\right)& O_{4\times 8}\\ F_{\mathrm{IP}}\left(t\right)& O_{8\times 2}& O_{8\times 4}& F_p\left(t\right)\end{array}\right]\left[\begin{array}{c}{x}_I\\ x_G\end{array}\right]+G\left[\begin{array}{c}{W}_I\\ W_G\end{array}\right],$

wherein in the formula,

$F_I\left(t\right)={\left[\begin{array}{cc}{F}_N\left(t\right)& F_S\left(t\right)\\ O_{9\times 9}& F_M\left(t\right)\end{array}\right]}_{21\times 21},$

F_N (t) is a 9-dimensional inertial navigation basic system matrix and O_3×9,

$F_S\left(t\right)={\left[\begin{array}{cccc}{C}_a& O_{3\times 3}& O_{3\times 3}& O_{3\times 3}\\ O_{3\times 3}& C_b^n\left(t\right)& O_{3\times 3}& O_{3\times 3}\\ O_{3\times 3}& O_{3\times 3}& C_{\omega }& O_{3\times 3}\\ O_{3\times 3}& O_{3\times 3}& O_{3\times 3}& C_b^n\left(t\right)\end{array}\right]}_{12\times 12},$

F_M(t)=[O_12×12], O_i×j is a null matrix with i rows and j columns, C_a and C_ω are an accelerometer coefficient error matrix and a gyroscope coefficient error matrix, respectively, C_bⁿ (t) is an attitude matrix at t moment,

$F_G\left(t\right)=\left[\begin{array}{cc}0& 1\\ 0& -\frac{1}{T_r}\end{array}\right],F_P\left(t\right)={\left[\begin{array}{cc}{O}_{4\times 4}& 2\pi ·I_{4\times 4}\\ K_{\mathrm{pll}}\text{/}T_1·I_{4\times 4}& 2\pi K_{\mathrm{pll}}T_2\text{/}T_1·I_{4\times 4}\end{array}\right]}_{8\times 8},$

T₁ and T₂ are parameters of the loop filter,

$F_{\mathrm{ID}}\left(t\right)={\left[\begin{array}{ccc}-{\overrightarrow{L}}_1^T·C_n^e·W& {\overrightarrow{L}}_1^T·C_n^e& O_{1\times 15}\\ -{\overrightarrow{L}}_2^T·C_n^e·W& {\overrightarrow{L}}_2^T·C_n^e& O_{1\times 15}\\ -{\overrightarrow{L}}_3^T·C_n^e·W& {\overrightarrow{L}}_3^T·C_n^e& O_{1\times 15}\\ -{\overrightarrow{L}}_4^T·C_n^e·W& {\overrightarrow{L}}_4^T·C_n^e& O_{1\times 15}\end{array}\right]}_{4\times 21},F_D\left(t\right)=-{\left[\begin{array}{cccc}{K}_{\mathrm{dll}}& 0& 0& 0\\ 0& K_{\mathrm{dll}}& 0& 0\\ 0& 0& K_{\mathrm{dll}}& 0\\ 0& 0& 0& K_{\mathrm{dll}}\end{array}\right]}_{4\times 4},F_{\mathrm{IP}}\left(t\right)={\left[\begin{array}{c}{O}_{4\times 4}2\pi ·I_{4\times 4}O_{4\times 13}\\ K_{\mathrm{pll}}\text{/}T_1·I_{4\times 4}2\pi K_{\mathrm{pll}}T_2\text{/}T_1·I_{4\times 4}O_{4\times 13}\end{array}\right]}_{8\times 21},$

{dot over (x)}₁ and {dot over (x)}_G are differential terms of x_I and x_G, respectively, G is a system-noise-driven matrix, W_I and W_G are system noise of INS, GNSS and tracking loops, T_r is a relevant time, K_pll is a carrier loop gain, I_i×j is a unit matrix with i rows and j columns, L^T=[L₁^T L₂^T L₃^T L₄^T] is a radial position vector matrix of satellite and receiver, W is a radial velocity error of satellite and receiver caused by INS position calculation errors, C_n^e is a conversion matrix of navigation coordinate system to earth-centered, earth-fixed coordinate system, K_dll is a code loop gain;

S13, establishing a measurement model of GNSS/INS deep coupling according to the system model of GNSS/INS deep coupling as:

θp=p_I−p_G=θp_I−(b_c+v_p+θp_dll),

θ{dot over (p)}={dot over (p)}_I−{dot over (p)}_G=θ{dot over (p)}_I−(d_c+v_{{dot over (p)}}+θ{dot over (p)}_pll),

wherein in the formula, θp and θ{dot over (p)} are observed quantities of the pseudo-range and pseudo-range rate of the satellite channel respectively, p_I and {dot over (p)}_I are the pseudo-range and the pseudo-range rate predicted by INS respectively, p_G and {dot over (p)}_G are the pseudo-range and the pseudo-range rate measured by GNSS respectively, b_c and d_c the clock bias and clock drift of the receiver respectively, v_p and v_{{dot over (p)}}are observation noise of the pseudo-range and pseudo-range rate respectively, θp_dll and θ{dot over (p)}_pll are the pseudo-range errors and pseudo-range rate errors of the tracking loop respectively, v is a measurement noise matrix, that is

