A novel method for cycle slip detection and restoration of BDS carrier phase

The chief purpose of this present study is to develop a new approach to cycle slip detection and restoration of the BeiDou Navigation Satellite System (BDS) carrier phase. Based on the RAIM algorithm in the application of aviation safety, we extend the application to ensure the precision of the carrier phase. To stand-alone receivers of a low-cost single frequency, carrier phase data is made different between adjacent epochs. Then least-square residuals were to design a RAIM scheme with differential carrier phase data to find out the epoch with cycle slip, and finally repair cycle slip by Chebyshev polynomial. Results indicate that the approach can detect cycle slip effectively and both big cycle slip and small cycle slip could be repaired.


Introduction
Traditional carrier phase differential positioning (BDS) applications have constraints in terms of positioning precision and dependability.One of the most crucial factors is cycle slip.The goal of this thesis is to address this problem using RAIM and the Chebyshev polynomial.The RAIM algorithm arose from the requirement for aircraft safety.The great capability of defect detection leads to an increase in RAIM research.Its use has permeated all elements of satellite navigation applications [1].RAIM's research objectives might include both pseudo-range measuring and carrier phase measurement.The differential carrier phase eliminates the need for real-time integer ambiguity resolution and high accuracy, making it more suitable for RAIM applications.A new cycle-slip detection approach is provided by treating the cycle slip as an aberrant value and detecting it using RAIM.Cycle-slip detection is possible using a dual-frequency receiver.While the large hardware needed raises the system cost [2][3].The triple difference approach requires base station and satellite differences; however, the result is poor for tiny cycle slips (less than 5).The pseudo-range / carrier phase approach is restricted to pseudo-range noise, which is insufficient for single-frequency receivers.In a dynamic context, polynomial fitting and wavelet transform methods are ineffective [4].This study presents a mathematical derivation and analysis of the epoch differential carrier phase, as well as an RAIM solution to identify cycle slip.Finally, the Chebyshev polynomial is used to fix the cycle slip.

The solution to cycle slip problem BDS carrier phase measurement over time is written as:
( ) where λ is the wavelength, k ϕ is the carrier phase, k ρ is the distance from the receiver to satellite, Integer ambiguity is eliminated in Equation ( 2).The cycle slip only influences the current differential carrier phase and causes a significant jump.While the following differential carrier phase will not be influenced by the other.So the work of cycle slip detection becomes easier and the only thing to do is to find the jump.

Cycle slip detection based on RAIM
When a failure occurs, the control terminal should identify it and update satellite data.The likelihood of a satellite fault is so minimal that RAIM is under the premise of no multiple faults [7].The least squares residuals approach is utilized to build the RAIM algorithm in this research.

The least squares residuals method
The measurement of satellite positioning can be written in the linearized form: The least squares solution of Equation ( 3) is [8,9]: The residual vector is: ( ) . If the alarm occurs under the condition of no-fault satellites, there will be a false alarm.Once a probability of a false alarm FA P is given, a threshold t can be calculated: If the error b exists in the satellite i , a non-central parameter λ is [10,11]: The horizontal location error is defined as ( ) , where 1i

A and 2i
A are the first two components of error respectively.Equation ( 8) is: The value of the Horizontal Protection Level (HPL) is: ( ) If the value of HPL is bigger than the given value, the RAIM algorithm is unable to use.

Cycle slip repairing by Chebyshev polynomial
When the cycle slip detection is done, the Chebyshev polynomial is used to repair the cycle slip.The basis for the location (time) of the cycle slip, using adjacent carrier phase without cycle slip to make polynomial fitting, calculates the size of the cycle slip.
The observation equation of the carrier phase is [12,13]: where ( ) F t is the fitting function, ε is the random error of carrier phase observation.The Chebyshev expansion of Equation ( 12) is: 0 ( ) ( ) where τ is observation time, i C is the polynomial coefficient, and i is the order.Suppose k T is the time of cycle slip, observation time is transformed from The polynomial coefficient is: The error equation of the carrier phase is: where B is the vector of i T (τ), C is the polynomial coefficient, and l is the carrier phase.The least squares solution of Equation ( 16) is: T τ , and k ϕ can be calculated by substituting Equation (15) into Equations ( 8)- (10).The size of the cycle slip: N Δ can be obtained by rounding the difference between ' where ' k ϕ is data with cycle slip and k ϕ is fitting data.

Algorithm test and analysis
The probability of a false alarm is 0.001% in most cases and the standard deviation of phase noise is 1cm .So related parameters are set respectively: .RAIM availability is judged by calculating the value of HPL and different cut-off elevation angles are selected.The cut-off elevation angle needs to be small (7.5°) enough to ensure the availability of the algorithm.Some data are selected without a cycle slip.The test result is shown in Figure 1: Test 1200 epoch and test statistic are below the threshold (dotted line), which indicates no cycle slip.From the tenth epoch, 1 cycle, 2 cycles, 5 cycles, 10 cycles, 20 cycles, and 100 cycles are added respectively.For the differential carrier phase, the least squares residuals method is used and a lower limit D T is selected as threshold.Once there is a cycle slip, the test statistic will exceed the threshold, which indicates that there is a cycle slip and the epoch with the cycle slip is found.The test result is shown in Figure 2: In Figure 2, the cycle slip can be detected without omission and the success rate reaches 100%.The epoch difference of the carrier phase is measured with high precision.Thus, the test statistic is relatively small and the detection threshold is also small.The test statistic is so sensitive to fault that once there is a cycle slip, the test statistic changes drastically, which exceeds the threshold.The cycle slip can be detected clearly and the corresponding epoch can be found.The Chebyshev polynomial was used to repair the cycle slip and the calculation results are shown in Table 1.ϕ ) is the same as the value of the cycle slip.Both small and large cycle slips can be calculated using this approach.

Conclusion
In this study, cycle-slip detection is achieved by distinguishing between neighboring epochs and then devising a least-squares-residuals approach to detect the cycle slip.This is done using the RAIM algorithm.The next step is to choose an appropriate cutoff elevation angle to guarantee algorithm availability and enable efficient cycle slip detection.Once the cycle-slip epoch is identified, the cycle slip is fixed using a binding Chebyshev polynomial.This research presents a system that reduces costs by working with standalone and single-frequency receivers.
be the threshold.There is a fault when ( threshold was exceeded, there would be a cycle clip.availability Judgment of RAIM availability needs to be done before implementing the least squares residuals algorithm.When the gross error of the carrier phase appears,

Figure 2 .
Figure 2. Test result with cycle slip.
is the velocity of light, t

Table 1 .
Calculation of cycle slip.