Advanced square‐root cubature Kalman filters based on singular value decomposition and sequential processing

IET Radar, Sonar & NavigationVolume 14, Issue 1 p. 10-18 Research ArticleFree Access Advanced square-root cubature Kalman filters based on singular value decomposition and sequential processing Zhifei Li, Corresponding Author Zhifei Li lizhifei17@nudt.edu.cn orcid.org/0000-0003-1246-741X College of Electronic Engineering, National University of Defense Technology, Hefei, 230037 People's Republic of ChinaSearch for more papers by this authorJianyun Zhang, Jianyun Zhang College of Electronic Engineering, National University of Defense Technology, Hefei, 230037 People's Republic of ChinaSearch for more papers by this authorJiegui Wang, Jiegui Wang College of Electronic Engineering, National University of Defense Technology, Hefei, 230037 People's Republic of ChinaSearch for more papers by this authorQingsong Zhou, Qingsong Zhou College of Electronic Engineering, National University of Defense Technology, Hefei, 230037 People's Republic of ChinaSearch for more papers by this author Zhifei Li, Corresponding Author Zhifei Li lizhifei17@nudt.edu.cn orcid.org/0000-0003-1246-741X College of Electronic Engineering, National University of Defense Technology, Hefei, 230037 People's Republic of ChinaSearch for more papers by this authorJianyun Zhang, Jianyun Zhang College of Electronic Engineering, National University of Defense Technology, Hefei, 230037 People's Republic of ChinaSearch for more papers by this authorJiegui Wang, Jiegui Wang College of Electronic Engineering, National University of Defense Technology, Hefei, 230037 People's Republic of ChinaSearch for more papers by this authorQingsong Zhou, Qingsong Zhou College of Electronic Engineering, National University of Defense Technology, Hefei, 230037 People's Republic of ChinaSearch for more papers by this author First published: 01 January 2020 https://doi.org/10.1049/iet-rsn.2019.0115AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Numerical accuracy of the cubature Kalman filter (CKF) is crucially degraded by the accumulated round-off errors. As a systematic solution to reduce the influence of round-off errors, the square-root filters are appealing. This study proposes a square-root CKF based on the singular value decomposition (SVD) approach to enhance the robustness against round-off errors. In addition, motivated by the impact of matrix inversion operation on the positive definiteness and symmetry of the error covariance matrix and numerical conditioning, the authors derive a sequential square-root CKF. The sequential CKF avoids matrix inversion operation involved in the original CKF and directly propagates square factor in each cycle step. The variation is a valid way to better preserve the positive definiteness and symmetry and numerical stability. To evaluate the proposed approaches, four numerical experiments are simulated. The results illustrate the numerical robustness and stability of the proposed approaches. 1 Introduction Since the Kalman filter (KF) [1] was discovered, it has achieved great success in various engineering disciplines. The KF requires to hold a few assumptions: the system and measurement models are known, the noise is uncorrelated Gaussian white-noise and the behaviour is linear [2]. If any of these assumptions are not satisfied in actual system, KF cannot work well. To address non-linear estimation problems, the extended Kalman filter (EKF) [3, 4] was proposed. However, EKF may have a severely degraded numerical accuracy for highly non-linear measurement function. To circumvent this shortcoming, the unscented Kalman filter (UKF) [5, 6] was derived to approximate the posterior probability density. Undoubtedly, a proper choice of weight values can affect the numerical stability of UKF. Another approximate Bayesian filter is the cubature Kalman filter (CKF) with third-order accuracy [7, 8]. Nevertheless, the accumulated round-off errors have severe impact on CKF and eventually might result in its divergence [9]. As a systematic solution to reduce the influence of round-off errors, the square-root filters are preferable for designing a robust filter [10, 11]. As the covariance matrix, reformed by multiplying its square-root factors together, is ensured to be positive-semi-definite [12], this benefit allows for the square-root filters computing with double precision and have better numerical conditioning. The key strategy of square-root methodology