- Email: [email protected]

ScienceDirect Advances in Space Research 62 (2018) 855–865 www.elsevier.com/locate/asr

A dynamic single epoch ambiguity resolution algorithm with straight trajectory constraints Ning Liu a,b,⇑, Qin Zhang a, Yangyang Yang a a

College of Geology Engineering and Geomatics, Chang’an University, Xi’an, China b State Key Laboratory of Geo-information Engineering, Xi’an, China

Received 6 December 2017; received in revised form 22 April 2018; accepted 20 May 2018 Available online 30 May 2018

Abstract The key problem of dynamic single epoch GPS positioning is the rapid decomposition of ambiguity. In some special dynamic positioning, the coordinates of the GPS receiver placed on the rover station often have the constraint condition, if this useful information is taken into account, it will help to determine the ambiguity accurately. For this reason, the straight motion trajectory of GPS receiver is used as the constraint condition for single epoch ambiguity resolution, in this algorithm, the wide lane ambiguities are calculated by using the MW combination, and the corresponding alternative combinations of wide lane ambiguity are established, which are determined according to the minimum residual sum of squares criterion ﬁrstly, it is used to lay the foundation for the subsequent L1 and L2 frequency ambiguity solution. And then the straight trajectory constraint condition and the phase observation equation are used to solve the integer ambiguity N1. When the constraint information of the trajectory is unknown, the space linear equation is established as the constraint condition according to the coordinate solution by wide lane relative positioning to solve the unknown problem. By designing the corresponding experiments, the results of this algorithm are proved to be feasible by using diﬀerent experimental schemes, such as without constraint condition, the constraint condition with known prior information, introducing dynamic constraint condition and so on, to solve the ambiguity, and to compare and analyze the eﬀect of linear constraint condition on single epoch ambiguity resolution. Ó 2018 COSPAR. Published by Elsevier Ltd. All rights reserved.

Keywords: Straight trajectory constraints; Single epoch; Integer ambiguity; MW combination; Minimum residual sum of squares criterion

1. Introduction In GPS dynamic measurement mode, the position of the rover station is changing at all times, and the case where the satellite signal is lost more frequently than static measurement, which brings more challenges to the integer ambiguity solution in phase observation data. While taking into account the real time requirement of dynamic positioning, how to quickly and accurately ﬁxed ambiguity becomes a key problem to realize the high precision

⇑ Corresponding author at: College of Geology Engineering and Geomatics, Chang’an University, Xi’an, Shaanxi 710054, China. E-mail address: [email protected] (N. Liu).

https://doi.org/10.1016/j.asr.2018.05.028 0273-1177/Ó 2018 COSPAR. Published by Elsevier Ltd. All rights reserved.

dynamic positioning (Liu et al., 2014). The single epoch ambiguity resolution refers to only using one epoch GPS observation data to solve the ambiguity parameter without cycle slip detection, when the GPS signal is unlocked, only the position solution of the current epoch is aﬀected, for subsequent epochs, its receiver position can be solved after re-capturing the satellite signal (Assiadi et al., 2014), so a lot of single epoch ambiguity decomposition algorithm came into being. Han et al. (1999) proposed a single epoch ambiguity resolution algorithm with ﬁxed baseline length as a constraint in GPS real time attitude determination, the results illustrated that single epoch ambiguity resolution success rate would achieve a 98.9% level. Mok (1999) proposed a GPS single epoch ambiguity resolution

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method with deformation constraints based on the deformation monitoring information is known, This method has good reliability and high success rate in ambiguity resolution. Cole et al. (2009) tested the method to use externally estimated position information that improve single epoch ambiguity resolution process. Teunissen (2010) and Giorgi (2012) develop new integer least-squares (ILS) theory for ambiguity resolution with nonlinear geometrical constraints, the results shown that the constraints model can improve to ﬁx the correct integer ambiguities. Based on the LAMBDA algorithm (Teunissen, 1995, 1997), the idea of solving the single epoch ambiguity using the baseline length as a constraint for attitude determination can be found in the work of Tang et al. (2005), Buist (2007), Giorgi (2010), Chen et al. (2012) and Gong et al.(2015). Most of the above single epoch ambiguity resolution algorithm are constrained by some exact prior information, so the successful rate and reliability of the results among these algorithms are higher. In GPS dynamic positioning, if the GPS receiver is mounted on a moving platform such as a train or an automobile and runs along a certain trajectory, its motion path will certainly conform to the linear equation of the road, the motion trajectory of the receiver can satisfy a certain function relation as the constraint condition for integer ambiguity decomposition. Therefore, a single epoch ambiguity resolution algorithm using the receiver trajectory as a constraint condition is proposed in this paper, the data processing idea of this algorithm is ﬁxed wide lane ambiguity parameter ﬁrstly, after that solving L1 and L2 frequency ambiguity parameter on the basis of wide lane ambiguity. For the determination of wide lane ambiguity, it is calculated by using the MW combination, and the corresponding alternative combinations of wide lane ambiguity are established, which are determined according to the minimum residual sum of squares criterion (V T V ¼ min) ﬁrstly, it is used to lay the foundation for the subsequent L1 and L2 frequency ambiguities solution. And then the straight trajectory constraint condition and the phase observation equation are used to solve the integer ambiguity N1, when the constraint information of the trajectory is unknown, the space linear equation is established as the constraint condition according to the coordinate solution by wide lane relative positioning. By designing the corresponding experiments, the results of this algorithm are proved to be feasible by using diﬀerent experimental schemes, such as without constraint condition, the constraint condition with known prior information, introducing dynamic constraint condition and so on, to solve the ambiguity, and to compare and analyze the eﬀect of linear constraint condition on single epoch ambiguity resolution.

