A Novel Precise GNSS Tracking Method Without Solving the Ambiguity Problem

Received June 3, 2020, accepted June 23, 2020, date of publication June 26, 2020, date of current version July 7, 2020.

Digital Object Identifier 10.1109/ACCESS.2020.3005147

A Novel Precise GNSS Tracking Method Without Solving the Ambiguity Problem WENHAO YANG, YUE LIU, AND FANMING LIU College of Automation, Harbin Engineering University, Harbin 150001, China

Corresponding author: Fanming Liu (fanmingliuheu@163.com) This work was supported by the National Natural Science Foundation of China under Grant 61633008.

ABSTRACT The Global Navigation Satellite System (GNSS) precise positioning has drawn increasing attention owing to the growing demand for accurate relative tracking of devices. The carrier phase becomes the most precise measurement available, the solution of carrier phase integer ambiguity is essential for achieving precise GNSS tracking. Methods of searching within the position domain show their advantage over the methods supported ambiguity fixing, e.g., far fewer epochs taken for obtaining the precise solution and proof against to cycle slips. However, the drawbacks of low computation efficiency and also the existence of multi-peak candidates restrict these methods to be utilized in modern GNSS tracking techniques. The novel tracking approach derived in this paper is based on the Segmented Simulated Annealing Modified Ambiguity Function Approach (SSA-MAFA) and the Relative Motion Tracking Method (RMTM). It focuses on reducing the computation time, which is the main drawback of the traditional Ambiguity Function Method (AFM) and giving out the precise relative tracking result efficiently without solving the integer ambiguity fixing problem. We use the SSA-MAFA search method to reduce the computation time, the Kernel Density Estimation (KDE) method to filter out the false peak candidates, and the RMTM method to obtain the precise relative motion vector between two adjacent epochs. Both static and kinematic experiments were carried out to evaluate the performance of the new approach. The static test shows that the RMTM method can give out a millimeter level of accuracy relative motion solution. The kinematic experiment shows that the precise tracking result can be obtained after handling only two epochs of data. Meanwhile, the tracking result of the proposed approach can be a centimeter-level of accuracy, even if the prior position is several meters far from the referenced value.

INDEX TERMS Ambiguity-free, MAFA, relative GNSS tracking, segmented simulate annealing.

I. INTRODUCTION

Nowadays, the need for obtaining a precise relative position is becoming more popular, especially in outdoor applications such as collision avoidance, self-driving cars, land surveying, structural health monitoring, and accurate agriculture [1].

One of the most precise measurements for relative positioning is the carrier phase [2]. However, the integer ambiguity problem should be solved before giving out an accurate fix solution [3]. There are two categories of algorithms that solve the integer ambiguity problem. The first category of algorithms search for the final solution in the ambiguity domain, and the second category of algorithms is seeking in the position domain.

The associate editor coordinating the review of this manuscript and approving it for publication was Shih-Wei Lin VOLUME 8, 2020

.

Among the first category algorithms, well-known examples are integer least squares (ILS), integer bootstrapping (IB), and integer rounding (IR) [4]. The least-squares ambiguity decorrelation adjustment (LAMBDA) is the optimal solution widely used nowadays owing to its computation efficiency based on the decorrelation between estimated ambiguities [5]–[7]. The RTKLIB was a representative open-source program to achieve the Real-Time Kinematic (RTK) techniques by using the LAMBDA method [8]. In [9], a path following an approach based on the LAMBDA method was proposed. It used the Extended Kalman Filter to give out the initial point so that the LAMBDA method can give out the precise ‘‘fix’’ solution. In [10], an RTK-based online structural monitoring system has been implemented for super-high building’s dynamic monitoring. In [11]–[13], the authors used a combination of RTK, and accelerometer to study

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W. Yang et al.: Novel Precise GNSS Tracking Method Without Solving the Ambiguity Problem

the GNSS-based technologies for bridge deformation monitoring. The result shows that the method can give out the accurate motion of the bridge. However, there are still some limitations of RTK-based dynamic tracking methods [14].

One drawback of the RTK technique is the requirement of an accurate approximate point. Without this, the LAMBDA method can not solve the ambiguities problem or fix the wrong ambiguity values, which leads to a ‘‘float’’ solution with only meter-level accuracy. Meanwhile, the LAMBDA method is not immune to cycle slips, which is a phenomenon of losing lock of satellites and failing to return a correct carrier phase observation [15]. When cycle slips happen, the LAMBDA method has to ignore the measurement from the satellites that have cycle slips and spend far too long to get a fixed ambiguity value [16].

The second category of algorithms that search in the position domain shows their advantages over the LAMBDA method, such as immune to the cycle slips, far few epochs taken for obtaining the precise solution, and no precise initial position is required. The representative algorithm is the Ambiguity Function Method (AFM) [17]. However, the drawback of low computation efficiency of this method makes it abandoned for use in the modern GNSS localization techniques. In [18], a novel method that significantly shortens the computation time of AFM was proposed. This method takes advantage of optimal dual-frequency observable combinations. However, more noise is introduced into the combinations data and yields a less accurate solution. In [19], an improved particle swarm optimization (IPSO) was used to decrease the computation time of AFM. However, a precise initial point and small amplitude variations should be guaranteed. In [20], the author presented a meaningful Modified Ambiguity Function Approach (MAFA). The result shows that the condition of parameter ‘‘integerness’’ is guaranteed even without the stage of integer search. However, a high accuracy approximate position is needed. Further researches on MAFA can be found in [21]–[23], the integer decorrelation procedure, and the solution to minimize the size of the search cube is used to improve the computation efficiency and accuracy. In [24], the authors gave out the necessary condition for obtaining the correct solution by utilizing the MAFA method.