is to decompose the error covariance matrix into square-root factors. Then the resulted factor can be directly propagated forward and updated in each cycle step. The CKF can utilise this processing approach to possess better numerical robustness. As developed in [7, 13], the QR decomposition can factorise into a product of . For the QR decomposition, the square-root factor of is not unique because a number of strategies exist to achieve it (e.g. Householder, Gram-Schmidt, modified Gram-Schmidt or Givens transformation) [14]. Moreover, when one performs the QR decomposition, the Cholesky factorisation is always involved. For instance, the square-root factor of the process noise covariance matrix or the measurement noise covariance matrix is required to ensure the implementation of the QR decomposition in [13]. If any of them is non-positive definite and/or non-symmetric, then the corresponding operation will cause an unexpected failure. The singular value decomposition (SVD) approach provides better estimation precision especially when the error covariance matrix becomes nearly singular [15, 16]. Some evidences, with respect to better estimation accuracy, exist in [16, 17]. Thus, we concentrate on how to design a CKF based on SVD approach. The previous SVD-based CKF (abbreviated as SVD-CKF) was discussed in [18, 19]. However, the approaches of [18, 19] merely apply SVD approach to replace Cholesky decomposition to factorise . As a result, SVD-CKF is still sensitive to round-off errors. Following this, we derive an efficient square-root SVD-CKF to further enhance the numerical robustness against round-off errors. The novel SVD-CKF directly propagates the SVD factors of rather than full matrix . Further, the novel SVD-CKF merely adopts SVD approach to decompose the time-varying noise covariance matrices. The variation provides better estimation accuracy and numerical stability. For CKF, a significant implementation is that it requires a square-root factor for recursion. The original CKF applies the Cholesky decomposition to factorise the error covariance matrix into the form of where is a lower triangular matrix (i.e. the Cholesky factor). The UD factorisation is another easily adopted approach to obtain the square-root factor. The idea of the UD factorisation is to factorise as where is a unit upper triangular matrix and is a diagonal matrix [20]. Both of the decomposition methodologies mentioned above can be accomplished easily for a symmetric positive definite . The symmetry and positive definiteness are two basic properties for . These two forced properties are helpful to obtain a more stable solution of Ricatti equation. In practice, however, numerically sensitive matrix inversion operation may catalyse losing these two properties [7]. If is non-positive definite and/or non-symmetric, CKF will have an unstable or even divergent output. Specifically, loss of positive definiteness may probably stop CKF operating continuously in MATLAB. In addition, as pointed out in [21], inversion calculation may cause the ill-conditioned CKF. Consequently, the numerical accuracy of CKF will be degraded. Sequential measurement processing is an important technique to handle the aforementioned issues. As shown in the sequential KF [22, 23], the great advantage of this technique is avoiding matrix inversion because all the calculations are scalar operations. Suppose that one obtains r independent measurements at time k, instead of processing the measurements as a vector, the sequential KF implements measurement-update equation a scalar measurement each time. Similarly, the technique can provide two significant benefits for CKF since matrix inversion is involved in the gain formula. The first is better preserving the symmetry and positive definiteness of . The second is preventing ill-conditioning to enhance numerical precision. These two benefits are obvious especially when the matrix becomes singular. Hence, CKF employs the technique to achieve sequential measurement processing (i.e. the sequential CKF). Furthermore, the idea of square-root filters is taken into consideration to further increase the numerical robustness of the sequential CKF. Apparently, the idea ensures that is symmetric and positive semi-definite, which is helpful to hold the two properties of . Thus, we propose a sequential square-root CKF to recursively update the square-root factor in each cycle step. The novel square-root CKF applies SVD approach/QR decomposition in the time-update procedure and adopts the sequential technique in the measurement-update procedure. This paper is organised as follows. Section 2 presents the original CKF. Section 3 proposes the square-root SVD-CKF to enhance the robustness against round-off errors. Section 4 proposes the square-root sequential CKF aimed at avoiding the matrix inversion operation and simultaneously preventing ill-conditioning. Section 5 illustrates the numerical stability and effectiveness of the proposed approaches by four numerical experiments. Section 6 concludes this paper. 