1. Introduction
As precise point positioning (PPP) was proposed by Zumberge et al. to realize positioning solution with only a stand-alone receiver, and it is used normally in double-frequency observations [
1,
2,
3]. However, the convergence time is typically 30 min [
2].
To reduce the convergence time in a single constellation Global Navigation Satellite System (GNSS), the simplest method is to reduce the level of pseudo-range noise by means of observation combinations theoretically [
4,
5,
6,
7,
8]. However, different observation combinations have a similar performance which is presented by Liu and Qin [
9,
10,
11].
Due to the fractional cycle biases (FCBs) in the Global Positioning System (GPS) observations are absorbed by the nondifferential ambiguity estimates, so their integer properties are destroyed [
12,
13]. FCB leads to greatly reducing the efficiency of PPP ambiguities searching, and makes it difficult for the filtering algorithm to converge in a short time. Moreover, FCB will also interfere with the filter update and brings in the wrong prior information.
The most fundamental method is ambiguity fixing. As early as 1999, Gabor and Nerem first proposed the algorithm for ambiguity-fixing in a stand-alone receiver [
14]. At that time, due to the existence of selective availability (SA) and the impact of satellite orbits and clock error accuracy on the narrow-lane ambiguity evaluation, they did not realize the PPP with ambiguity resolution (PPP-AR). Gao and Shen attempted to realize ambiguity pseudo-fixing [
4]. Subsequently, there are six PPP-AR methods [
15] where three existing PPP-AR methods are Ionosphere-free [
16,
17,
18]. They is a single difference between the Uncalibrated Phase Delay/Fractional Cycle Bias (UPD/FCB) model [
14], decoupled satellite clock (DSC) model [
12] and integer recovery clock (IRC) model [
19]. Ge et al. partially improved Gabor’s algorithm and proposed an ambiguity-fixing method based on FCB on the premise of the short-term stability of FCB [
13].
The narrow-lane FCB varies from 2 to 3 h [
16]. In the case of the FCB estimation in a stand-alone receiver, the estimated error of FCB is relatively large, so that the ambiguities can not be fixed accurately [
20]. Some researchers use observation models with the assistance of a continuously-operating reference station (CORS) [
21]. It is usually used to realize PPP with real-time kinematic (PPP-RTK) [
9,
15,
19,
22,
23,
24,
25,
26,
27,
28,
29]. The convergence time has been reduced to make it more practical.
PPP-AR needs real-time network corrections by the regional or global operating reference station network [
9,
28,
30,
31], compared with the float solution. When there is no network resource in special positioning scenarios, ambiguity-fixing becomes nearly impossible. In the post-processing PPP, Hu proposed that the narrow-lane FCB is updated once per 15 min, namely, narrow-lane FCB is calculated with 15-min segment data in a stand-alone receiver, and then the narrow-lane FCB is used to calculate the ambiguity fixed solution in this segment [
32]. The positioning solution is better than PPP with floating ambiguity. In the Real-time PPP, the estimated error of FCB is relatively larger and FCB cannot be estimated as same as that in post-processing PPP. The real-time PPP-AR can not be realized under no real-time network corrections. In this scenario, the convergence time of any other methods will be longer. However, compared with the float solution, whether is there an algorithm to reduce the convergence time under no real-time network corrections, especially when the performance of the float solution is poor?
To solve the above problem, we propose an ambiguity residual constraint-based precise point positioning with partial ambiguity resolution (PPP-PARC) to improve the convergence speed under no real-time network corrections. Experimental verification is conducted by real GPS data. The primary contributions of the paper are summarized as follows.
An improved FCB update strategy is presented to satisfy requirements of real-time PPP without the assistance of real-time network corrections.
An ambiguity residual constraint-based precise point positioning with partial ambiguity resolution (PPP-PARC) is proposed to fix partial ambiguities successfully so that convergence time can be reduced under no real-time network corrections.
A PPP experiment is operated to analyze the performance of the new algorithm in post-processing and real-time PPP.