That is, the prior position has to be inside the center of the correct Voronoi cell. In [25], the authors presented a novel search method based on the MAFA, and Segmented Simulate Annealing Method called SSA-MAFA. It can give out the precise position even if the initial coordinates are meters away from the actual position. The result also shows its computation efficiency over traditional AFM. However, the authors only conducted static experiments owing to the requirement of the accuracy relative motion of receivers between two adjacent epochs. Some other researches can be found in [26]–[29]. These methods construct new observations and eliminate the integer ambiguity item in the equations. Thus the tracking result can be given out without solving the ambiguity fixing problem. However, these methods are only a decimeter or meter level of accuracy, which is less accurate 118006

than the solutions by using the AFM class methods. Above all the AFM class methods, the experiments were conducted in the static scene. A few research have focused on the precise tracking in the dynamic scene by using the AFM class methods, even though these methods have many advantages, uniquely the immune to cycle slips. The only related research is presented in [30]. The authors used the AFM method and Regtrack approach to give out the relative tracking result.

However, low computation efficiency is still a big challenge, which is often the case in traditional AFM methods.

In the present paper, a novel ambiguity free method for precise tracking is proposed. The novel method is based on the Segmented Simulated Annealing Modified Ambiguity Function Approach (SSA-MAFA) and the Relative Motion Tracking Method (RMTM). The SSA-MAFA search method is used to reduce the computation time while the Kernel Density Estimation (KDE) method is used to filter out the false peak candidates by utilizing multi-epochs of search results. The RMTM method is used to obtain the precise relative motion vector of two receivers between two adjacent epochs, which is crucial for the KDE method.

The remainder of the paper is organized as follows—in Section II, the proposed algorithm is described in detail.

Experimental and analyses are presented in Section III. The paper ends with conclusion in Section IV. II. METHODOLOGY

The new approach derived in this paper is based on the SSA-MAFA search approach and the RMTM tracking motion method. It focuses on reducing the computation time and giving out the precise relative tracking result efficiently without solving the integer ambiguity fixing problem. The whole procedure is shown in Figure (1).

FIGURE 1. The whole procedure of the relative GNSS tracking approach.

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W. Yang et al.: Novel Precise GNSS Tracking Method Without Solving the Ambiguity Problem

A. RELATIVE MOTION TRACKING METHOD

The carrier phase is the most precise measurement available.

However, the integer ambiguities of carrier phase cycles should be determined, namely fixing the ambiguities problem. Fortunately, the integer ambiguity can be a constant number between two adjacent epochs of observations if there is no Cycle Slip (CS) happens. CS is a phenomenon of losing lock of satellites and failing to return a correct carrier phase observation. Double differential model between two adjacent epochs t1 and t2 (DDt1,2 ) can be formed to eliminate the integer ambiguity item:

can give out the relative motion vector 1blrb (t1 , t2 ) between receivers through time, shown as following, 1blrb = âˆ’Îť(BT WB)−1 BT W δ with 1 1 1 ∂∇1Ď rb ∂∇1Ď rb ∂∇1Ď rb  ∂x ∂y ∂z  2 2  ∂∇1Ď 2 ∂∇1Ď rb ∂∇1Ď rb rb   ∂y ∂z B =  ∂x  .. .. ..  . . .  n n n  ∂∇1Ď âˆ‚âˆ‡1Ď rb ∂∇1Ď rb rb ∂x ∂y ∂z   1 − ∇1Ď 1 + C∇1t ∇1φrb rb ďŁŻâˆ‡1φ 2 − ∇1Ď 2 + C∇1t  rb rb   δ=  ..   .



i i i ∇1φrb = 1φrb (t2 ) − 1φrb (t1 ) i = ∇1Ď rb + C∇1trb +

(1)

with i (ti ) 1φrb

=

i i 1Ď rb (ti ) + C1trb (ti ) + Îť1Nrb

+

(2)

where 1, single-differencing operation; ∇1, double differential operation accross receivers and time; P, pseudorange measurements (in meters); Ď geometrical range between receiver and satellite (in meters); C, speed of light; t, receiver clock biase; , multipath and other unmodel error sources (in meters); φ, carrier phase measurements (in meters); N , integer ambiguity (in cycles).

Considering that the satellites are far away from both receivers, the unit direction vectors (from receivers toward a satellite) can be assumed to be identical [31], shown in i can be replaced by: Figure (2). Thus 1Ď rb i 1Ď rb = blrb ¡ lĚ‚i

(3)

where blrb , true relative position between receiver r and b (called baseline vector); lĚ‚i , line of sight unit-vector of satellite i, then Equation (1) can be rewritten as: i ∇1φrb = blrb (t2 ) ¡ lĚ‚i (t2 ) − blrb (t1 ) ¡ lĚ‚i (t1 ) + C∇1trb

= 1blrb (t1 , t2 ) ¡ lĚ‚i + C∇1trb

(4)

where the error item is removed from Equation (4), as it is proved that the nominal accuracy of carrier phase measurement is in the granularity of a few milimeter [32]. With utilizing the weighted least squares adjustment (WLSA) algorithm, the double differential model across receivers and time VOLUME 8, 2020

 1   1   ..  .   1

(6)

(7)

n − ∇1Ď n + C∇1t ∇1φrb rb

where W, the weight matrix, it can be formed as, W = P−1

(8)

with P = Rt1 + Rt2  2 Ďƒ1  Ďƒ22  R= 

(9)  ..

. Ďƒn2

   

(10)

where Ďƒn , the standard deviation of phase-range measurement error of the n satellites. The following formulas can calculate it: Ďƒn2 = 2(a2 + b2 /sin2 El n + c2 ) + d 2

i and bl .

FIGURE 2. Illustration of relationship between 1Ď rb rb

(5)

(11)

where a, b, c, the error factor of carrier phase measurement (in meters); El, the elevation angels of satellite n; d, satellite clock stability(sec/sec).

After obtaining the relative motion solution, the validation procedure should be carried out. The procedure checks whether the number of satellites without cycle slips is down below four or not, and confirms that the solution of 1blrb does not exceed the relative motion threshold value. The Algorithm of the Relative Motion Tracking Method (RMTM) is shown in Algorithm (1). After then, a tracking algorithm can be obtained by using a simple dead reckoning method, e.g., adding the current relative motion solution to the last relative position estimate, without solving the ambiguity fixing problem.