2. Mathematical model of single epoch ambiguity resolution For dynamic relative positioning of single epoch algorithm, only the double diﬀerence carrier phase measure-

ments are used to calculate the unknown parameters (such as ambiguity, coordinate of rover station), which cannot be solved due to the rank-deﬁcient of the double diﬀerence equation. In general, the carrier phase and pseudorange are combined to solve the rank-deﬁcient problem, the mathematical model of double diﬀerence observation equation is deﬁned as follows (Ji et al., 2007): Au X þ BN ¼ Lu þ nu

ð1Þ

AP X ¼ L P þ n P

where Au and AP represent coeﬃcient matrix of carrier phase equation and pseudorange equation respectively; X is coordinate parameters; N is the double diﬀerence ambiguity, together with the corresponding coeﬃcient matrix B; Lu and LP are double diﬀerence carrier phase and pseudorange observables respectively, nu and nP are observable noises of carrier phase and pseudorange respectively; The model (1) is solved by applying the least squares criterion, and the normal equation of single epoch relative positioning can be derived as follows: # " # " AT/ P u Au þ ATP P P AP ATu P u B X ATu P u Lu þ ATP P P LP ¼ BT P u B N BT P u L u BT P u Au ð2Þ where P u and P P represent the weight matrix of carrier phase and pseudorange observations respectively; Since the accuracy of the pseudorange is much lower than the carrier phase, so the ambiguity ﬂoating solution calculated by Eq. (2) is poor, which makes it diﬃcult to solve the subsequent integer ambiguity. In practical applications, it is common to use relatively long wavelength observations, such as wide lane phase observations, are used to calculate the integer ambiguity. After the wide lane integer ambiguity is determined, according to the relationship between the wide lane ambiguity and the base ambiguity (N1 and N2), ambiguities N1 and N2 are ﬁnally determined respectively (Sjo¨berg, 1998; Deng et al., 2014). The corresponding mathematical model is given as: V u1 ¼ Au1 X þ BN 1 lu1 P u1 V u2 ¼ Au1 X þ BðN 1 N W Þ lu2

P u2

ð3Þ

where N W represent the wide lane integer ambiguity; Aui is coeﬃcient matrix; P ui is the weight matrix; lui is observation values minus computation values; V ui is residual vector ði ¼ 1; 2Þ; Similarly, based on the least squares criterion, the normal equation with ambiguity parameter N1 can be derived as follows: " # ATu1 P u1 Au1 þ ATu2 P u2 Au2 ATu1 P u1 B þ ATu2 P u2 B X T T T T N1 B P u1 B þ B P u2 B B P u1 Au1 þ B P u2 Au2 " # T T Au1 P u1 lu1 þ Au2 P u2 ðlu1 þ BN W Þ ¼ BT P u1 lu1 þ BT P u2 ðlu2 þ BN W Þ ð4Þ