$\left[\begin{array}{c}\mathrm{\delta \rho }\\ \delta \stackrel{.}{\rho }\end{array}\right]=H\left(t\right)\left[\begin{array}{c}{x}_I\\ x_G\end{array}\right]+v,H\left(t\right)={\left[\begin{array}{c}{\overrightarrow{L}}^TC_n^eO_{4\times 18}-I_{4\times 1}O_{4\times 1}-I_{4\times 4}O_{4\times 4}O_{4\times 4}\\ O_{4\times 3}O_{4\times 18}O_{4\times 1}{\mathrm{″I}}_{4\times 1}O_{4\times 4}O_{4\times 4}c\text{/}f_{L1}·I_{4\times 4}\end{array}\right]}_{8\times 35},$

f_L1 is a GPS L1 carrier frequency, and c is speed of light.

Further, the step S2 includes:

S21, calculating a state posterior distribution at {circumflex over (x)}_k−1 at k−1 moment by using a high-precision nonlinear filtering method;

wherein, {circumflex over (x)}_k−1˜N({circumflex over (x)}_k−1|k−1,P_k−1|k−1), {circumflex over (x)}_k−1 represents the state posterior distribution at k−1 moment, {circumflex over (x)}_k−1|k−1 represents the state distribution at k−1 moment speculated from the measurement at k−1 moment, P_k−|k−1 represents a covariance of the {circumflex over (x)}_k−1|k−1, x˜N(μ,σ²) represents that x obeys a Gaussian distribution with a mean of μ and a variance of σ²;

S22, generating cubature points required during the prediction process of the system state equation f(x) by using the standard cubature rules;

wherein,

$\{\begin{array}{c}{X}_{i,k-1k-1}^{+}=S_{k-1k-1}{\xi }_i+{\hat{x}}_{k-1k-1}\\ {\xi }_i={\sqrt{n_x}\left[I_{n_x}-I_{n_x}\right]}_i\end{array},$

wherein n the formula, x is a state variable of the GNS SANS deep coupling filtering model, and its dimension is n_x, I_nx is a n_x dimensional unit square matrix, X_{i,k−1|k−1} represents the i^th cubature point at k−1 moment speculated from the measurement at k−1 moment, ξ₁ is the i^th standard cubature point in CKF filtering, i=1, . . . , 2n_x, S_k−1|k−1 satisfies the P_k−1|k−1=S_k−1|k−1(S_k−1|k−1)^T.

Wherein, the high-precision nonlinear filtering method includes: setting the measurement equation of an iterated nonlinear filtering as:

x _k|k ^j+1 =x _k|k ^j+α_j{K_j(z_k−h(x_k|k^j)−(p_k|k−1 ⁻¹p_xz,k|k−1)^Tϑ_j)+ϑ_j},

wherein x_k|k^j is a state estimate value of the j^th measurement iteration at k moment, z_k is a measured value output by a phase discriminator of the multi-satellite channel at k moment, h(·) is a measurement equation of the deep coupling tracking loop, ϑ_j=x_k|k^j−{circumflex over (x)}_k|k−1 is an error between a posterior estimate and a priori estimate in the current iteration, K_j is a Kalman gain of the j^th iteration, α_j is an iterated step-size control coefficient set by accelerating convergence, satisfying 0 <α_j≤p_k|k−1⁻¹ is an inverse of P_k|k−1, P_xz,k|k−1 is an cross-covariance generated during a prediction process, a covariance matrix of state posterior estimation errors takes a result of the last iteration calculation.

Further, step S3 includes:

S31, calculating the state priori distribution at k moment by using the following formula:

$\{\begin{array}{c}{x}_k^{-}\sim N\left({\hat{x}}_{kk-1},P_{kk-1}\right)\\ {\hat{x}}_{kk-1}=\sum _{i=1}^{2n_x}w_if\left(X_{i,k-1k-1}^{+}\right)\\ P_{kk-1}=\sum _{i=1}^{2n_x}w_i\left(f\left(X_{i,k-1k-1}^{+}\right)-{\hat{x}}_{kk-1}\right){\left(f\left(X_{i,k-1k-1}^{+}\right)-{\hat{x}}_{kk-1}\right)}^T+Q_{k-1}\end{array},$

wherein in the formula, x_k⁻represents a state priori estimate at k moment, {circumflex over (x)}_k|k−1 is its mean, P_k|k−1 is its variance, {circumflex over (x)}_k|k−1 represents the state estimate at k moment speculated from the measurement and state at k−1 moment, P_k|k−1 represents the covariance of {circumflex over (x)}_k|k−1, w_i=1/2n_x, and Q_k−1 is the system noise variance matrix;