B. SSA-MAFA SEARCH METHOD OVERVIEW

By utilizing the double time differential model, the precise relative motion vector can be obtained. However, it is an open issue for obtaining the very first tracking position of the dead reckoning method in order to achieve a precise relative baseline tracking. In our previous research [25], 118007

W. Yang et al.: Novel Precise GNSS Tracking Method Without Solving the Ambiguity Problem

Algorithm 1 The procedure of the Relative Motion Tracking Method (RMTM) Input: satellite coordinates and carrier phase observations, both of two adajecent epochs ti and ti−1 ; carrier phase wavelength λ Output: the relative motion tracking result 1 calculate initial double differential clock bias error (∇1te ) across receivers and time; ◦ 2 for satellites in line of sight (elevation ≥ 15 ) do 3 if satellites appears in two adjacent epochs then 4 if no cycle slips happens then 5 calculate double differential carrier phase observations (∇1φ) across receivers and time ; 6

7

8 9 10

11 12 13 14

calculate design matrix B based on current satallite coordinates; p p calculate geometrical range ρrc ,ρr ,ρbc ,ρb based on current and previous satellite coordinates for both receivers; calculate wight matrix W based on Equation (8); calculate misclosure vector δ based on Equation (7); obtain relative motion vector 1blrb by solving the least squre adjustment problem with utilizing Equation (5); if validate 1blrb then update relative motion vector with current 1blrb ; return the relative motion vector 1blrb ; final ;

a novel search method based on the Modified Ambiguity Function Approach (MAFA) and the Segmented Simulated Annealing (SSA) was proposed to give out the precisely static position very efficiently. However, the SSA-MAFA method was only tested in the static experiment owing to the requirement of the precise relative motion solution between two adjacent epochs. We then introduce this method to the tracking algorithm to achieve a complete standalone precise localization system without solving the fixing ambiguity problem. For convenience, the overview of the SSA-MAFA method is introduced in this subsection.

The Segmented Simulate Annealing Modified Ambiguity Function Approach (SSA-MAFA) is based on the Modified Ambiguity Function Approach (MAFA). We use the Double Differential (DD) observation model to reduce or eliminate the error in the positioning system, shown as: 8=

1 ρ(Xc ) + a + e λ

(12)

where where 8, the carrier phase double differential measurement(in cycles); λ, the wave length of carrier phase (in meters); e, residual values in cycles (noise of the measurement); ρ(Xc ), double differential geometrical range with the true receiver position; a, double differential integer ambiguity in cycles.

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We then use the term ρ to represent the ρ(Xc ) in subsequent equations for simplicity, Equation (12) can be rewritten as: 1 ρ) − a (13) λ Considering the residual values e can be no more than half a cycle [33], and the ambiguity parameter a has an integer nature, then Equation (13) can be rewritten in the following form: 1 1 e = (8 − ρ) − (8 − ρ − e) λ λ 1 1 = (8 − ρ) − round(8 − ρ) (14) λ λ where round(.) is the rounding function to the nearest integer.

In [24], Equation (14) can be updated by using a differentiable and continuous function: e = (8 −

e = s− round(s) 1   arcsin[sin(πs)] s ∈ {s : cos(πs) ≥ 0} π = 1  − arcsin[sin(πs)] s ∈ {s : cos(πs) < 0} π 1 s = 8− ρ (15) λ The system equations can be expanded in the Taylor Form shown as: 1 e = 1 − Bb (16) λ with   δ1 δ2    1 =  . , (17)  ..  δn

1 1 δn = (8n − ρn (x0 )) − round(8n − ρn (x0 )) λ λ  ∂ρ ∂ρ1 ∂ρ1  1  ∂x ∂y ∂z   ∂ρ ∂ρ2 ∂ρ2   2      ∂x ∂y ∂z B=  . . .  .  . . . .   .  ∂ρn ∂ρn ∂ρn  ∂x ∂y ∂z

(18)

(19)

where e, the error vector (n × 1); b, the value of the increment to the approximate coordinates vector x0 ; B, the design matrix (n×3); 1, misclosure vector (n×1); x0 , the prior coordinates; ρ(x0 ), double differential geometrical range with utilizing a prior position and the satellite coordinates; n, the number of double differential observations.

Then the adjustment problem can be changed to: arg min(MAFV = eT We) b

(20)

with W represents the weight matrix, it can be formed as: W = (DRDT )−1

(21) VOLUME 8, 2020

W. Yang et al.: Novel Precise GNSS Tracking Method Without Solving the Ambiguity Problem

with  1  1 1 D = . λ  ..

−1 0 .. .

1 0  2 σ1  σ22  R= 

0 −1 .. .

··· ··· .. .

0

··· 

..

. σn2

   

0 0 .. .



    −1 n×n

(22)

(23)

where σn , standard deviation of phase-range measurement error of the n satellites. It can be calculated by using Equation(11).

The solution of the adjustment problem is the following parameter vector: b = −λ(BT WB)−1 BT W 1

(24)

The result can satisfy the mixed integer-real LS (MIRLS) problem even though no ambiguity parameter is solved.

Therefore, the solution is not only impossible to fix the wrong ambiguities but also susceptible to cycle slips, allowing a precise baseline determination. However, the search area becomes more abundant, as the accuracy of the prior position reduces. In this situation, if the condition(25) is satisfied by a group of observations, then the false global optimum appears.

8n −

1 1 ρn (xw )) − round(8n − ρn (xw )) λ λ 1 1 < (8n − ρn (xc )) − round(8n − ρn (xc )) (25) λ λ

where xw , the false Voronoi cell center; xc , the true Voronoi cell center. The Voronoi cell is a pull-in region, which features a single local optimal value among the points within the region equivalent to a set of integer ambiguities [34].

The Segment Simulated Annealing method (SSA) and Kernel Density Estimation(KDE) is used to eliminate the wrong candidate points and reduce the computation time. (refer to [25] for details on the Traditional Simulated Annealing (SA) method). The probability of accepting the worse solution in the SSA method is as follows. ( 1 new state is better P(1E) = (26) −1E e T new state is worse where 1E, the 1/MAFV value difference between the known state and new state, where the new state is obtained by adding the Gaussian random value to the known state. After the state transition, jump amplitude T can be updated by: Tk = d · Tk−1

(27)

It is proved in [22] that the optimal distance between two search points in the search grid can be half of the wavelength.