N. Liu et al. / Advances in Space Research 62 (2018) 855–865

2.1. Searching algorithm for wide lane ambiguity based on minimum residual sum of squares (V T V ¼ min) criterion Eq. (4) shows that if we want to obtain the correct solution of ambiguity N1, the precondition is to accurately ﬁx the wide lane ambiguity NW. For the determination of parameter NW, the ﬂoating solution and covariance matrix of NW can be obtained by solving the double diﬀerence wide lane phase observable equation, and the ﬁxed integer solution of NW can be obtained by LAMBDA method. The LAMBDA method is one of the classical method to ﬁx double diﬀerence ambiguity. However, the calculation process of this method is relatively complex, and it is necessary to obtain the ﬁxed integer ambiguity by decorrelation using Z-transformation, and condition search (Chang, 2005). Since the wavelength of the wide lane observation are longer than those of L1 and L2 observations, its ambiguity is easy to be ﬁxed. Therefore, it is somewhat complicated to use LAMBDA method to ﬁx wide lane ambiguity. Since the MW combination eliminates the clock error, the geometric distance between the satellite and the receiver, the atmospheric delay error, so the ﬂoating solution of wide lane ambiguity is calculated by the MW combination, which has higher accuracy. At the same time, we should not consider the ill-posed problem of single epoch wide lane double diﬀerence normal equation. so we use the MW combination to calculate the wide lane ambiguity, rounding the ambiguity to the nearest integers, and after that set as n cycles to construct an interval of the ambiguity, each candidate is returned to the wide lane double difference equation, and the least square principle is used to calculate the coordinate residuals V, the residual sum of squares calculated with each ambiguity candidate combination is compared, which corresponding to the minimum residual sum of squares is used as the optimal solution. If the number of satellites in common view for base station and rover station is m, suppose the interval n is equal to 3 cycles, the total number of wide lane ambiguity candidate combinations for one epoch would amount to be 823,543 in the case of m = 8, The computational cost is huge based on these ambiguity combinations (Corbett et al., 1995; Lin et al., 2006). In addition, the larger searching range of each wide lane ambiguity, the more ambiguity candidate combinations, the time required for ambiguity resolution will be longer, which will aﬀect the eﬃciency of dynamic positioning. Therefore, it is necessary to choose a reasonable search range for wide lane ambiguity. The observation accuracy of the carrier phase and the pseudorange is about 0.002 m and 0.3 m respectively, according to the propagation law of error, the accuracy of the wide lane ambiguity calculated by the MW combination is about 0.248 m. From the probability of statistical knowledge, the probability that accidental error is less than three times of root mean squared errors is 99.7%, three times of root mean squared errors for wide lane ambiguity calculated by MW combination is 0.744 m, which is less

857

than the wavelength of wide lane. After rounding the wide lane ambiguity to the nearest integers, the truth value of the ambiguity is very large probability in the interval of ±1 cycles, so we choose ±1 cycles as the candidate interval of wide lane ambiguity in this paper. Assuming that the number of satellites in common view is 8, the number of wide lane ambiguity candidate combinations for each epoch is 2187, which greatly reduces the number of candidate combinations and saves the ambiguity search time. 3. Single epoch ambiguity rapid decomposition with straight trajectory constraints As the trajectory of the train or car is characterized by continuity and smooth, it can be abstracted by a number of straight lines, circular curves and mitigation curves. At the same time, when calculus is used to calculate curvilinear integral in mathematics, the integral region can be divided into several small integral intervals, the corresponding integral interval curve can be considered a straight line when the integral interval is small enough. Therefore, the GPS receiver trajectory placed on a moving vehicle such as a train or a car can be abstracted as a straight line, so the straight linear equation is used to establish the constraint condition. 3.1. Establishment of static constraint equation If we know the WGS84 coordinates ½ X A Y A Z A and ½ X B Y B Z B of the points at both ends of a straight line AB, the linear equation of each epoch trajectory for the rover station is deﬁned as follows: C½ X

Y

Z T ¼ D

ð5Þ

where 0 YA YB XB XA ; C¼ 0 ZA ZB Y B Y A X A ðY A Y B Þ Y A ðX A X B Þ D¼ : Y A ðZ A Z B Þ Z A ðY A Y B Þ

T of The approximation coordinates X 0i Y 0i Z 0i rover station calculated by the relative positioning of the pseudorange are taken into Eq. (5), a static linear constraint equation can be derived as follows: C^x ¼ W

ð6Þ

where W ¼ D CX 0 . 3.2. Establishment of dynamic constraint equation The process of establishing the static constraint equation mentioned above, which is similar to the known trajectory of the rover station. However, it is impossible to obtain these prior information in advance, so that it is impossible to establish an accurate constraint equation for trajectory. In view of this situation, a dynamic con-

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straint equation is considered to solve the problem in this paper. The coordinate accuracy of the rover station with the double diﬀerence wide lane phase solution is higher than that of double diﬀerence pseudorange. When the constraint equation is established, the coordinates of the rover station obtained by using the double diﬀerence of wide lane can more accurately reﬂect the trajectory, so on the basis of choosing the MW combination to calculate the wide lane ambiguity, which is ﬁxed by using the minimum residual sum of squares criterion, and the coordinate of the rover station are obtained, And then the dynamic constraint equation is constructed with this coordinate. The basic idea of establishing dynamic constraint equation is described as follows: assuming that the L1 frequency ambiguities of the i + 1 epoch are now calculated, we need to judge whether each pair of wide lane ambiguity determined by the ﬁve epochs before i + 2 epoch and the i + 2 epoch is equal, if the wide lane ambiguities of these epochs are equal, the coordinates of the i + 2 and i epoch can be used to establish the space linear equation for constraint condition to calculate the i + 1 epoch ambiguity, increased an epoch data every time and until the calculation to the last epoch. In the recursive calculation until the last epoch, it is noteworthy that if the wide lane ambiguity of i + 2 epoch is diﬀerent from that of previous 5 epochs, this means that cycle slip or ﬁxed-error ambiguity may exist in the i + 2 epoch, the straight line equation established in this case will not be able to correctly determine the integer ambiguity N1 for the i + 1 epoch, when this happens, The integer ambiguity N1 of the i + 1 epoch can be calculated using the constraint equation established by the pseudorange.

