A MEMS-INS assisted GNSS vector tracking loop for robust navigation based on cubature Kalman Filter

The occlusion of shade leads to the intermittent reception of satellite signals, which subsequently results in a decline of the positioning accuracy of Global Navigation Satellite System (GNSS). In order to improve the accuracy of positioning and ensure the continuity of navigation results in weak signal environment, a cubature Kalman Filter (CKF)-based micro-electro mechanical-system (MEMS) grade inertial navigation system (INS) assisted GNSS vector tracking (VT) loop is proposed in this paper. Combining effective GNSS measurements and INS information, attitude, velocity, and position results are predicted by CKF, and the pseudo-range results from navigation filter are fed back into the VT loop. The field tests show that the proposed CKF-based MEMS-INS assisted GNSS VT loop can run continuously and stably in weak signal environment with higher navigation accuracy.


Introduction
With the development of unmanned and autonomous technology, credible location and positioning information are becoming more and more important and critical. 1,2 The Global Navigation Satellite System (GNSS) receiver has become one of the most popular navigation devices for providing precise position information. The GNSS receiver design mainly includes three processing steps: acquisition, tracking, and navigation solution. The carrier phase/frequency and code phase are finely estimated in the tracking link and send to navigation link, the navigation filter combines the ephemeris information to calculate the position, velocity, and time information. The traditional GNSS scalar (CT) tracking loop includes carrier loop and code loop, where the code loop is usually implemented by delay-locked loop (DLL), and the carrier loop includes phase-locked loop (PLL) and frequency-locked loop (FLL). All tracking loops are tracking different satellite signal independently and there is no interaction between channels. 3 In order to improve tracking stability, the more universally adaptive vector tracking (VT) loop was proposed. 4 Unlike the traditional CT algorithm, the VT loop can combine with navigation results to assist the tracking link so that the receiver obtains enough total signal power to successfully track the signal. 5,6 GNSS signal power is significantly attenuated in degraded environments such as indoors, tall buildings, leafy trees, the carrier-to-noise ratio satellite signal is reduced, a carrier-to-noise ratio less than 28 dB/Hz is regarded as weak signal generally. 7 If there is a low carrier-to-noise ratio signal, the VT loop is susceptible because each satellite channel is dependent. The combination of VT scheme and inertial navigation system (INS) is a new approach to construct a reliable navigation system. Because of its better stability and reliability, GNSS/INS integrated navigation system has been widely recognized as a promising alternative to standalone GNSS for navigation in weak signal environment. 8,9 Due to the high price of high precision gyros like laser gyro and optic gyro, micro-electro mechanical-system (MEMS)-grade accelerometers and gyroscopes are being used widely as low-cost Inertial Measurement Unit (IMU) sensors. Whereas, it reduces the accuracy of the GNSS/INS navigation system. It is keenly to achieve the same level of accuracy through advanced data fusion technology. [10][11][12][13][14] Kalman filter (KF) is used in a wide range of applications including communication, navigation, measurement, and control. [15][16][17] For navigation field, KF is not only used in optimizing the tracking loop but also acting as a fusion filter for GNSS and INS. The classical GNSS/INS data fusion uses the error propagation theory of the INS as the basis for the KF estimation algorithm. The data fusion of GNSS/INS is actually a nonlinear filtering problem, INS model will introduce linearization errors in the KF estimation. 18 In order to reduce linearization errors, advanced data fusion methods have been extensively investigated. Various nonlinear filtering techniques, such as the extended Kalman Filter (EKF), the unmarked Kalman Filter (UKF), and the cubature Kalman Filter (CKF) are used in the field of navigation. [19][20][21] The EKF approximates the nonlinear model by a Taylor series expansion, while it's computationally intensive due to the calculation of the Jacobi matrix and poor estimation accuracy by ignoring higher order terms. For GNSS/INS, EKF is not effective in event of weak observability and large initial yaw error. 22 The UKF is comparable to EKF in terms of computation complexity and has higher estimation accuracy than EKF, but it's less effective for highdimensional filtering. Because the dimension of GNSS/ INS integrated navigation is 15 or higher, CKF has higher accuracy and is more stable than UKF in navigation field. 23 However, researchers have only applied the CKF to GNSS/INS integrated system and have not combined the CKF navigation results with GNSS loop tracking.
To improve the accuracy of positioning and ensure the continuity of navigation results in weak signal environment, a CKF-based MEMS-INS assisted GNSS VT loop is proposed in this paper. Attitude, velocity and position results are predicted by CKF combining effective GNSS measurements and INS information, and pseudo-range and pseudo-range rate are fed back into the tracking loop. INS can navigate independently when GNSS satellites are less than four, when GNSS signals return to normal, GNSS/INS integration system would work properly and correct INS errors. The proposed architecture is assessed by field tests, and results demonstrate that the proposed algorithm can run continuously and stably in a weak signal environment, and the navigation accuracy is higher than traditional GNSS VT algorithm.
The rest of this paper consists of four sections. Section 2 introduces traditional VT structure; Section 3 explains the proposed CKF-based MEMS-INS assisted GNSS VT loop; Section 4 analyzes the test results in; and Section 5 draws the conclusion.