Thus, the search layers of the SSA method can be obtained by dividing the search length in the z direction with utilizing, VOLUME 8, 2020

e.g., 8cm as the split distance. The whole SSA search procedure is shown in Algorithm (2), and the SSA search procedure illustration on a specific search layer is shown in Figure (3). By dividing the search length in the z direction, the search region can be divided into several search layers, and the optimal distance between the closest search layers could be half of the wavelength [22]. As the illustration shown in Figure (3), the x and y value are maintained the same with the prior ones, and the z coordinate is settled to the particular search layer. The refined coordinates are then given out by solving Equation (24). Next, the refined coordinates are given out by using Equation (20), and Equation (26) is used to check if the current point is the optimum candidate and decide to make it the new search point of the next iteration. A new search point is obtained by adding the Gaussian random value to the determined search point and repeat the above procedure. When the final standard jump amplitude threshold is met, all the candidate solutions that appeared should be saved. In order to increase the opportunity of obtaining the exact candidate point, the probability of accepting the worse solution can be settled to a more considerable value, e.g., 0.3.

FIGURE 3. Illustration of SSA search procedure on a particular search layer.

C. KERNEL DENSITY ESTIMATION (KDE) METHOD FOR ELIMINATING THE FALSE OPTIMUM CANDIDATES

The purpose of the SSA search method is to reduce the search time as well as obtaining the most significant number of optimum candidate points in order to eliminate the influence of the existence of the false global optimum candidate [25]. Thus, there will be a large number of candidate points by using the SSA method dealing with one epoch of measurement data. As the accuracy of the MAFA method is the millimeter level for the correct solution, we can use the Kernel Density Estimation (KDE) method to filter out the false candidate points, the equation of KDE method is as follows, n

b fh (x) =

1 X x − xi K( ), nh h

(28)

i=1

where K , the non-negative kernel function; h, the bandwidth; n, total number of data samples.

One of the critical steps for using the KDE method is constructing the relative motion constraint of the rover receiver between any two adjacent epochs. The value of the bandwidth h depends directly on the accuracy of the relative 118009

W. Yang et al.: Novel Precise GNSS Tracking Method Without Solving the Ambiguity Problem

Algorithm 2 The Procedure of the SSA Search Algorithm Input: an approximate position Xp , covariance Qxp , decreasing coefficient d, final standard jump amplitude Td , number of the inner loop NIL Output: all the global optimum candidate coordinates in one epoch 1 calculate the search region, the number of search layer n based on Qx0 ; 2 calculate the initial standard jump amplitude T0 (advisably 0.1 to 0.5 of search region width); 3 for each search layer n do 4 obtain the initial search point X0 by setting the z coordinate of the prior position to the corresponding search layer; 5 while T0 ≤ Td do 6 for iteration ≤ NIL do 7 calculate the refined coordinates of X0 by solving Equation (24); 8 calculate the function value MAFV of the refined coordinates based on Equation (20); 9 if 1/MAFV > global optimum then 10 update the value of global optimum with 1/MAFV ; 11 update the coordinates of global optimum with the refined coordinates; 12 save the previous global optimum candidate point; 13 set the refined coordinates (with z value resettled back to the corresponding search layer) as the initial point Xi of next iteration ; else accept the refined coordinates (with z value resettled back to the corresponding search layer) as the initial point Xi of next iteration with a certain probability P;

14 15

update the initial search point X0 by adding Gaussian random value to Xi ;

16

update T0 with Equation (27)

17

save the current global optimum candidate point; resettle T0 to its initial value;

18 19 20

21

return the coordinates of all the global optimum candidate points; final ;

motion solution since the MAFA method can give out a very assuring accuracy for the correct point. As for the static situation or some situations of minor movement, the relative movement of receivers is none or a minimal value, the correct candidate points of different epochs of data are almost the same one. Thus h can be a very tiny number, e.g., 1 cm.

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In the dynamic situation, the KDE method should use the current candidate points, and the previous ones plus the relative motion vector between two adjacent epochs, as shown in Figure (4). Thus, the value of the bandwidth h should be more considerable, e.g., 5 to 10 cm, owing to the less accuracy of the relative motion solution.

FIGURE 4. Illustration of match status in the dymatic situation by using the SSA-MAFA search method and the KDE method.

We use the triangular as the kernel function and choose the value of h to be a tiny number, e.g., 5 cm, and make sure that only the true optimum candidate in different epochs (current candidates and previous candidates plus the relative motion vector between the two adjacent epochs) can obtain a more considerable Probability density function (PDF) value, as shown in Figure.4.

III. EXPERIMENTS AND ANALYSES

The static and kinematic experiments were carried out to verify the novel method. In the static experiment, we tested the accuracy of the relative motion solution by using the RMTM method. Next, the kinematic experiment was conducted to test the efficiency of obtaining precise coordinates by using the proposed method. In this test, we used the relative motion vector, and the KDE method to filter out the false candidate points, which exist in the search result with utilizing the SSA-MAFA method. The tracking accuracy of the proposed method is tested referenced to the ground truth, which is obtained by using the RTKLIB software (the least-squares ambiguity decorrelation adjustment (LAMBDA) method was used to calculate the ambiguity values), the filter type parameter was settled to combined mode. We also analyzed the computation efficiency of the newly proposed method, and the optimal parameters are given out based on the analysis data. All the data were post calculated by using a desktop computer with an Intel Core i7-6700T CPU, 16GB memory running Windows 10 operating system.

A. ACCURACY OF RELATIVE MOTION SOLUTION 1) THE STATIC EXPERIMENT

We carried out the static test at a parking lot at Queen’s University in Kingston, Ontario. We use two u-Blox Neo-M8T single frequency receivers as the rover receiver and base station, respectively. Both receivers were placed at the ground and connected to a laptop computer via USB, with a sample rate settled at 5Hz, as shown in Figure (5). The experiment information is listed in Table (1), the sky-plot and GDOP value during the sampling time are shown in Figure (6).

