An Improved Low-Cost Dual-Antenna GNSS Dynamic Attitude Determination Method in Complex Environments

Abstract

GNSS attitude determination has been widely adopted due to its high efficiency, absence of cumulative errors, and ease of installation. However, practical navigation and attitude determination systems often rely on low-cost receivers that struggle with substantial multipath effects, frequent cycle slips, and satellite signal loss, significantly impairing attitude determination accuracy in challenging urban environments. To address this issue, this contribution proposes a constrained dynamic prediction model (C-Dynamics), which enables more accurate initial coordinates and thereby increases the effectiveness of the constrained LAMBDA (CLAMBDA) technique. To evaluate the practical performance of C-Dynamics, two sets of real-world data collected from a vehicle platform were analyzed. The results demonstrate that C-Dynamics significantly enhances the accuracy of initial coordinate estimations across various environments. Compared with the lambda method, the CLAMBDA method + C-Dynamics method (CLAMBDA+CD) improves the fixing rate in the urban environment by 5.6%, and the accuracy of the heading angle, pitch angle, and baseline length improved by 66%, 70.9%, and 84.2%, respectively. Moreover, in challenging high obstruction environments, the fixing rate increased by 43.5%, while the accuracy of heading angle, pitch angle, and baseline length improved by 76.4%, 69.2%, and 94%, respectively. The proposed algorithm effectively addresses the low fixing rate and insufficient accuracy of the LAMBDA method in high obstruction environments and holds practical value for widespread adoption in the mass market.

1. Introduction

Attitude determination using the Global Navigation Satellite System (GNSS) offers key advantages, including all-weather capability, freedom from accumulated errors, and ease of installation, making it highly suitable for applications such as autonomous driving, drones, and tilt monitoring [1,2,3,4,5]. With advancements in low-cost GNSS receiver technology, researchers are increasingly focused on developing effective methods to leverage these affordable receivers for high-precision attitude determination, thereby expanding the potential applications of GNSS in various scenarios [6,7,8,9].

Early GNSS attitude determination techniques primarily relied on a single-antenna approach, where velocity vectors derived from time-differenced carrier phase (TDCP) measurements provided attitude information [10]. However, this method faces challenges in accurately estimating attitude in static or low-speed scenarios. Furthermore, TDCP is susceptible to disruptions from GNSS cycle slips and signal loss, which are particularly prevalent in urban environments [11]. To overcome these limitations, dual-antenna attitude determination strategies, commonly known as GNSS compasses [12], have gained widespread adoption. This approach involves mounting two antennas aligned with the vehicle’s directional axis. By resolving the baseline vector between the antennas, precise heading and pitch angles can be determined. Compared to single-antenna methods, dual-antenna strategies offer more continuous and stable attitude estimation results, making them better suited for applications in complex and dynamic environments.

Resolving ambiguities is a critical aspect of dual-antenna attitude determination. However, GNSS observations from low-cost antennas and receivers are often affected by multipath and cycle slips, particularly in complex vehicular environments. To enhance the robustness of the GNSS model and improve the success rate of ambiguity resolution, incorporating a fixed baseline length constraint can be beneficial [13,14]. Common strategies include the following: (1) The virtual observation method. This approach involves measuring the baseline length in advance and incorporating it into the parameter estimation process [15]. However, for short baselines, this can introduce additional linearization errors, potentially degrading the precision of ambiguity float solutions and extending initialization time [16,17]. (2) The baseline length verification method. This strategy uses the known baseline length as a verification check for ambiguity resolution. While effective in practice, it does not fully leverage the baseline information, as the baseline length is only used as a verification condition and does not directly participate in parameter estimation [18]. (3) The quadratically constrained integer least-squares (QC-ILS) method. This approach incorporates the baseline length information as an additional nonlinear constraint into the parameter estimation process, enhancing the ambiguity search and baseline validation procedures [19]. This approach allows for a more unified integration of baseline length information, regardless of whether it contains errors. The QC-ILS search algorithm, also known as the CLAMBDA algorithm, was proposed based on the Least-squares AMBiguity Decorrelation Adjustment (LAMBDA) algorithm. When the baseline length is precisely known, CLAMBDA operates as a strong constraint model [20]. In contrast, when there are errors in the baseline length, it functions as a weighted constraint model [21]. Compared with the virtual observation method and validation method, the CLAMBDA method strictly incorporates baseline information into the ILS criterion, resulting in better overall performance [22].

Currently, the most commonly used strategy in vehicle attitude determination is the single epoch mode, primarily aimed at avoiding the impact of cycle slips. For example, Wang et al. proposed a single epoch attitude determination method that combines a multi-baseline approach with the CLAMBDA algorithm, which reduces computational complexity [6]. Lu et al. collected multiple sets of data using a U-Blox low-cost receiver and validated that the single epoch CLAMBDA method can significantly improve the ambiguity fixing rate for single frequency, single system data [22]. However, the single-epoch mode does not utilize the correlation between adjacent epochs. Currently, multi-system, multi-frequency GNSS attitude determination is now a growing trend. Shu et al. conducted a detailed analysis of the attitude determination accuracy across different systems and confirmed that multi-system configurations offer stronger satellite geometry in challenging environments, leading to higher attitude determination accuracy [23]. In addition to incorporating baseline length into the ambiguity search, Zhang et al. introduced constraints such as heading angle or baseline vector into the ambiguity search process and compared the performance of different constraints. They found that the Vector-constrained LAMBDA method resulted in more accurate attitude determination [24]. Roth et al. also incorporated the attitude information provided by inertial and magnetic field sensors into the ambiguity search range in a special extended LAMBDA manner, which improved the reliability of the fixed solution [25]. Eling et al. added MEMS inertial sensors to a multiantenna GNSS system and conducted vehicle-based dynamic experiments in GNSS-friendly and GNSS-challenged environments [26]. However, additional hardware can sometimes complicate the implementation of positioning systems [27], especially under low-cost conditions where not all hardware is highly reliable [28]. To avoid redundant computations, this article focuses on attitude determination without using additional sensors. Existing research has shown that accurate initial coordinates in complex environments can effectively improve positioning accuracy and accelerate convergence speed [29,30,31]. It is reasonable to infer that accurate initial coordinates in attitude determination have the same effect. Conventional attitude determination frequently employs the Single Point Position (SPP) solution as the initial coordinates [32,33]. Nonetheless, the precision of the SPP is typically limited to the meter scale, and its performance may deteriorate further in challenging environmental conditions. Consequently, to address the issue of insufficient initial coordinate accuracy in attitude determination within intricate environments, this article proposes a method for attitude determination that imposes constraints in both the ambiguity domain and the coordinate domain. The performance of dual-antenna GNSS attitude determination is evaluated through vehicle experiments conducted in various urban environments.

