
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12458-9
10.1016/j.heliyon.2024.e36427
e36427
Research Article
Extended Kalman filter-based robust roll angle estimation method for spinning vehicles
Feng Lu ac
Wu Peng wupeng@ccsu.edu.cn
bcd⁎
Zheng Linhua a
Tong Haibo c
Shi Haonan c
Wang Yong e
a College of Electronic Science and Technology, National University of Defense Technology, Changsha, 410073, China
b Xi'an Key Laboratory of Integrated Transport Big Data and Intelligent Control, Xi'an, 710064, China
c School of Electronic Information and Electrical Engineering, Changsha University, Changsha, 410022, China
d Hunan Province Navigation and Attitude Measurement Integrated Application Engineering Technology Research Center, Changsha, 410022, China
e Hunan TGONG Precision Technology Co., Changsha, 410201, China
⁎ Corresponding author. Xi'an Key Laboratory of Integrated Transport Big Data and Intelligent Control, Xi'an, 710064, China. wupeng@ccsu.edu.cn
20 8 2024
30 8 2024
20 8 2024
10 16 e3642728 12 2023
6 7 2024
15 8 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Attitude measurement is a basic technique for monitoring vehicle motion states and safety. The spin motion of a vehicle couples the attitude angles with each other, which has an impact on the navigation and control of the vehicle. Global navigation satellite system (GNSS) signals-based roll angle measurement methods are important for vehicle attitude measurement. Most of existing studies use continuous signal power, but the case of loop lock loss leading to discontinuous power reception has not been considered. A robust estimation method for the roll angle based on the Tukey weight function is proposed to improve the measurement accuracy in cases of discontinuous reception. The characteristics of the GNSS signals, the geometric relationship between the signal power and roll angle of the vehicle are discussed. By installing a GNSS receiver with a single patched antenna on a rotating platform with a controllable rolling speed, the proposed method was verified by experiments. The robust estimation errors of different weight functions are analyzed. According to the characteristics of the gross measurement errors, a robust estimation method of multisatellite power observations is proposed to obtain a high-precision and stable estimation of the vehicle roll angle. The results show that the proposed algorithm can improve the accuracy of roll angle estimation even with gross measurement errors. As a result of the experiments, the estimation errors of the algorithm are 6.57° at a confidence level of 68 % and 15.49°at the confidence level of 95 %. In contrast, they are 11.38° and 37.31° for the traditional LS method. Moreover, the estimation accuracy of the algorithm is not significantly correlated with the vehicle rotational speed. Therefore, the vehicle roll angle can be estimated with high accuracy under a variety of rotational speeds.

Keywords

Roll attitude measurement
GNSS signal power
Robust estimation
Tukey weight function
==== Body
pmc1 Introduction

The attitude parameters of a vehicle are important in navigation. The attitude measurement of vehicle refers to the determination of a vehicle's orientation in three-dimensional space. These attitude parameters typically include heading angle, pitch angle, and roll angle. This technology is one of the key techniques in aviation, aerospace, navigation, and terrestrial navigation. Conventional methods for acquiring vehicle attitude information include inertial navigation systems, geomagnetic sensors, and satellite navigation receivers [1,2]. Conventional inertial navigation systems have the advantages of low power consumption, good autonomy and easy integration [3]. However, when used in vehicle attitude measurements, measurement errors gradually accumulate over time, and the vehicle attitude cannot be accurately measured continuously. Moreover, initial calibration is necessary, and its cost is much greater than those of other attitude measurement methods. In addition, the volume of inertial devices is large, so a large installation space is required in vehicles, and the high overload of vehicles during launch often prevents inertial devices from working normally. In contrast to those of inertial devices, the measurement errors of geomagnetic sensors do not accumulate over time. Moreover, geomagnetic sensors work independently of the vehicle motion state. Thus, such sensors have advantages in terms of anti-overload effects and application costs. However, their operation is easily affected by the external magnetic fields and the metal structure of the vehicle, and the interference resistance is poor. Thus, these devices are usually used as auxiliary sensors for attitude measurements and are combined with other sensors [[4], [5], [6]]. The global navigation satellite system (GNSS) has high positioning accuracy and strong interference resistance ability, and it has the advantages of low costs and high stability in overload circumstances. Thus, the GNSS is widely used in vehicle navigation and attitude measurement [7]. Currently, flight control systems are widely equipped with GNSS receiving equipment. Therefore, users can obtain high-precision position, navigation, and time information with GNSS due to its fast signal acquisition and stable tracking. Moreover, the GNSS, which has low power consumption, low structural complexity, and high-cost effectiveness, can be used to obtain high-precision attitude measurements.

To ensure the flight stability of sounding rockets, intelligent ammunition and other vehicles, their rotations are usually maintained during motion. A change in the roll angle will also directly affect the measurement of other attitude angles and subsequently affect the operation of the vehicle's navigation and control system [8]. Therefore, measuring the roll angle is a priority for determining the vehicle's attitude parameters. The spinning of the vehicle makes the gyroscopic effect and Magnus effect more obvious and makes it difficult to use traditional rolling attitude measurement methods.

Since the 1990s, researchers have used the GNSS carrier phase to measure vehicle heading and rolling attitudes [[9], [10], [11]]. The measurement methods with the GNSS signal as observations can be divided into single-antenna and multiantenna types [12]. In multiantenna methods, multiple antennas are usually installed in the same cross section of a rotating vehicle, and the vehicle's rolling attitude is identified through the characteristics carrier phase difference [13,14]. The integer ambiguity in carrier phase tracking is a key problem in the application of this method. These methods require multiple receiving antennas to be installed on the side of the vehicle, which requires additional space and has a negative impact on the structural design of the vehicle. Additionally, the signals received by multiple antennas can interfere with each other. These factors limit the practical application of the GNSS multiantenna vehicle attitude measurement method.