2 Cubature Kalman filter Consider the discrete time non-linear dynamic system with addictive noise (1) (2) where is the state of the dynamic system at time k; is the state transition matrix; is the control input matrix; is the control input vector; is the measurement vector at time k; is the known function; and are independent process and measurement Gaussian noise with zero means and covariances and , respectively. The CKF adopts a third-degree spherical-radial cubature rule to numerically compute the Gaussian-weighted integrals. This idea allows for achieving third-order accuracy. A significant property of CKF is that it does not entail any free parameters [24]. The CKF can be summarised as follows: INITIALISATION . Set and . TIME UPDATE . (i) Factorise the error covariance matrix. (3) (ii) Calculate the cubature points . (4) where , and . [1] represents a set of points, for example (iii) Calculate the propagated cubature points (5) (iv) Calculate the predicted state and error covariance matrix (6) (7) MEASUREMENT UPDATE . (i) Factorise the predicted error covariance matrix (8) (ii) Calculate the cubature points (9) (iii) Calculate the propagated cubature points (10) (iv) Calculate the predicted measurement, innovation covariance matrix and cross-covariance matrix (11) (12) (13) (v) Calculate the updated state, gain matrix and posterior error covariance matrix (14) (15) (16) 3 SVD factorisation-based CKF 3.1 SVD-based CKF Consider the SVD factorisation [25]: a matrix of rank r can be written as where , are unitary matrices and is the conjugate transpose of . Here denote singular values of . In the original CKF realm, the Cholesky decomposition is employed to obtain the square-root factor. However, the error covariance matrix may lose symmetry and/or positive definiteness, causing unstable or even divergent behaviour. Besides, a crucial fact is that when the error covariance matrix is symmetric positive semi-definite, the Cholesky factorisation exists at least one upper triangular matrix [26]. Considering the aforementioned facts, the SVD-CKF [18] integrates a more robust factorisation method (i.e. SVD) into CKF to maintain a high level of numerical stability. The SVD approach does not need to satisfy the positive definiteness condition. This is a great benefit to ensure the code continuous operation in MATLAB. Compared with the CKF, the variation of previous work on the SVD-CKF is presented as follows: INITIALISATION . Apply SVD approach for . Set , and . TIME UPDATE (17) (18) MEASUREMENT UPDATE (19) (20) The readers can refer [18] for explicit derivation about the SVD-CKF mentioned above. Although the SVD-CKF enhances its numerical robustness by adopting SVD approach, it is still sensitive to round-off errors, because it recursively updates the error covariance matrix rather than its square-root factor . Undoubtedly, the direct propagation of the square-root factor is a valid solution to reduce the influence of round-off errors. The square-root filters are motivated by two considerations: (i) the product can never be indefinite even in the existence of round-off errors; (ii) the numerical conditioning of is better than that of [12]. The sensitivity to uncertainties in the input data and round-off errors is characterised by the condition number . The condition number is defined as , where and are the maximum eigenvalue and minimum eigenvalue of , respectively. While , is said to be ill-conditioned, and approaches a singular matrix [22]. The benefit of square-root approach is obvious by comparing with (i.e. ). This means operation with may encounter numerical difficulties if , but the square-root approach is still valid until . In other words, the square-root filters yield doubled-order precision. Instead of propagating , hence, we propagate its square root factor . 