Improved square root adaptive cubature Kalman filter

IET Signal ProcessingVolume 13, Issue 7 p. 641-649 Research ArticleFree Access Improved square root adaptive cubature Kalman filter Lei Zhang, Lei Zhang School of Automation, Nanjing University of Science and Technology, Nanjing, 210094 People's Republic of ChinaSearch for more papers by this authorSheng Li, Corresponding Author Sheng Li lisheng@njust.edu.cn School of Automation, Nanjing University of Science and Technology, Nanjing, 210094 People's Republic of ChinaSearch for more papers by this authorEnze Zhang, Enze Zhang College of Information Engineering, Yangzhou University, Yangzhou, 225009 People's Republic of ChinaSearch for more papers by this authorQingwei Chen, Qingwei Chen School of Automation, Nanjing University of Science and Technology, Nanjing, 210094 People's Republic of ChinaSearch for more papers by this authorJian Guo, Jian Guo School of Automation, Nanjing University of Science and Technology, Nanjing, 210094 People's Republic of ChinaSearch for more papers by this author Lei Zhang, Lei Zhang School of Automation, Nanjing University of Science and Technology, Nanjing, 210094 People's Republic of ChinaSearch for more papers by this authorSheng Li, Corresponding Author Sheng Li lisheng@njust.edu.cn School of Automation, Nanjing University of Science and Technology, Nanjing, 210094 People's Republic of ChinaSearch for more papers by this authorEnze Zhang, Enze Zhang College of Information Engineering, Yangzhou University, Yangzhou, 225009 People's Republic of ChinaSearch for more papers by this authorQingwei Chen, Qingwei Chen School of Automation, Nanjing University of Science and Technology, Nanjing, 210094 People's Republic of ChinaSearch for more papers by this authorJian Guo, Jian Guo School of Automation, Nanjing University of Science and Technology, Nanjing, 210094 People's Republic of ChinaSearch for more papers by this author First published: 01 September 2019 https://doi.org/10.1049/iet-spr.2018.5029Citations: 4AboutSectionsPDF 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 onFacebookTwitterLinked InRedditWechat Abstract In this study, an improved square root adaptive cubature Kalman filter (ISRACKF) is proposed to improve the filter performance in terms of accuracy, computation efficiency, and robustness. Through the evaluated measure of non-linearity value, the cubature rule under different accuracy levels can be adaptively selected in the dynamic process or measurement model. In this way, high accuracy can be maintained without sacrificing computation efficiency. Furthermore, the maximum correntropy criterion cost function can help improve the robustness of ISRACKF. The measure of non-Gaussianity value is utilised to control the computation complexity of the robust iterative process as well. The stability proof of estimated state error and covariance is given. The comparison results of the target tracking problem and integrated navigation system demonstrate the superior performance of the proposed ISRACKF in this study. 1 Introduction Owing to the practicability and flexibility, non-linear filters have been widely studied in recent decades [1-12]. Motivated by the solution of intractable numerical integrals within the filter framework, various non-linear filters are proposed. The typical filters are the extended Kalman filter (EKF) [1, 2], unscented Kalman filter (UKF) [3-5], Gauss–Hermite quadrature filter (GHQF) [6, 7], sparse grid quadrature filter (SGQF) [8-10] and cubature Kalman filter (CKF) [11, 12]. The EKF is only accurate to a first-order Taylor expansion polynomial and its Jacobian matrix is hard to evaluate in a complex system, which limits the application of EKF. Compared with the EKF, the UKF, GHQF, SGQF, and CKF are more preferable choices as they can be accurate to a second-order Taylor expansion polynomial or higher. The latter three non-linear filters are based on the assumption of Gaussian probability density function (PDF) distribution. They are distinct from each other due to different integration rules. Based on the unscented transform (UT) rule, it is indicated that sigma points are needed in the UKF. However, if the dynamic system dimension is higher than three, the parameter satisfying is negative. It will lead to the negative definite weight which could render an unstable filter process. The GHQF utilises the Gauss-Hermite quadrature (GHQ) rule to conduct the filter process with the quadrature sampling points that it can be accurate to arbitrary order Taylor expansion polynomial. The GHQ rule is composed of the direct tensor product where