An Efficient UD Factorization Implementation of Kalman Filter for RTK Based on Equivalent Principle

Abstract

Real-time kinematic (RTK) is a technique frequently utilized to provide real-time highly precise positioning services for mobile Internet-of-Things (IoT)-embedded terminals from intelligence appliances and smartphones to autonomous drones and self-driving vehicles. To fully utilize hardware resources, the internal GNSS chips or modules equipped in IoT terminals should satisfy the traits of energy efficiency and low computational complexity. As the number of global navigation satellite system (GNSS) increases, the continuous accumulation of high-dimensional rounding errors, the rough system model, and seriously distorted observations will result in divergence and considerable processing burden in the conventional Kalman filter (KF) process. Computational efficiency is significant in the reduction in the power consumption and intensifies the positioning performance of GNSS receivers. Here, a new filter strategy based on UD factorization, where U stands for the unit upper-triangular factor and D indicates the diagonal factor, is proposed for RTK positioning to enhance the numerical stability and reduce the computational effort. The equivalent principle was applied to turn double-difference (DD) observations into zero-difference (ZD) observations. Then, the UD-factorization-based Kalman filter (UD-KF) is proposed as a way to sequentially provide accurate real-time estimations of the filter states and variance–covariance (VC) matrix. Both static and dynamic tests were carried out with single-frequency data from a GPS to evaluate the performance of UD-KF. The results of the zero-baseline test show that UD-KF can obtain smaller RMS of the estimated parameters as the noise of DD observations was twice that of the ZD observations. A short baseline test showed that, compared to the regular filter approach with DD observations, UD-KF achieved a shorter computation time with a higher data utilization rate for both filtering and fixing stages, with an average improvement of 32% and 18%. Finally, a dynamic test showed that the UD-KF can avoid the undesirable effect of satellite changes. Therefore, compared to KF with DD observations, the UD-KF with equivalent ZD observations can enhance the robustness as well as improve the positioning accuracy of RTK positioning.

1. Introduction

At present, the continuous improvement of GNSS, i.e., GPS, Galileo, BDS, and GLONASS, and regional navigation systems including NavIC, QZSS, etc., has promoted the application of positioning services in various fields, such as dynamic navigation [1], engineering survey [2,3], and space and atmospheric science [4]. GNSS data processing and analysis software has become an essential focus among scientists and engineers worldwide. Many famous software packages have been released, including GAMIT/GLOBK [5], Bernese [6], PANDA [7], RTKLIB [8], gLab [9], and PPPH [10], to name but a few [8,9,11,12,13,14]. Recently, new developments have been aimed at parallel programming routines on cluster platforms and cloud computing based on multinode and multicore processors [15,16]. These investigations made significant contributions to the efficient processing of massive GNSS data. However, the traditional processing models based on DD are highly complex and increase the noise of observation models but have low data utilization [1,17].

Battery-operated IoT positioning sensors, smartphones [18], and autonomous vehicles [19] often employ GNSS due to its coverage and ease of use. However, GNSS chipsets are power-hungry, incurring a considerable decrease in the IoT sensor battery lifetime, an aspect that vendors are trying to tackle through the development of low-powered GNSS chipsets [20]. A large number of on-chip resources (i.e., CUP and RAM) were allocated for the RTK positioning solution in GNSS receivers. Mass-produced IoT sensors must develop an efficient and less-complicated mathematical model to ensure the real-time and persistence of RTK services [21].

Differencing and zero-differencing algorithms have been algebraically proved to be equivalent [17]. The equivalence principle can now be easily explained and accepted. The equivalence principle has been developed by leaps and bounds since then, and its unique advantages in improving data utilization have attracted continuous attention.

Least square (LS) and KF are two frequently applied approaches for positioning, navigation, and timing (PNT) functions embedded in GNSS receivers. The trade-off between accuracy and time consumption is the priority of navigation devices due to the limited storage capacity. KF suffers from numerical deterioration as the underlying assumptions about prior knowledge of state noise and infinite precision arithmetic are not necessarily met in practice [22,23]. Actually, due to the effects of high-dimensional accumulated round-off and the differential operation, there is no guarantee that the VC matrix maintains positive definiteness [24,25]. Benefiting from sequentially processing, UD-factorization-based KF (UD-KF) is the most computationally economic among all square-root decomposition methods adopted to keep the VC matrix’s positive definiteness. UD factorization can provide accurate real-time estimations for system states, and their VC matrix has improved computation efficiency as well numerical stability [22]. It has been widely used in deep-space navigation, high-dynamic target tracking, and other applications [26,27,28]. However, it is difficult to directly apply UD-KF to RTK since it is limited by the correlation of the DD stochastic model [29].

In this paper, a filter strategy based on UD factorization with a low computation burden for RTK is proposed for the first time. The equivalent ZD measurements with a lower noise level are sequentially used by UD-KF to accelerate the filter progress and ensure positioning accuracy.

The following sections are organized as follows: In Section 2, the conventional DD observation equations are outlined, followed by the analysis of its stochastic model. Furthermore, the equivalence ZD model is formulated, and the UD-KF strategy is derived for efficient and numerical stable computations. Moreover, the algorithm implementation is discussed. As described in Section 3, to demonstrate the accuracy and efficiency of the presented methodology for single-frequency RTK, the GPS L1 datasets collected from the Curtin GNSS Research Center and the Hongkong GNSS continuously operating reference stations’ (CORS) network were used for static tests, and a dataset collected by a vehicle experiment was also processed. Finally, some concluding remarks are given in Section 4.