3.3. Solving ambiguity N1 under trajectory constraints After establishing the constraint condition and taking into account Eq. (3), the mathematical model of single epoch ambiguity resolution with straight constraints is derived as follows: V u1 ¼ Au1^x þ BN 1 Lu1 V u2 ¼ Au2^x þ BðN 1 N W Þ Lu2

ð7Þ

C^x ¼ W The least squares criterion is applied to Eq. (7), then the corresponding normal equation is given as: 2 T 3 Au1 P u1 Au1 þ ATu2 P u2 Au2 ATu1 P u1 B þ ATu2 P u2 B C T 6 T 7 4 B P u1 Au1 þ BT P u2 Au2 BT P u1 B þ BT P u2 B 0 5 2

3

X^ 6 7 4 N1 5 ¼ K

2

C

ATu1 P u1 Lu1 6 T 4 B P u1 Lu1

0 þ þ

3

0

ATu2 P u2 ðLu1 þ BN W Þ 7 BT P u2 ðLu2 þ BN W Þ 5 W ð8Þ

The ﬂoating solution and the variance matrix of N1 are obtained by solving Eq. (8). And then based on the LAMBDA method, the integer ambiguity N1 is ﬁxed and N1 is substituted into the carrier phase observation equation, at this time, only the coordinate of rover station are unknown, which can be obtained by the calculation of each epoch. 4. Experiments and result analysis 4.1. Experimental design and data collection In this experiment, two Unistrong GPS receivers are used for data collection with 1 s sampling interval. One GPS receiver is placed on the roof of the teaching building as the base station, and the other GPS receiver is mounted on a linear guide rail ﬁxed on the roof as the rover station. The receiver of rover station can be moved back and forth on the guide rail, the experiment is shown in Fig. 1. The relevant information of the experimental data is shown in Table 1. Using the algorithm proposed in this paper to deal with the collected data, and the cut-oﬀ elevation angle is set to 15°. The coordinate of rover station obtained from the precise relative positioning with commercial software HGO is taken as true value, which is used to verify the feasibility of the algorithm for solving the ambiguity, the trajectory of the rover station calculated by HGO is shown in Fig. 2. The satellite G31 with the highest elevation angle is selected as the reference satellite in the calculation, and the data are processed by the double diﬀerence observation equation. The integer ambiguity results of seven satellitepairs calculated by the HGO software are taken as truth values, results are listed in Tables 2 and 3. At the same time, based on the single epoch ambiguity resolution algorithm proposed in this paper, the following ﬁve diﬀerent schemes are used to calculate the data, and the result of ambiguity resolution is compared to verify the feasibility of the algorithm. The scheme A: By using the phase observation of L1 and L2 frequencies, the double diﬀerence equations of wide lane are combined with the double diﬀerence equations of pseudorange. Based on the LAMBDA method, the wide lane ambiguity is solved, and then the ambiguity parameter N1 are solved on the basis of wide lane ambiguity. The scheme B: The ﬂoating solution of wide lane ambiguity is calculated by MW combination, which is ﬁxed based on minimum residual sum of squares criterion. And then the ambiguity parameter N1 are solved on the basis of wide lane ambiguity. The scheme C: Similarly, the minimum residual sum of squares search algorithm is used to ﬁx the wide lane ambiguity. By establishing the plane line equation as the static constraint condition, the ambiguity parameter N1 are solved by using Eq. (7) and the LAMBDA method. The scheme D: The wide lane ambiguity ﬁxing algorithm is the same as that of scheme C, but the space linear

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859

Fig. 1. Experiment (left. base station; right. rover station).

Table 1 The relevant information of the experimental data. Date and local time (yy/mm/dd) (hh:mm)

Receiver type

Sampling interval

Number of common view satellites

Total epochs

2013/3/23 08:59–09:12 2013/3/25 08:52–09:03

Unistrong

1s

8

749

Unistrong

1s

8

714

Table 3 Ambiguity truth value of seven satellite-pairs (March 25).