the number of sampling points is large. So the GHQF would suffer from the curse of dimensionality. From [9], it is proven that UKF is the subset of SGQF. The SGQF can maintain the performance of GHQF and alleviate the computation burden effectively. The CKF with cubature rule will guarantee the positive weight and its number of sampling points is . To further improve the filter accuracy, the high degree CKF [13] was proposed which could reach close performance to the GHQF with less computation burden. The high degree cubature rule and sparse grid quadrature (SGQ) rule can achieve high accuracy at the cost of computation efficiency. This issue promotes the development of adaptive non-linear filters with high-computation efficiency. In [14], an adaptive SGQF is proposed which selects the SGQ rule under different accuracy levels by the adaptive comparison criterion. However, the high degree SGQ rule needs to be calculated before being determined which brings unnecessary cost. Besides, an adaptive CKF (ACKF) [15, 16] was proposed which can tune the number of points adaptively online. The proposed ACKF with the idea of directional uncertainty is insightful, but the element of the scaling matrix is not easy to be predetermined especially in a complex system. This may limit its application. Therefore, the adaptive non-linear filter needs to be improved. Inspired by [17-19], the measure of non-linearity value (MNLV) and measure of non-Gaussianity value (MNGV) could be the effective indexes to improve the performance of a non-linear filter in the dynamic system. On the one hand, through the evaluated MNLV from measure function, the adaptive non-linear filter can be carried out with an appropriate selection of integration rules under different accuracy levels. On the other hand, in the filter process, the uncertain noise characteristic or disturbance may cause the non-Gaussian PDF distribution which will affect the accuracy. The MNGV can be used to assess real-time PDF distribution. In [20-24], the M-estimation method is considered as an effective robust technology. Using the weight cost function, the valuable weight corresponding to the error can be acquired. Then the measurement information is reformulated. By virtue of newly formed information, the filter accuracy can be improved with stronger robustness. These facts contribute to the main work of this study, which an improved square root ACKF (ISRACKF) is proposed. The MNLV in the dynamic process model and measurement model can help determine the appropriate cubature rule online, which decreases unnecessary computational load effectively. In the filter framework of ISRACKF, the maximum correntropy criterion (MCC) cost function of the M-estimation method is adopted to improve the robustness. Moreover, the MNGV can be used to terminate a robust iterative process while the PDF approximates Gaussian distribution. It would bring benefits to the reduction of unnecessary computation. Furthermore, the idea of a square-root filter structure would strengthen the numerical stability. The ISRACKF is an alternative non-linear filter that owns the advantages of high accuracy with less computation burden and strong robustness. The rest of paper is organised as follows. In Section 2, the MNLV is introduced and the ACKF is addressed as well. By the introduction of MCC robust cost weight function and MNGV, the specific flow of ISRACKF is completed in Section 3. Section 4 presents the stability proof of the proposed ISRACKF. In Section 5, two numerical examples are utilised to demonstrate the validity of the proposed filter. The conclusions are drawn in Section 6. 2 MNLV-based ACKF 2.1 Evaluation of MNLV Consider the non-linear discrete dynamic system given by (1)where is the state vector and is the measurement vector, refers to the transition matrix of process noise, the process noise satisfies and the measurement noise satisfies . In the filter process, the state mean and covariance are calculated by the Gaussian type numerical integral. The integral can be classified in the following expression: (2)Following the cubature rule, the above integral could be approximated as follows: (3)where and refer to the sampling points and corresponding weight, and N is the number of sampling points. In the discrete dynamic system, the non-linearity is varying at different times, which can be assessed by the MNLV. Various methods can be utilised to update the MNLV. Each of them has its own merit in the appropriate application. Owing to the separate evaluation of MNLV in the process model and measurement model, the local MNLV method is considered