The subsequent sections of this article are structured as follows: Section 2 offers a comprehensive derivation of the precision attitude determination methodology applicable to low-cost dual-antenna GNSS configurations, taking into account baseline length constraints and an improved predictive model for vehicle motion equations. Section 3 and Section 4 outline the configuration of the testing platform and the data acquisition process, as well as the experimental validation of the proposed algorithm. Finally, Section 5 and Section 6 provide a discussion and conclusion of this study, respectively.

2. GNSS Attitude Determination

The GNSS attitude determination employs a double differenced observation model. This model effectively mitigates receiver clock errors, as well as ionospheric and tropospheric delays, particularly for short baselines. Upon determining the baseline vector within the Earth-Centered Earth-Fixed (ECEF) coordinate system, it is subsequently converted into the East-North-Up (ENU) coordinate system to ascertain the coordinates ${\left(\begin{array}{lll}\Delta e\hfill & \Delta n\hfill & \Delta u\hfill \end{array}\right)}^T$. The heading angles ($\psi$) and pitch angles ($\theta$) can then be calculated as follows:

(1)$\left\{\begin{array}{l}\psi ={\tan }^{-1}\left({\displaystyle \frac{\Delta e}{\Delta n}}\right)\hfill \\ \theta ={\tan }^{-1}\left({\displaystyle \frac{\Delta u}{\sqrt{\Delta e^2+\Delta n^2}}}\right)\hfill \end{array}\right.$

2.1. CLAMBDA Model

The CLAMBDA method integrates the nonlinear baseline length constraint into the ambiguity estimation process to broaden the search domain for the quadratic ambiguity model, with the objective of identifying the ambiguity vector that minimizes the mixed quadratic form. When the baseline length assumes a known constant, the model is classified as a strong constraint model. Conversely, when baseline length errors are accounted for, the model is referred to as a weighted constraint model. Given that measurement errors are inherent in practical applications, this study adopts the weighted constraint model. The corresponding mathematical formulation is presented as follows:

(2)$\begin{array}{l}E(y)=Aa+Bb,D(y)=Q_{yy},a\in {\mathbb{Z}}^n,b\in {ℝ}^3\\ E(l)=\Vert b\Vert ,D(l)={\sigma }_l^2\end{array}$

where, $y$ represents the GNSS observations; $a$ denotes the ambiguity parameters; $b$ is the baseline component parameter; $A$ and $B$ are the corresponding coefficient matrices; $l$ denotes the measured baseline length; $E(\cdot )$ and $D(\cdot )$ represent the expectation and variance operators, respectively. Equation (2) employs the least squares criterion to execute the orthogonal decomposition of the quadratic form:

(3)$\begin{array}{l}\underset{a\in {\mathbb{Z}}^n,b\in {ℝ}^3}{\min }\left\{\Vert y-Aa-Bb{\Vert }_{Q_{yy}}^2+{\sigma }_l^{-2}{(I-\Vert b\Vert )}^2\right\}\\ ={\Vert v\Vert }_{Q_{yy}}^2+\underset{a\in {\mathbb{Z}}^n,b\in {ℝ}^3}{\min }\left\{{\Vert \widehat{a}-a\Vert }_{Q_{\widehat{a}\widehat{a}}}^2+{\Vert \widehat{b}(a)-b\Vert }_{Q_{\widehat{b}(a)\widehat{b}(a)}}^2+{\sigma }_l^{-2}{(l-\Vert b\Vert )}^2\right\}\\ ={\Vert v\Vert }_{Q_{yy}}^2+\underset{a\in Z^n}{\min }\left\{{\Vert \widehat{a}-a\Vert }_{Q_{\widehat{a}\widehat{a}}}^2+\underset{b\in {ℝ}^3}{\min }\left({\Vert \widehat{b}(a)-b\Vert }_{Q_{\widehat{b}(a)\widehat{b}(a)}}^2+{\sigma }_l^{-2}{(l-\Vert b\Vert )}^2\right)\right\}\\ =\Vert v{\Vert }_{Q_{yy}}^2+\underset{a\in Z^n}{\min }F(a)\end{array}$

where, $v=A\widehat{a}+B\widehat{b}-y$; $\hat{\mathrm{b}}(\mathrm{a})$ is the fixed baseline vector solution when the ambiguity is fixed to $a$. Equation (3) can be considered as a quadratic constraint integer least squares model (QC-ILS). It is evident that the residual quadratic form ${‖v‖}_{Q_{yy}}^2$ is a constant, so Equation (3) is equivalent to minimizing the mixed quadratic form $F(a)$, i.e.,

(4)$F(a)=\Vert \widehat{a}-a{\Vert }_{Q_{\widehat{a}\widehat{a}}}^2+\underset{b\in {ℝ}^3}{\min }\left({\Vert \widehat{b}(a)-b\Vert }_{Q_{\widehat{b}(a)\widehat{b}(a)}}^2+{\sigma }_l^{-2}{(l-\Vert b\Vert )}^2\right)=\min$

