This application is the national phase entry of International Application No. PCT/CN2017/080383, filed on Apr. 13, 2017, which is based upon and claims priority to Chinese Patent Application No. 201610575270.4, filed on Jul. 19, 2016, the entire contents of which are incorporated herein by reference.
The present invention relates to filtering techniques, and particularly to a cubature Kalman filtering method suitable for high-dimensional GNSS/INS deep coupling.
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.
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} b_{107 }] 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:
$$
wherein in the formula,
$$
F_{N }(t) is a 9-dimensional inertial navigation basic system matrix and O_{3×9},
$$
F_{M}(t)=[O_{12×12}], O_{i×j }is a null matrix with i rows and j columns, C_{a }and C_{ω} are an accelerometer coefficient error matrix and a gyroscope coefficient error matrix, respectively, C_{b}^{n }(t) is an attitude matrix at t moment,
$$
T_{1 }and T_{2 }are parameters of the loop filter,
$$
{dot over (x)}_{1 }and {dot over (x)}_{G }are differential terms of x_{I }and x_{G}, respectively, G is a system-noise-driven matrix, W_{I }and W_{G }are system noise of INS, GNSS and tracking loops, T_{r }is a relevant time, K_{pll }is a carrier loop gain, I_{i×j }is a unit matrix with i rows and j columns, {right arrow over (L)}^{T}=[{right arrow over (L)}_{1}^{T }{right arrow over (L)}_{2}^{T }{right arrow over (L)}_{3}^{T }{right arrow over (L)}_{4}^{T}] is a radial position vector matrix of satellite and receiver, W is a radial velocity error of satellite and receiver caused by INS position calculation errors, C_{n}^{e }is a conversion matrix of navigation coordinate system to earth-centered, earth-fixed coordinate system, K_{dll }is a code loop gain;
S13, establishing a measurement model of GNSS/INS deep coupling according to the system model of GNSS/INS deep coupling as:
θp=p_{I}−p_{G}=θp_{I}−(b_{c}+v_{p}+θp_{dll}),
θ{dot over (p)}={dot over (p)}_{I}−{dot over (p)}_{G}=θ{dot over (p)}_{I}−(d_{c}+v_{{dot over (p)}}+θ{dot over (p)}_{pll}),
wherein in the formula, θp and θ{dot over (p)} are observed quantities of the pseudo-range and pseudo-range rate of the satellite channel respectively, p_{I }and {dot over (p)}_{I }are the pseudo-range and the pseudo-range rate predicted by INS respectively, p_{G }and {dot over (p)}_{G }are the pseudo-range and the pseudo-range rate measured by GNSS respectively, b_{c }and d_{c }the clock bias and clock drift of the receiver respectively, v_{p }and v_{{dot over (p)}}are observation noise of the pseudo-range and pseudo-range rate respectively, θp_{dll }and θ{dot over (p)}_{pll }are the pseudo-range errors and pseudo-range rate errors of the tracking loop respectively, v is a measurement noise matrix, that is
$$
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,σ^{2}) represents that x obeys a Gaussian distribution with a mean of μ and a variance of σ^{2};
S22, generating cubature points required during the prediction process of the system state equation f(x) by using the standard cubature rules;
wherein,
$$
wherein n the formula, x is a state variable of the GNS SANS deep coupling filtering model, and its dimension is n_{x}, I_{n}_{x }is a n_{x }dimensional unit square matrix, X_{i,k−1|k−1 }represents the i^{th }cubature point at k−1 moment speculated from the measurement at k−1 moment, ξ_{1 }is the i^{th }standard cubature point in CKF filtering, i=1, . . . , 2n_{x}, S_{k−1|k−1 }satisfies the P_{k−1|k−1}=S_{k−1|k−1}(S_{k−1|k−1})^{T}.
Wherein, the high-precision nonlinear filtering method includes: setting the measurement equation of an iterated nonlinear filtering as:
x
_{k|k}
^{j+1}
=x
_{k|k}
^{j}+α_{j}{K_{j}(z_{k}−h(x_{k|k}^{j})−(p_{k|k−1 }^{−1}p_{xz,k|k−1})^{T}ϑ_{j})+ϑ_{j}},
wherein x_{k|k}^{j }is a state estimate value of the j^{th }measurement iteration at k moment, z_{k }is a measured value output by a phase discriminator of the multi-satellite channel at k moment, h(·) is a measurement equation of the deep coupling tracking loop, ϑ_{j}=x_{k|k}^{j}−{circumflex over (x)}_{k|k−1 }is an error between a posterior estimate and a priori estimate in the current iteration, K_{j }is a Kalman gain of the j^{th }iteration, α_{j }is an iterated step-size control coefficient set by accelerating convergence, satisfying 0 <α_{j}≤p_{k|k−1}^{−1 }is an inverse of P_{k|k−1}, P_{xz,k|k−1 }is an cross-covariance generated during a prediction process, a covariance matrix of state posterior estimation errors takes a result of the last iteration calculation.