in this study. The local MNLV method is based on the idea of a mean square error method [17-19]. It evaluates the distance between the non-linear function of the dynamic process model or measurement model and a set of linear functions. Here, the calculation of MNLV in the process model is taken as an example. The calculation of MNLV in the measurement model is similar to the process model, where the difference is the non-linear propagated function. Based on the cubature rule, the sampling points are obtained , is from . Through the non-linear function in the process model, the transformed sampling points are obtained. (4)Then, the state mean and error covariance can be updated (5) (6) (7) (8)The local MNLV of a process model can be assessed as follows: (9)where is the trace calculation of matrix. Remark 1.The local MNLV can be evaluated in the process model or measurement model, respectively, while the global MNLV needs to combine all the state vectors in two models. Thus, the computation complexity of local MNLV is lower, especially in the partially linear model. Additionally, the local MNLV is within the range [0, 1] which is easy to be compared with the threshold. What is more, the measuring method considers the influence of noise covariance that the local MNLV is more convinced. 2.2 MNLV-based ACKF Based on the cubature rule, the Gaussian type integral in (2) is divided into two parts: the spherical integral and radial integral. It should be noted that the arbitrary degree cubature rule could be formed by various methods of dealing with two integrals [13, 16]. The high-degree rule could achieve high accuracy as well as a large computation burden. For practical application, the trade-off between the accuracy level and computation complexity should be considered. In this study, the third-degree and fifth-degree cubature rules (3-CR and 5-CR) are nested in the proposed filter. Then expression (3) can be rewritten as follows. The 3-CR (10)The 5-CR (11)where , , is the state mean, , is the state error covariance, is the unit column vector and the element equals to 1, is the vector dimension of a model. Fig. 1Open in figure viewerPowerPoint Algorithm 1: ACKF By combining the evaluation of MNLV and different cubature rules, the ACKF is designed which is shown in Algorithm 1 (Fig. 1). (12) (13) (14) (15) (16) (17) In the Gaussian-type integrals, the accuracy and numerical stability are the important indexes of the approximation method. The stability criterion of the integral is an effective method to assess numerical stability [25] (18)where is the weight of sampling points, is the total number of sampling points. When the stability parameter is >1.0, the truncation error would occur that the corresponding integration rule is not stable. Otherwise, if the stability parameter maintains 1.0, the integration formula is totally stable. Here, the stability parameters of the UT rule, 3-CR and 5-CR are calculated as follows: (19) (20) (21)It can be found that different rule will lead to different numerical stability. When the system dimension is higher than three, the integral process using the UT rule is not stable. The integral process using two cubature rules is stable which shows the numerical stability of the proposed ACKF in this study. 3 Improved square root ACKF 3.1 M-estimation-based robust ACKF By virtue of two cubature rules under different accuracy levels, ACKF could alleviate the curse of dimensionality. However, the uncertainty of noise or disturbance will give rise to the non-Gaussian PDF distribution which could affect the evaluation of MNLV and accuracy in the filter process. The M-estimation-based robust filter is an efficient method to deal with the non-Gaussian PDF distribution [20-24]. The key part of the M-estimation method is the weight cost function. Huber's cost function is widely used. However, it cannot provide valuable information for the large residual [20, 24]. The correntropy function is useful in many applications including robust regression, non-linearity tests and so on [26-28]. To improve the robustness, the MCC [21-24] was proposed, which is a smoothed maximum a posteriori estimation method. The MCC cost function-based filter is effective in dealing with non-Gaussian PDF distribution. Fig. 2Open in figure viewerPowerPoint Weight distribution of different weight functions In this study, one cost function should be selected to improve the ACKF. So the comparison of cost functions is given. To be clear, the Institute of Geodesy and Geophysics (IGG) cost function is also considered as well. The result is shown in Fig. 2. For