To more quickly find the minimum value of $F(a)$, two additional functions can be incorporated into the search and contraction strategies to improve the efficiency of ambiguity search and avoid most meaningless solutions. According to the properties of the Rayleigh quotient, the baseline quadratic form under the constrained model satisfies the following inequality [20]:

(5)${\displaystyle \frac{{\lambda }_{\min }}{\left(1+{\lambda }_{\min }{\sigma }_l^2\right)}}{(l-\Vert \widehat{b}(a)\Vert )}^2⩽G(a)⩽{\displaystyle \frac{{\lambda }_{\max }}{\left(1+{\lambda }_{\max }{\sigma }_l^2\right)}}{(l-\Vert \widehat{b}(a)\Vert )}^2$

where, ${\lambda }_{\min }$ and ${\lambda }_{\max }$ represent the minimum and maximum eigenvalues of $Q_{\widehat{b}(a)\widehat{b}(a)}$, $G(a)=\underset{b\in {ℝ}^3}{\min }{\Vert \widehat{b}(a)-b\Vert }_{Q_{\widehat{b}(a)\widehat{b}(a)}}^2$, respectively; the relationship between $F(a)$ and the two additional functions is:

(6)$F_1(a)⩽F(a)⩽F_2(a)$

where, $F_1(a)$ and $F_2(a)$ are constructed as:

(7)$\begin{array}{l}{F}_1(a)=E(a)+{\displaystyle \frac{{\lambda }_{\min }}{\left(1+{\lambda }_{\min }{\sigma }_l^2\right)}}{(l-\Vert \widehat{b}(a)\Vert )}^2\\ F_2(a)=E(a)+{\displaystyle \frac{{\lambda }_{\max }}{\left(1+{\lambda }_{\max }{\sigma }_l^2\right)}}{(l-\Vert \widehat{b}(a)\Vert )}^2\end{array}$

where, $E(a)=\Vert \widehat{a}-a{\Vert }_{Q_{\widehat{a}\widehat{a}}}^2$.when the size of the search space is ${\chi }^2$, the spatial relationship between the integer ambiguity vectors of $F$, $F1$, and $F2$ is as follows:

(8)${\Omega }_{F_2}\left({\chi }^2\right)\subset {\Omega }_F\left({\chi }^2\right)\subset {\Omega }_{F_1}\left({\chi }^2\right)$

It is evident from the above formula that computing $F1$ and $F2$ is simpler than calculating $F(a)$, which significantly improves search efficiency. Therefore, $F1$ and $F2$ can be used to quickly reduce the search space ${\chi }^2$, and then the minimum $F(a)$ can be found within this reduced space. The specific process is as follows: Begin by updating the search space ${\chi }^2$ based on $F_2(a)$. Next, traverse all candidate ambiguity vectors within the space where $F_1(a)$ is satisfied. Finally, find the minimum value of $F(a)$ within the space obtained from the enumeration.

2.2. Improve Constrained Dynamic Prediction Model

Traditional attitude determination methods often use a single epoch mode, where the SPP solution is used as the initial coordinate. Alternatively, the SPP solution is used as the initial coordinates only at the initialization time, and the dynamics model is used to predict the initial coordinates at the subsequent moments [34]. In this paper, a constrained dynamic prediction model is proposed based on the dynamic model. The following equation represents the transfer matrix between ambiguity-float position parameters and ambiguity parameters:

(9)$\left[\begin{array}{c}{r}_{t+1}\\ \nabla \Delta N_{t+1}^1\\ ⋮\\ \nabla \Delta N_{t+1}^n\\ \nabla \Delta N_{t+1}^{n+1}\\ ⋮\end{array}\right]=\left[\begin{array}{c}{r}_{\mathrm{con}.}\\ \nabla \Delta N_t^1\\ ⋮\\ \nabla \Delta N_t^n\\ \nabla \Delta {\varphi }_{t+1}^{n+1}-{\displaystyle \frac{\nabla \Delta P_{t+1}^{n+1}}{\lambda }}\\ ⋮\end{array}\right]\begin{array}{c},\left[\begin{array}{c}{Q}_{r_{t+1}}\\ Q_{\nabla \Delta N_{t+1}^1}\\ ⋮\\ Q_{\nabla \Delta N_{t+1}^n}\\ Q_{\nabla \Delta N_{t+1}^{n+1}}\\ ⋮\end{array}\right]=\left[\begin{array}{c}{Q}_{r_{\mathrm{t}}}\\ Q_{\nabla \Delta N_t^1}\\ ⋮\\ Q_{\nabla \Delta N_t^n}\\ {\displaystyle \frac{Q_{\nabla \Delta N_0}}{{\lambda }^2}}\\ ⋮\end{array}\right]+\left[\begin{array}{c}{Q}_{\Delta r_{t+1,t}}\\ Q_N\\ ⋮\\ Q_N\\ 0\\ ⋮\end{array}\right]\end{array}$

where, $t$ and $t+1$ represent two consecutive epochs; $r_{t+1}$ denotes the initial position at epoch $t+1$; $r_{\mathrm{con}.}$ represents the updated initial position; $Q_{r_{t+1}}$ indicates the covariance matrix of the epoch $t+1$; and $Q_{\Delta r_{t+1,t}}$ refers to the updated covariance between the consecutive epochs.