2. Methods

2.1. RTK Mathematic Model

The original ZD carrier phase measurements φ (in cycles) and code measurements P (in meters) for n common-in-view satellites can be defined as [30]:

$$\{\begin{array}{l}{\lambda }^i{\phi }_{\mathrm{b}}^i={\rho }_{\mathrm{b}}^i+(c{t}_{\mathrm{b}}-c{t}^i)-\lambda (N_{\mathrm{b}}^i+{\delta }_{\mathrm{b}}-{\delta }^i)+T_{\mathrm{b}}^i-I_{\mathrm{b}}^i+e_{\mathrm{b}}^i\\ P_{\mathrm{b}}^i={\rho }_{\mathrm{b}}^i+(c{t}_{\mathrm{b}}-c{t}^i)+T_{\mathrm{b}}^i+I_{\mathrm{b}}^i+{\epsilon }_{\mathrm{b}}^i\\ {\lambda }^i{\phi }_{\mathrm{r}}^i={\rho }_{\mathrm{r}}^i+(c{t}_{\mathrm{r}}-c{t}^i)-\lambda (N_{\mathrm{r}}^i+{\delta }_{\mathrm{r}}-{\delta }^i)+T_{\mathrm{r}}^i-I_{\mathrm{r}}^i+e_{\mathrm{r}}^i\\ P_{\mathrm{r}}^i={\rho }_{\mathrm{r}}^i+(c{t}_{\mathrm{r}}-c{t}^i)+T_{\mathrm{r}}^i+I_{\mathrm{r}}^i+{\epsilon }_{\mathrm{r}}^i\end{array}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(i=1,\cdots ,n)$$

(1)

where i is the index of satellites, r and b denote the rover and base receiver, respectively, λ is the carrier wavelength, ρ is the geometric range between receiver and satellite, c₀ = 299,792,458 m/s is the speed of light, t denotes the receiver and clock errors, N and δ are the phase ambiguity and initial receiver and satellite phase biases, T and I are the tropospheric and ionospheric delay on the signal propagation path, and e and ε are the remaining unmodeled errors on phase and code, including multipath noise, system noise, etc. The hardware code delay lumps into the clock errors and is therefore omitted here. Additionally, the phase hardware delay diminishes when DD equations are built. The stochastic model of Equation (1) is expressed as follows:

$$R_{\mathrm{ZD}}=({\mathit{P}}_{\mathrm{ZD}}{)}^{-1}=diag({\sigma }_{\mathrm{b},1}^2,\cdots ,{\sigma }_{\mathrm{b},n}^2,{\sigma }_{\mathrm{r},1}^2,\cdots ,{\sigma }_{\mathrm{r},n}^2)=\left[\begin{array}{cccccc}{\sigma }_{\mathrm{b},1}^2& & & & & \\ & \ddots & & & & \\ & & {\sigma }_{\mathrm{b},n}^2& & & \\ & & & {\sigma }_{\mathrm{r},1}^2& & \\ & & & & \ddots & \\ & & & & & {\sigma }_{\mathrm{r},n}^2\end{array}\right]$$

(2)

where σ² = a²/sin²E is based on satellite-elevation-dependent weight model with empirical coefficient a = 3 mm for carrier phase and a = 0.3 m for code [31], E = 15° is the cut-off elevation, and observations with elevation less than 15° are discarded, and R_ZD denotes the observation covariance matrix of ZD observations. Taking i-th satellite as a reference, the functional model of DD can be expressed as:

$$\{\begin{array}{l}\nabla \Delta {\lambda }^{\mathrm{m}}{\phi }_{\mathrm{br}}^{mi}=\nabla \Delta {\rho }_{\mathrm{br}}^{mi}-\lambda \nabla \Delta N_{\mathrm{br}}^{mi}+\nabla \Delta T_{\mathrm{br}}^{mi}-\nabla \Delta I_{\mathrm{br}}^{mi}+\nabla \Delta e_{\mathrm{br}}^{mi}\\ \nabla \Delta P_{\mathrm{br}}^{mi}=\nabla \Delta {\rho }_{\mathrm{br}}^{mi}+\nabla \Delta T_{\mathrm{br}}^{mi}+\nabla \Delta I_{\mathrm{br}}^{mi}+\nabla \Delta {\epsilon }_{\mathrm{br}}^{mi}\end{array} (m=1,2,\cdots ,n \mathrm{and} m\ne i)$$

(3)

where ∇Δ is the DD operator and the effect of T, and I can be ignored for a short baseline. The single-difference (SD) transformation matrix can be defined as:

$${\mathrm{S}}_{n\times n}=\left[\begin{array}{cccccc}-1& \cdots & 0& 1& \cdots & 0\\ ⋮& \ddots & ⋮& ⋮& \ddots & ⋮\\ 0& \cdots & -1& 0& \cdots & 1\end{array}\right]$$

The DD transformation matrix can be defined as:

$${\mathrm{D}}_{n-1\times n}=\left[\begin{array}{cccccc}1& 0& \cdots & -1& \cdots & 0\\ 0& 1& \cdots & -1& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮& \ddots & ⋮\\ 0& 0& \cdots & \underset{ith }{\underbrace{-1}}& \cdots & 1\end{array}\right]$$