Further, step S3 includes:
S31, calculating the state priori distribution at k moment by using the following formula:
$$
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},
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,
$$
wherein in the formula, z_{k}^{−} represents a measurement likelihood estimate at k moment, {tilde over (z)}_{kk−1 }is its mean, P_{zz,k|k−1 }is its variance, {tilde over (z)}_{k|k 1 }is a measurement at k moment predicted from the state at k−1 moment, h(x) is the measurement equation, w_{i}=1/2n_{x}, and R_{k }is the measurement noise variance matrix;
S35, calculating a posterior distribution function of a state quantity x;
wherein,
$$
wherein in the formula, x_{k}^{1 }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=[w_{1 }w_{2 }] 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}, Ξ_{k}^{+}=L_{k+1}(L_{k+1})^{T}, wherein B is a transformation matrix to be solved, {tilde over (X)}_{i,k|k}^{+}is an updated cubature point error matrix; further Ξ_{k}^{+}=BL_{k}(L_{k})^{t}B^{T}, B=L_{k+1}Γ(L_{k})^{−1}, wherein Γ is an arbitrary orthogonal matrix that satisfies ΓT^{T}=I_{n}_{x}, 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)}_{k|k}+{tilde over (X)}_{i,k|k}^{+}, 0 ≤i≤2n_{x}.
Further, an equation of the GNSS code tracking phase-discrimination pseudo-range error ρ_{dll }is
δ{dot over (ρ)}_{dll}=−K_{dll}δρ_{dll}+δV_{aid}+K_{dll}Q,
wherein θρ_{dll }is an output of the code loop filter {dot over (ρ)}_{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
$$
wherein, ϕ_{dll }is the phase error of the carrier loop, f_{pll }is the frequency error of the carrier loop, K_{pll }is a loop gain, and θf_{aid }is a Doppler frequency error assistance provided by the INS to the carrier tracking loop after filtering output calibration, T_{1 }and T_{2 }are parameters of the loop
filter, that is, the transfer function of the loop low-pass filter is
$$
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.
As shown in
S1, constructing a high-dimensional GNSS/INS deep coupling filter model, as shown in
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,
x_{I}=[θP θV ψ k_{α}b_{α} k_{ω }b_{ω], }
x_{G}=[b_{c }d_{c }δρ_{dll }δϕ_{pll }δf_{pll}],
and wherein in the formula, [δP δV ψ k_{a }b_{a }k_{ω} b_{107 }] 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 }δρ_{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.
Wherein an equation of the GNSS code tracking phase-discrimination pseudo-range error ρ_{dll }is
δ{dot over (ρ)}_{dll}=−K_{dll}δρ_{dll}+δV_{aid}+K_{dll}Q,
wherein δρ_{dll }is an output of the code loop filter, {dot over (ρ)}_{dll }is a differential of the ρ_{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
$$
wherein, ϕ_{pll }is the phase error of the carrier loop, f_{pll }is the frequency error of the carrier loop, K_{dll }is a loop gain, and δf_{aid }is a Doppler frequency error assistance provided by the INS to the carrier tracking loop after filtering output calibration, T_{1 }and T2 are parameters of the loop filter, that is, the transfer function of the loop low-pass filter is
$$
S12, establishing a system model of GNSS/INS deep coupling according to the state variables x_{I}, and x_{G }as:
$$
wherein in the formula,
$$
F_{N }(t) is a 9-dimensional inertial navigation basic System matrix and O_{3×9},
$$
F_{M}(t)=[O_{12×12}], O_{i×j }is a null matrix with i rows and j columns, C_{α} and C_{ω} are an accelerometer coefficient error matrix and a gyroscope coefficient error matrix respectively, C_{b}^{n}(t) is an attitude matrix at t moment,
$$
T_{1 }and T_{2 }are parameters of the loop filter,
$$
{dot over (x)}_{I }and {dot over (x)}_{G }are differential terms of x_{I }and x_{G }respectively, G is a system-noise-driven matrix, W_{I }and W_{G }are system noise of INS, GNSS and tracking loops, T_{r }is a relevant time, K_{pll }is a carrier loop gain, I_{i×j }is a unit matrix with i rows and j columns, {right arrow over (L)}^{T}=[{right arrow over (L)}_{1}^{T} {right arrow over (L)}_{2}^{T} {right arrow over (L)}_{3}^{T} {right arrow over (L)}_{4}^{T}] is a radial position vector matrix of satellite and receiver, W is a radial velocity error of satellite and receiver caused by INS position calculation errors, C_{n}^{e }is a conversion matrix of navigation coordinate system to earth-centered, earth-fixed coordinate system, K_{dll }is a code loop gain.