117.047962

longtitude (degree

117.047960

117.047958

117.047956

117.047954

Satellite-pairs

N1 (cycle)

N2 (cycle)

G31-G16 G31-G30 G31-G29 G31-G32 G31-G20 G31-G14 G31-G25

7 6 0 1 2 3 2

13 6 4 1 33 1 2

117.047952

117.047950 39.232970

39.232975

39.232980

39.232985

39.232990

39.232995

latitude (degree)

Fig. 2. Trajectory of rover station.

Table 2 Ambiguity truth value of seven satellite-pairs (March 23). Satellite-pairs

N1 (cycle)

N2 (cycle)

G31-G16 G31-G30 G31-G29 G31-G32 G31-G20 G31-G14 G31-G25

20 7 1 4 0 10 14

35 34 4 5 2 11 5

equation is used as the static constraint condition, and the ambiguity parameter N1 are solved by using Eq. (7) and the LAMBDA method. The scheme E: Firstly, the same algorithm as the scheme C is used to ﬁx the wide lane ambiguity, then the ﬁxed solution of wide lane ambiguity are used to calculate the coordinates of rover station by relative position, and the space linear equation is established as the dynamic constraint

condition. Finally, the combined Eq. (7) and the LAMBDA method are used to solve the ambiguity parameter N1. It is worth noting that in the scheme C and D, the HGO software is ﬁrstly used to obtain precise coordinates of the points on both ends of the guide rail. And then the plane linear constraint equation and the space linear constraint equation are established respectively. That is, scheme C and scheme D are additional strong constraints to solve ambiguity. 4.2. Verifying the feasibility of V T V ¼ min criterion search algorithm to determine the wide lane ambiguity Among the ﬁve schemes mentioned above, the schemes B, C, D and E use the V T V ¼ min criterion proposed in this paper to determine the wide lane ambiguity, which is diﬀerent from the scheme A. Therefore, the comparison between the scheme A and the scheme B is carried out ﬁrst. The dynamic data of March 23 and March 25 were processed by schemes A and B respectively, and the three satellitespairs of G31-G16, G31-G29 and G31-G25 were taken as the examples for comparative analysis. The ﬂoating solutions of wide lane ambiguity were calculated by using

G31-G16

-14.0 -14.5 -15.0 -15.5 -16.0 0

100

200

300

400 500 epoch number

600

700

rounding the value (cycle)

N. Liu et al. / Advances in Space Research 62 (2018) 855–865

rounding the value (cycle)

860

-5.5 -6.0 -6.5 -7.0 0

800

-4.2 -4.4 -4.6 -4.8 -5.0

rounding the value (cycle)

0

100

200

300

400 500 epoch number

600

700

-9.0 -9.5 -10.0 0

100

200

300

400 500 epoch number

600

700

800

a

300

400 500 epoch number

600

700

800

G31-G29

-3.2 -3.4 -3.6 -3.8 -4.0 0

G31-G25

-8.5

200

-3.0

800

-8.0

100

-2.8

rounding the value (cycle)

G31-G29

rounding the value (cycle)

rounding the value (cycle)

-3.8 -4.0

G31-G16

-5.0

100

200

300

400 500 epoch number

600

2.0

700

800

G31-G25

1.5 1.0 0.5 0.0 -0.5 -1.0 0

100

200

300

400 500 epoch number

600

700

800

b

Fig. 3. The rounded integers of wide lane ambiguity in three groups satellite-pairs (a. March 23; b. March 25).

scheme B, and then rounding them to the nearest integers, the rounded integers are shown in Fig. 3. It can be seen from the ﬁgure, the ﬂoating solution of most epochs for wide lane ambiguity in satellite-pairs G31-G16 and G31-G29 is calculated by MW combination, which is the true value after it is directly rounded, the rounding value and the true value of a few epochs for wide lane ambiguity appeared integer deviation, but the range of deviation is within 1 cycle, the ﬂuctuation of the wide lane ambiguity in the satellite-pairs G31-G25 is obvious, the reason is that the elevation of satellite G25 is low, which may leads to the larger error in the observation value and aﬀects the accuracy of the ﬂoating solution. But the rounding value and the truth value of the ambiguity in satellite-pairs G31-G25 appeared within 1 cycle deviation on March 23, and for the same satellite-pairs on March 25, only two epochs whose calculation result deviation exceeds 1 cycle. From this we can see that for most epochs, the ﬁxed solution of wide lane ambiguity can be obtained based on the V T V ¼ min criterion search algorithm. If you expand the search range when searching for wide lane ambiguity, the search successful rate should be improved accordingly, but the search calculation load will increase signiﬁcantly and the search eﬃciency will be reduced. For the dynamic positioning solution, while ensuring the successful rate of ambiguity resolution, we also need to ensure a certain search eﬃciency, It shows that the search range of wide lane ambiguity determined by this paper is reasonable. The successful rate of single epoch ambiguity resolution information (Xu et al., 2015, Li