The stochastic model can be expressed as:

$$\{\begin{array}{l}{\mathit{R}}_{\mathrm{SD}}{=(\mathit{P}}_{\mathrm{SD}}{)}^{-1}=\mathit{S}\cdot {\mathit{R}}_{\mathrm{ZD}}\cdot {\mathit{S}}^T=diag({\sigma }_{\mathrm{b},1}^2+{\sigma }_{\mathrm{r},1}^2,\cdots ,{\sigma }_{\mathrm{b},n}^2+{\sigma }_{\mathrm{r},n}^2)=\left[\begin{array}{ccc}{\sigma }_{\mathrm{b},1}^2+{\sigma }_{\mathrm{r},1}^2& & \\ & \ddots & \\ & & {\sigma }_{\mathrm{b},n}^2+{\sigma }_{\mathrm{r},n}^2\end{array}\right]\\ {\mathit{R}}_{\mathrm{DD}}=({\mathit{P}}_{\mathrm{DD}}{)}^{-1}=\mathit{D}\cdot {\mathit{R}}_{\mathrm{SD}}\cdot {\mathit{D}}^T=\left[\begin{array}{cccc}{\sigma }_{\mathrm{b},1}^2+{\sigma }_{\mathrm{r},1}^2+{\sigma }_{\mathrm{b},i}^2+{\sigma }_{\mathrm{r},i}^2& {\sigma }_{\mathrm{b},i}^2+{\sigma }_{\mathrm{r},i}^2& \cdots & {\sigma }_{\mathrm{b},i}^2+{\sigma }_{\mathrm{r},i}^2\\ {\sigma }_{\mathrm{b},i}^2+{\sigma }_{\mathrm{r},i}^2& {\sigma }_{\mathrm{b},2}^2+{\sigma }_{\mathrm{r},2}^2+{\sigma }_{\mathrm{b},i}^2+{\sigma }_{\mathrm{r},i}^2& \cdots & {\sigma }_{\mathrm{b},i}^2+{\sigma }_{\mathrm{r},i}^2\\ ⋮& ⋮& \ddots & ⋮\\ {\sigma }_{\mathrm{b},i}^2+{\sigma }_{\mathrm{r},i}^2& {\sigma }_{\mathrm{b},i}^2+{\sigma }_{\mathrm{r},i}^2& \cdots & {\sigma }_{\mathrm{b},n-1}^2+{\sigma }_{\mathrm{r},n-1}^2+{\sigma }_{\mathrm{b},i}^2+{\sigma }_{\mathrm{r},i}^2\end{array}\right]\end{array}$$

(4)

where P_SD and P_DD are the weight matrix of the SD and DD model, respectively. R_SD and R_DD are the stochastic models for SD and DD measurements. Equation (4) only holds for the phase measurements, and the stochastic model of code can be obtained by analogy. Equation (4) shows that the correlation is introduced into R_DD by adding the SD variance of the i-th satellite on off-diagonal elements of the CV matrix [32], which aggravates the computation burden and memory intensity [33]. The sequential KF is usually used to conquer these challenges in applications such as GNSS/INS integration and multi-GNSS precise-point positioning (PPP) [34,35,36] to enhance computational efficiency. These cases prove that sequential operation takes the mutual independence of observation (or combined observation) as a precondition [37,38]. Therefore, observations’ decorrelation is needed for the DD model of RTK.

2.2. Proposed Methodology

2.2.1. Measurement Decorrelation

The linearization of Equation (1) is expressed as [39]:

$$\mathit{V}=\left[\begin{array}{cc}\mathit{A}& \mathit{B}\end{array}\right]\left[\begin{array}{c}\mathit{x}\\ \mathit{N}\end{array}\right]+\left[\begin{array}{cc}\mathit{C}& \mathit{D}\end{array}\right]\left[\begin{array}{c}{\mathit{t}}_{\mathrm{rec}}\\ {\mathit{t}}^{\mathrm{sat}}\end{array}\right]-\mathit{L}, {\mathit{P}}_{\mathrm{ZD}}$$

(5)

where ${\mathit{P}}_{\mathrm{ZD}}={\mathit{R}}_{\mathrm{ZD}}^{-1}$ is the weight matrix of ZD measurements, and x is the coordinate vector of the kinematic rover, which will be calculated in each epoch. $\mathit{N}=\left[\underset{\mathrm{base}}{\underbrace{N_{\mathrm{b}}^1,N_{\mathrm{b}}^2,\cdots ,N_{\mathrm{b}}^n}},\underset{\mathrm{rover}}{\underbrace{N_{\mathrm{r}}^1,N_{\mathrm{r}}^2,\cdots ,N_{\mathrm{r}}^n}}\right]$ is ZD ambiguity vector, t_rec is the receiver clock error vector including base and rover, t^sat is the clock error vector of satellites, L is the observation minus computation (OMC) residual vector, and observational vector V is the residual vector of ZD measurements. A, B, C, and D are coefficient matrices [40]. The transformation matrix M₁ is employed to eliminate t^sat according to equivalence principle [41]:

$${\mathit{M}}_1=\mathit{I}-\mathit{D}{({\mathit{D}}^T\mathit{P}\mathit{D})}^{-1}{\mathit{D}}^T{\mathit{P}}_{\mathrm{ZD}}$$

where I is an identity matrix. The equivalent SD form of Equation (5) is:

$${\mathit{V}}^{\prime }=\left[\begin{array}{cc}{\mathit{A}}^{\prime }& {\mathit{B}}^{\prime }\end{array}\right]\left[\begin{array}{c} \mathit{x}\prime \\ \mathit{N}\prime \end{array}\right]+{\mathit{C}}^{\prime }{\mathit{t}}_{\mathrm{rec}}-{\mathit{L}}^{\prime }, {\mathit{P}}_{\mathrm{ZD}}$$

(6)

where, V′ = M₁V, A′ = M₁A, B′ = M₁B and C’ = M₁C, L′ = M₁L. x′ and N′ are the equivalent SD coordinate corrections and ambiguities. Furthermore, to eliminate t_rec, the transformation matrix M₂ can be denoted as:

$${\mathit{M}}_2=\mathit{I}-{\mathit{C}}^{\prime }{({{\mathit{C}}^{\prime }}^{T}P{\mathit{C}}^{\prime })}^{-1}{{\mathit{C}}^{\prime }}^T, {\mathit{P}}_{\mathrm{Z}\mathrm{D}}$$