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W. Yang et al.: Novel Precise GNSS Tracking Method Without Solving the Ambiguity Problem

FIGURE 5. The static experiment setup.

TABLE 1. The experimental information of the static experiment.

FIGURE 7. Error result of relative motion solution (a) Error in East direction. (b) Error in North direction. (c) q Error in Up direction. (d) Error of relative motion solution, where Dist =

errE2 + errN2 + errU2 .

the ocean near Sussex England. The experimental information is listed in Table (2).

TABLE 2. The experimental information of the dynamic experiment.

In the test, one of the moving receivers can be treated as a static reference station (even if it is not). It causes all of the relative motion to be attributed to the second receiver.

The distance between the two receivers is fixed (2.82m), but the orientation between them varies as the platform is moving. Therefore, the accumulation by using the dead reckoning with the relative motion solution should be a perfect circle, and the radius should equal to the distance between the two receivers. The experiment result is shown in Figure (8).

FIGURE 6. GDOP and Skyplot during the data sampling periods (a) GDOP value and number of satellites. (b) The skyplot of visible satellites, the gray trace means the elevation of the satellite is less than 15â&#x2014;Ś .

As both receivers were holding static, the relative motion between two receivers should be none. Thus, any value other than zero in the relative motion solution should be considered as an error. The error result is shown in Figure (7).

The relative motion vector solution all passes the validation procedure owing to there are no cycle slips between any two adjacent epochs of measurements in this test, as shown in Figure (6). The error result shows that the solution of the relative motion vector can be a millimeter level accuracy for each epoch, which plays a very significant role in using the KDE method to eliminate the false candidate points.

2) THE DYNAMIC EXPERIMENT

In the dynamic experiment, two Emlid Reach NEO-M8T receivers were used and mounted at each end of a kayak (2.82m apart) running a 5 Hz sample rate while kayaking in VOLUME 8, 2020

FIGURE 8. The tracking result by utilizing the dead reckoning method only with the relative motion solution and the corresponding horizontal error. (a) The tracking result for one receiver towards another (both receivers are moving). (b) The horizontal error result, referring to the q distance between two receivers: 2.82 m, where HE = errN2 + errE2 .

It is shown that the LAMBDA method can be the most accurate solution for the tracking solution. For the RMTM method, the error result of each epoch shown in Figure (7), has a determinative impact effect on the final accumulation tracking result. The error of the tracking result by using the RMTM method can be all under 10 cm in the most time.

Considering the tracking solution used a dead reckoning method only, the precision of the relative motion solution 118011

W. Yang et al.: Novel Precise GNSS Tracking Method Without Solving the Ambiguity Problem

TABLE 3. Parameters of the SSA-MAFA method for obtaining the optimum candidate points.

between two adjacent epochs in the dynamic experiment can be said still very accurate.

B. EFFICIENCY OF OBTAINING THE CORRECT OPTIMUM POINT

This test was carried out to prove the efficiency of obtaining the correct optimum point by using the SSA-MAFA method and relative motion solutions. It was conducted on 2019/11/25 21:25:05 (GPS time) at the University of Calgary, Alberta, Canada. We used an uBlox-F9 receiver as the rover receiver and mounted it on top of a car, as shown in Figure (9).

A Trimble-R9 receiver was taken as the base station placed at a fixed position where the receiver could obtain an excellent quality of satellite signals. The sample rate of both receivers was settled to 1 Hz for GPS and GLONASS L1 frequency signals. The precise position of the base station in the ECEF frame was (-1641890.0811, -3664879.3446, 4939969.4285).

of which are the most significant ones. As the SSA-MAFA method is a multi-layer search method, the search layers near the center of the correct Voronoi cell could obtain the same accuracy solution. Thus there are the same accuracy candidates in the search result. Meanwhile, it shows that the novel method can obtain the accuracy solution even in the case of a single epoch of data, which means the novel method can be immune to cycle slips. On the other hand, there are still some false candidate points that can yield similar 1/MAFV value with the correct point, as shown in Figure (10).c.

Nevertheless, the false candidate point may give out a more significant value than the correct one owing to the existence of false global optimum point [25]. The relative motion solution and the KDE method can be used to filter out the rest false candidate points and give out the correct solution, that is calculating the optimal candidates for each epoch data by using the SSA-MAFA method and calculating the new position for the optimum candidates of the previous epoch by using the relative motion solution between two adjacent epochs. The result should match the current optimum candidates, or the candidate is the false optimum one and should be filtered out. The value of 1/MAFV mainly depends on the position solution as well as the unmodeled noise in the measurement data. If the coordinates are precisely the actual value, the 1/MAFV value should be significant, e.g., 100. However, even if there is a millimeter error in the

FIGURE 9. The experiment setup. (a) Experimental car. (b) A uBlox-F9 receiver as the rover receiver, was mounted on top of the car.

The Extended Kalman Filter (EKF) method was used to obtain the initial point for the SSA-MAFA method of each epoch, the details of the used parameters of the SSA-MAFA method are shown in Table (3), and the error of the initial points in each epoch are shown in Table (4). It shows that the accuracy of the initial points by using the EKF method can be decimeter level or meter level.

TABLE 4. The initial point error in East, North, and Up direction, where q Dist = erre2 + errn2 + erru2 .

The search results by handling four epochs of data are shown in Figure (10). It shows that the search result can obtain the correct optimum candidate point and the 1/MAFV value 118012

FIGURE 10. 1/MAFV of optimum candidate points by using the SSA-MAFA method in different epochs (a) Epoch 1. (b) Epoch 2. (c) Epoch 3. (d) Epoch 4. Where 1/MAFV is the reciprocal of Equation (20) and the value in the bracket is the error of the candidate point referenced to the ground truth position.

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position solution, the 1/MAFV value may drop to less than 1.