S32, calculating the cubature point error matrix {tilde over (X)}_i,k|k−1⁻ of the prediction process using the following formula, and defining Ξ_k=P_k|k−1 −Q/_k−1 as an error variance of the Sigma point statistical linear regression (SLR) in a priori PDF approximation process;

{tilde over (X)} _i,k|k−1 ⁻ =X _i,k|k−1 −{tilde over (x)} _k|k−1, 0≤i ≤2n_x,

• • wherein X_i,k|k−1=f(X_{i,k 1k 1} ⁺) is the cubature point after X_{i,k−1|k−1}⁺ propagates through the system equation;

S33, taking the cubature point after the propagation of the system equation as a cubature point of the measurement update process;

S34, using CKF measurement update to calculate a likelihood distribution function of the measured value;

wherein,

$\{\begin{array}{c}{z}_k^{-}\sim N\left({\hat{z}}_{kk-1},P_{\mathrm{zz},kk-1}\right)\\ {\hat{z}}_{kk-1}=\sum _{i=1}^{2n_x}w_ih\left(X_{i,kk-1}\right)\\ P_{\mathrm{zz},kk-1}=\sum _{i=1}^{2n_x}w_i\left(h\left(X_{i,kk-1}\right)-{\hat{z}}_{kk-1}\right){\left(h\left(X_{i,kk-1}\right)-{\hat{z}}_{kk-1}\right)}^T+R_k\end{array},$

wherein in the formula, z_k⁻ represents a measurement likelihood estimate at k moment, {tilde over (z)}_kk−1 is its mean, P_zz,k|k−1 is its variance, {tilde over (z)}_{k|k 1} is a measurement at k moment predicted from the state at k−1 moment, h(x) is the measurement equation, w_i=1/2n_x, and R_k is the measurement noise variance matrix;

S35, calculating a posterior distribution function of a state quantity x;

wherein,

$\{\begin{array}{c}{x}_k^{+}\sim N\left({\hat{x}}_{kk},P_{kk}\right)\\ {\hat{x}}_{kk}={\hat{x}}_{kk-1}+K_k\left(z_k-{\hat{z}}_{kk-1}\right)\\ P_{kk}=P_{kk-1}-K_kP_{\mathrm{zz},kk-1}K_k^T\\ P_{\mathrm{xz},kk-1}=\sum _{i=1}^{2n_x}w_i\left(f\left(x_{i,k-1k-1}\right)-{\hat{x}}_{kk-1}\right){\left(h\left(x_{i,kk-1}\right)-{\hat{z}}_{kk-1}\right)}^T\end{array},$

wherein in the formula, x_k¹ is a posterior estimate of the state variable at k moment, its mean and variance are {tilde over (x)}_k|k and P_k|k, respectively, K_k=P_xz,k|k−1(P_zz,k|k−1)₋₁ is a Kalman gain matrix, P_xz,k|k−1 is a cross-covariance between a posterior estimate of the state variable and the measurement likelihood estimate;

S36, defining an error caused by the Sigma point approximation to the posterior distribution as Ξ_k⁺=P_k|k,w=[w₁ w₂ ] is a weight of a CKF cubature point SLR, the SLR of the priori distribution at k moment should accurately capture the mean and covariance of the state, and consider the effects of system uncertainty and noise, then

{tilde over (X)} _i,k|k−1 ⁻ W=0,

{tilde over (X)} _i,k|k−1 ^−diag( w)X_i,k|k−1^−T=P_k|k−1−Q_k,

wherein in the formula, {tilde over (X)}_i,k|k−1 is a cubature point error matrix of the prediction process, Σ=diag(W) represents that the matrix Σ is constructed using W diagonal elements, in an SLR of a similar posterior distribution, the cubature point can at least accurately match the mean and variance of the state, namely,

{tilde over (X)} _i,k|k ⁺ w=0,

{tilde over (X)} _i,k|k ⁺diag(W)({tilde over (X)}i,k|k⁺)^Ti=P_k|k,

wherein in the formula, {tilde over (X)}_i,k|k⁺ is an updated cubature point;

S37, assuming both Ξ_k⁻ and Ξ_k⁻ are symmetric positive definite matrices, and {tilde over (X)}_i,k|k=B·{tilde over (X)}_i,k|k−1⁻, then Ξ_k⁻=L_k(L_k)^T, μ_k⁺=L_k+1(L_k+1)^T, wherein B is a transformation matrix to be Ξ_k⁺=BL_k(L_k)^TB^T, B=L_k+1Γ(L_k)⁻¹, wherein Γ is an arbitrary orthogonal matrix that satisfies ΓT^T=I_nx, when Γ is taken as unit matrix, B=L_k+1(L_k)⁻¹ is obtained;

S38, obtaining an updated cubature point based on the posterior state estimate mean {circumflex over (x)}_k|k and updated cubature point error matrix as X_i,k|k⁺={circumflex over (x)}_kk+{tilde over (X)}_i,k|k⁺0 ≤i≤2n_x.