Equation (6) can be transformed into equivalent DD form as:

$$\underset{\mathit{V}}{\underbrace{{\mathit{V}}^{″}}}=\underset{\mathit{H}}{\underbrace{\left[\begin{array}{cc}{\mathit{A}}^{″}& {\mathit{B}}^{″}\end{array}\right]}}\underset{\mathit{X}}{\underbrace{\left[\begin{array}{c} \mathit{x} ″\\ \mathit{N} ″\end{array}\right]}}+\underset{Z}{\underbrace{{\mathit{L}}^{″}}}, {\mathit{P}}_{\mathrm{ZD}}$$

(7)

where V″ = M₂V′, A″ = M₂A′, B″ = M₂B′, and L″ = M₂L′, x″ and N″ are the equivalent DD baseline correction vector and equivalent DD ambiguities. The ambiguities of the base receiver and the reference satellite should be set to 0 as the rank deficiency of M₁ and M₂ is n and 1, respectively. Then, the number of parameters that need to be estimated can be kept consistent before and after equivalent transformation. Advantages can be found for the formulated equivalence model: (1) More measurements can be retained for adjustment; and (2) P_ZD remains unchanged after equivalent transformation, which ensures the independence of measurements.

The parameters can be estimated by KF with the following state vector:

$$\mathit{X}=\left[ \mathit{x}″, \mathit{N}″\right]$$

(8)

Defining the VC matrix of the system state, system noise, measurement noise, and design matrix as P, Q, R, and H, the KF equations can be expressed as:

$$\{\begin{array}{l}{\mathit{P}}_{k|k-1} = {\mathit{\Phi }}_{k,k-1}{\mathit{P}}_{k-1}{\mathit{\Phi }}_{k,k-1}^T+{\mathit{\Gamma }}_{k-1}{\mathit{Q}}_{k-1}{\mathit{\Gamma }}_{k-1}^T\\ {\mathit{K}}_k={\mathit{P}}_{k|k-1}{\mathit{H}}_k^T({\mathit{H}}_k{\mathit{P}}_{k|k-1}{\mathit{H}}_k^T+{\mathit{R}}_k{)}^{-1}\\ {\mathit{P}}_k=(\mathit{I}-{\mathit{K}}_k{\mathit{H}}_k){\mathit{P}}_{k|k-1}\\ {\mathit{X}}_{k|k-1}={\mathit{\Phi }}_{k,k-1}{\hat{\mathit{X}}}_{k-1}\\ {\hat{\mathit{X}}}_k={\hat{\mathit{X}}}_{k|k-1}+{\mathit{K}}_k({\mathit{Z}}_k-{\mathit{H}}_k{\hat{\mathit{X}}}_{k|k-1})\end{array}$$

(9)

where K is the Kalman gain, and Φ is the state transform matrix, and Z_k is the specific measurement vector at epoch k. The roundoff errors and extremely noisy measurements can lead to a negative-definite P_k|k−1, which can lead to the result of ${({\mathit{H}}_k{\mathit{P}}_{k|k-1}{\mathit{H}}_k^T+{\mathit{R}}_k)}^{-1}$ being wrong or unsolvable and eventually introduce huge errors into the estimation of K and X. According to the independent and identically distributed property of Equation (7), the recursion of P and K can be optimized through UD factorization to form an enhanced filter strategy.

2.2.2. UD-KF Strategy

Assuming that there are m measurements, and that g states need to be estimated at a certain epoch, a measurement–covariance set can be expressed as Z~R = diag(r₁,…, r_m). Additionally, the prior state–covariance pair can be expressed as $\tilde{\mathit{X}} ~\mathit{P}$. An upper-triangular matrix U with all 1 on diagonal elements and a diagonal matrix D are applied to realize U-D factorization. The special structure of U and D ensures P will always be positive-definite by defining mnemonics as [42]:

$$\{\begin{array}{l}\widehat{\mathit{P}}={\mathit{P}}_k , \widehat{\mathit{U}}={\mathit{U}}_k , \widehat{\mathit{D}}={\mathit{D}}_k\\ \tilde{\mathit{P}}={\mathit{P}}_{k|k-1}, \tilde{\mathit{U}}={\mathit{U}}_{k|k-1} , \tilde{\mathit{D}}={\mathit{D}}_{k|k-1} \end{array}\Rightarrow \{\begin{array}{l}\tilde{\mathit{P}}=\tilde{\mathit{U}}\tilde{\mathit{D}}{\tilde{\mathit{U}}}^{\mathit{T}}\\ \widehat{\mathit{P}}=\widehat{\mathit{U}}\widehat{\mathit{D}}{\widehat{\mathit{U}}}^{\mathit{T}}\end{array}$$

(10)

Once U and D are determined, P can be determined accordingly. To illustrate this purpose, the one-step prediction of P is rewritten as:

$$\widehat{\mathit{P}}=\tilde{\mathit{P}}-\tilde{\mathit{P}}{\mathit{H}}^T(\mathrm{r}+\mathit{H}\tilde{\mathit{P}}{\mathit{H}}^T){\mathit{H}}^T\tilde{\mathit{P}}$$