Huber, the tuning parameter is 1.345. For IGG, two tuning parameters are 1.0 and 3.0, respectively. For MCC, the bandwidth is 3.0. It is indicated that the large error cannot lead to valuable weight information in Huber's cost function. In addition, Huber's and IGG's cost functions maintain the constant value 1.0 under small error while the MCC's cost function could maintain reasonable weight distribution. Thus, MCC weight cost function is selected. The transformed non-linear regression is established based on the dynamic model (1) (22)where , , is the measurement noise covariance. Then, the MCC weight cost function is introduced [21-24] (23)where , refers to the component of in (13). is the bandwidth of the Gaussian kernel, which is used to adjust the error weight distribution. Then, the weight matrix is formed to modify noise covariance so the newly formed noise covariance could be used to continue the filter process (24) Lemma 1.Assume positive definite matrices , if the condition exists, then the inequality is concluded: . Proof.The condition implies that . Take the inverse on both sides, then . With the matrix inversion lemma, it yields that (25)Then , the proof is finalised. □ Lemma 2.The newly formed under MCC robust filter satisfies the inequality: . Proof.From (23), it is found that all the diagonal elements of the weight matrix satisfy . So it is concluded that . Apply Lemma 1, set , , , then it implies: , the proof is finalised. □ 3.2 Improved square root ACKF In the filter process, the MNGV is used to assess the similarity of the true PDF and Gaussian PDF approximation. In Section 3.1, the MCC cost function of the M-estimation method is utilised and that the filter needs several iterations. By virtue of the real-time MNGV, the iterative process can be controlled. If the MNGV is large, which refers to the high non-Gaussianity. The iterative process needs to be continued. Otherwise, if the MNGV is small, which refers to close Gaussian PDF distribution. The iterative process can be terminated. The appropriate iterations benefits for maintaining high filter accuracy without sacrificing computation efficiency. In this study, the MNGV of a dynamic system is obtained from the normalised moment-matching method [19]. The specific expression is shown as follows: (26) (27)where and refer to the component of and , N depends on the cubature rule determined by the ACKF. To improve numerical stability, the idea of a square-root filter is introduced in the proposed filter. It can preserve the symmetry of the covariance matrix and the stability could be enhanced. Fig. 3Open in figure viewerPowerPoint Algorithm 2: ISRACKF Based on the above results, the proposed ISRACKF is completed. The ISRACKF is shown in Algorithm 2 (Fig. 3). Remark 2.With the determined sampling points and weight of the cubature rule from ACKF part, the unnecessary calculation could be reduced. What is more, the newly formed measurement noise satisfies which could be proved by applying Lemma 2 iteratively. 4 Stability analysis of the ISRACKF The error property of the proposed ISRACKF is analysed in this section. Firstly, the state is expanded by a Taylor series (28)where refers to the derivation of around , . The predicted state mean can be expanded in the same way. Taking the high-term error into consideration [29], the predicted model error and measurement error are given (29) (30)where , , and are problem-dependent scaling matrices, and account for the linearisation error, , . and provide the extra degree of freedom. For simplicity, take (31) (32)then the state error and covariance can be rewritten as follows: (33) (34) (35) Lemma 3.If two positive definite matrices exist, then . Proof.Assume the positive definite matrices , then apply the matrix inversion lemma gives (36)Set and to be the identity matrices with appropriate dimensions, (36) can be transformed into (37)As and are positive definite, then . It implies that , insert it into (37), then , the proof is finalised. □ Theorem 1.Based on Lemma 2, suppose there exist some positive real constants satisfying the bounds (38)Then the estimated error covariance is bounded. Proof.Follow the error expression in (33) and (34), the gain matrix is obtained (39)The estimated model error and error covariance can be shown as follows: (40) (41)Inserting (39) into (41), we obtain (42)Set , , apply Lemma 3, then (43)So the inequality is obtained, the proof is finalised. □ Theorem 2.Based on Theorem 1, assume there are some positive real constants as follows: (44) Then the estimated error is bounded. Proof.Select the discrete Lyapunov function (45)With