3.2 Square-root SVD-CKF To further enhance numerical accuracy and reduce the influence of round-off errors, we derive a novel efficient implementation, abbreviated as SRCKF, under the original SVD-CKF. The SRCKF recursively updates its SVD factors rather than full matrix . In essence, it iteratively calculates the square-root factor in each recursion step. The SRCKF can be summarised as follows: INITIALISATION . Apply SVD approach for . Set , and . TIME UPDATE . (i) Given , estimate the predicted state by (18), (5) and (6). (ii) Calculate the square-root factor of the predicted error covariance matrix. Our goal is to find the factors and from (7) such that . However, there exists the danger of numerical accuracy deterioration, which has been illustrated in numerical examples [27]. Thus, we define a following matrix, and obtain corresponding SVD factors (21) where are the SVD factors of such that . The orthogonal matrix is the resulted SVD factor. The weighted, centred matrix is (22) Then, we can obtain (23) MEASUREMENT UPDATE . (i) Skip (9)–(12) to calculate the predicted measurement and cross-covariance matrix. (ii) Calculate the gain matrix (24) where are the SVD factors of such that . The orthogonal matrix is the resulted SVD factor. The weighted, centred matrix is (25) According to (24) and (25), the gain matrix can be derived as follows: (26) (iii) Calculate the updated state as in (14). (iv) Calculate the square-root factor of the posterior error covariance matrix. The SRCKF utilises the Joseph stabilised version [22] to calculate the SVD factors of the posterior error covariance matrix rather than (16). Such symmetric form will be more stable and robust. Besides, it can ensure that is always positive definite, as long as is positive definite (27) where the orthogonal matrix is the resulted SVD factor. The linearised measurement matrix is (28) where can be obtained easily by only calculating the reciprocal of the entries on the main diagonal of . Then, we can get (29) The detailed proof of (23), (26), (28) and (29) is presented in Appendix 1. Remark 1.When and are constant over time, the SVD factors can be pre-computed at the initial step. Meanwhile, if they are symmetric and positive definite, the corresponding matrix product can be replaced by the Cholesky factor. For example, is utilised rather than , where is the Cholesky factor of such that . For the time-varying noise covariances, nevertheless, we only adopt the SVD approach. As if any of them becomes non-positive definite and/or non-symmetric in any recursion step, then corresponding practical operation will exist an unexpected failure in MATLAB (i.e. function chol). 4 Sequential square-root CKF For CKF, a symmetric and positive definite error covariance matrix is significant to guarantee its numerical stability. As discussed in [7], however, matrix inversion computation may affect the two properties (i.e. symmetry and positive definiteness) of the error covariance matrix. Additionally, in practice, some factors contribute to the ill-conditioned KF, e.g. large uncertainties in the values of the matrix parameters , or ; ill-conditioning of the intermediate result for inversion in the gain formula; poor machine precision [21]. Similarly, is required to calculate the gain matrix in CKF. If becomes nearly singular, then corresponding inversion operation results in the ill-conditioned CKF. To address the aforementioned issues, hence, we concentrate on how to derive an equivalent presentation of CKF without matrix inversion. The sequential measurement processing is an effective strategy to avoid matrix inversion because it processes measurement vector one scalar each time. Here, the sequential technique is extended to CKF. Meanwhile, as discussed in Section 3.1, the square-root filters can guarantee the symmetry and positive semi-definiteness of the error covariance matrix and possess better numerical conditioning. Due to this fact, the idea of the square-root filters is considered here to further enhance numerical stability. To perform the square-root filter, we reformulate the error covariance matrix as the form of matrix product. Next, the square-root factor of can be updated recursively. In the following section, we derive the sequential square-root CKF in two cases, abbreviated as SECKFD and SECKFC, respectively. Case 1: The sequential square-root CKF is derived on the condition that is a diagonal matrix (SECKFD). Its detailed procedure can be summarised as follows: INITIALISATION . Apply SVD approach for . Set , and . TIME UPDATE . The TIME UPDATE procedure is identical to the SRCKF. If is