(11)

where r is the scalar diagonal element of R. ${\mathit{f}}_{n\times 1}={\tilde{\mathit{U}}}^T\mathit{H}$ is defined as the data vector and ${\mathit{v}}_{n\times 1}=\tilde{\mathit{D}}{\tilde{\mathit{U}}}^T\mathit{H}=\tilde{\mathit{D}}f$ as the residual vector. Equation (11) can be factorized into U-D factorization as:

$$\widehat{\mathit{U}}\widehat{\mathit{D}}{\widehat{\mathit{U}}}^T=\tilde{\mathit{U}}\tilde{\mathit{D}}{\tilde{\mathit{U}}}^T-\tilde{\mathit{U}}\tilde{\mathit{D}}{\tilde{\mathit{U}}}^T{\mathit{H}}^T{(\mathit{R}+\mathit{H}\tilde{\mathit{U}}\tilde{\mathit{D}}{\tilde{\mathit{U}}}^T{\mathit{H}}^T)}^{-1}\mathit{H}\tilde{\mathit{U}}\tilde{\mathit{D}}{\tilde{\mathit{U}}}^T=(\mathit{U}\overline{\mathit{U}})\overline{\mathit{D}}{\left(\mathit{U}\overline{\mathit{U}}\right)}^T$$

(12)

where $\alpha =r+\mathit{H}\tilde{\mathit{U}}\tilde{\mathit{D}}{\tilde{\mathit{U}}}^T{\mathit{H}}^T$, $\overline{\mathit{U}}\overline{\mathit{D}}{\overline{\mathit{U}}}^T=\mathit{D}-\frac{1}{\alpha }{\mathit{vv}}^T$. As the differential operation is canceled in Equation (12), the recursion of $\widehat{\mathit{P}}$ avoids numerical instability [43]. The process of UD-KF is shown in Figure 1, and the specific implementation of UD factorization is given in

Table 1. A stepwise implementation of UD factorization.

Step	Stage	Implementation
1	Initialization	$f={\mathit{U}}^T\mathit{H}$$, \mathit{v}=\mathit{Df}$$, {\alpha }_1=r+f_1{v}_1$$, d_1=d_{1}r/{\alpha }_1$$, b_1=v_1$
2	U-D factor Update	for j = 2:n $\hspace{1em} {\alpha }_j={\alpha }_{j-1}+f_j{v}_j$$, {d^}_j={\tilde{d}}_j{\alpha }_{j-1}/{\alpha }_j$$, b_j=v_j$$, p_j=-f_j/{\alpha }_{j-1}$   for i = 1: j − 1 $\hspace{1em}\hspace{1em}\hspace{1em}{\hat{U}}_{ij}={\tilde{U}}_{ij}+b_i{p}_j$, $\hspace{1em}\hspace{1em}\hspace{1em}{b}_i{=b}_i+U_{ij}{v}_j$   end end $K=b/a_n$
3	State Update	for j = 1:g $\hspace{1em} x_j=x_j+Kz$ end

step by step.

As shown in Figure 1, the scalar element of R is sequentially imported, and the following superiorities can be drawn [44]: (1) X and K are implicitly obtained without an additional computational burden; (2) because 0 ≤ α_{j − 1}/α_j≤ 1 and ${\widehat{d}}_j$ is iteratively obtained by multiplication with its former values, the numeric accuracy is enhanced and cancellation-type errors are avoided [45]; and (3) computation efficiency is improved, as Z is proposed in a component-wise manner, and the updating of α_j requires sequential multiplication and superposition of newly constructed f and v.

Figure 2 summarizes the flow of UD-KF, and the positioning solution is divided into the filtering stage and the fixing stage [46]. The filter is initialized by LS based on conventional DD measurements to form the VC matrix and weighting matrix of X at the first epoch [47]. Φ can be defined as an identity matrix without cycle slip. Q can be defined as a zero matrix. A real-time cycle slip detection strategy is applied for data preprocessing [48]. While cycle slip is detected on i-th satellite, adequately large process noise (1000 cycles is a reasonable empirical value) must be added to Q_ii to guarantee stability and sensitivity [8,49]. To verify the proposed UD-KF, KF and DD measurements (DD-KF) are simultaneously implemented to obtain the float solution $(\mathsf{\Delta }\widehat{x},\mathsf{\Delta }\widehat{N})$ [30]. The least-square ambiguity decorrelation adjustment (LAMBDA) is applied to fix ambiguities [50].

3. Results and Discussion

To explore the effect of the UD-KF, a zero-baseline test, a test with different stationary baselines, and a vehicular dynamic experiment were conducted utilizing single-frequency GPS data. All results were obtained through postprocessing performed on an Intel Core i7 2.30 GHz notebook PC with 16 GB RAM running Windows 10.

3.1. Zero-Baseline Test

A dataset with 30 s intervals and E = 15° collected by station CUT00 on 1 January 2021, at Curtin University, Australia was analyzed. As shown in Figure 3, two zero-baselines (CUT0-CUT2, CUT0-CUT3) were formed, as a common antenna (TRM 59,800.00 SCIS) connects to three receivers, CUT0(Trimble NETR9), CUT2(Trimble NETR9), and CUT3(JAVAD TRE_G3TH_8). Figure 4 depicts the available number of satellites and relative dilution of precision (RDOP) in the partial observation period. The average of visible satellites and RDOP were 8 and 0.7, respectively, indicating an ideal environment for relative positioning.

The positioning error on the east (E), north (N), and up (U) components is illustrated in Figure 5. A rapid convergence appeared as the preset Q was tuned at the initial stage. The RMS and STD for the positioning error are summarized in

Table 2. The RMS and STD of the positioning error.

	Processing Model	CUT0-CUT2			CUT0-CUT3
		E	N	U	E	N	U
RMS (mm)	UD-KF	0.5	0.2	0.3	0.3	0.6	0.5
	DD-KF	3.5	1.0	1.8	13.0	2.3	7.6
STD (mm)	UD-KF	0.3	0.2	0.3	0.3	0.5	0.5
	DD-KF	1.9	0.7	1.7	6.2	1.6	6.2

.

