Center for Mapping

ON-THE-FLY AMBIGUITY RESOLUTION TECHNIQUE FOR THE INTEGRATED SYSTEM

 

High-accuracy, kinematic differential positioning with GPS requires On-The-Fly ambiguity resolution, independent of the receiver motion.  As a result, this technique is ideally suited for real-time cm-level positioning. With dual-frequency GPS receivers OTF ambiguity resolution is very fast (ambiguities are resolved within a few epochs of time). The OTF method establishes a search space using differential phase and pseudorange positioning.  The correct solution within the search space is identified using least-squares search or ambiguity covariance methods. These methods are generally limited to short and medium baselines.

 

A robust GPS and INS integration, as implemented in the AIMS concept, provides fast and reliable correction of cycle-slips and losses of lock affecting GPS measurements, and allows for fast and robust OTF ambiguity recovery over long baselines. The real-time GPS software provides initial L1 and L2 double‑differenced phase ambiguities using search technique, and precise platform location as starting conditions for the integrated Kalman filter. This software is designed to run during the initial INS static alignment and for a few minutes after take‑off. This assures that the initial ambiguity resolution software will properly accommodate possible losses of lock during initial airplane maneuvers, and valid estimates of both ambiguities and the coordinates will be delivered for fast filter start-up. The accuracy of this real‑time GPS positioning system was found to range from mm‑level to a few centimeters, depending on satellite geometry, baseline length, and multipath environment, as shown in Table 1. Figures 1 and 2 present differences in the positioning solution determined from two base stations that average around 1 cm (max 3 cm in vertical direction).

 

 

 

Baseline length/multipath conditions		PDOP
		Low (below 3)	High (3-5)
Long baseline	Low multipath	1-2  cm	2-4 cm
Short baseline	Medium multipath	1-2 cm	2-3 cm
Short baseline	Low multipath	mm-level	cm-level

 

 

Table 1. GPS-only positioning accuracy.

 

 

 

 

 

Figure 1. Differences in East, North, and Up rover coordinates determined from two base 

                 stations (baseline – rover separation 0.4 km, medium multipath).

 

 

 

 

 

Figure 2. Differences in East, North, and Up rover coordinates determined from two base

                 stations  (baseline – rover separation 1.5 km, low multipath).

 

 

The speed and reliability of the OTF ambiguity resolution algorithm was evaluated by introducing one-second losses of lock to all of the satellites every 60 seconds. The total number of samples was 80. The time required to successfully recover the ambiguities was evaluated as a function of the number of satellites and satellite geometry (PDOP). For all of the tests, ambiguity resolution was accomplished at the 95% probability level within an average of 10 epochs for short baselines (1–2 km) and 45 epochs for longer baselines (~16 km), as presented in Table 2. These results demonstrate the high accuracy and reliability of our GPS real-time system.

 

 

Baseline length/multipath conditions
Long baseline	Low multipath
Short baseline	Medium multipath
Short baseline	Low multipath

 

 

Table 2. OTF ambiguity resolution speed.

 

Initial ambiguity estimation does not, however, address the entire ambiguity resolution problem. The ambiguities for satellites whose tracking starts during the flight must be resolved OTF, using positioning estimates from the GPS/INS Kalman filter. Similarly, cycle-slips must be detected and repaired. Figure 3 presents the block diagram of the INS-aided OTF ambiguity resolution in AIMS.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3. INS-aided OTF ambiguity resolution.

 

As long as the level of double-differenced residuals obtained from the phase observable and the filter prediction is below 0.2l₁, no cycle-slips are assumed; otherwise, the ambiguity resolution mechanism is activated. The INS prediction of positions and ionospheric effects are used to approximate the new integer ambiguities if cycle-slips are suspected. However, if at least three valid double-differences are available, the search is not necessary, and cycle-slips are fixed based on the known base and rover positions. In the case of total loss of lock, the four highest satellites are selected, and the search loop is activated. The ambiguity candidates are obtained from the double-differenced phase equation, where all terms, except for the ambiguity, are known from the filter prediction step. The standard deviation of the ambiguity candidate provided by the filter is obtained from the covariance matrix of the predicted positions and ionospheric estimates. This information is subsequently used to build the search interval for the best ambiguity. Positioning error growth in AIMS is about 10 cm for the horizontal components, and about 20 cm for the vertical one, after a 50-second loss of GPS lock, which still enables instantaneous ambiguity recovery after the GPS signal is recovered (Figures 4–5). The validation of the best integer candidate ambiguity is based on the ratio test (Eq. 1–2).

 

                                                                       (1)

 

where:                                                                             (2)

                                       

 - double-differenced L1, L2 and widelane residuals,

 - the smallest sum of squared residuals,

 - next to the smallest sum of squared residuals,

n – number of candidate solutions,

m – number of satellites at the epoch.

 

 

Figure 4. Positioning error growth during GPS data outage.

 

 

 

 

 

Figure 5. Differences between filter-predicted L1 double differences and GPS double

                   differences,  when no GPS data is applied in the filter.

 

In addition, an early rejection criterion is applied to the position candidates that are related to the ambiguity candidates within the search interval. If the candidate position is outside of the circle, centered at the prediction point, with radius equal to 3*Ö(DX²+DY²+DZ²), the candidate point is removed from the candidate list. DX, DY, and DZ denote separation in the X, Y, and Z coordinates, respectively, between the filter-predicted and the candidate position. This criterion reduces the number of candidates by 25–40%, which significantly shortens ambiguity search time. The search loop in our OTF ambiguity resolution scheme was tested for speed and reliability of ambiguity recovery as a function of the quality of ambiguity approximation provided by INS prediction. Our tests showed that all of the cycle slips could be fixed instantly if positioning quality was not worse that 1 m per coordinate. In all of the tests performed, the system locked to the correct ambiguities.

 

Tight GPS/INS integration, combined with ionospheric error modeling, provides robust OTF ambiguity resolution, and solves the problem of cycle-slips when the ratio of the number of slips on L1 and L2 is close to 77/60, and widelane change is equal to 1–2 cycles. Under such circumstances the traditional GPS technique of using geometry-free combination and epoch-by-epoch widelane may fail. Those are, for example, cases with slips of 4 cycles on L1 and 3 on L2, 5 on L1 and 4 on L2, or 9 on L1 and 7 on L2. In the AIMS GPS/INS system, if a geometry-free combination fails, the level of L1 residuals based on the Kalman filter prediction will easily indicate a potential cycle-slip. In general, GPS/INS integration significantly enhances the power of cycle-slip detection and fixing, providing high-quality, reliable bridging during GPS losses of lock, allowing for instantaneous ambiguity recovery after tracking is resumed.