Fortunately, the incorrect coordinates that far from the real value usually can not obtain the 1/MAFV value larger than 0.3, as shown in Figure (10). In order to reduce the number of false candidate points, the threshold 1/MAFV value of obtaining the candidate points can be settled to 0.3.

The search result of each epoch and the relative motion solution in the 3D scene is shown in Figure (11).

FIGURE 12. Probability Density Function Value (PDF) of optimum candidate points in different epochs (a) PDF value of candidate points in Epoch 2. (b) PDF value of candidate points in Epoch 3.

FIGURE 11. Match status by using the SSA-MAFA method and relative motion solution, where the dot is the candidate points, the circle is the bandwidth of the KDE method, and the value in the bracket is the error of the correct solution referenced to the ground truth.

The solution by using the SSA-MAFA method and the relative motion solution can be very assuring. The result shows that there are 13 optimum candidates in the first epoch (some are actually close to each other), and only two possible optimum candidates left in the second epoch. In the third epoch, the correct solution is given out, and the error component in East, North and Up direction is 0.000 m, -0.007 m, and 0.009 m, respectively referenced to the ground truth.

As shown in Figure (12), after handling only two epochs of data, the correct candidate point can obtain a more significant Probability Density Function Value (PDF) than the others.

The PDF value of the correct point grows while more epochs of data are handled, even if the initial points are meters away from the ground truth. It should be noticed here, the error of the relative motion solution may introduce a centimeter-level of error in the result, such as the relative motion vector solution of Epoch 3 to Epoch 4, shown in Figure (11). Thus, the bandwidth of the KDE method should be settled in no small value, e.g., 5 cm, in order to yield a more significant PDF value for the correct optimum point.

C. TRACKING ACCURACY OF THE PROPOSED APPROACH

The setup of the tracking accuracy test was the same as the experiment (III-B). In this test, we first use the SSA-MAFA method and RMTM approach to obtain the precise initial point. Then we use the relative motion solution and the dead reckoning method to give out the complete tracking VOLUME 8, 2020

result without solving the integer ambiguity fixing problem.

The ground truth was post obtained by using the RTKLIB software with filter type settled to combined mode (forward and backward), and the LAMBDA method was used to calculate the ambiguity values. The tracking result is shown in Figure (13).

FIGURE 13. Tracking result by using the SSA-MAFA search method and Relative Motion Tracking Method (RMTM).

It shows that the RMTM method can give out an accuracy relative trajectory rather than the precise absolute positioning result. The absolutely tracking result of the new proposed method is more accuracy than the RMTM method owing to the obtaining of the initial tracking point.

As shown in Figure (14), the error of the proposed method becomes significant in both the horizontal and up-direction due to the accumulation error of each epoch calculation using the RMTM method. Although the error in the up direction for two methods are in different accuracy level, the shape of the error curve is similar, which shows that the initial inaccuracy point has little effect in calculating the relative motion solution in the up direction. However, it is a different 118013

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TABLE 5. Search time of SSA-MAFA (in seconds) for one search layer, where the search domain in xy direction is ±3 m; d , decreasing coefficient of the standard jump amplitude; NIL, number of inner loops for the search procedure with one standard jump amplitude.

FIGURE 15. The computation efficiency of the Reletive Motion Tracking Mehtod (RMTM).

FIGURE 14. The tracking error result. (a) The up direction error result. (b) The horizontal error result referenced to the ground truth, where the ordinate of the proposed method is on theqleft, and the RMTM only method is on the right. Horizontal Error =

errN2 + errE2 .

story for the horizontal error result. Thus, the SSA-MAFA method can be used to reinitiate the precise tracking point when the failure of calculating the relative motion solution, in order to give out the continuously precise tracking result.

D. THE COMPUTATION EFFICIENCY

The computation time of the new approach can be divided into two parts: the computation time of the Relative Motion Tracking method (RMTM) and the search time of the SSA-MAFA method. As is shown in Figure (15), the RMTM method is very efficient for giving out the relative motion vector between any two adjacent epochs of data, and it is less efficient than the LAMBDA method, especially when the LAMBDA method fixes and holds the ambiguity item.

The search time of one particular search layer by using the SSA-MAFA method based on different parameters is given out in Table (5).

It shows that the search time of the SSA-MAFA method closely links to the parameter NIL and d. With a small value 118014

of these parameters, the search time can be shorter. However, when the value of the parameter d is settled down to 0.8, there is a chance for the SSA-MAFA method to fail to obtain the correct candidate point in the search result. Therefore, the parameters d = 0.9 and NIL = 50 can be used as the optimal parameters considering both efficiency and accuracy.

The whole search time is still a big concern for the new search method, which mainly depends on the number of search layers in z direction. In other words, a more accurate initial z coordinate means more computation efficiency.

Taking for an example, when the search width in z direction is 1.5 meters and making use of the other parameters in Table.3, the search time for one epoch of data by using the novel method is about 1.4 seconds. For comparison, we give out the computation efficiency of the traditional AFM method under the same search domain in xyz direction (±3 m in xy direction and ±0.75 m in z direction). Considering the search accuracy of the SSA-MAFA method (millimeter level of accuracy), the search step length is settled to 1cm. The calculated time for one epoch of data by using the AFM method is about 2 minutes 15 seconds. The computation time of the novel method has a considerable improvement over the traditional AFM method. In the meantime, the search time should be understood as only a conservative result from the preliminary test as the code optimization should reduce the search time significantly owing to the non-interference of the search on different layers, and it should be our next research goal.

IV. CONCLUSION

The carrier phase is the most precise measurement available for precise GNSS positioning, especially for outdoor tracking applications, such as land surveying, structure deformation monitoring, and precision agriculture. The solution of VOLUME 8, 2020

W. Yang et al.: Novel Precise GNSS Tracking Method Without Solving the Ambiguity Problem

carrier phase ambiguity is the first place of consideration in these applications. Methods of searching within the position domain show their advantage over the methods supported ambiguity fixing, e.g., far fewer epochs taken for obtaining the precise solution and proof against to cycle slips.

However, these methods have a drawback of low computation efficiency. Meanwhile, there may be several maxima points needed to be filtered out in order to identify the correct point.