In the single-antenna attitude measurement mode, the antenna receiving the satellite signal is usually installed on the side of a cylindrical vehicle, and it moves in a circular motion on the cross section of the vehicle. The received signal contains a large amount of rolling information about the vehicle and can be used to measure the rolling attitude of it. At present, in research on vehicle attitude measurements using a single antenna, the phase locking loop (PLL) is mainly used to track changes in the GNSS carrier phase, code phase and received signal power. The advanced spinning-vehicle navigation component that was designed by Doty's team uses the PLL to eliminate the impact of vehicle rotation on the received signal; additionally, the vehicle roll frequency is estimated according to the change in the rotation phase [15,16]. Kim's team used the orthogonal correlation method to estimate the vehicle rotation frequency and phase by tracking and observing the carrier phase and carrier frequency offset caused by rotation [17,18]. Bahder et al. proposed a method of using a single antenna to track the carrier phase of seven satellites to determine vehicle positions and attitudes. This method is based on the phase change in the received signal determined by the position and direction of the satellite signal [19]. Shen Qiang et al. analyzed the relationship between the GNSS signal power received by a single antenna and the rolling attitude of the vehicle and tracked the change in the signal power by means of an FLL-assisted PLL to obtain the rolling attitude of the vehicle. With increasing rolling frequency, the estimation error of the roll angle may significantly increase [20,21]. In Ref. [22], a method was proposed in which a UKF was used to estimate the roll angle with a model of the GNSS-received signal power, and this method achieved better accuracy under the simulation conditions. However, this method has not been analyzed in practice.

According to existing methods of roll angle estimation, which utilize the carrier phase, the carrier tracking loop may experience signal interruption due to blockage by the vehicles (e.g., an aircraft or spacecraft) or a rapid decrease in antenna gain caused by the antenna turning towards a satellite located behind it. Such interruptions can lead to significant losses in the carrier tracking loop, resulting in substantial errors in roll angle estimation. When the power of the signal is lower than the acquisition sensitivity of the receiver, the rolling attitude estimation loop cannot obtain the observations, resulting in discontinuous rolling attitude estimation or divergence of the estimation error. Although the assisted GNSS method proposed in Ref. [23] can perform the vehicle positioning under rolling conditions, the roll attitude estimate is still affected. In conclusion, the previous studies are mainly focus on methods such as phase-locked loops to extract the roll angle of the vehicle. In improving measurement accuracy, only continuously received GNSS signals are used as observations. The problem of discontinuous signal reception and its impact on estimation accuracy has not been mentioned. This paper will mainly focus on this discontinuous reception problem.

First, a microstrip antenna installed on the side of a cylindrical rolling vehicle is used to receive GNSS signals. The gain pattern of the antenna shows a consistent change as the vehicle rolls. The received signal power is taken as the observation. The received signal power is obtained by coherent integration in the baseband processing of the receiver. To track signals with a wider dynamic range, the carrier tracking loop in baseband processing is usually designed to be less sensitive, so the tracking of signals is more prone to out of lock. This makes the signal power discontinuous. There will be obvious gross errors in the roll angle estimation when discontinuous signal power is received.

Second, considering the correlation between the law of power change and the roll angle of the vehicle, a real-time estimation model of the roll angles for multiple satellites is established in this paper. Polynomial fitting is used to obtain the analytic function expression between the signal power and the rolling angle of the vehicle. Then, the independent fitting polynomial of each satellite is used as the EKF measurement equation, and the roll angle is estimated by the received single-satellite signal power. We use the Tukey weight mode to perform robust processing on the roll angle estimates after analyzing the characteristics of several kinds of weight schemes. The robust estimation is use to solve the influence of gross measurement errors on the estimation results. With multiple satellite power observations, this method can effectively reduce the influence of gross errors in the observations and improve the accuracy of roll angle estimation.

Finally, the algorithm proposed in this paper is verified on a rolling experimental platform equipped with a GNSS receiver. The experimental results verify the feasibility of the proposed algorithm. The errors in the measured results are analyzed, and Approaches based on the active selection of high-quality observations are discussed for further studies.

2 Analysis of the signal power received by a single GNSS antenna during rolling

The GNSS signals received by the rolling vehicle are shown in Fig. 1, these signals are received by the rectangular microstrip antenna mounted on the side of the cylindrical rolling vehicle. The antenna receives signals from different satellites, and the signal incidence directions change with the rotation of the vehicle.Fig. 1 Diagram of the GNSS signal received by a rolling vehicle.

Fig. 1

Fig. 2 shows the conversion from the Earth-centred Earth-fixed (ECEF) coordinate system to the vehicle rolling coordinate system. The positions of the satellites and receivers can be calculated using ephemeris received or with external inputs. It is assumed that the position in the ECEF coordinate system is s(n)(x(n),y(n),z(n)) and that the centroid coordinate of the vehicle is OENU(x(u),y(u),z(u)). The elevation angle αecef formed by the line of sight (LOS) projection and the vector OX and the azimuth angle βecef formed by the LOS projection and the XOY plane can be calculated.Fig. 2 Definition of the rolling coordinate system.