When using the single-epoch mode, $r_{t+1}=r_{\mathrm{con}.}=r_{\mathrm{SPP}}$, $Q_{r_t}={30}^2 {\mathrm{m}}^2$, $Q_{\Delta r_{t+1,t}}=0$, meaning that the SPP solution is used as the initial position, and the position covariance matrix is reset, while the covariance between the position and the ambiguity is reset to 0. When dynamic prediction is employed, $r_{t+1}=r_{\mathrm{con}. }=r_t+\Delta r_{t+1,t}$, $Q_{r_{t+1}}=Q_{r_{\mathrm{con}. }}=Q_{r_t}+Q_{\Delta r_{t+1,t}}$, where $r_t$ and $Q_{r_t}$ represent the position and covariance matrix calculated at epoch $t$, $\Delta r_{t+1,t}$ denotes the predicted position difference between epochs $t+1$ and $t$ based on the results from epoch $t$, and $Q_{\Delta r_{t+1,t}}$ represents the updated covariance between consecutive epochs.

The improved dynamic prediction model determines the initial position by analyzing the covariance of the predicted position. When the mean variance of the position state exceeds a specified threshold, constraints on the initial coordinates are applied using baseline length, heading angle, and pitch angle. The specific process is as follows:

(10)$r_{t+1}=r_{\mathrm{con}. }=r_b+S^{-1}\left[\begin{array}{c}\sin {\alpha }_{t+1}\\ \cos {\alpha }_{t+1}\\ \sin {\beta }_{t+1}\end{array}\right]l\hspace{1em}\hspace{1em}\begin{array}{c}\mathrm{var}>l^2\end{array}$

where, $r_b=(x_b,y_b,z_b)$ represents the base station coordinates; $S$ denotes the coordinate transformation matrix between the ECEF and ENU coordinate systems; ${\alpha }_{t+1}$ and ${\beta }_{t+1}$ represent the heading angle and pitch angle at epoch $t+1$; and $\mathrm{var}$ is the average variance of the position state. The threshold is defined as $l^2$ due to the method’s minimum positional accuracy being $l$. To obtain more accurate values for the heading and pitch angles, it is essential to fully utilize the velocity vector changes between two consecutive epochs. When the previous epoch is successfully fixed, the calculation of ${\alpha }_{t+1}$ and ${\beta }_{t+1}$ is expressed by the following equation:

(11)$\left\{\begin{array}{cc}if \mathrm{TDCP} is available\hfill & {\alpha }_{t+1}={\alpha }_t+△{\alpha }_{t+1,t}^{TDCP}, {\beta }_{t+1}={\beta }_t+△{\beta }_{t+1,t}^{TDCP}\\ if \mathrm{Doppler} is available\hfill & {\alpha }_{t+1}={\alpha }_t+△{\alpha }_{t+1,t}^{\mathrm{Doppler}}, {\beta }_{t+1}={\beta }_t+△{\beta }_{t+1,t}^{\mathrm{Doppler}}\end{array}\right.$

where, $△{\alpha }_{t+1,t}^{TDCP}$ represents the angular change of the velocity vector between two epochs calculated by TDCP. In instances where the TDCP is unable to perform calculations, the Doppler method is employed to determine the variations in the velocity vector. The calculation of $△{\alpha }_{t+1,t}$ is based on velocity estimates from both the Doppler and TDCP methods, obtained by calculating the velocity vector change between two epochs. The velocity estimates from the Doppler and TDCP methods can achieve dm/s level to cm/s-level accuracy [35]. When both the Doppler method and TDCP method fail, let $△{\alpha }_{t+1,t}=0$, which means that the current epoch inherits the fixed attitude of the previous epoch. Concurrently, the covariance matrix for the position needs to be updated to an empirical value, e.g., $Q_{r_{t+1}}={0.5}^2 {\mathrm{m}}^2$. When the accuracy of the attitude from the previous epoch cannot be guaranteed, meaning the previous epoch is not fixed, the direction of the motion vector determined by the TDCP or Doppler method can be directly utilized as the heading angle and pitch angle at that time $t+1$.

(12)$\left\{\begin{array}{cc}if \mathrm{TDCP} is available\hfill & {\alpha }_{t+1}={\alpha }_{t+1}^{TDCP}, {\beta }_{t+1}={\beta }_{t+1}^{TDCP}\\ if \mathrm{Doppler} is available\hfill & {\alpha }_{t+1}={\alpha }_{t+1}^{\mathrm{Doppler}}, {\beta }_{t+1}={\beta }_{t+1}^{\mathrm{Doppler}}\end{array}\right.$

The accuracy of the initial coordinates obtained by the above equation is slightly lower, so it is necessary to enlarge the position covariance matrix ($Q_{r_{t+1}}=1^2 {\mathrm{m}}^2$) to maintain the same precision as the position parameters. If the previous epoch cannot be fixed and both TDCP and Doppler fail, the SPP solution is used as the initial coordinate. Concurrently, it is essential to evaluate the SPP solution. When the baseline length obtained by SPP exceeds $3l$, indicating that the SPP accuracy is insufficient, the initial coordinates are set equal to the reference station coordinates, and the position covariance matrix is reset to an empirical value, e.g., $Q_{r_{t+1}}=3^2 {\mathrm{m}}^2$, while the covariance between position and ambiguity is reset to zero. Although the initial coordinates obtained by this method differ from the true coordinates by a fixed baseline length, compared with the coordinates obtained by SPP, this method can ensure that the initial coordinates remain within a controllable accuracy range. Furthermore, the initial coordinates obtained by this method meet the initial conditions required by the Kalman filter [36,37].

(13)$\mathrm{E}({\mathrm{r}}_{t+1})={\hat{\mathrm{r}}}_{t+1}$

where, ${\hat{\mathrm{r}}}_{t+1}$ denotes the posterior state estimate of the position.