For UD-KF, the mean RMS of the ENU components for baselines CUT0-CUT2 and CUT0-CUT3 was (0.5 mm, 0.2 mm, 0.3 mm) and (0.3 mm, 0.6 mm, 0.5 mm). For DD-KF, the mean RMS of CUT0-CUT2 and CUT0-CUT3 was (3.5 mm, 1.0 mm, 1.8 mm) and (13.0 mm, 2.3 mm, 7.6 mm). The mean STD of UD-KF for CUT0-CUT2 and CUT0-CUT3 was (0.3 mm, 0.2 mm, 0.3 mm) and (0.3 mm, 0.5 mm, 0.5 mm), respectively, while the mean STD of DD-KF was (1.9 mm, 0.7 mm, 1.7 mm) and (6.2 mm, 1.6 mm, 6.2 mm). The results show that the UD-KF is more accurate than the DD-KF in all three directions.

A conclusion can also be drawn that UD-KF converges significantly faster than DD-KF. In the CUT0-CUT2 test, UD-KF quickly converged within 2 mm, while DD-KF failed to converge within 2 mm in the first ten hours. In the CUT0-CUT3 test, UD-KF converged after a short period of fluctuation, while DD-KF converged only after the 15th hour and retained a bias in the eastward direction. The reason CUT0-CUT2 achieved smaller RMS and STD than CUT0-CUT3 for both processing models is mainly due to the similarities and differences in receiver types used at both ends of different baselines.

As UD-KF limited position error to the millimeter level as expected, it is an effective substitute for DD-KF. Despite similar positioning accuracy obtained by both models, UD-KF performed better than conventional DD-KF [50].

3.2. Static Test

3.2.1. Position Accuracy Analysis

As shown in Figure 6, the GPS L1 datasets with E = 15° were collected by 7 GNSS CORS stations located in Hongkong during 00 h−01 h on 1 January 2021. Six baselines with various distances were used; the processing schemes are listed in

Table 3. The processing schemes for different baselines.

No.	Baseline		Distance (km)	Interval	Processing Model
	Base	Rover
1	HKKT	HKSC	15.613	1 s, 5 s	UD-KF, DD-KF
2	HKKS	HKSC	18.303	1 s, 5 s	UD-KF, DD-KF
3	HKLM	HKSC	11.634	1 s, 5 s	UD-KF, DD-KF
4	HKOH	HKSC	12.211	1 s, 5 s	UD-KF, DD-KF
5	HKPC	HKSC	11.418	1 s, 5 s	UD-KF, DD-KF
6	HKST	HKSC	9.232	1 s, 5 s	UD-KF, DD-KF

.

As shown in Figure 7, the RMS and STD of position error on ENU for UD-KF and DD-KF were all within 10 cm, which implies they performed similarly.

Table 4. RMS and STD improvement rate on ENU.

	Baseline	Interval	E			N			U
			UD	DD	Improvement	UD	DD	Improvement	UD	DD	Improvement
RMSE (cm)	HKSC-HKKT	1 s	2.73	6.02	54.65%	2.34	2.82	17.02%	8.07	4.54	−77.75%
		5 s	1.19	3.39	64.90%	1.314	4.34	69.72%	2.61	3.54	26.27%
	HKSC-HKKS	1 s	0.67	0.74	9.46%	0.88	1.87	52.94%	2.43	1.251	−94.24%
		5 s	1.69	0.75	−125.33%	2.09	1.65	−26.67%	2.70	1.24	−117.74%
	HKSC-HKLM	1 s	1.14	3.50	67.43%	0.93	0.81	−14.81%	2.89	1.24	−133.06%
		5 s	1.07	2.92	63.36%	1.06	1.01	−4.95%	2.80	1.12	−150%
	HKSC-HKOH	1 s	0.69	3.07	77.52%	1.07	2.19	51.14%	2.40	2.79	13.98%
		5 s	0.65	2.35	72.34%	0.78	2.23	65.02%	1.75	2.44	28.28%
	HKSC-HKPC	1 s	1.12	3.83	70.76%	1.37	1.36	−0.73%	3.51	3.95	11.14%
		5 s	1.01	4.72	78.60%	1.19	1.09	−9.17%	3.36	4.39	23.46%
	HKSC-HKST	1 s	1.15	0.91	−26.37%	1.14	0.89	−28.09%	4.012	6.08	34.01%
		5 s	1.21	1.18	−2.54%	1.43	0.71	−101.41%	4.16	6.58	36.78%
STD (cm)	HKSC-HKKT	1 s	2.61	4.83	45.96%	2.28	2.52	9.52%	7.97	4.35	−83.22%
		5 s	0.96	2.52	61.90%	1.01	4.32	76.62%	2.58	3.34	22.75%
	HKSC-HKKS	1 s	0.64	0.54	−18.52%	0.82	1.51	45.70%	2.43	0.74	−228.38%
		5 s	1.69	0.49	−244.90%	1.60	1.12	−42.86	2.70	0.77	−250.65%
	HKSC-HKLM	1 s	1.14	1.72	33.72%	0.84	0.77	−9.09%	2.82	0.73	−286.30%
		5 s	1.05	1.37	23.36%	0.96	0.93	−3.23%	2.76	0.71	−288.73%
	HKSC-HKOH	1 s	0.65	1.48	56.08%	0.94	1.79	47.49%	2.25	1.12	−100.89%
		5 s	0.65	1.15	43.48%	0.72	1.77	59.32%	1.75	1.66	−5.42%
	HKSC-HKPC	1 s	1.04	3.53	70.54%	1.35	1.07	−26.17%	3.45	3.812	9.50%
		5 s	1.01	3.88	73.97%	1.18	0.85	−38.82%	3.36	4.09	17.85%
	HKSC-HKST	1 s	1.13	0.90	−25.56%	1.099	0.88	−24.89%	3.94	4.611	14.55%
		5 s	1.06	1.00	−6.00%	1.42	0.69	−105.80%	4.06	4.54	10.57%

provides corresponding statistics. Negative values represent performance degradation, while positive values represent an improvement.