Fig. 2

Then, the transformation matrix from the coordinate system O−XYZ to the topocentric coordinate system OENU−ENU of the vehicle can be represented as Eq. (1).(1) L=[−sinβecefcosβecef0−sinαecefcosβecef−sinαecefsinβecefcosβecefcosαecefcosβecefcosαecefsinβecefsinαecef]

The position of satellite n in the OENU−ENU coordinate system can be expressed as senu(n)(xenu(n),yenu(n),zenu(n)), and it can be calculated by Eq. (2).(2) [xenu(n)yenu(n)zenu(n)]=L⋅{[x(n)y(n)z(n)]−[x(u)y(u)z(u)]}

To calculate the roll angle, the rolling coordinate system is defined as OENU−XRYRZR. It is assumed that the roll angle calculation plane is a cross section passing through the centre of mass of the vehicle. The origin of the rolling coordinate system is the centre of mass of the vehicle OENU. The axis of the roll coordinate system OENUZR is defined as the direction of the vehicle's rolling axis. For flying vehicles, the direction is usually consistent with the radial movement direction of the vehicle. We assume that the elevation angle of the vehicle motion direction vector in OENU−ENU is αv and the azimuth angle is βv. The coordinate system OENU−XRYRZR can be obtained by coordinate rotation, as axis OENUU rotates in the direction of OENUZR, and axes OENUXR and OENUYR can be obtained accordingly.

The coordinate transformation matrix from OENU−ENU to OENU−XRYRZR can be represented as Eq. (3).(3) R=[−sinβvcosβv0−sinαvcosβv−sinαvsinβvcosβvcosαvcosβvcosαRsinβvsinαv]

Then, the position of satellite n in the rolling coordinate system can be obtained as sR(n)(xR(n),yR(n),zR(n)), that can be calculated by Eq. (4).(4) [xR(n)yR(n)zR(n)]=R⋅[xenu(n)yenu(n)zenu(n)]

In the rolling coordinate system, the direction OENUXR is defined as the vehicle roll angle 0∘. When the microstrip patch antenna is mounted on the side of the cylindrical vehicle, the roll angle is determined by OENUP and OENUXR,where P is the phase centre of antenna.

The signal power used in this algorithm represents the combined power from orthogonal branches of the IQ demodulation process within the carrier tracking loop of the GNSS receiver. This power directly correlates with the antenna's gain [24]. When a microstrip antenna is used to receive GNSS signals, a change in the direction of the incident signal will change the gain of the antenna. Fig. 3 shows the gain pattern of the microstrip patch antenna for the GNSS receiver.Fig. 3 (a) Gain direction diagram of the microstrip antenna of E-plane, (b) Gain direction diagram of the microstrip antenna H-plane.

Fig. 3

The antenna gain is usually calculated by the incident angle θA and ϕA of the signal [25].Fig. 3(a) shows the change of antenna gain when ϕA changes in the E-plane, and θA=90∘. Fig. 3(b) shows the change of antenna gain when θA changes in the H-plane, and ϕA=0∘.When the signal incidence direction is perpendicular to the surface of the microstrip antenna (θA=90∘ and ϕA=0∘)， the maximum gain is reached. Otherwise, the signal amplitude declines. When the incident signal is received from back of the antenna, the received signal gain is minimized. In addition, when the receiving plane of the antenna turns behind the vehicle, the signal reception will be blocked by it, and corresponding attenuation will occur.

The power variation in the received GNSS signal is analyzed in the rolling coordinate system. Fig. 4 shows the variation in the incident signal direction in the vehicle rolling coordinate system.Fig. 4 Analysis of the signal power received by a single antenna.

Fig. 4

Since the power variation law of the received signal is shown in Fig. 4, the rolling coordinate system of the vehicle is not represented in the figure, and only the change in the antenna coordinate system is given. When the vehicle rotates, the phase centres of the antenna are P and P′ at two epochs, and the antenna coordinate system rotates from P−XaYaZa to P−Xa′Ya′Za′. The radial motion direction of the vehicle is assumed to be the direction of the rolling axis. Then, the axes OENUZR and PZa(P′Za′) are in the same direction. In the figure, S(n) represents the location of the satellite, which is in the plane XaPZa.We drop perpendiculars from point S(n) to vectors PXa and P′Xa′, and the perpendicular feet are denoted as A and B, respectively. At this time, the angle ϕ (ϕ′) between the LOS and the normal vector PXa (or P′Xa′) of the antenna receiving plane is the signal incidence angle, which is composed of incident angles θA and φA in the antenna analysis. Therefore, the following expressions can be obtained:(5) sinϕ=|S(n)AS(n)P|

(6) sinϕ′=|S(n)BS(n)P′|

In plane XaPZa, S(n)A⊥PXa and PZa⊥PXa; thus, S(n)A//PZa. When the antenna is fixed on the side of the vehicle, the plane in which the antenna normal vector is located is the same plane as the cross section passing through the antenna phase centre. PXa (P′Xa′) is in the cross section, and PZa (P′Za′)is perpendicular to the plane. Then S(n)A is also perpendicular to the plane XaPYa.Therefore S(n)A is vertical with respect to vector AB.

And both vectors are perpendicular to the plane XaPZa. Therefore, S(n)A⊥AB. According to the sine theorem, for ΔS(n)AB, the following expression Eq. (7) can be derived.(7) |S(n)B|sin∠S(n)AB=|S(n)A|sin∠S(n)BA

Moreover, ∠S(n)AB=90∘>∠S(n)BA. Thus, |S(n)B|>|S(n)A|. By substituting this relation into Equations (5), (6), when {ϕ,ϕ′}∈[−π2,π2], ϕ>ϕ′ can be obtained. That is, when the vehicle rolls, only when the LOS is in the plane formed by the normal vector of the antenna receiving plane and OENUZa does the incident angle formed by the signals of satellite n reach the minimum value. In combination with Fig. 3, this means that the received signal has the maximum gain at this point. Moreover, when the phase centre of the antenna rotates along the cross section of the vehicle, the received signal power changes periodically with the signal incidence angle ϕ.