symmetric, positive definite and constant over time, (30) is recommended to have cheaper computational complexity. We denote the QR decomposition as (30) where is the Cholesky factor of such that . MEASUREMENT UPDATE . (i) Initialisation (31) (ii) For , r is the dimension of the measurement vector, perform the following procedure. (a) Define as the ith element of , as the Cholesky factor of such that , as the ith row of , as the variance of the ith measurement and as the ith row of . Notice that here (b) Calculate the ith gain matrix (32) (c) Calculate the ith square-root factor of the posterior error covariance matrix (33) where the orthogonal matrix is the resulted SVD factor. Then, we can get (34) (d) Calculate the ith updated state (35) where is the ith element of . (iii) Set the square-root factor of the posterior error covariance matrix and the updated state as (36) (37) The detailed proof of (32) and (34) is presented in Appendix 2. Case 2: In this case, the sequential square-root CKF is derived on the condition that is a constant over time and positive definite matrix (SECKFC). One can perform the SVD decomposition of such that , where and are a diagonal matrix of eigenvalues and an orthogonal matrix, respectively [21]. We define a normalised measurement vector as (38) where . TIME UPDATE . The TIME UPDATE procedure is identical to the SECKFD. MEASUREMENT UPDATE . (i) Initialisation is same as in (31). (ii) For , r is the dimension of the measurement vector, perform the following procedure. (a) Define as the ith element of , as the Cholesky factor of such that , as the ith row of , as the variance of the ith normalised measurement and as the ith row of . Notice that here (b) Calculate the ith gain matrix. (39) (c) Calculate the ith square-root factor of the posterior error covariance matrix. (40) where the orthogonal matrix is the resulted SVD factor. Then, we can get (41) (d) Calculate the ith updated state (42) where is the ith element of . (iii) Set the square-root factor of the posterior error covariance matrix and the updated state as in (36) and (37). Remark 2.The sequential processing only makes sense if certain conditions hold: (1) the measurement noise covariance is a diagonal matrix; (2) if condition (1) cannot be satisfied, should be constant over time and positive definite. Then one can perform the SVD decomposition off-line and use the sequential processing technique. However, if is time varying (but not a diagonal matrix), one has to perform the SVD decomposition at each step of SECKFC. There will be a lot of computational effort to avoid a matrix inversion. More importantly, must be positive definite at each step to ensure the SECKFC implementation. If cannot satisfy this strict condition (i.e. cannot be factorised into ), SECKFC will be invalid. 5 Numerical experiments In this section, the SRCKF and SECKF proposed in this paper are compared with the original CKF and the SVD-CKF by simulation. First, we perform a non-linear estimation problem and an ill-conditioned problem to illustrate their performance and numerical robustness, respectively. Then, we make a comparison between SRCKF and SECKF to testify the sequential processing condition. Finally, we evaluate the SECKF against CKF by demonstrating the impact of matrix inversion operation on the two properties of the error covariance matrix. All simulations are implemented in the same precision (64-bit floating point) in MATLAB-2016b running on a PC with processor Intel(R) Core(TM) i5-7200U CPU 2.5 GHz and with 4 GB RAM. Example 1.A manoeuvring target moves with constant velocity (CV) in the -plain. True initial state . The trajectory is shown in Fig. 1, where the target goes through a and two turns. This target is observed by a radar with scan time delivering range and bearing measurements with standard deviations and , respectively. We use a CV dynamic model to track the target, where the is tuned to make filters follow the turns [28]. The dynamic model is given as follows: where , and the notation denotes Kronecker product. The initial condition , and The comparison results are the root mean square error (RMSE) over Monte Carlo runs. The RMSE of position is defined as follows: where the superscript j denotes quantity on the jth run, and denotes the ith element of . The definition of is similar to the RMSE in position. Fig. 1Open in figure viewerPowerPoint Trajectory of target The comparison results are shown in Figs. 2 and 3. These figures show that all algorithms