3. Test Platform and Data Collection

The dual-antenna GNSS attitude determination algorithm was evaluated through dynamic experiments conducted on a terrestrial vehicle. The experimental configuration, illustrated in Figure 1, features two GNSS antennas securely affixed to the roof of the vehicle. Then, the low-cost A8D receiver and the high-precision measurement receiver Septentrio Polar X5s are connected to two antennas simultaneously using a power divider to receive data from both antennas. The market price of the low-cost A8P receiver is approximately USD70. The baseline established by the antennas is aligned within the vehicle’s longitudinal symmetry plane, with the primary antenna positioned at the rear and the secondary antenna at the front. The distance between the two antennas was measured multiple times using a tape measure, and the average value obtained was utilized as the prior baseline length. The associated measurement uncertainty was maintained as a constant value of 2 cm. The coordinates of the primary antenna, which functions as the attitude reference station, were acquired in real-time kinematic (RTK) positioning, utilizing differential data supplied by Qianxun Inc.(Shanghai, China) to ensure the accuracy of the reference station’s coordinates. The reference trajectory was obtained by processing data collected by the high-precision measurement receiver Septentrio Polar X5s using GrafMov Version 8.40, a proprietary software developed by NovAtel Inc. (Calgary, Alberta, Canada). Given that true ambiguity remains indeterminate, we assess the baseline ambiguity solutions to verify the accuracy of the ambiguity resolution. An ambiguity resolution is deemed correct if the discrepancy between the calculated baseline length and the true baseline length is within a threshold of 3 cm.

Figure 2 illustrates the driving routes for both experiments. Table 1 presents fundamental details regarding the two experiments. Experiment 1 was executed on 2 August 2024, in Wuhan, Hubei, with an average baseline length measured at 1.14 m. This experiment encompassed a range of driving behaviors, including straight driving, turning, stopping at traffic signals, and navigating on and off elevated roadways. The covered observation conditions included high-occlusion static environments, open dynamic sky environments, and dynamic urban settings. Data collection spanned over one hour, covering a total driving distance exceeding 20 km. Experiment 2 took place on 6 August 2024, also in Wuhan, Hubei, utilizing the same vehicle and equipment as Experiment 1. This experiment commenced in a high-occlusion dynamic environment and concluded with static observations in an open parking lot, with an average baseline length of 1.19 m. Data collection for this experiment extended over two hours. Figure 3 depicts the variations in the number of satellites and the Position Dilution of Precision (PDOP) throughout the two experiments. Notably, the data from Experiment 1 indicates a lack of observations during epochs 1900 to 2040 due to the vehicle’s passage through a tunnel, resulting in the absence of displayed data for those epochs.

Additionally, the signal-to-noise ratio (SNR) serves as an indicator of the quality of carrier observations, which is predominantly affected by the observational environment and the elevation angle of the satellites. Setting an appropriate SNR threshold helps reduce the negative effects caused by poor observation results. Figure 4 illustrates the fluctuations in SNR of visible satellites as recorded by the antennas in both Experiment 1 and Experiment 2, focusing on a selection of representative satellites observable over extended durations. Given that the primary antenna is situated approximately 1 m from the secondary antenna and both utilize identical antenna types, the SNR values are notably similar. Consequently, this analysis is confined to the SNR data obtained from the secondary antenna. As depicted in Figure 4, during Experiment 1, several satellites exhibited elevated SNR levels, and these values exhibited sinusoidal fluctuations prior to epoch 1600, attributable to signal refraction or non-line-of-sight conditions [38]. In terms of implementation details, satellites exhibiting elevation angles below 15° or SNR values lower than 30 dB-Hz were temporarily omitted from consideration. Corrections for satellite clock errors and relativistic effects were applied utilizing broadcast ephemerides. Following this, double-difference observation equations were established, and cycle slip detection was conducted on the carrier phase observations. A variety of techniques were employed for cycle slip detection, including the loss of lock indicator (LLI), the absence of geometric combinations, and the TDCP method [39].

4. Experimental Results and Analysis

In this part, two independent experiments will be carried out to test the performance of the proposed enhanced constrained dynamic prediction model (referred to as the C-Dynamics method). Initially, we examine the influence of initial coordinates on attitude accuracy by comparing the initial coordinates and attitude accuracy derived from various initial coordinate obtaining methods. Subsequently, the second set of experimental data is used to assess the CLAMBDA+C-Dynamics method (referred to as the CLAMBDA+CD method) against the conventional LAMBDA method and the CLAMBDA method, validating the efficacy of the CLAMBDA+CD method.

4.1. Analysis of Different Methods for Obtaining Initial Coordinates

In order to compare the different initial coordinate acquisitions and to assess the efficacy of the C-Dynamics method proposed in this article, the section conducts a comparative analysis of the single epoch method, the dynamic method, and the C-Dynamics method across various typical scenarios derived from Experiment 1. It is important to note that all ambiguity resolution was performed utilizing CLAMBDA methods.

Figure 5 illustrates the initial baseline length, representing the distance between the initial coordinates and the base station as calculated by three different methods. As Figure 5 shows, the initial baseline length in the single epoch model frequently exhibits discrepancies from the true value, especially under challenging environments before 1600 epochs, during which the vehicle was stationary beneath a building for static observation. The dynamic model displayed substantial noise in its baseline length prior to 200 epochs but gradually stabilized as it progressed. Conversely, the C-Dynamics model maintained consistent stability throughout the experiment. Table 2 illustrates the root mean square (RMS) of the initial baseline length errors of the three methods in different environments. It is evident that in high obstruction environments, both the single epoch method and the dynamic method exhibit substantial initial baseline length errors, which can reach several meters or even exceed ten meters. In contrast, the C-Dynamics method controls the error within a smaller range through constraints. In other open and urban environments, although the initial baseline length errors of the three methods are reduced, the C-Dynamics method consistently demonstrates superior performance, yielding smaller errors compared to the other two methods.