According to the analysis above, the GNSS signal power received by a single patched antenna will change periodically while the vehicle rolls. The feature can be used in roll attitude estimation. However, the rotation of the vehicle also makes the acquisition and tracking in the baseband signal processing of the receiver unstable. When carrier-to-tracking loop is out of lock, the periodically changing signal power cannot be output by the receiver, which will cause an error in the attitude estimation. When using the signal power to estimate the roll angle of the vehicle, different satellite positions also affect the maximum gain, which affects the accuracy of the estimation.

In the carrier tracking loop of the GNSS receiver, the parameters are selected to adapt to a larger dynamic range. Then, the tracking accuracy decreases, and the carrier tracking loop is more likely to be out of lock. Fig. 7 shows the discontinuous signal reception caused by lower tracking accuracy during GNSS signal acquisition.

Fig. 5(a) shows a set of signal power with a duration of 50000 ms, indicating that the discontinuities reception occurs in satellites. Fig. 5(b) shows the details of the discontinuous reception for Satellite 4. The receiver tracking loop transitions from stable tracking to out-of-lock. During this period, the signal power output by the receiver changes from obvious periodicity to a decreasing periodic fluctuation amplitude, and it finally loses its periodic characteristics. According to a series of experiments, discontinuous reception occurs randomly during the whole receiving process. This leads to significant errors in the roll angle estimation using signal power as an observation. To solve this problem, a method that can effectively resist random errors is needed to improve the accuracy of vehicle roll angle estimation.Fig. 5 Discontinuity of the received signal power, (a) received signal power of 3 satellites, (b) detail in received signal power of Satellite 4.

Fig. 5

3 Roll angle estimation model based on the power of the GNSS signal

To estimate the roll angle using the signal power characteristics, it is necessary to establish a correlation model between the received signal power amplitude and the vehicle roll angle. The geometric relationship between the two is shown in Fig. 5.

The symbols used in Fig. 6 are the same as those given previously. Because OENUS(n)≫OENUP, it can be assumed that OENUS(n)=PS(n). Since PS(n) is in the plane ZaOENUXa, the maximum gain will be obtained in a rolling period. Moreover, the elevation angle of satellite S(n)(xR(n),yR(n),zR(n)) in the rolling coordinate system OENU−XRYRZR is αR(n), and the azimuth angle is βR(n), which can be obtained by Eq. (8).(8) {αR(n)=arctg[zR(n)(xR(n))2+(yR(n))2+(zR(n))2]βR(n)=arctg[yR(n)xR(n)]

If the antenna reaches P at tm, the following expressions can be obtained:(9) φ(n)(tm)=βR(n)

Taking the roll frequency fr as a prior condition, the roll angle can be calculated from(10) φ(n)(t)=φ(n)(tm)+2πfr(t−tm)=βR(n)+2πfr(t−tm)

the antenna gain reaches its maximum value in a rolling period at tm, and the vehicle roll angle coincides with the azimuth angle of the satellite. Then, the roll angle at t can be expressed by Eq. (10).Fig. 6 Signal incidence direction in the rolling coordinate system.

Fig. 6

Fig. 7 Correlation model of the power amplitude and roll angle.

Fig. 7

Fig. 7 shows the correlation between the amplitude of the GNSS signal power and the roll angle of the vehicle. The figure shows that the vehicle roll angle and the signal power are correlated during a rolling period, and these two parameters are periodic time functions. To obtain the function relating the signal power and roll angle, the signal propagation model when the power amplitude reaches the maximum value is analyzed in the rolling coordinate system.(11) y(n)(t)=f[φ(n)(t)]0≤t≤1fr

In Eq. (11), y(n)(t) denotes the signal power from satellite n, and f(·) is the function representing the relationship between the signal power and roll angle. φ(n)(t) denotes the roll angle determined by signal power from satellite n. The variable t is limited to 0 to 1fr, which means that the function represents only the relationship between the signal power and the roll angle in a cycle.

Then, the mapping relationship between the roll angle and signal power can be obtained through time consistency. To obtain an accurate analytic expression of the GNSS signal power and vehicle roll angle, the periodic change in the signal power is fitted to the sinusoidal polynomial of the rolling angle represented by Eq. (11). Sinusoidal polynomials are widely used to express concepts in mechanics, acoustics, electrical engineering and other disciplines. Physical phenomena such as vibrations, waves, oscillations, and rotations are often reduced to sinusoidal problems. When a sinusoidal polynomial is used to fit the values, the fundamental frequency and harmonic characteristics of the values can be better reflected. In this algorithm, a roll period of the signal power is extracted. The roll angle within a cycle can be obtained by Eq. (9) to Eq. (10), which is mapping with the signal power. Considering the complexity of the algorithm, a third-order sinusoidal polynomial is selected, and higher-order polynomials can obtain better fitting accuracy. The fitting polynomial can be expressed as follows:(12) f[φ(n)(t)]=a0+a1sin[b1φ(n)(t)+c1]+a2sin[b2φ(n)(t)+c2]+a3sin[b3φ(n)(t)+c3]

Eq. (12) is an expansion of Eq. (11), where ai, bi and ci are the coefficients of the polynomial. The functional analytic expression shown in Eq. (12) is used as the measurement equation in the EKF.

4 Robust EKF roll angle estimation method based on the Tukey weight mode

Due to the nonlinearity of Eq. (12), which we take as measurement equation, the extended Kalman filter (EKF) method is applied. It is often used to solve the estimation problem under such nonlinear conditions.