2. Partial Ambiguity Fixing
2.1. Non-Integer Ambiguity
The conventional ionosphere-free combining observation model is
where
,
, and
are observations with different frequencies,
and
are the signal frequencies of observations.
is the distance between receiver and satellite.
is receiver clock error.
is satellite clock error.
is the projection of tropospheric zenith wet path delay.
is the ambiguity of carrier phase.
is the wavelength of carrier phase.
,
,
, and
are hardware delay of pseudorange and carrier phase in receiver and satellite.
The hardware delay with the uncalibrated phase is represented by Uncalibrated Hardware Delay (UHD) [
12,
13,
16], where the integer part is represented by UPD and the fractional part is represented by FCB.
The UPDs can be absorbed into the ambiguities, there is no influence for both the integer characteristics of ambiguities and the positioning solution. However, When the FCBs are absorbed into the ambiguities, the integer characteristics of the ambiguities are destroyed.
To converge quickly, the most direct way is the ambiguity pseudo-fixing. It does not consider the carrier hardware delay, the ambiguities are fixed into integers coercively. However, there may be one cycle deviation between the estimated ambiguity and the real ambiguity. If the positioning accuracy is not required to be so high, the ambiguity pseudo-fixing may accelerate convergence and reach decimeter-level positioning.
Another method is ambiguity-fixing based on FCB.
After the single difference between satellites,
The variation of wide-lane UPD with a single difference between satellites is relatively stable for several months [
13]. However, the variation of narrow-lane UPD with a single difference between satellites is not stable within a day, so the ambiguities can be fixed by dividing a day into several stable segments for narrow-lane UPD with a single difference between satellites.
2.2. Ambiguity Residual Constraint-Based PPP-PAR
The flow chart of Precise Point Positioning can be described as shown in
Figure 1. The PPP with float resolution needs a long convergence time. In order to converge fast, adding the GNSS system is a feasible method, and another method is ambiguity resolution which does not need to add a data source. However, conventional ambiguity resolution needs FCB estimated by the network. When the communication network is poor, there is no assistance in real-time network corrections.
We tried to estimate FCB by a stand-alone receiver. Due to the FCB estimated by a stand-alone receiver being worse than that estimated by CORS network, not all ambiguities can be fixed, so partial ambiguity-fixing with strict constraints is more practical. Compared with conventional PPP-PAR, FCB estimated by a stand-alone receiver should have a different update period. Meanwhile, there are more strict constraints in screening fixed ambiguity.
6. Conclusions
Ambiguity resolution is a critical prerequisite for positioning solutions. However, some of the ambiguities may have biases in the case of FCB estimation with a stand-alone receiver, PPP with partial ambiguity resolution is needed. The research is focused on a partial ambiguity fixed solution PPP without the assistance of the real-time network corrections.
Firstly, an ambiguity-fixing method for FCB estimated by a stand-alone receiver was proposed in both post-processing PPP and real-time PPP. Considering the fast convergence, the update period was changed from 15 min to 5 min. Then, the influencing factors of the ambiguity success rate in a fixed-solution were analyzed. It found that ambiguity residuals are related with the ambiguity success rates. The smaller the floating ambiguity residual is, the better the performance of ambiguity fixing is. In the case of the FCB estimation in a stand-alone receiver, the combined floating ambiguity residuals were added as the constraints to guarantee the partial ambiguity-fixing.
Subsequently, the controlled variable experiments were used to determine the ambiguity residual threshold in both post-processing PPP and real-time PPP. The result shows that the ambiguity residual threshold is 0.2 cycles. Finally, the performance of the algorithm was analyzed by independent experiments with 22 stations in both post-processing PPP and real-time positioning.
When the float solution is not good at the GPS stations, the average convergence time of fixed solutions is reduced by 15.8% and 26.4% in post-processing and real-time positioning, respectively. However, if the float solution is stable at the GPS stations, the PPP-PARC method has similar performance as the float solution.
The ambiguity residual constraint-based PPP-PAR method has room for optimization. In the future, we will continue analyzing the PPP-PARC method in detial and optimize the performance.