The heading angle, pitch angle, and baseline length calculated by the three methods are illustrated in Figure 6. It is evident that the heading angle, pitch angle, and baseline length calculated by the single epoch method exhibit significant deviations in poor observation environments. Conversely, the Dynamics mode shows fewer large errors but still struggles with prolonged non-convergence in challenging environments. Meanwhile, it can be seen that the initial baseline length of the C-Dynamics method is constrained to be near the true value. Furthermore, the attitude results obtained from both the single epoch and Dynamics methods exhibit significant fluctuations when the initial baseline length deviates excessively, indicating that the precision of the initial coordinates has a direct impact on the accuracy of the final attitude determination outcomes. Table 3 presents the fixing rate and the errors in attitude and baseline length for the three methods. The fixing rate of the single epoch method reached only 52.6%, while the Dynamics method improved the fixing rate by 14.8%. The C-Dynamics method, by enhancing the accuracy of the initial coordinates, further increased the fixing rate to 84.4%, representing a 31.8% improvement compared to the single epoch method. In addition, the C-Dynamics method significantly improved the accuracy of the attitude and baseline length, with improvements of 72%, 58.7%, and 93.1% in heading angle, pitch angle, and baseline length accuracy, respectively, compared to the single epoch method.

By comparing the estimated values with the prior measured values, the baseline length error is calculated, and the performance of different methods is shown through the cumulative distribution function (CDF) of baseline length error, as illustrated in Figure 7. Each method is represented by distinct curves. The figure illustrates that the C-Dynamics method exhibits a lower and more concentrated overall error, suggesting that it yields a more precise estimation of baseline length.

To further reveal the differences among the three methods, Figure 8 illustrates the errors associated with heading angle, pitch angle, and baseline length estimation for each method. It is evident that the C-Dynamics method has fewer overall errors compared to the single epoch and Dynamics methods, with no frequent outliers. Consequently, the C-Dynamics method demonstrates enhanced accuracy in both attitude and baseline length estimations. To further delineate the characteristics of the various methods, Figure 9 presents the three-dimensional baseline vectors calculated by each method. For a vehicle platform traveling on the ground, the optimal three-dimensional baseline vector should approximate a nearly horizontal circular formation. However, the baseline vectors generated by the single epoch method show significant deviations from this ideal shape, with many outliers appearing across the available epochs. In contrast, the dynamics method exhibits fewer outliers and shows an improvement in performance. The C-Dynamics method has the fewest outliers, and its baseline vector forms a shape closest to the ideal horizontal circular shape, thereby further substantiating its superiority regarding both attitude and baseline length accuracy.

Table 4 presents the RMS errors of heading angle, pitch angle, and baseline length estimated using the dual-antenna GNSS scheme in three typical observation environments. The dataset encompasses a variety of environments; during epochs 0 to 1600, the vehicle was stationary beneath a building for static observation, as indicated by the red dot in the left sub-figure of Experiment 1 in Figure 2. It is evident that the observation point was encircled by three tall structures with glass facades, resulting in a significant presence of non-line-of-sight and indirect signals; this scenario is categorized as a high obstruction static environment. Epochs 2400 to 2600 represent an open dynamic environment with no obstruction, thereby providing ample carrier phase observations. Epochs 4400 to 5200 correspond to a typical urban dynamic environment, as shown in the right sub-figure of Experiment 1 in Figure 2, where the vehicle traversed between buildings. It can be seen that, with the exception of the high obstruction static environment, the accuracy of the heading angle is generally greater than that of the pitch angle.

In high obstruction static environments, as illustrated in Figure 9, the C-Dynamics method converges faster compared to the other two methods, and the other two methods nearly fail to attain fixed. According to Table 5, the C-Dynamics method has a fixing rate of 66.8%, while the Dynamics method achieves only a 9.6% fixing rate, and the traditional single epoch method fails to fix the solution at all. This demonstrates that the C-Dynamics method effectively enhances both the fixing rate and accuracy in high obstruction environments. In open dynamic environments, there is little difference between the three methods. As illustrated in Table 4 and Table 5, all three methods achieve a 100% fixing rate, with the respective accuracies for heading angle, pitch angle, and baseline length reaching 0.1°, 0.4°, and 0.003 m, respectively. This finding suggests that each method is viable and demonstrates high precision under optimal observational conditions. In urban environments, compared to open environments, the accuracy of heading angle, pitch angle, and baseline length declines. The single epoch method has more fixed failures, with heading angle, pitch angle, and baseline length accuracies of 5.7°, 6.3°, and 0.2 m, respectively. Although the Dynamics and C-Dynamics methods have the same fixing rate of 99.7%, the C-Dynamics method still slightly outperforms the Dynamics method in accuracy, achieving 0.36°, 0.88°, and 0.007 m for heading angle, pitch angle, and baseline length, respectively, compared to the Dynamics method’s 0.38°, 0.92°, and 0.007 m. While the Dynamics method’s accuracy is lower than that of C-Dynamics, it shows a significant improvement over the single epoch method.

4.2. Analysis of Different Attitude Determination Methods

The findings from Experiment 1 demonstrate that the single epoch method is markedly less effective than the Dynamics method regarding both fixing rate and accuracy. Therefore, in Experiment 2, the initial coordinates were obtained using the Dynamics method. Compared with Experiment 1, Experiment 2 adopted a different driving route and tested different observation environments. The primary aim of Experiment 2 was to analyze the effects of the LAMBDA method, CLAMBDA method, and CLAMBDA+CD method on attitude determination. Figure 2 and Figure 3 illustrate the route map of the experiment and the variations in the number of satellites and PDOP, respectively. Figure 10 presents the computed heading angles, pitch angles, and baseline lengths for the three methods, while Table 6 summarizes the overall fixing rate, attitude, and baseline length errors for these methods. The traditional LAMBDA method had a fixing rate of only 94.9%. In comparison, the fixation rates of CLAMBDA and CLAMBDA+CD methods reach 98.4% and 99.6%, respectively, with further improvements in attitude and baseline length accuracy.