4.1 State update

In the EKF roll angle estimation method based on sinusoidal polynomial fitting, the roll frequency of the vehicle is considered constant. The filtering model is established in constant-velocity mode. The state variable is defined by xˆk=[φk,ωrk], and the state equation and the covariance matrix are as Eq. (13) ∼ Eq. (15).(13) {xˆk−=Axˆk−1+Wk−1Pk−=AkPk−1AkT+Qk

where xˆk− represents the prior estimate of the state variable, φk and ωrk represent the roll angle and angular frequency of rolling, respectively.(14) φk=ωrkt+φ0

φ0 is the initial roll angle. The process noise Wk=[0wk]T is considered as a zero-mean white noise, and its variance is σ2. A=[1Δt01] is the transition matrix from state k−1 to k.(15) E[wk]=0,Cov(wk)=E[wkwkT]=Qk

where QK=[Δt33Δt22Δt22Δt]⋅σ2 is the process noise covariance matrix. Δt is the sampling interval (1 ms in this paper).

4.2 Measurement update

The power of the GNSS signal is taken as the observation in the paper, and an accurate mathematical analytic expression of the measurement equation can be obtained by sinusoidal polynomial fitting according to Eq. (12). The posterior state and the estimation error covariance matrix are expressed by Eq. (16).(16) {Kk=Pk−CkT(CkPk−CkT+VkRkVkT)−1xˆk=xˆk−+Kk[yk−h(xˆk−,0)]Pk=(I−KkCk)Pk−

where Kk is the filter gain. where yk is the signal power of a satellite in epoch k. h(·) is a function of the state variable xk,and Eq. (12) shows its functional form.(17) yk=h(xk,vk)

vk is the measurement noise of the observation. When the signal power is processed independently, the noise is 0-mean Gaussian white noise superimposed on the propagation path, as is the noise power. As described above, the observation can be expressed as a function of φk; that is, the relationship between the observation and the state variable can be established by Eq. (17). However, this equation is a nonlinear function, so it needs to be linearized by the Taylor expansion. Eq. (18) ∼ Eq. (22) show the linearized forms of the measurement equation.(18) h(xk,vk)=f[φ(n)(t)]+vk

(19) Ck=∂hx,v∂x|x=xˆk−,vk=0=∂fφ∂φ|φ=φk−,vk=0,∂fφ∂φ|φ=φk−,vk=0⋅∂φωr∂ωr

(20) Vk=∂h(x,v)∂v|x=xˆk−,vk=0

The measurement noise covariance matrix is(21) E[vk]=0,Cov(vk)=E[vkvkT]=Rk

and(22) Cov[wk,vj]=E[wkvjT]=0

4.3 Robust estimation method for the roll angle based on the Tukey weight mode

With the EKF, the real-time vehicle roll angle in any epoch can be calculated. As mentioned above, discontinuous reception introduces gross errors into the observations, which in turn leads to significant deviations in the roll angle estimations. Robust estimation is an effective method for preventing gross measurement errors. It is based on M-estimation. The key idea is to evaluate the rationality of the valuation and assign the weight mode according to the difference between the assumed model and the actual effective valuation. Weight functions, including Huber, IGG, Andrews, and Tukey, are commonly used for weight assignment [[26], [27], [28], [29], [30]].

The Huber weight mode performs weight reduction for doubtful observations without setting an elimination zone. Although the number of available observations increases, the anti-error effect weakens, and gross errors still affect the estimation results to a certain extent. In the IGG scheme, LS estimation is used for normal observations to improve the estimation efficiency. However, when the variance of the observations is large, the variance of the robust estimation results can also increase. The Andrews weight mode sets a selection area and reduces the weights of all observations which can effectively avoid the problems of the Huber and IGG schemes. However, the weight reduction of suspicious observations is relatively slow, which has a certain impact on the accuracy of the estimation results. Smooth and rapid weight reduction for suspicious observations is adopted in the Tukey weight mode, and abnormal observations are eliminated. Thus, better accuracy is obtained in robust estimation with the Tukey weight function. According to the gain of the microstrip antenna and the analysis of the signal power observations, the Tukey weight mode is selected to perform a robust estimation of the roll angle.

If the signal power observations of n satellites participate in the roll angle estimation, then there is an n-dimensional roll angle estimation vector for each observation epoch. The gross error in the roll angle estimation of n satellites can be identified by the robust estimation method, and the weight can be assigned to obtain an accurate estimation of the roll angle in the current epoch. The estimation process is as follows.(23) {φˆLrb=(BTp‾L−1B)−1BTp‾L−1ΦkVLrb=BφˆLrb−Φkp‾L=p‾L−1ΓL

In Eq. (23), Φk=[φk(1),φk(2),⋯φk(n)]T is a set of values φk(n), which are the roll angles estimated by the EKF at epoch k. B is an n×1-dimensional coefficient matrix, and the observations from each satellite are considered to have the same coefficients. p‾ is the equivalent weight function adjusted by iteration during robust estimation. Because each observation is independent, the original weight function matrix is set as an identity matrix in this paper. Vrb is the observation residual vector whose elements are virb. Γ=[w1,w2,⋯wn]T is the weight reduction factor. To distinguish the iterative process of robust estimation from the EKF update process, the number of robust estimation iterations is represented as L, and the termination condition is that the estimation differences at adjacent epochs is less than a selected threshold. φˆLrb is the roll angle in epoch k when the iterative process of robust estimation ends. In calculating the weight reduction factor, the residual vector needs to be standardized, and the variance factor is defined as Eq. (24).(24) σˆ2=(Vrb)Tp‾Vrbr

where r=n−m is the number of redundant observations. Then the standardized residual is |viσ|. In the Tukey weight mode, the weight reduction factor is calculated by Eq. (25).(25) wn={[1−(|vnσ|⋅1c)2]2|vnσ|≤c0|vnσ|>c

where c is generally 4.685. By setting the convergence condition of the robust roll angle robust estimation, the variance factor and the weight of each observation can be modified in one observation epoch, and the estimation results with improved accuracy can be obtained.