Therefore a few research have a focus on using the AFM class method for precise GNSS tracking.

In the present paper, a novel approach is derived focused on giving out the precise relative tracking result without solving the integer ambiguity fixing problem and reducing the computation time. The proposed approach is based on the Segmented Simulated Annealing Modified Ambiguity Function Approach (SSA-MAFA) and the Relative Motion Tracking Method (RMTM). The SSA-MAFA method is a novel search method based on the least squares adjustment (LSA) algorithm. It uses the suitable constraints in the function model to ensure the integer nature of ambiguities even it does not calculate the full-cycle ambiguities directly.

In the meantime, the Segmented Simulated Annealing (SSA) method solves the low computation efficiency problem, which often the case of traditional AFM class methods. The Kernel Density Estimation (KDE) method is used to eliminate the false candidate points. The RMTM method uses the constant nature of the ambiguity term in two adjacent epochs to eliminate it, and it calculates the relative motion of two receivers by constructing a double differential model across receivers and time. The mathematical models expressed by some formulas are presented. The algorithms of the computational process have been proposed. As shown in the experiment section, the RMTM method can give out a millimeter level of accuracy for the relative motion vector between two adjacent epochs. By utilizing the SSA-MAFA method, the precise tracking point can be obtained after handling two epochs of data even if the initial point of each epoch is meters away from the ground truth. After taking the first initial tracking point, the tracking result can be a centimeter-level of accuracy in the kinematic scene by using the SSA-MAFA method and the RMTM method.

The computation time of the RMTM method can be very efficient compared with the least-squares ambiguity decorrelation adjustment (LAMBDA) method. However, the computation efficiency of the novel tracking method is still hard to be said a real-time one owing to the search time of the SSA-MAFA method depends on the number of search layers in z direction. The SSA-MAFA method indeed shows its advances in reducing the computation time over the traditional AFM class method, but only with an accurate z coordinate of the initial point can the search time reach a real-time level. As the SSA-MAFA method is a multi-layer mutual independent search algorithm, the total search time can then be decreased by using the code program optimization to achieve a parallel search. On the other hand, methods for obtaining a more accurate z coordinate of the initial point VOLUME 8, 2020

should be a more efficient way to reduce the total computation time of the proposed method, which can be the next research goal.

ACKNOWLEDGMENT

The data of Experiment(III-A2) utilized in this paper are derived from open datasets contributed by Tim Everett, blogger of the rtklibexplorer website: http://rtkexplorer.com/ downloads/gps-data REFERENCES [1] R. Banalagay, ‘‘Ph.D. Forum abstract: Accurate and robust low-cost GNSS localization,’’ in Proc. 15th ACM/IEEE Int. Conf. Inf. Process. Sensor Netw. (IPSN), Apr. 2016, pp. 1–2.

[2] X. He, X. Zhang, L. Tang, and W. Liu, ‘‘Instantaneous real-time kinematic decimeter-level positioning with BeiDou triple-frequency signals over medium baselines,’’ Sensors, vol. 16, no. 1, p. 1, Dec. 2015.

[3] X.-W. Chang, X. Yang, and T. Zhou, ‘‘MLAMBDA: A modified LAMBDA method for integer least-squares estimation,’’ J. Geodesy, vol. 79, no. 9, pp. 552–565, Dec. 2005.

[4] K. Nowel, S. Cellmer, and D. Kwaśniak, ‘‘Mixed integer–real least squares estimation for precise GNSS positioning using a modified ambiguity function approach,’’ GPS Solutions, vol. 22, no. 1, p. 31, Jan. 2018.

[5] P. Joosten and C. Tiberius, ‘‘LAMBDA: FAQs,’’ GPS Solutions, vol. 6, nos. 1–2, pp. 109–114, Nov. 2002.

[6] P. J. G. Teunissen, ‘‘A new method for fast carrier phase ambiguity estimation,’’ in Proc. IEEE Position, Location Navigat. Symp. PLANS, Apr. 1994, pp. 562–573.

[7] P. Teunissen, P. De Jonge, and C. Tiberius, ‘‘The LAMBDA method for fast GPS surveying,’’ in Proc. Int. Symp. GPS Technol. Appl., Bucharest, Romania, Sep. 1995, pp. 1–8.

[8] T. Takasu and A. Yasuda, ‘‘Development of the low-cost RTK-GPS receiver with an open source program package RTKLIB,’’ in Proc. Int. Symp. GPS/GNSS, Nov. 2009, pp. 4–6.

[9] W. Travis and D. M. Bevly, ‘‘Trajectory duplication using relative position information for automated ground vehicle convoys,’’ in Proc. IEEE/ION Position, Location Navigat. Symp., 2008, pp. 1022–1032.

[10] C. Ogaja, C. Rizos, J. Wang, and J. Brownjohn, ‘‘Toward the implementation of on-line structural monitoring using RTK-GPS and analysis of results using the wavelet transform,’’ in Proc. 10th FIG Int. Symp.

Deformation Meas., Mar. 2001, pp. 19–22.

[11] V. Ashkenazi, A. Dodson, T. Moore, and G. Roberts, ‘‘Real time otf gps monitoring of the humber bridge,’’ Surveying World, vol. 4, no. 4, pp. 26–28, 1996.

[12] G. Roberts, X. Meng, and A. Dodson, ‘‘The use of kinematic gps and triaxial accelerometers to monitor the deflections of large bridges,’’ in Proc. 10th Int. Symp. Deformation Meas., FIG, Mar. 2001, pp. 19–22.

[13] G. W. Roberts, E. Cosser, X. Meng, and A. Dodson, ‘‘High frequency deflection monitoring of bridges by GPS,’’ J. Global Positioning Syst., vol. 3, nos. 1–2, pp. 226–231, Dec. 2004.

[14] N. Shen, L. Chen, J. Liu, L. Wang, T. Tao, D. Wu, and R. Chen, ‘‘A review of global navigation satellite system (GNSS)-based dynamic monitoring technologies for structural health monitoring,’’ Remote Sens., vol. 11, no. 9, p. 1001, Apr. 2019.