The CDF of baseline length errors for Experiment 2 is presented in Figure 11. The figure indicates that the CLAMBDA+CD method has overall smaller and more concentrated errors. Additionally, the maximum absolute errors (MAE) for the LAMBDA, CLAMBDA, and CLAMBDA+CD methods demonstrate a sequential decrease, being 4.65 m, 4.125 m, and 0.675 m, respectively. It is noteworthy that when the error surpasses 1.5 m, the disparity between the CLAMBDA and LAMBDA methods becomes negligible, indicating that the enhancements offered by the CLAMBDA method in challenging environments are not substantial.

To further reveal the differences in estimation results among the LAMBDA method, CLAMBDA method, and CLAMBDA+CD method, Figure 12 illustrates the errors in heading angle, pitch angle, and baseline length computed by the three methods. It can be seen that the CLAMBDA+CD method has fewer abrupt errors and higher accuracy compared to the other methods. Figure 13 presents the three-dimensional baseline vectors for the three methods, which helps to clarify the characteristics of each method. The LAMBDA method has more error clusters around circles, while the CLAMBDA+CD method has the fewest outliers, indicating that the CLAMBDA+CD method performs better than the other two methods.

A detailed analysis of different observation environment segments better highlights the differences among the methods. In Experiment 2, epochs 0 to 200 correspond to a high obstruction dynamic environment, as indicated by the small inset in the leftmost part of Figure 2, which depicts a densely wooded area with significant obstruction. Epochs 2200 to 4500 represent a typical urban dynamic environment. Epochs 5500 to 8500 involve vehicle observations in a static state in an open parking lot.

Table 7 lists the RMS values of heading angle, pitch angle, and baseline length errors estimated using the dual GNSS antenna scheme under the three observation conditions described above. Table 8 shows the attitude fixing rates of the three methods in different environments. It can be observed that in the heavily shaded and low visibility sky segments, the LAMBDA method has a fixing rate of only 46.5%. The CLAMBDA method only improves this by 5.5%, with no significant enhancement in attitude accuracy. However, with the CLAMBDA+CD method, there is a substantial increase in ambiguity fixing rate, improving by 43.5% compared to the LAMBDA method, and the accuracies of heading angle, pitch angle, and baseline length improve by 75%, 70.2%, and 93.8%, respectively. In an open static environment, all three methods achieve full fixing, with the heading angle, pitch angle, and baseline length errors of the CLAMBDA+CD method being roughly comparable to those of the CLAMBDA method and slightly higher than those of the LAMBDA method. In a typical urban environment, the LAMBDA method has a fixing rate of 94.3%; the CLAMBDA method reaches 99.5%, showing a significant improvement. With the CLAMBDA+CD method, the fixing rate is further increased to 99.9%, with heading angle, pitch angle, and baseline length accuracies improving by 66%, 70.1%, and 84.2% compared to the LAMBDA method. It is evident that the CLAMBDA+CD method compensates for the limited improvement of the CLAMBDA method in adverse conditions and also provides some improvement in better observation environments.

5. Discussion

To address the challenge of attitude determination in challenging environments, this paper presents a refined vehicle dynamics prediction model that significantly improves the initial coordinate accuracy by using baseline length constraints and velocity information. Compared with the single epoch method, this method retains the correlation between the previous and next epochs and constrains the initial coordinates using available information, thereby significantly improving the convergence performance and overall attitude accuracy in complex environments. Compared with the LAMBDA method, the proposed method uses the baseline length to constrain the ambiguity domain to enhance the robustness of the GNSS model and improve the success rate of ambiguity resolution.

While the suggested approach significantly enhances the robustness and reliability of attitude determination in highly obstructed environments, there is still potential for improvement in the model. Additionally, the use of multi-frequency and multi-system approaches is becoming increasingly popular. This paper currently only employs a dual-frequency multi-system. Therefore, future efforts will concentrate on further optimizing the proposed model and incorporating available multi-frequency signals to enhance result reliability. Furthermore, dynamic experiments should be conducted using more theoretical methods, such as Monte Carlo simulations of radio propagation conditions, to provide more comprehensive and dependable test outcomes.

6. Conclusions

To evaluate the effectiveness of the algorithm, two sets of kinematic experiments were conducted in urban environments. Experiment 1 demonstrates that enhanced initial coordinate accuracy significantly improves both the fixing rate and the precision of attitude determination. This effect is particularly pronounced in challenging environments, where the fixing rate of the single epoch method declines sharply, while the C-Dynamics method can still maintain relatively excellent performance. The second experiment involved a comparative analysis of various attitude determination methods, demonstrating the effectiveness of the CLAMBDA+CD method proposed across diverse observational environments. The results indicate that in a conventional urban environment, the CLAMBDA method shows a significant improvement in attitude accuracy compared to the LAMBDA method, and the heading and pitch angle accuracy increase by 69.6% and 62.4%, respectively. However, in more complex obstruction environments, the efficacy of the CLAMBDA method is less pronounced, resulting in enhancements of only 43.6% and 2.4% in the accuracy of heading and pitch angles, respectively. By contrast, this problem can be well solved by the CLAMBDA+CD method, achieving improvements of 76.4% and 69.2% in the accuracy of heading and pitch angles, respectively. Overall, the proposed algorithm markedly enhances the usability of low-cost dual-antenna attitude determination in challenging environments, thereby broadening the applicability of GNSS attitude determination algorithms across various contexts and holding significant potential for widespread market implementation.

Author Contributions

Conceptualization, K.X.; Methodology, X.L.; Software, K.X.; Validation, K.X., X.L., J.Z. and C.L.; Formal analysis, J.Z.; Resources, L.C.; Data curation, K.X. and J.T.; Writing—original draft, K.X.; Writing—review & editing, J.T., X.L. and L.C.; Supervision, C.L.; Funding acquisition, J.T. and C.L. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors on request.

Conflicts of Interest

The authors declare no conflict of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Schematic diagram of the experimental platform for dual-antenna GNSS attitude estimation.