5 Experimental verification and performance analysis of the algorithm

To verify the effectiveness and accuracy of the proposed roll angle estimation method, a series of experiments were carried out. The experiments used a rotation experiment platform. A microstrip antenna is installed on the side of cylindrical part of the platform. Fig. 8 shows the structure of the experimental platform. The experiments were conducted between July and August 2023. An autonomously developed GNSS receiver was employed for the experiment. B3I signals in BDS, which has a nominal frequency of 1268.520 MHz,were collected. In the experiments, signals of satellites NO.1 to NO.5 were acquired with the sampling rate 1 KHz, with each collection lasting for a minimum of 200 s.Fig. 8 Rotational experiment platform.

Fig. 8

The experimental platform is moved by program-controlled motors, and the rotational speed can be set within the range of 0∼25 revolutions per seconds by the software. The GNSS receiver is placed inside the platform. When the rotational speed is set, the cylindrical part of the platform starts to rotate under control of the motor. And the GNSS receiver starts to work. The signal power received from each satellite through independent channels is output by the receiver. Then the signal power is transmitted to a PC via a serial port. Specialized data processing software can be used to parse this data and store it in commonly used data file formats. The simulating of this algorithm was then carried out on the PC.

Moreover, a Hall sensor is used to set the standard for the roll angle estimation. It is used to observe the vehicle rolling state synchronously. The Hall sensor is fixed on the side of the rotating platform, and the magnet is mounted on the cylindrical part of the platform. When the cylindrical part rotates, the magnet, which is detected by the Hall sensor, rotates with it.

Two receivers are used in the experiments. These receivers are used for GNSS signal acquisition in the proposed algorithm and provide a clock for the Hall sensor. This ensures that the estimation algorithm and Hall detection are strictly synchronized. The Hall sensor determines the vehicle rotational speed and roll angle from the time interval between the two passes of the magnet. Since there is a fixed difference between the installation position of the Hall sensor and the position of the vehicle roll angle of 0°, the initial angle difference is measured by the gradient sensor and high-precision digital protractor after the sensor is installed.

Fig. 9 shows the roll angle estimation results of the EKF, when signals received from a satellite. The rotational speed is 10 r/s. Comparing the top and bottom figures, it is not difficult to find the impact of discontinuous reception on roll angle estimation. When the signal is received discontinuously, as shows in Fig. 9(a) (during 7000 ms–10000 ms, 16000 ms–19000 ms, etc.), large errors appear in the EKF roll angle estimation results, as show in Fig. 9(b), obtained by using this signal power as the observation. While during other stable signal reception periods, the roll angle estimation results are stable.Fig. 9 Roll angle estimation of the EKF at 10r/s by single satellite. (a) Shows the signal power of Satellite 2 in 10s, and (b) shows the relevant roll angle estimation of the EKF.

Fig. 9

Under the same receiving conditions, signals from multiple satellites are received by independent channels. The signal power and corresponding roll angle estimation results are shown in Fig. 10. As figure (a) shows the signal power of Satellite 2 and Satellite 3 remains continuous throughout the observation epoch, while the signal of Satellite 4 experiences significant attenuation starting from 500 ms. As figure (b) shows, when the signal is continuous, the roll angle estimation results from all satellites exhibit consistent. However, when Satellite 4 experiences discontinuous reception, its roll angle estimation results cannot maintain consistency with the others. Therefore, if the traditional Least Squares (LS) method is adopted to overcome the influence of observation noise using signals of multisatellite, the estimation results will be affected by gross errors during discontinuous reception, leading to deviations.Fig. 10 Roll angle estimation of the EKF at 10r/s by multiple satellites. (a) Shows the signal power of 3 satellites in 2s, and (b) shows the roll angle estimation of the EKF, relatively.

Fig. 10

When using the roll angle detected synchronously by the Hall sensor as the standard, to analyze the stability of the algorithm over a longer period, the roll angle estimation results will be output every second. This approach is also more aligned with practical application requirements, as it can reduce the delay of data reading, writing, and storage to a certain extent. Thus, the following analysis will be carried out for a rate of one output per second.

After the roll angle is estimated by the EKF method designed in this paper, a robust estimation method with a Tukey weight function is used to solve the problems caused by discontinuous reception. In the experiment, the robust extended Kalman filter method is compared with the least squares method. The performances of different weight modes, such as Huber, Andrews and IGG, are also compared with the Tukey weight mode for robust estimation.

In Fig. 11, the robust estimation method with the Tukey weight mode (R-Tukey) is compared with the LS method. In the experiment, three rotational speeds were selected, and the roll angle estimation results were obtained in 50 s under the same experimental conditions. The rotational speed is 2r/s in Figs. 11(a), 10r/s in Figs. 11 (b), and 20r/s in Fig. 11 (c). The figure shows that the error of the roll angle estimation using the proposed algorithm is significantly smaller than that of the LS method.Fig. 11 Roll angle estimation of R-Tukey method and LS method, respectively. The rotational speed is 2r/s (a), 10r/s (b), and 20r/s (c).

Fig. 11

Fig. 12 shows the error of the robust estimation using different weight modes under different rotational speeds. The rotational speed is 2r/s in Figs. 12(a), 10r/s in Figs. 12(b), and 20r/s in Fig. 12 (c). It can be seen from the figure that when the signal is received continuously, the errors of the robust estimation results given by several weight modes are close to each other. When discontinuous reception occurs, the Tukey weight model obtain a smaller error. This means that the algorithm proposed in this paper is effective in addressing discontinuous reception problems.Fig. 12 Roll angle estimation of various weight functions, respectively. The rotational speed is 2r/s (a), 10r/s (b), and 20r/s (c).