[15] T. Takasu and A. Yasuda, ‘‘Cycle slip detection and fixing by MEMSIMU/GPS integration for mobile environment RTK-GPS,’’ in Proc. 21st Int. Tech. Meeting Satell. Division Inst. Navigat. (ION GNSS), Savannah, GA, USA: Citeseer, Sep. 2008, pp. 64–71.

[16] D. Chen, S. Ye, C. Xu, W. Jiang, S. Li, P. Xia, and P. Jiang, ‘‘A method for the repair of cycle slip using double-differenced velocity estimation for GNSS RTK positioning,’’ Adv. Space Res., vol. 63, no. 9, pp. 2809–2821, May 2019.

[17] C. C. Counselman and S. A. Gourevitch, ‘‘Miniature interferometer terminals for Earth surveying: Ambiguity and multipath with global positioning system,’’ IEEE Trans. Geosci. Remote Sens., vol. GE-19, no. 4, pp. 244–252, Oct. 1981.

[18] S. Han and C. Rizos, ‘‘Improving the computational efficiency of the ambiguity function algorithm,’’ J. Geodesy, vol. 70, no. 6, pp. 330–341, Mar. 1996.

118015

W. Yang et al.: Novel Precise GNSS Tracking Method Without Solving the Ambiguity Problem

[19] X. Li, G. Huang, Q. Zhang, and Q. Zhao, ‘‘A new GPS/BDS tropospheric delay resolution approach for monitoring deformation in super high-rise buildings,’’ GPS Solutions, vol. 22, no. 3, p. 90, Jul. 2018.

[20] S. Cellmer, P. Wielgosz, and Z. Rzepecka, ‘‘Modified ambiguity function approach for GPS carrier phase positioning,’’ J. Geodesy, vol. 84, no. 4, pp. 267–275, Apr. 2010.

[21] S. Cellmer, ‘‘The real time precise positioning using mafa method,’’ in Proc. Int. Conf. Environ. Eng., Vilnius, Lithuania, vol. 8, May 2011, pp. 1310–1314.

[22] S. Cellmer, K. Nowel, and D. Kwasniak, ‘‘The new search method in precise GNSS positioning,’’ IEEE Trans. Aerosp. Electron. Syst., vol. 54, no. 1, pp. 404–415, Feb. 2018.

[23] K. Nowel, S. Cellmer, and D. Kwasniak, ‘‘A minimum size of the search cube in the MAFA-ILS method,’’ in Proc. 10th Int. Conf. Environ. Eng., Aug. 2017, pp. 1–7.

[24] S. Cellmer, ‘‘A graphic representation of the necessary condition for the MAFA method,’’ IEEE Trans. Geosci. Remote Sens., vol. 50, no. 2, pp. 482–488, Feb. 2012.

[25] W. Yang, Y. Liu, and F. Liu, ‘‘An improved ambiguity-free method for precise GNSS positioning with utilizing single frequency receivers,’’ Sensors, vol. 20, no. 3, p. 856, Feb. 2020.

[26] S. Zhao, Y. Chen, and J. A. Farrell, ‘‘High-precision vehicle navigation in urban environments using an MEM’s IMU and single-frequency GPS receiver,’’ IEEE Trans. Intell. Transp. Syst., vol. 17, no. 10, pp. 2854–2867, Oct. 2016.

[27] T. Wang, Z. Yao, and M. Lu, ‘‘Combined difference square observationbased ambiguity determination for ground-based positioning system,’’ J. Geodesy, vol. 93, no. 10, pp. 1867–1880, Oct. 2019.

[28] T. Wang, Z. Yao, and M. Lu, ‘‘On-the-fly ambiguity resolution based on double-differential square observation,’’ Sensors, vol. 18, no. 8, p. 2495, Aug. 2018.

[29] T. Wang, Z. Yao, and M. Lu, ‘‘On-the-fly ambiguity resolution involving only carrier phase measurements for stand-alone ground-based positioning systems,’’ GPS Solutions, vol. 23, no. 2, p. 36, Apr. 2019.

[30] W. Hedgecock, M. Maroti, A. Ledeczi, P. Volgyesi, and R. Banalagay, ‘‘Accurate real-time relative localization using single-frequency GPS,’’ in Proc. 12th ACM Conf. Embedded Netw. Sensor Syst. SenSys, 2014, pp. 206–220.

[31] W. Hedgecock, M. Maroti, J. Sallai, P. Volgyesi, and A. Ledeczi, ‘‘Highaccuracy differential tracking of low-cost GPS receivers,’’ in Proc. 11th Annu. Int. Conf. Mobile Syst., Appl., Services MobiSys, 2013, pp. 221–234.

[32] G. Blewitt, ‘‘Basics of the GPS technique: Observation equations,’’ in Geodetic Applications of GPS, B. Johnson, Ed. Gothenburg, Sweden: Nordic Geodetic Commission, 1997, pp. 10–54.

[33] C. Mekik and O. Can, ‘‘Multipath effects in RTK GPS and a case study,’’ J. Aeronaut., Astronaut. Aviation, Ser. A, vol. 42, pp. 231–240, Oct. 2010.

[34] A. Hassibi and S. Boyd, ‘‘Integer parameter estimation in linear models with applications to GPS,’’ IEEE Trans. Signal Process., vol. 46, no. 11, pp. 2938–2952, Nov. 1998.

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WENHAO YANG is currently pursuing the Ph.D. degree with Harbin Engineering University (HEU), China. He has studied as a Visiting Student with Queen’s University, Kingston, Canada, for a year. His current research interest includes precise GNSS positioning technology.

YUE LIU is currently pursuing the Ph.D. degree with Harbin Engineering University (HEU), China. She has studied as a Visiting Student with the University of Calgary, AB, Canada, for a year. Her current research interest includes GNSS precise point positioning technology.

FANMING LIU received the Ph.D. degree from Harbin Engineering University, in 2005. He is currently a Professor with Harbin Engineering University. His main research interest includes passive positioning technology.

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