UD-KF performed better for short baselines (HKSC-HKST/HKSC-HKLM/HKSC-HKOH/HKSC-HKPC). Taking HKSC-HKOH as an example, when a switch was made from DD-KF to UD-KF, the RMS of the 1 s interval dataset was improved by (+2.38 cm, +1.12 cm, +0.39 cm), while the RMS of the 5 s interval dataset was improved by (+1.7 cm, +1.45 cm, +0.69 cm). For HKSC-HKPC, the positioning accuracy was improved on E and U components, but a 0.7% degradation appeared on N. Similarities can also be found in HKSC-HKKT/HKSC-HKLM. It should be noted that the RMS and STD in the upwards direction of baseline HKSC-HKLM were twice as large as in the DD-KF scheme, indicating a larger residual assignment. A reasonable explanation is that UD-KF cannot only enhance the positioning accuracy but also redistribute the positioning residual in different directions.

For the longer baselines, the position accuracy of the 5 s interval dataset degraded by (−0.94 cm, −0.44 cm, −1.46 cm) for HKSC-HKKS and improved by (+2.2 cm, +3.026 cm, +0.93 cm) for HKSC-HKKT. The possible reason for this discrepancy is that although UD-KF benefits positioning accuracy by removing the correlation of the measurements, the ZD observation may still be affected by atmospheric errors that have not been eliminated equivalently [38]. As a consequence, UD-KF achieved better positioning accuracy without distance correlation.

3.2.2. Redundancy and Ambiguity Dilution of Precision (ADOP) Analysis

To provide sufficient test data, the 1 s interval dataset used in the former section is analyzed here. The filtering and ambiguity fixing efficiency are also analyzed.

In Equation (6), the matrix M₁ has a rank deficit of n. This implies that to convert the unknown parameters of the rover to the relative parameters, we should set the coordinate parameters, ZD ambiguities, and clock error of the base receiver to zero. In Equation (7), the matrix M₂ has a rank deficit of 1. This means that the SD ambiguities of the reference satellite should be set to zero to keep the parameters independent, and the other equivalent SD ambiguities can be converted to equivalent DD ambiguities. This means that the number of states that need to be estimated for UD-KF and DD-KF should be the same [1,51].

The computational efficiency is closely related to model redundancy. Higher redundancies mean higher data utilization. Here, the redundancy is expressed as the difference between observations and unknown states. Taking phase and code measurements of n satellites into account, according to Equations (6) and (7), the proof can be summarized as follows:

For UD-KF: The number of observation equations is: 4n. The number of the unknown parameters is 3 + (n − 1).

For DD-KF: The number of observation equations is 2(n − 1). The number of unknown parameters is 3 + (n − 1).

As a result, the redundancies for UD-KF and DD-KF are 3n − 2 and n − 4, respectively.

ADOP is introduced to evaluate the GNSS model strength for ambiguity resolution, and is defined as [52]:

$$\mathrm{ADOP}={\sqrt{\left|{\mathit{Q}}_{\nabla \mathsf{\Delta }\widehat{N}}\right|}}^{\frac{1}{v}}$$

(13)

where ν is the number of ambiguities to be estimated, |·| is the determinant operator, and ${\mathit{Q}}_{\nabla \mathsf{\Delta }\widehat{N}}$ is formed by design matrix and noise matrix:

$$\mathit{Q}={({\mathit{H}}^T{\mathit{R}}^{-1}\mathit{H})}^{-1}$$

(14)

According to Equations (13) and (14), the ADOP is determined by the satellite geometric distribution H and measurement noise R. Compared to DD-KF, UD-KF maintained superiority in the following two aspects: (1) the noise level of ZD measurements applied in UD-KF was not enlarged by difference operation, and (2) the higher redundancy of UD-KF can improve the geometric distribution.

Figure 8 shows the ADOP of each baseline with more than six satellites stably observed. An ADOP less than 1 cycle implies a good observation environment. For DD-KF, the ADOP showed a downward trend and converged near 0.4 cycles as time increased. For UD-KF, ADOP changed under different baselines with a consistent trend, rapidly dropping down and stably converging near 0.05 cycles. The ADOP performance of UD-KF was better than that of DD-KF. This is mainly because of the high data utilization and the lower noise level due to employing the equivalent ZD measurements.

3.2.3. Computation Efficiency Analysis

The time consumption of the filtering and fixing stage at each epoch is shown in Figure 9 and Figure 10, respectively. Here, the 1-h observation data with a 1 s sample rate was adopted (a total of 3600 epochs). The ‘Total Filter Time’ and ‘Total Fix Time’ were the time consumption at the ‘filter stage’ and ‘fix stage’ for the whole hour, respectively. The blue and red dots represent the calculation time of UD-KF and DD-KF, respectively. Although the time burden of filtering and fixing was on the order of milliseconds, UD-KF took significantly less time than DD-KF [53].

Table 5. Total time consumption statistics.