Fig. 12

For quantitative analysis of the proposed algorithm, several experiments were carried out. Ten groups of data were collected at three rotational speeds, and the data acquisition time of each group was more than 60 s Table 1 is the statistical analysis for the roll angle estimation errors with a confidence level of 68 % and 95 %.

At various rotational speeds, it can be seen from the statistical results in Table 1, the error is the smallest in the method with Tukey weight function. When the gross measurement errors are used in traditional LS method by multisatellite, the roll angle estimation results are interfered by these errors, and they have not been processed in any way. Then, the estimation error is larger than the others. When robust estimation is introduced and the weight mode is used to fuse the roll angle estimations of different satellites, the estimation error can be significantly reduced. In the robust estimation process, the setting of the weight mode in the normal region and elimination region will affect the accuracy of the estimations. Although the performances of the Tukey and Hubber weight mode are similar, the elimination area is not set in the Huber mode, and the gross error of the observation still affects the estimation in a long term. According to the experimental results, the estimation accuracy of the proposed algorithm does not change with rotational speed, so the algorithm can estimate the roll angle at different rotational speeds.Table 1 Error analysis of the roll-attitude estimation algorithm.

Table 1Weight functions	errors at 2 r/s (°)	errors at 10 r/s (°)	errors at 10 r/s (°)	Average error （°） 68 %	Average Error （°） 95 %	
68 %	95 %	68 %	95 %	68 %	95 %	
Tukey	7.24	12.37	3.50	11.26	8.98	22.83	6.57	15.49	
Andrews	10.95	21.25	4.52	19.73	11.13	26.15	8.87	22.37	
Huber	7.50	15.67	3.81	11.89	9.07	24.34	6.79	17.30	
IGG	10.03	16.34	4.69	13.53	9.83	26.22	8.18	18.70	
LS	13.66	38.50	12.66	35.58	7.81	37.86	11.38	37.31	

6 Conclusion

A roll angle estimation method based on a robust EKF is proposed in this paper, which can improve the accuracy of the roll angle estimation under the condition that the gross measurement errors are inevitable.

The features of GNSS signals received by a single microstrip antenna mounted on the side of a rolling vehicle are studied in this paper. The law of signal power variation is analyzed, and the roll angle of the vehicle is estimated according to this law. The relationship between the received signal power and the rolling attitude is analyzed via geometric analysis. The influence of the GNSS receiver carrier tracking loop on signal reception is analyzed, that has not been considered in related research. The design intended to meet the dynamic demand of signals reduces the tracking accuracy of the receiver, which leads to discontinuous reception of signal power observations and subsequent roll angle estimation. An extended Kalman filter is designed at the first step, and the vehicle roll angle can be estimated by the received signal powers of multiple satellites. Then, according to the characteristics of the measurement errors in multisatellite estimation, the Tukey weight mode is used to process the roll angles estimated by multiple satellites.

The performance of the proposed algorithm is verified by setting up a semi-physical simulation platform that can simulate the rotation of a vehicle. The experimental results show that the robust EKF algorithm designed in this paper can effectively estimate the vehicle roll angle. According to the statistical results, at the confidence level of 68 %, the estimation error of the algorithm in this paper is 6.57°, and 15.49°at the confidence level of 95 %. Compared with that of the traditional LS method, they are 11.38° and 37.31°. Moreover, the estimation accuracy of the algorithm is not significantly correlated with the vehicle rotational speed.

Currently, discontinuous GNSS signals are passively received. It can be further studied that by actively identifying discontinuous reception phenomena and selecting better-quality observations, the estimation accuracy of the algorithm can be further improved.

Data availability statement

Data will be made available on request from the corresponding author.

CRediT authorship contribution statement

Lu Feng: Writing – original draft, Validation, Software, Methodology, Data curation, Conceptualization. Peng Wu: Methodology. Linhua Zheng: Methodology. Haibo Tong: Validation. Haonan Shi: Validation. Yong Wang: Validation.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This paper was supported by the Key Research and Development Projects of the Hunan Provincial Department of Science and Technology in 2022 (2022GK2026 ), the 10.13039/501100004735 Hunan Natural Science Foundation Project (2022JJ30636 ), the Science and Technology Plan Project of the Hunan Provincial Department of Natural Resources (2023-78), the Science and Technology Plan Project of the Hunan Provincial Department of Natural Resources (2022-21 ), the Aid Program for Science and Technology Innovative Research Teams in Higher Educational Institutions of Hunan Province, the Open Fund of the Xi'an Key Laboratory of Integrated Transport Big Data and Intelligent Control (10.13039/501100002837 Chang'an University ) (300102343515 ).
==== Refs
References