Further, an equation of the GNSS code tracking phase-discrimination pseudo-range error p_dll is

δ{dot over (p)}_dll=−K_dllδp_dll+δV_aid+K_dllQ,

wherein θp_dll is an output of the code loop filter {dot over (p)}_dll is a differential of the p_dll, K_dll is a code loop gain, θV_aid is a speed assistance provided by the INS to the code tracking loop after filtering output calibration, Q is the system noise caused by loop thermal noise and interference.

Further, equations of the carrier loop errors are

$\delta {\stackrel{.}{\phi }}_{\mathrm{pll}}=2\pi \left(\delta f_{\mathrm{pll}}+\delta f_{\mathrm{aid}}\right),\delta {\stackrel{.}{f}}_{\mathrm{pll}}=\left[2\pi \frac{T_2}{T_1}\left(\delta f_{\mathrm{pll}}+\delta f_{\mathrm{aid}}\right)+\frac{\delta {\phi }_{\mathrm{pll}}}{T_1}\right]·K_{\mathrm{pll}},$

wherein, φ_dll is the phase error of the carrier loop, f_pll is the frequency error of the carrier loop, K_pll is a loop gain, and θf_aid is a Doppler frequency error assistance provided by the INS to the carrier tracking loop after filtering output calibration, T₁ and T₂ are parameters of the loop

filter, that is, the transfer function of the loop low-pass filter is

$F\left(s\right)=\frac{T_2s+1}{T_1s}.$

Advantageous effects: compared with the prior art, the present invention has the following significant advantages:

(1) initializing and generating the cubature point by using a higher precision nonlinear filtering method;

(2) updating the cubature point based on matrix transformation, which improves the consistency of the prediction process variance in the state estimation process, and improves the stability of the filter;

(3) using the state error parameter of the tracking loop as the state parameter of the integrated filter to eliminate the correlation between the filter measurement and the tracking loop errors, which improves the accuracy and stability of the INS auxiliary quantity calibration;

(4) using two kinds of strategies to generate the cubature point of CKF, that is, the cubature point is generated by using a time-consuming and high-precision method during initialization, and the subsequent cubature point update process is realized based on the variance of the state estimation error at the previous moment, which improves the utilization of the cubature point.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a system block diagram according to one embodiment of the present invention.

FIG. 2 is a schematic diagram of high-dimensional GNSS/INS deep coupling filtering model.

DETAILED DESCRIPTION OF THE EMBODIMENTS

As shown in FIG. 1, the cubature Kalman filtering method suitable for high-dimensional GNSS/INS deep coupling of the present embodiment includes the following steps:

S1, constructing a high-dimensional GNSS/INS deep coupling filter model, as shown in FIG. 2.

Step S1 specifically includes:

S11, setting the INS state variable of the subsystem of the INS as x_I and the GNSS state variable of the subsystem of the GNSS as x_G, respectively, wherein,

θp=p_I−p_G=θp_I−(b_c+v_p+θp_dll),

θ{dot over (p)}={dot over (p)}_I−{dot over (p)}_G=θ{dot over (p)}_I−(d_c+v_{{dot over (p)}}+θ{dot over (p)}_pll),

and wherein in the formula, [δP δV ψ k_a b_a k_ω b₁₀₇ ] are respectively and successively a 3D position error, a velocity error, an attitude error, an accelerometer coefficient error, an accelerometer coefficient null bias, a gyroscope coefficient error and a constant value drift; [b_c, d_c, δp_dll δf_pll are respectively and successively a receiver clock bias, a clock drift, a phase-discrimination pseudo-range error of the code tracking loop, a phase error of the carrier tracking loop, and a frequency error of the carrier tracking loop.

Wherein an equation of the GNSS code tracking phase-discrimination pseudo-range error p_dll is

δ{dot over (p)} _dll =−K _dll δp _dll +δV _aid +K _dll Q,

wherein δp_dll is an output of the code loop filter, {dot over (p)}_dll is a differential of the p_dll K_dll is a code loop gain, δV_aid is a speed assistance provided by the INS to the code tracking loop after filtering output calibration, Q is the system noise caused by loop thermal noise and interference.

Wherein equations of the carrier loop errors are

$\delta {\stackrel{.}{\phi }}_{\mathrm{pll}}=2\pi \left(\delta f_{\mathrm{pll}}+\delta f_{\mathrm{aid}}\right),\delta {\stackrel{.}{f}}_{\mathrm{pll}}=\left[2\pi \frac{T_2}{T_1}\left(\delta f_{\mathrm{pll}}+\delta f_{\mathrm{aid}}\right)+\frac{\delta {\phi }_{\mathrm{pll}}}{T_1}\right]·K_{\mathrm{pll}},$

wherein, φ_pll is the phase error of the carrier loop, f_pll is the frequency error of the carrier loop, K_dll is a loop gain, and δf_aid is a Doppler frequency error assistance provided by the INS to the carrier tracking loop after filtering output calibration, T₁ and T2 are parameters of the loop filter, that is, the transfer function of the loop low-pass filter is

$F\left(s\right)=\frac{T_2s+1}{T_1s}.$

S12, establishing a system model of GNSS/INS deep coupling according to the state variables x_I, and x_G as:

$\left[\begin{array}{c}{\stackrel{.}{x}}_I\\ {\stackrel{.}{x}}_G\end{array}\right]=\left[\begin{array}{cccc}{F}_I\left(t\right)& O_{21\times 2}& O_{21\times 4}& O_{21\times 8}\\ O_{2\times 21}& F_G\left(t\right)& O_{2\times 4}& O_{2\times 8}\\ F_{\mathrm{ID}}\left(t\right)& O_{4\times 2}& F_D\left(t\right)& O_{4\times 8}\\ F_{\mathrm{IP}}\left(t\right)& O_{8\times 2}& O_{8\times 4}& F_p\left(t\right)\end{array}\right]\left[\begin{array}{c}{x}_I\\ x_G\end{array}\right]+G\left[\begin{array}{c}{W}_I\\ W_G\end{array}\right],$

wherein in the formula,

$F_I\left(t\right)={\left[\begin{array}{cc}{F}_N\left(t\right)& F_S\left(t\right)\\ O_{9\times 9}& F_M\left(t\right)\end{array}\right]}_{21\times 21},$

F_N (t) is a 9-dimensional inertial navigation basic System matrix and O_3×9,

$F_S\left(t\right)={\left[\begin{array}{cccc}{C}_a& O_{3\times 3}& O_{3\times 3}& O_{3\times 3}\\ O_{3\times 3}& C_b^n\left(t\right)& O_{3\times 3}& O_{3\times 3}\\ O_{3\times 3}& O_{3\times 3}& C_{\omega }& O_{3\times 3}\\ O_{3\times 3}& O_{3\times 3}& O_{3\times 3}& C_b^n\left(t\right)\end{array}\right]}_{12\times 12},$

F_M(t)=[O_12×12], O_i×j is a null matrix with i rows and j columns, C_a and C_ω are an accelerometer coefficient error matrix and a gyroscope coefficient error matrix respectively, C_bⁿ(t) is an attitude matrix at t moment,

$F_G\left(t\right)=\left[\begin{array}{cc}0& 1\\ 0& -\frac{1}{T_r}\end{array}\right],F_P\left(t\right)={\left[\begin{array}{cc}{O}_{4\times 4}& 2\pi ·I_{4\times 4}\\ K_{\mathrm{pll}}\text{/}T_1·I_{4\times 4}& 2\pi K_{\mathrm{pll}}T_2\text{/}T_1·I_{4\times 4}\end{array}\right]}_{8\times 8},$

T₁ and T₂ are parameters of the loop filter,

$F_{\mathrm{ID}}\left(t\right)={\left[\begin{array}{ccc}-{\overrightarrow{L}}_1^T·C_n^e·W& {\overrightarrow{L}}_1^T·C_n^e& O_{1\times 15}\\ -{\overrightarrow{L}}_2^T·C_n^e·W& {\overrightarrow{L}}_2^T·C_n^e& O_{1\times 15}\\ -{\overrightarrow{L}}_3^T·C_n^e·W& {\overrightarrow{L}}_3^T·C_n^e& O_{1\times 15}\\ -{\overrightarrow{L}}_4^T·C_n^e·W& {\overrightarrow{L}}_4^T·C_n^e& O_{1\times 15}\end{array}\right]}_{4\times 12},F_D\left(t\right)=-{\left[\begin{array}{cccc}{K}_{\mathrm{dll}}& 0& 0& 0\\ 0& K_{\mathrm{dll}}& 0& 0\\ 0& 0& K_{\mathrm{dll}}& 0\\ 0& 0& 0& K_{\mathrm{dll}}\end{array}\right]}_{4\times 4},F_{\mathrm{IP}}\left(t\right)={\left[\begin{array}{ccc}{O}_{4\times 4}& 2\pi ·I_{4\times 4}& O_{4\times 13}\\ K_{\mathrm{pll}}\text{/}T_1·I_{4\times 4}& 2\pi K_{\mathrm{pll}}T_2\text{/}T_1·I_{4\times 4}& O_{4\times 13}\end{array}\right]}_{8\times 21},$

{dot over (x)}_I and {dot over (x)}_G are differential terms of x_I and x_G respectively, G is a system-noise-driven matrix, W_I and W_G are system noise of INS, GNSS and tracking loops, T_r is a relevant time, K_pll is a carrier loop gain, I_i×j is a unit matrix with i rows and j columns, L^T=[L₁^TL₂^TL₃^TL₄^Tis a radial position vector matrix of satellite and receiver, W is a radial velocity error of satellite and receiver caused by INS position calculation errors, C_n^e is a conversion matrix of navigation coordinate system to earth-centered, earth-fixed coordinate system, K_dll is a code loop gain.