et al., 2017)in scheme A and scheme B is counted, and the results are shown in Table 4. It can be seen from Table 4 that the successful rate of the wide lane ambiguity resolution can reach more than 99%, which indicates that the algorithm is feasible, and lays the foundation for resolution the ambiguity N1 and N2 in single epoch. 4.3. Verifying the eﬀect of introducing constraints on the ambiguity resolution The measured dynamic data of March 23 were processed using the mentioned above schemes B, C, D and E respectively. The satellite G31 is selected as the reference satellite in the calculation process, and the three satellites-pairs of G31-G16, G31-G29 and G31-G25 were also taken as the examples for comparative analysis. The statistical histogram of the diﬀerence between the ﬂoating solution and the true value of ambiguity parameters N1 solved by the four diﬀerent schemes is shown in Fig. 4. As Fig. 4 shows, the diﬀerence between the ﬂoating solution and the truth value of the ambiguity N1 obtained by using four diﬀerent data processing schemes is basically consistent with the normal distribution. It is found by comparison that when the ambiguity N1 is solved without adding constraint conditions (scheme B), the number of epochs whose diﬀerence between the ﬂoating solution and the true value is around zero less than that of the other three schemes, the distribution is more discrete. When the space straight line constraint is added (scheme D), the number of

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Table 4 The ﬁxed solution information of ambiguity. Observation time

Total epochs

Scheme A

Scheme B

Successful epochs

Successful rate

Successful epochs

Successful rate

March 23 March 25

749 714

670 655

89.45% 91.7%

749 711

100% 99.57%

220

320

G31-G16

scheme B scheme C scheme D scheme E

200 180 160

G31-G29

240

140

200

120

number

number

scheme B scheme C scheme D scheme E

280

100

160 120

80 60

80

40

40

20 0 -2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

0 -2.0

2.0

-1.5

-1.0

-0.5

difference (cycle)

0.0

0.5

1.0

1.5

2.0

difference (cycle)

a

b 480 440

G31-G25

scheme B scheme C scheme D scheme E

400 360 320

numbe

280 240 200 160 120 80 40 0 -1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

difference (cycle)

c Fig. 4. The diﬀerence between the ﬂoating solution and truth value of ambiguity N1 on March 23 (a. satellite-pairs G31-G16; b. satellite-pairs G31-G29; c. satellite-pairs G31-G25).

epochs whose diﬀerence between the ﬂoating solution and the truth value in the vicinity of zero is the largest. When the dynamic linear constraint is added (scheme E), the number of epochs whose diﬀerence between the ﬂoating solution and the truth value near the zero is greater compared to the situation when plane straight line constraint is added (scheme C). At the same time, the diﬀerence distribution of scheme C and E is also more concentrated. So it can be seen that adding constraints can improve the precision of ambiguity ﬂoating solution, among them, the best solution is obtained when the space straight line is used, the main reason is that the space linear constraint is

restricted to the three directions of X, Y and Z, while the plane linear constraint only constrains two directions, and the constraint is weaker than the three dimensional constraint. The dynamic constraints also use the space straight line to establish the equation, the single epoch wide lane relative position needs to be obtained the coordinate of rover station ﬁrst, and then the constraint equation is established. The equation is inevitably aﬀected by the precision of wide lane relative positioning, the static space linear constraint equation is established by using a priori precise coordinates, that is, the ambiguity solution under strong

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Table 5 Mean and RMS of the diﬀerence between the ﬂoating solution and the truth value for four diﬀerent schemes (March 23). Satellites-pairs

Mean (cycle)

RMS (cycle)

Scheme B

Scheme C

Scheme D

Scheme E

Scheme B

Scheme C

Scheme D

Scheme E

G31-G16 G31-G25 G31-G29

0.193 0.253 0.214

0.090 0.062 0.014

0.013 0.014 0.002

0.064 0.012 0.018

0.618 0.443 0.554

0.517 0.198 0.224

0.325 0.048 0.187

0.349 0.131 0.200

360

220

G31-G16

scheme B scheme C scheme D scheme E

200 180 160

G31-G29

280 240

140 120

200

number

number

scheme B scheme C scheme D scheme E

320

100

160

80

120

60 80 40 40

20 0 -2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

0 -1.5

2.0

-1.0

-0.5

0.0

0.5

1.0

1.5

difference (cycle)

difference (cycle)

a

b 440

G31-G25

scheme B scheme C scheme D scheme E

400 360 320

number

280 240 200 160 120 80 40 0 -1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

difference (cycle)

c Fig. 5. The diﬀerence between the ﬂoating solution and truth value of ambiguity N1 on March 25 (a. satellite-pairs G31-G16; b. satellite-pairs G31-G29; c. satellite-pairs G31-G25).