S13, establishing a measurement model of GNSS/INS deep coupling according to the system model of GNSS/INS deep coupling as:
δρ=ρ_{I}−ρ_{G}=δρ_{I}−(b_{c}+v_{ρ}+δρ_{dll}),
δ{dot over (ρ)}={dot over (ρ)}_{I}−{dot over (ρ)}_{G}=δ{dot over (ρ)}_{I}−(d_{c}+v_{{dot over (ρ)}}+δ{dot over (ρ)}_{pll}),
wherein in the formula, δρ and δ{dot over (ρ)} are observed quantities of the pseudo-range and pseudo-range rate of the satellite channel respectively, ρ_{I }and {dot over (ρ)}_{I }are the pseudo-range and the pseudo-range rate predicted by INS respectively, ρ_{G }and {dot over (ρ)}_{G }are the pseudo-range and the pseudo-range rate measured by GNSS respectively, b_{c }and d_{c }are the clock bias and clock drift of the receiver respectively, v_{ρ} and v_{ρ} are observation noise of the pseudo-range and pseudo-range rate respectively, δρ_{dll }and δ{dot over (ρ)}_{pll }are the pseudo-range errors and pseudo-range rate errors of the tracking loop respectively, v is a measurement noise matrix, that is
$$
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, σ^{2 }) represents that x obeys a Gaussian distribution with a mean of μ and a variance of σ^{2}.
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 }^{−1}p_{xz,k|k−1})^{T}ϑ_{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, α_{j } 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,
$$
wherein in the formula, x is a state variable of the GNSS/INS deep coupling filtering model, and its dimension is n_{x}, I_{n}_{x}, is a n_{x }dimensional unit square matrix, X_{i,k 1|k}^{+} represents the i^{th }cubature point at k−1 moment speculated from the measurement at k−1 moment, ξ_{i }is the i^{th }standard cubature point in CKF filtering, i=1, . . . , 2n_{x}, S_{k−1|k−1}satisfies the P_{k−1|k−1}=S_{k−1|k−1 }(S_{k−1|k−1})^{T}.
S3, performing CKF filtering by using novel cubature point update rules.
Step S3 includes:
S31, calculating the state priori distribution at k moment by using the following formula:
$$
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}^{−}=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,
$$
wherein in the formula, z_{k}^{−} represents a measurement likelihood estimate at k moment, {tilde over (z)}_{k|k−1 }is its mean, P_{zz,k|k−1 }is its variance, {tilde over (z)}_{k|k−1 }is a measurement at k moment predicted from the state at k−1 moment, h(x) is the measurement equation, w_{i}=1/2n_{x}, and R_{k }is the measurement noise variance matrix;
S35, calculating a posterior distribution function of a state quantity x;
wherein,
$$
wherein in the formula, x_{k}^{+} is a posterior estimate of the state variable at k moment, its mean and variance are {circumflex 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=[w_{1 }. . . w_{2n}_{x }] 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,k|k}^{+}is an updated cubature point error matrix; further Ξ_{k}^{+}=BL_{k}(L_{k})^{T}B^{T}, B=L_{k+1}Γ(L_{k})^{−1}, wherein Γ is an arbitrary orthogonal matrix that satisfies ΓΓ^{T}=I_{n}_{x}, 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)}_{k|k}+{tilde over (X)}_{i,k|k}^{+}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.
Number | Date | Country | Kind |
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2016 1 0575270 | Jul 2016 | CN | national |
Filing Document | Filing Date | Country | Kind |
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PCT/CN2017/080383 | 4/13/2017 | WO | 00 |
Publishing Document | Publishing Date | Country | Kind |
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WO2018/014602 | 1/25/2018 | WO | A |
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7149629 | Scherzinger | Dec 2006 | B1 |
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9062978 | Win | Jun 2015 | B2 |
10564296 | Hide | Feb 2020 | B2 |
20090132164 | Goodall | May 2009 | A1 |
20180120111 | Petillon | May 2018 | A1 |
20200124421 | Kang | Apr 2020 | A1 |
Number | Date | Country |
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105737823 | Jul 2016 | CN |
106291645 | Jan 2017 | CN |
2015081424 | Jun 2015 | WO |
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Number | Date | Country | |
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20190129044 A1 | May 2019 | US |