Traditional VT structure
The traditional VT Loop is shown in Figure 1. The detailed implementation uses for reference the software defined receiver (SDR) based on VT was proposed in a previous study. 24 The digital intermediate frequency (IF) signal of different channel is mixed, correlated, integrated, and dumped, the output results are then sent to loop discriminator to obtain the phase error of the carrier and code replicas. The carrier phase/frequency and code phase errors are linearly transformed to pseudo-range rate error and pseudo-range error, which are sent to the KF for achieving position, velocity, and time information. Combining with the KF positioning results and ephemeris data, the predicted pseudo-range is used to control code numerically controlled oscillator (NCO) and fed back to each tracking loop. To do this, satellite ephemeris data must be known in advance. In this paper, the ephemeris data is obtained by conventional tracking first.
The pseudo-range error Dr is calculated as where Dt is the phase shift between the received code and the local copied code, k represents the current time, f C=A is the code chipping rate (1.023 MHz for GPS L1 C/A), c is the speed of light.
The code frequency f code, k at time k is expressed as 24 Figure 1. The block schematic diagram of traditional VT structure.
where r is the pseudo-range between the GNSS receiver and the visible satellite, T is the sampling interval. The pseudo-range r at time k is expressed as where p r, k and p s are the receiver position and the satellite s position respectively, Db is GNSS receiver clock bias, and Db s is satellite clock bias, I k is the ionospheric delay, T k is the tropospheric delay, e r is the measurement noise.

The proposed CKF-based MEMS-INS assisted GNSS VT loop
As is shown in Figure 2, the proposed CKF-based MEMS-INS assisted GNSS VT loop includes GNSS, the INS, and the integration navigation filter. On the basis of CKF, GNSS measurement and INS information are integrated to predict attitude, velocity, and position. INS can navigate independently when GNSS signals are invalid, GNSS/INS integration system would work properly and correct INS errors when GNSS signals return to normal. Note that the GNSS navigation in Figure 2 represents the GNSS navigation results, including position, velocity, and time information.
When the carrier to noise ratio is lower than the predetermined threshold, the GNSS fails and tracking loop may loss of locking. INS operates independently when GNSS is invalid. The carrier-to-noise ratio is calculated as where S = I P 2 + Q P 2 , I P is the result of coherent integration of the in-phase branch and Q P is the result of coherent integration of the quadra-phase branch. E(S) is mean of the satellite signal and D(S) is variance.

GNSS/INS with classical KF
Traditional GNSS/INS combination tool is KF, including state model and measurement model.
The state model of KF 25 is where X k is the state matrix, F kÀ1 is the state transition matrix, W b includes w b g and w b a , w b g is gyroscope angular velocity measurement white noise, w b a is accelerometer specific force measurement white noise.
where f is the 331 attitude error matrix, dv is the 331 velocity error matrix, dp is the 331 position error matrix, e b is the 331 gyro bias vector matrix, $ b is the 331 accelerometer bias vector matrix, dl b is the 331 lever arm position error vector matrix between INS and GNSS, dt is the time out-sync error, 0 represents zero matrix. In equation (7) The measurement model of KF is where Z k is the measurement matrix,ṽ INS ,ṽ GNSS ,p INS , p GNSS represent velocity and position results from INS and GNSS. H k is the measurement transition matrix, v b eb represents the angular velocity of rotation from earth frame to body frame. V includes V v and V p , V v is satellite receiver speed measurement white noise, V p is satellite receiver position measurement white noise, a n is average linear acceleration of the carrier, 1 represents unit matrix.