Baseline	Total Filer Time (ms)		Improvement	Total Fix Time(ms)		Fix Rate		Improvement
	UD-KF	DD-KF		UD-KF	DD-KF	UD-KF	DD-KF
HKSC-HKST	599.37	844.83	29.05%	3037.49	3715.58	98.47%	100%	18.25%
HKSC-HKKT	525.31	1073.34	51.06%	2801.20	3589.02	93.14%	94.42%	21.95%
HKSC-HKPC	518.57	777.03	33.26%	2729.49	3634.99	98.25%	98.92%	24.91%
HKSC-HKLM	541.10	741.99	27.07%	2803.85	3189.85	83.58%	84.47%	12.10%
HKSC-HKOH	535.73	746.97	28.28%	2858.32	3395.42	98.56%	100%	15.82%
HKSC-HKKS	528.23	725.42	27.18%	2798.22	3354.70	98.39%	100%	16.59%

gives the total time consumption for 3600 epochs of each baseline. The total filtering time was within 600 ms for UD-KF, with an average of 541.385 ms, while the filtering time was more than 700 ms, with an average of 818.26 ms, for DD-KF. The proposed strategy improved the computational efficiency, especially the time burden of HKSC-HKKT is reduced by 51.06%.

The total fixing time of UD-KF was less than 3100 ms, for which the minimum was 2729.49 ms and the maximum was 3037.49 ms. However, the minimum of DD-KF was 3189.85 ms, and the maximum was 3715.58 ms. UD-KF achieved efficiency improvement of (18.25%, 21.95%, 24.91%, 12.10%, 15.82%, 16.59%) on each baseline. Additionally, the success rate of the ambiguity fixing of UD-KF and DD-KF remained at a high level, although more ambiguities needed to be fixed by UD-KF.

3.3. Kinematic Test

The dynamic efficiency and accuracy of the UD-KF were statistically analyzed by conducting a test in the urban environment of Wuhan, China, on 29 July 2021, from 6:04 to 8:55 UTC, along the trajectory shown in Figure 11.

A temporary base station was set up nearby the terminal, whose coordinate was determined through PPP with 2 h static data. The rover with a 1 Hz output rate was mounted on the roof of the experimental vehicle. During the test, 6–10 satellites were stably available. Furthermore, the postprocessing results of DD-KF were treated as reference coordinates. If the three-dimensional error of positioning was less than 20 cm, this epoch was regarded as a successful positioning [54].

Figure 12a shows the resulting trajectory in an open-sky environment. The vehicle motion will change due to acceleration and deceleration. Figure 12b shows the result in the street scene, in which the signal was seriously blocked by buildings and vegetation. The yellow dots and red dots denote the trajectory of UD-KF and DD-KF, respectively. The positioning difference between the two models is shown in Figure 13. Apart from a few epochs, the difference was less than 20 cm for the U component and 10 cm for the E and N component. The interrelating RMS and STD in ENU were (1.14 cm, 1.06 cm, 1.48 cm) and (1.12 cm, 1.05 cm, 1.48 cm), respectively. This implies that UD-KF maintained consistency with DD-KF under dynamic conditions. The calculation time burden of UD-KF and DD-KF for filtering was (1063.51 ms, 1424.24 ms) and (2969.69 ms, 3413.47 ms) for fixing ambiguities, respectively, an improvement of (25.33%, 13.00%).

A change in satellites leads to the change of model redundancy, which can cause different calculation burdens. We further evaluated whether the UD-KF could provide robust efficiency improvement when satellites change; the top and middle of Figure 14 show the results of (DD-KF time burden minus UD-KF time burden) at each epoch for different stages. The corresponding number of satellites is shown at the bottom of Figure 14.

There were seven to nine satellites that could be observed by the base receiver and rover receiver separately, but common-view satellites vary greatly due to obstacles and blockage. The position of the rover relies entirely on the time update of KF when less than four satellites are available.

The positive values of the time difference in Figure 14 (top and middle) imply that UD-KF experienced a smaller computation burden at each epoch than DD-KF. At each epoch, the time consumption was reduced on average by 0.11 ms at the filtering stage and 0.13 ms at the fixing stage. Meanwhile, the UD-KF was insensitive to satellite change, remaining stable during the time difference of each stage while the number of satellites changed drastically. It can be inferred that UD-KF can steadily improve efficiency and avoid the impact on the environment, which further shows its robustness in the urban environment [55].

4. Conclusions

In order to tackle the dilemma of a restricted processing capacity of GNSS IoT devices, in this contribution we proposed a new and efficient equivalent ZD algorithm for RTK. First of all, conventional DD measurements were decorrelated to equivalent ZD measurements by the equivalent principle. Additionally, the sequential strategy for RTK was constructed by the UD factorization implementation. The most outstanding feature of UD-KF is that it has a rigorous model that can enhance numeric stability and efficiency with high data utilization and availability. The results show that the proposed method can achieve faster convergence and higher positioning accuracy efficiently compared with the conventional DD-KF. The following conclusions were obtained:

(1)

For the zero-baseline test, UD-KF achieved a better positioning accuracy than DD-KF, which validates the reliability and feasibility of the new proposed algorithm. If the same type of receivers is used at both ends of the baseline, UD-KF can reduce the RMS of the position error by 86% on E, 80% on N, and 83% on U, as compared to the conventional DD-KF.

(2)

For the static test, UD-KF achieved better position accuracy for short baselines. The increase in baseline distance does not affect the positioning performance of UD-KF. The average performance improvement of RMS for six baselines was 69% on E and 27% on N, and more errors occurred on the U component. The computational efficiency was improved by 25–50% at the filtering stage and 15–25% at the fixing stage.

(3)

For the dynamic test, the robustness of UD-KF was verified by the reduction in time consumption, which kept stable when satellites in view changed dramatically. The UD-KF achieved an accurate position in typical urban environments and accelerated the filtering and fixing process by (0.11 ms, 0.13 ms), respectively, for each epoch.

Our future work will extend the proposed method to multifrequency and multi-GNSS positioning, considering the computational efficiency and exploring its application in the field of deep combination with vector tracking to improve the precision and reliability of dynamic positioning in the urban environment.