Figure 2. Vehicle traveling routes for Experiment 1 (top) and Experiment 2 (bottom).

Figure 2. Vehicle traveling routes for Experiment 1 (top) and Experiment 2 (bottom).

Figure 3. Plots of satellite number and PDOP variation for Experiment 1 (left) and Experiment 2 (right).

Figure 4. SNR variation of some satellites in Experiment 1 (left) and Experiment 2 (right).

Figure 5. Comparison of initial baseline lengths between initial coordinates and base stations for single epoch, Dynamics, and C-Dynamics methods.

Figure 6. Comparison of vehicle heading angle, pitch angle, and baseline length estimated by single epoch, Dynamics, and C-Dynamics methods.

Figure 7. Cumulative distribution function of baseline length error.

Figure 8. Comparison of vehicle heading angle, pitch angle, and baseline length errors estimated by single epoch, Dynamics, and C-Dynamics methods.

Figure 9. Baseline vectors for single epoch (a), Dynamics (b), and C-Dynamics (c) methods.

Figure 10. Comparison of vehicle heading angle, pitch angle, and baseline length estimated by LAMBDA, CLAMBDA, and CLAMBDA+CD methods.

Figure 11. Cumulative distribution function of baseline length error.

Figure 12. Comparison of vehicle heading angle, pitch angle, and baseline length errors estimated by LAMBDA, CLAMBDA, and CLAMBDA+CD methods.

Figure 13. Baseline vectors for the LAMBDA (a), CLAMBDA (b), and CLAMBDA+CD (c) methods.

Table 1

Basic processing configuration.

Experiment Parameters	Specifications
Experiment area	Wuhan
Receiver information	Kisilicon A8D receiver
Systems	GPS + Galileo + BDS2 + BDS3
Frequencies	L1 + L5; E1 + E5a; B1I + B2I; B1I + B2a
Sampling interval	1s
Elevation threshold	15°
Ionosphere/troposphere	Ignored
Stochastic model	Signal-to-Noise Ratio
Weight of code and phase	1:100
Prior precision of baseline length	2 cm
SNR threshold	30 db-Hz

Table 2

RMS values of the initial baseline lengths errors(m) based on the dual antenna scheme in three typical observation conditions.

Environment	Single Epoch	Dynamics	C-Dynamics
Static obstruction sky	15.643	4.013	0.775
Dynamic open sky	2.069	0.319	0.127
Dynamic urban	5.749	0.384	0.366

Table 3

Comparison of fixation rates and RMS values of the heading ($\psi$), pitch ($\theta$), and length ($l$) errors for single epoch, Dynamics, and C-Dynamics approaches.

Mode	Fixing Rate	$\mathit{\psi }$ (°)	$\mathit{\theta }$ (°)	$\mathit{l}$ (m)
Single epoch	52.6%	48.579	34.538	6.262
Dynamics	67.4%	19.958	26.718	2.455
C-Dynamics	84.4%	13.57	14.279	0.435

Table 4

RMS values of the heading ($\psi$), pitch ($\theta$), and length ($l$) errors based on the dual antenna scheme in three typical observation conditions.

Environment	Single Epoch			Dynamics			C-Dynamics
	$\mathit{\psi }$ (°)	$\mathit{\theta }$ (°)	$\mathit{l}$ (m)	$\mathit{\psi }$ (°)	$\mathit{\theta }$ (°)	$\mathit{l}$ (m)	$\mathit{\psi }$ (°)	$\mathit{\theta }$ (°)	$\mathit{l}$ (m)
Static obstruction sky	86.436	52.989	29.855	36.338	27.925	3.271	22.786	21.698	0.724
Dynamic open sky	0.113	0.424	0.003	0.105	0.475	0.002	0.109	0.481	0.002
Dynamic urban	5.749	6.252	0.205	0.384	0.915	0.008	0.366	0.876	0.007

Table 5

Ambiguity fixing success rate of different methods.

Environment	Single Epoch	Dynamics	C-Dynamics
Static obstruction sky	0%	9.6%	66.8%
Dynamic open sky	100%	100%	100%
Dynamic urban	96%	99.7%	99.7%

Table 6

Comparison of fixation rates and RMS values of the heading ($\psi$), pitch ($\theta$), and length ($l$) errors for the three methods: LAMBDA, CLAMBDA, and CLAMBDA+CD.

Mode	Fixing Rate	$\mathit{\psi }$ (°)	$\mathit{\theta }$ (°)	$\mathit{l}$ (m)
LAMBDA	94.9%	7.268	6.986	3.912
CLAMBDA	98.4%	3.021	4.183	0.246
CLAMBDA+CD	99.6%	1.089	1.595	0.02

Table 7

RMS values of the heading ($\psi$), pitch ($\theta$), and length ($l$) errors based on the dual antenna scheme in three typical observation conditions.

Environment	LAMBDA			CLAMBDA			CLAMBDA+CD
	$\mathit{\psi }$ (°)	$\mathit{\theta }$ (°)	$\mathit{l}$ (m)	$\mathit{\psi }$ (°)	$\mathit{\theta }$ (°)	$\mathit{l}$ (m)	$\mathit{\psi }$ (°)	$\mathit{\theta }$ (°)	$\mathit{l}$ (m)
Dynamic obstruction sky	28.731	25.479	1.904	16.212	24.865	1.348	6.754	7.856	0.114
Static open sky	0.391	1.108	0.002	0.113	0.285	0.002	0.107	0.261	0.001
Dynamic urban	0.833	2.532	0.019	0.253	0.951	0.006	0.282	0.737	0.003

Table 8

Ambiguity fixing success rate of different methods.

Environment	LAMBDA	CLAMBDA	CLAMBDA+CD
Dynamic obstruction sky	46.5%	52%	90%
Static open sky	100%	100%	100%
Dynamic urban	94.3%	99.5%	99.9%

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