GNSS/INS with CKF
Similar to classical KF, the CKF consists of two parts: prediction and correction. The algorithm structure of CKF is shown in Figure 3.
Cubature point calculation. By the spherical radial criterion, we can find that 26 x i k = S k j i +x k , i = 1, 2, :::, 2n ð11Þ where P k is initial covariance matrix;x k is initial state matrix; n is the number of dimensions of the state matrix and take it as 15; j i is the set of volume points, as follows: where 1 ½ represents the unit matrix.
Update of CKF. One-step prediction of the state mean value One-step prediction of the variance of state values wherex k + 1=k is the one-step predicted value of the state quantity, P k + 1=k is the covariance array of one-step predicted state quantities, Q is the system state noise variance.
The mean value of the one-step predictive measure is The variance of the one-step prediction measure is whereŷ k + 1 is a one-step prediction of the measured value, P y k + 1 is the variance of the one-step prediction measure, R is measuring noise variance for the system.
The covariance between the one-step predicted state value and the one-step predicted quantitative value is The cubature Kalman gain is The posterior mean and posterior variance of the target state arê x k + 1 =x k + 1=k + K k + 1 (y k + 1 Àŷ k + 1 ) ð19Þ

Results and discussions
An electric vehicle is chosen to implement and test the algorithm. The GNSS/MEMS-INS system integrated a low-cost STIM-300 MEMS-based IMU with a GNSS IF signal collector. The GPS L1 signal is collected by the GNSS IF signal collector with IF is 3.996 MHz and the sampling frequency is 16.369 MHz. The noise parameters of STIM-300 are shown in Table 1. In order to get accurate reference information, a real-time kinematic (RTK) equipment was also mounted on the electric vehicle during the field test. The RTK equipment consists of positioning and orientation board (OEM617) developed by Beijing Beidou star navigation Co., Ltd., a GNSS base station, and radio links for data communication. The reference system performance parameters are shown in Table 2. The experimental setup is shown in Figure 4. The environment of field test is shown in Figure 5. In Figure 6(a), the code frequency of traditional CT changes greatly, especially after entering the shade of trees, the code frequency change increased significantly and the code loop is completely out of lock. The code frequency changes of latter two methods are little, and the tracking result of GNSS/INS is slightly better than traditional VT, because the code frequency fed by pseudo-range from navigation result has higher tracking accuracy. In Figure 6(b), the carrier frequency change is similar as code frequency, the carrier frequency waves significantly under the shade of trees. With the exception of the traditional CT method which is completely out of lock, the other two methods are only occasionally out of lock and do not affect signal re-locking. The maximum Doppler shifts of all five GPS satellites are shown in Table 3.    The precision of position solution is necessary to assess the reliability of service and the precision parameters of navigation system. 27 The Root Mean Squared Error (RMSE) of position are given in Table 4, and it is defined by where RMSE E and RMSE N are the RMSE of east and north position. The calculation of RMSE is shown in      equation (22), y i is the position result obtained by the different algorithms, x i is the position result of the reference system. As the RMSE of position for different algorithm from Table 4, it can be seen that the CKF-based MEMS-INS assisted GNSS VT Loop has the highest positioning accuracy compared to other navigation results.

Conclusions
A CKF-based MEMS-INS assisted GNSS Vector tracking Loop is proposed in this paper. The proposed algorithm is adopted to address the accuracy loss caused by interrupted GNSS signal. For tracking processing step, the code frequency is predicted by new pseudo-range from navigation filter in code tracking loop. For navigation processing step, the effective GNSS measurements corrected for MEMS-INS error by CKF. Field experiments demonstrate the effectiveness and superiority of the algorithm. Experimental results show that the proposed algorithm can run continuously and stably in weak signal environment with higher navigation accuracy.

Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.