S13, establishing a measurement model of GNSS/INS deep coupling according to the system model of GNSS/INS deep coupling as:

θp=p_I−p_G=θp_I−(b_c+v_p+θp_dll),

θ{dot over (p)}={dot over (p)}_I−{dot over (p)}_G=θ{dot over (p)}_I−(d_c+v_{{dot over (p)}}+θ{dot over (p)}_pll),

wherein in the formula, δp and {dot over (p)} are observed quantities of the pseudo-range and pseudo-range rate of the satellite channel respectively, p_I and {dot over (p)}_I are the pseudo-range and the pseudo-range rate predicted by INS respectively, p_G and {dot over (p)}_G are the pseudo-range and the pseudo-range rate measured by GNSS respectively, b_c and d_c are the clock bias and clock drift of the receiver respectively, v_p and v_p are observation noise of the pseudo-range and pseudo-range rate respectively, θp_dll and δp_pll are the pseudo-range errors and pseudo-range rate errors of the tracking loop respectively, v is a measurement noise matrix, that is

$\left[\begin{array}{c}\delta \rho \\ \delta \stackrel{.}{\rho }\end{array}\right]=H\left(t\right)\left[\begin{array}{c}{x}_I\\ x_G\end{array}\right]+v,H\left(t\right)={\left[\begin{array}{ccccccc}{\overrightarrow{L}}^TC_n^e& O_{4\times 18}& -I_{4\times 1}& O_{4\times 1}& -I_{4\times 4}& O_{4\times 4}& O_{4\times 4}\\ O_{4\times 3}& O_{4\times 18}& O_{4\times 1}& -I_{4\times 1}& O_{4\times 4}& O_{4\times 4}& c/f_{L1}·I_{4\times 4}\end{array}\right]}_{8\times 35},$

f_L1 is a GPS L1 carrier frequency, and c is speed of light.

S2, generating an initialization cubature point for the constructed filter model by using standard cubature rules.

Step S2 specifically includes:

S21, calculating a state posterior distribution {circumflex over (x)}_{k 1} at k−1 moment by using a high-precision nonlinear filtering method;

wherein, {circumflex over (x)}_k−1˜N({circumflex over (x)}_k−1|k−1, P_k−1|k−1 {circumflex over (x)}_k−1 represents the state posterior distribution at k−1 moment, {circumflex over (x)}_k−1|k−1 represents the state distribution at k−1 moment speculated from the measurement at k−1 moment, P_k−1|k−1 represents a covariance of the {circumflex over (x)}_k−1|k−1, x˜N(μ, σ² ) represents that x obeys a Gaussian distribution with a mean of μ and a variance of σ².

Wherein, the high-precision nonlinear filtering method includes: setting the measurement equation of an iterated nonlinear filtering as:

x _k|k ^j+1 =x _k|k ^j+α_j{K_j(z_k−h(x_k|k^j)−(p_k|k−1 ⁻¹p_xz,k|k−1)^Tϑ_j)+ϑ_j},

where x_k|k^j is a state estimate value of the j^th measurement iteration at k moment, z_k is a measured value output by a phase discriminator of the multi-satellite channel at k moment, h(·) is a measurement equation of the deep coupling tracking loop, ϑ_j=x_k|k^j−{circumflex over (x)}_k|k−1 is an error between a posterior estimate and a priori estimate in the current iteration, K_j is a Kalman gain of the j^th iteration, a, is an iterated step-size control coefficient set by accelerating convergence, satisfying 0<α_j≤1, P_k|k−1⁻¹ is an inverse of P_k|k−1, P_{xz, k|k−1} is an cross-covariance generated during the prediction process, the covariance matrix of the state posterior estimation errors takes the result of the last iteration calculation.

S22, generating cubature points required during the prediction process of the system state equation f(x) by using the standard cubature rules;

wherein,

$\{\begin{array}{c}{X}_{i,k-1k-1}^{+}=S_{k-1k-1}{\xi }_i+{\hat{x}}_{k-1k-1}\\ {\xi }_i={\sqrt{n_x}\left[I_{n_x}-I_{n_x}\right]}_i\end{array},$

wherein in the formula, x is a state variable of the GNSS/INS deep coupling filtering model, and its dimension is n_x, I_nx, is a n_x dimensional unit square matrix, X_{i,k 1|k}⁺ represents the i^th cubature point at k−1 moment speculated from the measurement at k−1 moment, ξ₁ is the i^th standard cubature point in CKF filtering, i=1, . . . , 2n_x, S_k−1|k−1satisfies the P_k−1|k−1=S_k−1|k−1 (S_k−1|k−1)^T.

S3, performing CKF filtering by using novel cubature point update rules.