the error expression in (33) and (34), it implies (46)From (35), it implies (47)As satisfies the following inequality: (48)Inserting (48) into (47), we obtain (49)Take the inversion of (49) on both sides, then the first term in the right side of (46) satisfies (50)As , then is concluded. It can be noted that is bounded by (35). Similarly, is also bounded by (39). So the assumptions , are set for simplicity. Take the trace of the second term on the right side of (46) that leads to the following results: (51)Substitute (50) and (51) into (46), it is concluded that (52)Apply the stochastic stability lemma in [30], the predicted state error is bounded. From (33), the model error is expressed. With the bounded assumptions in (42), the model error is bounded as well. □ 5 Numerical examples In this section, the validity of the proposed filter is demonstrated by two numerical examples. The examples are run with the Matlab 2013 in the computer with CORE i3–4010U 1.7 GHz CPU and 4.00 GB memory. 5.1 Target tracking problem In this subsection, a typical target tracking problem [11-13, 15, 16] is utilised. The discrete dynamic model of the tracking problem is given as follows: (53) (54)where the state vector is , refer to the position, refer to the velocity, refers to the turn rate in the 2D plane, is the sampling interval, and are the Gaussian white noise with covariance and , , , and . The parameters of the simulation are set as follows. The true initial state is (55)covariance is ((56)) The initial state estimation is randomly generated by the normal distribution . To compare the performance, four non-linear filters are implemented including the UKF, third-degree CKF, fifth-degree CKF, and proposed ISRACKF. What is more, two cases of different noise distribution property are considered. Case 1: time-varying measurement noise Case 2: contaminated Gaussian distribution noise In case 1, it satisfies: , (57) (58)and . The bandwidth of ISRACKF is 4.0. In case 2, it satisfies: , , , , and are both selected as 0.7 while is 0.2 in the simulation. In this case, to compare the affection of the bandwidth, two ISRACKFs are considered. The bandwidth is 4.0 in the ISRACKF1 while it is 2.0 in the ISRACKF2. The simulation results are based on 100 Monte–Carlo runs. The MNGV is selected as 0.2. The total run time is 100 s. The sampling interval is 1 s. The root mean square error (RMSE) is an effective metrics to compare the performance [11, 15, 16]. The cumulative results are used to demonstrate the position part and velocity part , respectively. Fig. 4Open in figure viewerPowerPoint RMSE of position, velocity and turn rate in case 1 (a) RMSE of position, (b) RMSE of velocity, (c) RMSE of turn rate From Fig. 4 and Table 1, it can be found that the ISRACKF is more accurate than the other three non-linear filters under the time-varying measurement noise. In the filter process, the MNLV is continually evaluated which could be compared with the threshold. As a consequence, the cubature rule is determined in the dynamic process model or measurement model. Fig. 5 shows the selection of the cubature rule in the filter process. Moreover, case 1 concerns the time-varying noise property, the MCC-based robust filter structure could help improve the accuracy. According to the comparison results, the ISRACKF is more preferable with high accuracy, strong robustness, and less computation complexity. Table 1. RMSE results in case 1 Pos, m Velocity, m/s Turn rate, rad/s UKF 121.9886 13.0461 0.3928 third-CKF 117.4962 12.6818 0.3826 fifth-CKF 110.7621 12.5655 0.3802 ISRACKF 109.2641 12.4545 0.3766 Fig. 5Open in figure viewerPowerPoint Dynamic and measurement MNLV with the degree switch in case 1 (a) Dynamic MNLV, (b) Measurement MNLV, (c) Dynamic selection of cubature rule, (d) Measurement selection of cubature rule Fig. 6Open in figure viewerPowerPoint RMSE of position, velocity and turn rate in case 2 (a) RMSE of position, (b) RMSE of velocity, (c) RMSE of turn rate Based on the results in Fig. 6 and Table 2, it is obvious that two ISRACKFs outperforms the other non-linear filters. Two ISRACKFs both own the property to select the appropriate cubature rule to decrease the computation burden. The distinct between them is the embedded bandwidth (Fig. 7). The ISRACKF1 with a larger bandwidth can obtain higher accuracy than the ISRACKF2. On account of the expression of MCC cost function in (23), it is concluded that the larger bandwidth leads to the larger weight value for the large residual. This is beneficial to the filter improvement under non-Gaussian distribution noise case. However, the utilisation