constraint. Therefore, the accuracy of the ambiguity ﬂoating solution calculated by the scheme E is slightly lower than that obtained by the scheme D. Moreover, we compute the statistical results for assessing the four diﬀerent schemes, the means and root mean squared errors (RMS) precision of the diﬀerence between the ﬂoating solution and the truth value are shown in Table 5. Under the condition of adding plane linear constraint and space linear constraint, both means and RMS for three satellites-pairs are obviously improved, the RMS of the diﬀerence for ambiguity is much less than 0.5 cycle in static and dynamic space

linear constraints, it is shown that the accuracy of the ambiguity ﬂoating value can be greatly improved by introducing the constraint condition into the solution. Among them, the precision of ambiguity ﬂoating solution with static spatial line constraint is the most obvious, followed by the dynamic space linear constraint, and ﬁnally the plane line constraint. In order to further verify the impact of constraints on the accuracy of integer ambiguity resolution, The dynamic data collected in March 25 are also solved with the scheme B, C, D, and E. The diﬀerence between the ﬂoating solution

N. Liu et al. / Advances in Space Research 62 (2018) 855–865

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Table 6 Mean and RMS of the diﬀerence between the ﬂoating solution and the truth value for four diﬀerent schemes (March 25). Satellites-pairs

Mean (cycle)

G31-G16 G31-G25 G31-G29

RMS (cycle)

Scheme B

Scheme C

Scheme D

Scheme E

Scheme B

Scheme C

Scheme D

Scheme E

0.197 0.142 0.092

0.142 0.120 0.085

0.066 0.009 0.035

0.084 0.022 0.059

0.564 0.391 0.490

0.480 0.186 0.180

0.294 0.048 0.169

0.301 0.081 0.182

Table 7 Successful rate of integer ambiguity N1 for four diﬀerent schemes. March 23 Scheme Scheme Scheme Scheme Scheme

B C D E

March 25

Total epochs

Successful epochs

Successful rate

Total epochs

Successful epochs

Successful rate

749

726 744 748 747

96.93% 99.33% 99.87% 99.47%

714

695 709 711 710

97.34% 99.30% 99.56% 99.44%

x

0.012

residual error (m)

residual error (m)

0.012 0.008 0.004 0.000 -0.004

x

0.008 0.004 0.000 -0.004

0

100

200

300

400

500

600

700

0

100

200

epoch number y

0.01

400

500

600

0.00 -0.01 -0.02 -0.03

0.01 0.00 -0.01 -0.02 -0.03 -0.04

0

100

200

300

400

500

600

700

0

100

200

epoch number

300

400

500

600

700

epoch number

0.010 Z

0.000 -0.005 -0.010

Z

0.005

residual error (m)

0.005

residual error (m)

700

y

0.02

residual error (m)

residual error (m)

0.02

-0.04

300

epoch number

0.000 -0.005 -0.010

-0.015 0

100

200

300

400

500

600

700

epoch number

-0.015

0

100

200

300

400

500

600

700

epoch number

a

b

Fig. 6. The residual error of coordinate (a. March 23; b. March 25).

and the true value of ambiguity N1 resolution by four schemes is calculated according to the distribution interval, and the results for three satellites-pairs is shown in Fig. 5. The corresponding means and root mean squared errors (RMS) precision of the diﬀerence are shown in Table 6.

From Fig. 5 and Table 6, it can be seen that the degree of discretization for the ambiguity ﬂoating solution becomes low and the accuracy is improved after adding constraints. The results show that the ambiguity resolution with space linear constraint is the best, which is consistent

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N. Liu et al. / Advances in Space Research 62 (2018) 855–865

with the data analysis in March 23. At the same time, the ambiguities of all the satellite-pairs in two days (March 23 and March 25), which are solved by the above four schemes, and are statistically analyzed to verify the inﬂuence of the introduction of the constraint conditions on the successful rate of ambiguity resolution. The results of statistics are presented in Table 7. It can be seen from Table 7 that the single epoch ambiguity calculation has lowest successful rate without constraint, and the highest successful rate of ambiguity resolution under static space linear constraints. The measured data of two days are processed by adding the dynamic space linear constraint equation, and the corresponding successful rate of integer ambiguity N1 resolution can reach more than 99%, it shows that when the prior information of rover station trajectory is unknown, it is feasible to establish the space linear constraint equation in real time by using the coordinate of rover station calculated by wide lane relative positioning to participate in the ambiguity solution, it is further explained that the introduction of space linear trajectory constraints can change the structure of the normal equation of the single epoch dynamic positioning, so that the ambiguity ﬂoating solution can be easily distinguished and the successful rate of single epoch ambiguity solution can be eﬀectively improved. On the other hand, in order to further analyze the calculation accuracy of coordinate, using the ﬁxed ambiguity in scheme E to calculate the coordinates of rover station by single epoch, and compare the calculated results with the HGO solution results, the coordinate error sequence in the X, Y and Z directions are shown in Fig. 6. When the ambiguity is ﬁxed correctly, the coordinate accuracy of rover station in X, Y and Z directions by single epoch resolution can reach the centimeter level, and its error value mainly concentrates within 2 cm.