Step S3 includes:

S31, calculating the state priori distribution at k moment by using the following formula:

$\{\begin{array}{c}{x}_k^{-}\sim N\left({\hat{x}}_{kk-1},P_{kk-1}\right)\\ {\hat{x}}_{kk-1}=\sum _{i=1}^{2n_x}w_if\left(X_{i,k-1k-1}^{+}\right)\\ P_{kk-1}=\sum _{i=1}^{2n_x}w_i\left(f\left(X_{i,k-1k-1}^{+}\right)-{\hat{x}}_{kk-1}\right){\left(f\left(X_{i,k-1k-1}^{+}\right)-{\hat{x}}_{kk-1}\right)}^T+Q_{k-1}\end{array},$

wherein in the formula, x_k⁻represents a state priori estimate at k moment, {circumflex over (x)}_k|k−1 is its mean, P_k|k−1 is its variance, {circumflex over (x)}_k|k−1 represents the state estimate at k moment speculated from the measurement and state at k−1 moment, P_k|k−1 represents the covariance of {circumflex over (x)}_k|k−1, w_i=1/2n_x, and Q_k−1 is the system noise variance matrix;

S32, calculating the cubature point error matrix X_i,k|k−1 of the prediction process using the following formula, and defining Ξ _k^−=P_k|k−1 Q_k−1 as an error variance of the Sigma point statistical linear regression (SLR) in a priori PDF approximation process;

{tilde over (X)} _i,k|k−1 ⁻ =X _i,k|k−1 −{tilde over (x)} _k|k−1, 0≤i≤2n_x,

wherein X_i,k|k−1=f(X_{i,k 1k 1} ⁺) is the cubature point after X_{i,k−1|k−1}⁺ propagates through the system equation;

S33, taking the cubature point after the propagation of the system equation as a cubature point of the measurement update process;

S34, using CKF measurement update to calculate a likelihood distribution function of the measured value;

wherein,

$\{\begin{array}{c}{z}_k^{-}\sim N\left({\hat{z}}_{kk-1},P_{\mathrm{zz},kk-1}\right)\\ {\hat{z}}_{kk-1}=\sum _{i=1}^{2n_x}w_ih\left(X_{i,kk-1}\right)\\ P_{\mathrm{zz},kk-1}=\sum _{i=1}^{2n_x}w_i\left(h\left(X_{i,kk-1}\right)-{\hat{z}}_{kk-1}\right){\left(h\left(X_{i,kk-1}\right)-{\hat{z}}_{kk-1}\right)}^T+R_k\end{array},$

wherein in the formula, z_k⁻ represents a measurement likelihood estimate at k moment{tilde over (z)}_{k k−1} is its mean, P_zz,k|k−1 is its variance, {tilde over (z)}_k|k−1 is a measurement at k moment predicted from the state at k−1 moment, h(x) is the measurement equation, w_i=1/2n_x, and R_k is the measurement noise variance matrix;

S35, calculating a posterior distribution function of a state quantity x;

wherein,

$\{\begin{array}{c}{x}_k^{+}\sim N\left({\hat{x}}_{kk},P_{kk}\right)\\ {\hat{x}}_{kk}={\hat{x}}_{kk-1}+K_k\left(z_k-{\hat{z}}_{kk-1}\right)\\ P_{kk}=P_{kk-1}-K_kP_{\mathrm{zz},kk-1}K_k^T\\ P_{\mathrm{zk},kk-1}=\sum _{i=1}^{2n_x}w_i(f(X_{i,k-1k-1}\left(-{\hat{x}}_{kk-1}\right){\left(h\left(X_{i,kk-1}\right)-{\hat{z}}_{kk-1}\right)}^T\end{array},$

wherein in the formula, x_k⁺ is a posterior estimate of the state variable at k moment, its mean and variance are {tilde over (x)}_k|k, and P_k|k, respectively, K_k=P_xz,k|k−1(P_{zz,k k−1})⁻¹ is a Kalman gain matrix, P_xz,k|k−1 is a cross-covariance between a posterior estimate of the state variable and the measured likelihood estimate;

S36, defining an error caused by the Sigma point approximation to the posterior distribution as Ξ_k⁺=P_k|k,w=[w₁ w₂ ] is a weight of a CKF cubature point SLR, the SLR of the priori distribution at k moment should accurately capture the mean and covariance of the state, and consider the effects of system uncertainty and noise, then

{tilde over (X)} _i,k|k−1 ⁻ W=0,

{tilde over (X)} _i,k|k−1 ^−diag( w)X_i,k|k−1^−T=P_k|k−1−Q_k,

wherein in the formula, {tilde over (X)}_i,k|k−1 is a cubature point error matrix of the prediction process, Σ=diag(W) represents that the matrix Σ is constructed using W diagonal elements, in an SLR of a similar posterior distribution, the cubature point can at least accurately match the mean and variance of the state, namely,

{tilde over (X)} _i,k|k ⁺ w=0,

{tilde over (X)} _i,k|k ⁺diag(W)({tilde over (X)}i,k|k⁺)^Ti=P_k|k,

wherein in the formula, {tilde over (X)}_i,k|k⁺ is an updated cubature point;

S37, assuming both Ξ_k⁻ and Ξ_k⁻ are symmetric positive definite matrices, and {tilde over (X)}_i,k|k=B·{tilde over (X)}_i,k|k−1⁻, then Ξ_k⁻=L_k(L_k)^T, μ_k⁺=L_k+1(L_k+1)^T, wherein B is a transformation matrix to be solved, {tilde over (X)}_{i, kk}⁺is an updated cubature point error matrix; further Ξ_k⁺=BL_k(L_k)^TB^T, B=L_k+1Γ(L_k)⁻¹, wherein Γ is an arbitrary orthogonal matrix that satisfies ΓΓ^T=I_nx, when Γ is taken as unit matrix, B=L_k+1(L_k)⁻¹ is obtained;

S38, obtaining an updated cubature point based on the posterior state estimate mean i_k|k and updated cubature point error matrix as X_i,k|k⁺={circumflex over (x)}_kk+{tilde over (X)}_i,k|k^{+b 0 ≤i≤}2n_x.