examined in this well-conditioned example have almost the same estimation accuracy. During the turn manoeuvring phase, the CV model cannot well match the turn manoeuvring. As a result, a relatively large estimation error exists at three-time instants. To analyse the numerical difference of each algorithm, the statistics (i.e. mean and standard derivation) of and are collected in Table 1, respectively. The statistics confirm the algebraic equivalence between all forms of CKF discussed in this section. Table 1. Comparison of examined methods performance and CPU time Method Mean Standard derivation CPU, s , m , m/s , m , m/s CKF 55.055 25.988 103.554 56.641 0.022 SVD-CKF 55.056 25.988 103.556 56.641 0.030 SRCKF 55.055 25.989 103.557 56.641 0.033 SECKFD 55.144 25.988 103.676 56.634 0.031 Fig. 2Open in figure viewerPowerPoint RMSE errors of position Fig. 3Open in figure viewerPowerPoint RMSE errors of velocity The difference in CPU time averaged over Monte Carlo simulations indicates that CKF operates faster than its any other counterpart. It is reasonable because the Cholesky decomposition has cheaper computation than SVD factorisation. The SRCKF is the slowest among the SVD-based algorithms due to more SVD factorisations in each cycle step. Example 2.Notice that Example 2 represents an ill-conditioned problem which is used to explore numerical robustness. The dynamic motion of a satellite travelling in a circle orbit is given as follows [29]: where , and . Assume , but where denotes unit round-off error (i.e. ). We repeat the set of numerical experiments for the various ill-conditioning parameter such that . The comparison results are the norm of accumulated root mean square error (ARMSE) over Monte Carlo simulations. The ARMSE in each component of is defined as follows: where is the discrete time of the dynamic system. The numerical results are summarised in Table 2. Having analysed the outcomes in Table 2, we conclude that CKF has the worst performance among the examined algorithms. When is large, all algorithms work well and possess the same accuracy. If , CKF cannot operate continuously in MATLAB because the error covariance matrix is non-positive definite. Compared with the CKF, the SVD-CKF can better manage this ill-conditioned situation. This result demonstrates that the SVD approach has better robustness especially when the matrix is nearly singular. If , however, the SVD- CKF produces a 'NAN', which is a sign of failure. Table 2. Norm of absolute error at each component of the state, i.e. Method (m), , while growing ill-condition CKF 0.0617 0.0549 0.0578 — — — — — — — — — — — SVD-CKF 0.0617 0.0549 0.0578 0.0562 0.0565 0.0575 0.0570 NAN NAN NAN NAN NAN NAN NAN SRCKF 0.0617 0.0549 0.0578 0.0562 0.0565 0.0575 0.0570 0.0565 0.0564 0.0553 0.0589 0.0566 0.0544 0.0585 SECKFD 0.0617 0.0549 0.0578 0.0562 0.0565 0.0575 0.0570 0.0565 0.0733 0.0567 0.0604 0.0579 0.0556 0.0575 Accumulated round-off errors have severe impact on CKF, even though the inherently stable SVD-based strategy cannot avoid the degradation of performance. In contrast, the SRCKF and SECKF can work precisely until very small ill-conditioning parameter values. This outcome is consistent with the theoretical derivations presented in Sections 3 and 4. The square-root version of CKF combining SVD approach or sequential technique is more efficient to reduce the influence of round-off errors. Example 3.Consider the satellite dynamic system in Example 2 with Define a diagonal measurement noise covariance . The true measurement noise covariance can be modelled by a staircase function The initial condition , and . Here, the comparison results are the ARMSE in each component of together with the CPU time averaged over Monte Carlo simulations. In addition, we exploit the average normalised estimation error squared (ANEES) where The comparison results are summarised in Table 3. Note that because is time varying, SECKF includes two cases (i.e. SECKFD and SECKFC). As collected in Table 3, SECKF and SRCKF have the same estimation accuracy for ARMSE. Undeniably, is not positive definite at time , i.e. cannot be factorised into . More precisely, we cannot obtain the normalised measurement vector and its corresponding diagonal matrix . In addition, the difference in CPU time indicates for a time-varying (but not a diagonal matrix) , SECKF is more computationally expensive than SRCKF. This conclusion is reasonable because if time , SECKF has to perform the matrix decomposition in each cycle