number of 5-CR in the ISRACKF1 is more than that in the ISRACKF2, which means the ISRACKF1 needs more computation load. Thus, the appropriate bandwidth should be selected in the actual application that the ISRACKF can show its best performance with minimum computation load (59) Table 2. RMSE results in case 2 Pos, m Velocity, m/s Turn rate, rad/s UKF 154.7195 15.2813 0.3471 third-CKF 143.6552 13.0682 0.3292 fifth-CKF 134.8043 13.0129 0.3265 ISRACKF1 133.1427 12.8121 0.3220 ISRACKF2 133.3298 12.8871 0.3222 Fig. 7Open in figure viewerPowerPoint Dynamic and measurement MNLV with the degree switch in case 2 (a) Dynamic MNLV, (b) Measurement MNLV, (c) Dynamic selection of cubature rule, (d) Measurement selection of cubature rule 5.2 Integrated navigation system As a practical application, the non-linear filter is the key technology in the multi-sensor integrated navigation system. The continuous dynamic model [31] in (59), which can be discretised to a similar expression in (1). The measurement model is given in (60). Based on the commercial software navigation toolbox developed by the GPSoft corporation, the flight trajectory, the intertial measurement unit (IMU) sampling data and global positioning system (GPS) data can be generated. The simulated flight trajectory is shown in Fig. 8(60) Fig. 8Open in figure viewerPowerPoint Flight trajectory The state vector is , is the attitude error, is the velocity error, refers to the position error on the latitude, longitude, and height, and are the gyro and accelerometer errors. In the simulation, the update rate of IMU is , the gyroscope constant bias is 0.02, the angular random walk is 0.002, the accelerometer constant bias is , the velocity random walk is . The position root-mean-square of GPS is 3 m with update rate . The measurement noise is the contaminated Gaussian distribution, , , , is 0.7. The bandwidth of the ISRACKF is 3.0. In order to be more comparative, the square root ACKF (SRACKF) without robust technology is added. The comparison results are the RMSEs handled by 100 Monte–Carlo runs. In Fig. 9, the yaw error, velocity x -axis error, and longitude error can basically represent the filter results in the attitude, velocity, and position. The comparison demonstrates that the SRACKF can obtain close accuracy to the fifth-degree CKF with less 5-CR. Whereas, the manoeuvring of flight and the non-Gaussian noise distribution have an influence on the filter performance. The ISRACKF could deal with the problem well while the other filters are unable to do it. Besides, Fig. 10 shows the improvement of MNLV in the ISRACKF that the computation burden of high degree cubature rule is decreased. So the cost of ISRACKF is less than the SRACKF. In view of higher accuracy, lower calculation burden and stronger robustness, the validity of ISRACKF are verified in the multi-sensor integrated navigation system. Fig. 9Open in figure viewerPowerPoint Attitude, velocity and position error in the integrated navigation system (a) Yaw error, (b) Velocity y -axis error, (c) Longitude error Fig. 10Open in figure viewerPowerPoint Dynamic MNLV with the degree switch in the navigation system (a) Dynamic MNLV, (b) Selection of cubature rule 6 Conclusions In this study, an ISRACKF non-linear filter is proposed to solve the non-linear state estimation problems. By virtue of the updated MNLV, the trade-off between accuracy and computational burden can be achieved. The introduction of local MNLV in the independent non-linear process or measurement model could effectively decrease the computation complexity of the whole system. The MCC robust weight cost function could enhance the robustness. It can optimise the MNLV value, which promotes the appropriate selection of the cubature rule. Numerical examples demonstrate that the ISRACKF exhibits better performance than some non-linear filters, which means it is a preferable selection in some applications. 7 Acknowledgments The authors would like to thank the editors and reviewers for their valuable comments and suggestions which could improve the present and future work. This work was supported in part by the National Natural Science Foundation of China under grant nos. 61673217, 61673219, and 61673214, the Innovation Fund Project for Graduate Student of Jiangsu Province under grant no. KYLX16 – 0450, the Jiangsu Key research and development plan (Modern Agriculture) under grant nos. BE2017370-2 and BE2018004-3. 8 References 1Fang, J., Yang, S.: 'Study on innovation adaptive EKF for inn-flight alignment of airborne POS', IEEE Trans. Instrum. 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