The above are only preferred embodiments of the present invention, and the scope of the present invention is not limited thereto. Therefore, equivalent changes made according to the claims of the present invention are still within the scope of the present invention.

Claims

1. A cubature Kalman filtering method suitable for high-dimensional GNSS/INS deep coupling, wherein the cubature Kalman filtering method comprises: S1, constructing a high-dimensional GNSS/INS deep coupling filter model to obtain a constructed filter model;S2, generating an initialization cubature point for the constructed filter model by using standard cubature rules;S3, performing CKF filtering on the constructed filter model by using novel cubature point update rules.

2. The cubature Kalman filtering method suitable for high-dimensional GNSS/INS deep coupling according to claim 1, wherein the step S1 comprises: S11, setting an INS state variable of a subsystem of an INS as xI and a GNSS state variable of a subsystem of a GNSS as xG , respectively, wherein, xI=[δP δV ψ ka ba kω b107 ],xG=[bc dc δpdll δφpll δfpll],and wherein in the formula, [δP δV ψ ka ba kω b107 ] are respectively and successively a 3D position error, a velocity error, an attitude error, an accelerometer coefficient error, an accelerometer coefficient null bias, a gyroscope coefficient error and a constant value drift; [bc dc δpdll δφpll δfpll ] are respectively and successively a receiver clock bias, a clock drift, a phase-discrimination pseudo-range error of a code tracking loop, a phase error of a carrier tracking loop, and a frequency error of a carrier tracking loop;S12, establishing a system model of the high-dimensional GNSS/INS deep coupling according to the INS state variable xI and the GNSS state variable xG as:

3. The cubature Kalman filtering method suitable for high-dimensional GNSS/INS deep coupling according to claim 1, wherein the step S2 comprises: S21, calculating a state posterior distribution {circumflex over (x)}k−1 at k−1 moment by using a high-precision nonlinear filtering method;wherein, {circumflex over (x)}k−1˜N({circumflex over (x)}k−1|k−1,Pk−1|k−1), {circumflex over (x)}k−1 represents the state posterior distribution at k−1 moment, {circumflex over (x)}k−1|k−1 represents the state distribution at k−1 moment speculated from the measurement at k−1 moment, Pk−|k−1 represents a covariance of the, {circumflex over (x)}k−1|k−1, x˜N(μ,σ2) represents that x obeys a Gaussian distribution with a mean of μ and a variance of σ2;S22, generating cubature points required during the a prediction process of the a system state equation f(x) by using the standard cubature rules;wherein,

4. The cubature Kalman filtering method suitable for high-dimensional GNSS/INS deep coupling according to claim 3, wherein the high-precision nonlinear filtering method comprises: setting the a measurement equation of an iterated nonlinear filtering as: xk|kj+1=xk|kj+αj{Kj(zk−h(xk|kj)−(pk|k−1 −1pxz,k|k−1)Tϑj)+ϑj},where xk|kj is a state estimate value of the jth measurement iteration at k moment, zk is a measured value output by a phase discriminator of a multi-satellite channel at k moment, h(·) is a measurement equation of a deep coupling tracking loop, ϑj=xk|kj−{circumflex over (x)}k|k−1 is an error between a posterior estimate and a priori estimate in the current iteration, Kj is a Kalman gain of the jth iteration, αj is an iterated step-size control coefficient set by accelerating convergence, satisfying 0 <αj≤Pk|k−1−1 is an inverse of Pk|k−1 , Pxz,k|k−1 is an cross-covariance generated during a prediction process, a covariance matrix of state posterior estimation errors takes a result of the last iteration calculation.

5. The cubature Kalman filtering method suitable for high-dimensional GNSS/INS deep coupling according to claim 1, wherein the step S3 comprises: S31, calculating the a state priori distribution at k moment by using the following formula:

6. The cubature Kalman filtering method suitable for high-dimensional GNSS/INS deep coupling according to claim 2, wherein an equation of the phase-discrimination pseudo-range error pdll of the code tracking loop is δ{dot over (p)}dll=−Kdllδpdll+δVaid+KdllQ, wherein δpdll is an output of a code loop filter, {dot over (p)}dll is a differential of the pdll, Kdll is a code loop gain, δVaid is a speed assistance provided by the INS to the code tracking loop after a filtering output calibration, Q is system noise caused by loop thermal noise and interferences.

7. The cubature Kalman filtering method suitable for high-dimensional GNSS/INS deep coupling according to claim 2, wherein equations of errors of the carrier tracking loop are