1 Le V.D.T. Nguyen A.T. Nguyen L.H. Dang N.T. Tran N.D. Han J. Effectiveness analysis of spin motion in reducing dispersion of sounding rocket flight due to thrust misalignment Int. J. Aeronaut. Space Sci 22 5 2021 1194 1208 10.1007/s42405-021-00383-x
2 Yang D. Xiong Y. Ren Q. Wang X. Nutation instability of spinning solid rocket motor spacecraft Chin. J. Aeronaut. 30 4 2017 1363 1372 10.1016/j.cja.2017.06.005
3 Yuan D.D. Yi W.J. Guan J. Sun L. Zhang H.R. Study of projectile roll-attitude measurement method based on unscented kalman filter J. Ballist. 29 2 2017 8 12 10.3969/j.issn.1004-499X.2017.02.002
4 Yang W. Wang Z. Shen C. Liu Y. Liu S. Design of a roll angle measuring sensor IEEE Access 2020 115159 115166 10.1109/ACCESS.2020.3004365
5 A M. Y E.M. S E. Passive attitude estimation using gyroscopes and all-accelerometer imu 2016 7th International Conference on Mechanical and Aerospace Engineering 2016 ICMAE) 368 376 10.1109/ICMAE.2016.7549568
6 Sebastien C. Volker F. Dominique B. Projectile attitude and position determination using magnetometer sensor only Proc. SPIE 2005 49 58 10.1117/12.602099
7 Nguyen D.N. Nguyen T.A. Investigate the relationship between the vehicle roll angle and other factors when steering Model. Simulat. Eng. 2023 1 15 10.1155/2023/6069078
8 Chen Z. Li H. Wei Y. Zhou Z. Lu M. Gnss antispoofing method using the intersection angle between two directions of arrival (ia-doa) for multiantenna receivers GPS Solut. 27 1 2023 10.1007/s10291-022-01345-w
9 Zhao H. Su Z. Liu F. Li C. Li Q. Liu N. Extraction and filter algorithm of roll angular rate for high spinning projectiles Math. Probl Eng. 2019 1 15 10.1155/2019/3181727
10 Krasuski D.W.K. Estimation of rotation angles based on gps data from a ux5 platform Measurement Automation Monitoring 61 11 2015 516 520
11 Garcia Guzman J. Prieto Gonzalez L. Pajares Redondo J. Sanz Sanchez S. Boada B. Design of low-cost vehicle roll angle estimator based on kalman filters and an iot architecture Sensors 18 6 2018 1800 10.3390/s18061800 29865271
12 Li N. Zhao L. Li L. Jia C. Integrity monitoring of high-accuracy gnss-based attitude determination GPS Solut. 22 4 2018 120 10.1007/s10291-018-0787-x
13 Im H.C. Lee S.J. Gps signal tracking on a multi-antenna mounted spinning vehicle by compensating for the spin effects Int. J. Control Autom. Syst. 16 2 2018 867 874 10.1007/s12555-016-0705-3
14 Zharkov M.V. Veremeenko K.K. Antonov D.A. Kuznetsov I.M. Attitude determination using ambiguous gnss phase measurements and absolute angular rate measurements Gyroscopy and navigation 9 4 2018 277 286 10.1134/S2075108718040090
15 Doty J.H. Advanced spinning-vehicle navigation - a new technique in navigation of munitions 2001 Proceedings of the 57th Annual Meeting of the Institute of Navigation 745 754
16 Doty J.H. Anderson D.A. Bybee T. A demonstration of advanced spinning-vehicle navigation 2004 National Technical Meeting of The Institute of Navigation (2004)573-584 573 584
17 Kim J. Cho J. Hwang D. Lee S. A roll rate estimation method using gnss signals for spinning vehicles Journal of Institute of Control, Robotics and Systems 14 7 2008 689 694 10.5302/J.ICROS.2008.14.7.689
18 Kim J.W. Kang H.W. Hwang D. Lee S.J. Signal tracking method of gnss receivers for spinning vehicles Int. J. Control Autom. Syst. 10 3 2012 529 535 10.1007/s12555-012-0309-5
19 Bahder T.B. Attitude determination from single-antenna carrier-phase measurements J. Appl. Phys. 91 7 2002 4677 4684 10.1063/1.1448871
20 Shen Q. Li M. Gong R. Gps positioning algorithm for a spinning vehicle with discontinuous signals received by a single-patch antenna GPS Solut. 21 4 2017 1491 1502 10.1007/s10291-017-0623-8
21 Deng Z. Shen Q. Deng Z. Roll angle measurement for a spinning vehicle based on gps signals received by a single-patch antenna Sensors 18 10 2018 3479 10.3390/s18103479 30332769
22 Yang L. Li H. Du X. Roll attitude measurement technique based on gps signal power IEEE International Conference on Unmanned Systems (ICUS), Beijing, China 2019 280 284 10.1109/ICUS48101.2019.8995938
23 Li R. Wu P. Feng L. Tong H. Ren Z. Assisted-gnss positioning algorithm based on one-way fuzzy time information Heliyon 9 9 2023 e20318 10.1016/j.heliyon.2023.e20318
24 Hong Ju-Hyeon Kim Yeon-Jung Kyung Ryoo Chang Roll Angle Estimation for Smart Projectiles Using a Single Patch Antenna 43 9 2020 1 9 10.2514/1.G005083
25 C S. M N.D. M J. Dual-band circularly polarized annular ring patch antenna for GPS-aided GEO-augmented navigation receivers IEEE Antenn. Wireless Propag. Lett. 21 9 2022 1737 1741 10.1109/LAWP.2022.3178980
26 Wang L. Ma J. Zhao X. Li X. Zhang K. Jiao Z. Adaptive robust unscented kalman filter-based state-of-charge estimation for lithium-ion batteries with multi-parameter updating Electrochim 2022 140760 10.1016/j.electacta.2022.140760
27 Shevlyakov G. Smirnov P. Robust estimation of the correlation coefficient: an attempt of survey Aust. J. Stat. 40 1&2 2016 147 156 10.17713/ajs.v40i1&2.206
28 Fang C. Liu J. Tian Y. Lu J. Robust state estimation of active distribution networks based on improved igg weight function IEEE 2020 225 228 10.1109/ACPEE48638.2020.9136278
29 Lotfy A. Abdelfatah M. El-Fiky G. Improving the performance of gnss precise point positioning by developed robust adaptive kalman filter The Egyptian Journal of Remote Sensing and Space Science 25 4 2022 919 928 10.1016/j.ejrs.2022.09.005
30 Varvani Farahani A. Abolfathi S. Sliding mode observer design for decentralized multi-phase flow estimation Heliyon 8 2 2022 e08768 10.1016/j.heliyon.2022.e08768