5. Conclusions In this paper, by solving the GPS dynamic data, it is shown that the single epoch ambiguity resolution algorithm with trajectory constraints is feasible. The wide lane ambiguity is ﬁxed ﬁrst in this algorithm, and then the data processing scheme of ambiguity parameters N1 is solved. In determining the wide lane ambiguity, the combination of MW is used to calculate the corresponding integer value. In addition, an alternative combination of ambiguity is established, and the wide lane ambiguity is determined according to the V T V ¼ min criterion, which lays the foundation for subsequent ambiguity N1 resolution. At the same time, the actual results of this algorithm are veriﬁed with the observed data. Four diﬀerent data processing schemes are solved N1 by using without constraint, adding plane constraint, static space linear constraint and dynamically establishing space line constraint, And compared the results obtained from the four diﬀerent schemes, the results show that adding constraints can improve the successful

rate of ambiguity resolution. Under the condition of adding space linear constraints, the three-dimensional constraint strength is higher than the two-dimensional plane constraint, so the ambiguity resolution eﬀect is better. Among them, In the dynamic construction of the constraint equation, the rover station coordinates obtained by the wide lane relative positioning are used to construct the trajectory constraint condition dynamically, the calculation is simple and the precision is reliable, which makes up for the lack of unknown motion trajectory information. Acknowledgements The research in this paper was supported by China Postdoctoral Science Foundation Funded Project (No.: 2016M592733) and State Key Laboratory of Geoinformation Engineering (No.: SKLGIE2015-M-2-4), It was also partially supported by the Fundamental Research Funds for the Central Universities (No.: 310826151049). References Assiadi, M., Edwards, S.J., Clarke, P.J., 2014. Enhancement of the accuracy of single-epoch GPS positioning for long baselines by local ionospheric modelling. GPS Solut. 18 (3), 453–460. Buist, P.J., 2007. The baseline constrained LAMBDA method for single epoch, single frequency attitude determination applications. Proc. ION GPS 2007 (3), 2962–2973. Chang, X.W., Yang, X., Zhou, T., 2005. MLAMBDA: a modiﬁed LAMBDA method for integer least-squares estimation. J. Geod. 79 (9), 552–565. Chen, W., Qin, H., 2012. New method for single epoch, single frequency land vehicle attitude determination using low-end GPS receiver. GPS Solut. 16 (3), 329–338. Cole, A., Wang, J.L., Dempster, A.G., et al., 2009. Single epoch integer ambiguity resolution after pseudorange adjustments. In: International Global Navigation Satellite Systems Society IGNSS Symposium 2009, Qld, Austrilia, 1–3 December. Corbett, S.J., Cross, P.A., 1995. GPS single epoch ambiguity resolution. Survey Rev., 257(33), 149–160. Deng, C.L., Tang, W.M., Liu, J.N., et al., 2014. Reliable single-epoch ambiguity resolution for short baselines using combined GPS/BeiDou system. GPS Solut. 18 (3), 375–386. Giorgi, G., 2010. The multivariate constrained LAMBDA method for single-epoch, single-frequency GNSS-based full attitude determination. Proc. ION GNSS 2010 (9), 1429–1439. Giorgi, G., Teunissen, P.J.G., Verhagen, S., et al., 2012. Integer ambiguity resolution with nonlinear geometrical constraints. In: Vii HotineMarussi Symposium on Mathematical Geodesy, pp. 39–45. Gong, A., Zhao, X., Pang, C., et al., 2015. GNSS single frequency, single epoch reliable attitude determination method with baseline vector constraint. Sensors 15(12), 30093–30103. Han, S., Rizos, C., 1999. Single-epoch ambiguity resolution for real-time GPS attitude determination with the aid of one-dimensional optical ﬁber gyro. GPS Solut. 3 (1), 5–12. Ji, S.Y., Chen, W., Zhao, C.M., et al., 2007. Single epoch ambiguity resolution for Galileo with the CAR and LAMBDA methods. GPS Solut. 11 (4), 259–268. Li, Q., Zhang, L., Wu, J., et al., 2017. A novel constrained ambiguity resolution approach for Beidou attitude determination. Adv. Space Res. 60, 2423–2436. Lin, S.G., Tzeng, D.B., 2006. Single epoch kinematic GPS positioning technique in short baseline. J. Surv. Eng. 132 (2), 52–57.

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