step. There will be a lot of computational effort to avoid a matrix inversion. Table 3. Comparison of the absolute errors and CPU time for SRCKF and SECKF Method , m , m , m/s , m/s CPU, s SRCKF 0.6184 0.1431 0.0607 0.0345 0.0089 SECKF 0.6184 0.1431 0.0607 0.0345 0.0136 An efficient method is also required to deliver a relative quality measure, typically in form of the credibility of estimation. The credibility is characterised via ANEES. The closer to 1 the ANEES is, the more credible the estimator [30]. If ANEES is much greater than 1, the actual estimation error is much larger than what the estimator believes (i.e. the estimator is too optimistic), vice versa [28]. With respect to the self-assessment of SECKF and SRCKF (ANEES shown in Fig. 4), their values are within a reasonable range except at time . Evidently, for SECKF, ANEES is much greater than 1 at time (ANEES = 2.481). This is a too optimistic outcome for SECKF state estimation. The actual estimation error is much larger than what SECKF believes. In this case, SECKF is invalid. In conclusion, if is a time-varying and non-diagonal matrix, we recommend adopting SRCKF rather than SECKF. Fig. 4Open in figure viewerPowerPoint Average normalised estimation error squared Example 4.In this example [22], we testify the impact of matrix inversion operation on the symmetry and positive definiteness of the error covariance matrix by comparing the original CKF with SECKF. Suppose that we have an LTI system with For a finite word-length digital computer, suppose that the measurement covariance , , but . The rounded values of the CKF gain and posterior error covariance matrix at time k can be given by Now, we implement the measurement-update equation using SECKF. We start out with We only need to iterate one time through (32) and (34) because the dimension of the measurement vector is 1. The rounded values of parameters in (32) and (34) can be given by Having analysed the outcomes in Example 4, we conclude that has lost the positive definiteness in CKF. Hence, we can anticipate that the estimation precision of CKF will be rapidly deteriorated. In contrast, the is still symmetric and positive definite in SECKF. This result verifies the benefit that can be gained by using SECKF. More precisely, implementation without the matrix inversion operation and direct propagation of the square-root factor are helpful to hold the two properties of the error covariance matrix. 6 Conclusion In this paper, we derive two efficient square-root versions of CKF based on SVD approach and the sequential processing, respectively. Compared with the original CKF, theoretical derivations and simulation results demonstrate that these two new approaches possess better numerical robustness and stability. This benefit indicates that the square-root CKF combining the SVD approach or sequential technique is a preferable choice to design the robust CKF. Moreover, for CKF, implementation without the matrix inversion operation and direct propagation of the square-root factor are valid to better preserve the symmetry and positive definiteness of the error covariance matrix, which is helpful to enhance its numerical stability. Further research is devoted to developing the SVD-based information filtering and high-degree CKF. Additionally, another open problem is exploring the numerical insights of SVD approach. 8 Appendix 8.1 Appendix 1 In this appendix, we prove (23), (26) (28) and (29) presented in Section 3.2. Considering the property of orthogonal matrix, we obtain the following formulas. 8.1.1 Calculation of (43) 8.1.2 Calculation of (44) Substituting (44) into (15) yields (45) 8.1.3 Calculation of The Joseph stabilised version of the posterior error covariance matrix is given as [22] (46) where denotes the measurement matrix, for example For CKF, the can be rewritten as (47) where is the linearised measurement matrix of non-linear function . 8.1.4 Calculation of We can directly obtain from (52), i.e. (48) then substituting (23) into (48) yields (49) 8.2 Appendix 2 In this appendix, we prove (32) and (33) presented in Section 4. The Taylor series expansion of around is performed, and its higher-order terms are omitted. (50) where is the Jacobian matrix of . According to (50), and can be derived as (51) (52) Based on (51) and (52), (15) and (16) (i.e. and ) can be rewritten as (53) (54) Now, we can process the measurement one scalar each time using the sequential filter. 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