Cycle Slip Detection of Single-Frequency Measurements in Drone Platforms ^†

Abstract

For the precise operation of Unmanned Aerial Vehicles (UAV), Real-Time Kinematic (RTK) techniques of Global Navigation Satellite System (GNSS) have been used as a positioning source. In a typical drone platform with a low-cost multi-frequency receiver, there are several dual-frequency measurements and a small set of single-frequency measurements at one time. In this paper, considering the measurement characteristics, we introduce a novel method that aims to detect a cycle slip of a small set of single-frequency measurements using dual-frequency measurements of other satellites. The performance of the proposed single-frequency cycle slip detection methods is compared with a conventional Doppler-based approach using flight test data of a drone platform.

1. Introduction

The use of a carrier phase is mandatory for precise positioning using GNSS, such as real-time kinematic (RTK) positioning or precise point positioning (PPP) [1,2,3,4]. To implement a carrier phase-based positioning, an integer ambiguity must be resolved, and its continuity should be reliably monitored. If a discontinuity of the carrier phase called a cycle slip occurs, a new integer ambiguity of the carrier phase should be found [5,6], or a cycle slip should be repaired [7,8].

In the case of single-frequency measurements, it is difficult to eliminate nuisance terms in measurements like dual-frequency measurements. Here, a typical approach is to compare a range increment from a Doppler frequency and one from carrier phase measurements, referred to as Doppler-aided cycle slip detection (DACSD) [9,10]. While Doppler measurements are immune to cycle slips, they have a high noise level that makes it difficult to detect small cycle slips. Instead of using Doppler measurements, some other approaches used an inertial measurement unit, which would make a cycle slip detection problem more complicated [11,12].

Nowadays, dual-frequency GNSS receivers are typically installed in a small UAV platform. However, a lower-grade GNSS antenna and receiver on a small UAV platform and a dynamic motion with vibrations during flights often lead to unexpected cycle slips and poor detection performance. Also, a fair amount of single-frequency measurements, along with some dual-frequency measurements, are typically observed from a dual-frequency receiver on a small UAV platform.

In the case of mixed single- and dual-frequency measurements, rather than applying a Doppler-aided cycle slip detection method like the prior works, this study proposes a novel single-frequency cycle slip detection method based on UAV velocity and receiver clock drift estimates from dual-frequency carrier measurements, providing a lower noise level in cycle slip detection test statistics.

The structure of the paper is as follows. Section 2 presents the prior DACSD and introduces a novel cycle slip detection algorithm for the mixed single- and dual-frequency measurements. Section 3 presents flight test results, followed by the conclusions.

2. Cycle Slip Detection Methods and Proposed Approach for Single-Frequency Measurements

In this section, we briefly review DACSD as a prior representative method and suggest a cycle slip detection algorithm for single-frequency measurement that utilizes only the carrier phase measurement.

2.1. Doppler-Aided Cycle Slip Detection Method for Single-Frequency Measurements

GNSS measurement models of a receiver are

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\lambda }_i{\varphi }_i& =r-I_i+T+(c\delta t_u-c\delta t^s)+{\lambda }_i{N}_i+{\epsilon }_{\varphi },\hfill \\ \hfill d_i& =\frac{d}{dt}\left({\varphi }_i\right)=-\frac{1}{{\lambda }_i}\left[\dot{r}+(c\dot{\delta t_u}-c\dot{\delta t^s})+{\epsilon }_d\right]\hfill \end{array}\end{array}$$

(1)

where $\varphi$ represents a carrier measurement, r is the actual distance between a satellite and a user, I is an ionospheric delay error, and T is a tropospheric delay error. GNSS receiver and satellite clock errors are $\delta t_u$ and $\delta t^s$, respectively. c is the speed of light. $\lambda$ is the wavelength of a carrier, and N is an integer ambiguity in a carrier phase measurement. $\epsilon$ is the noise of each measurement. d is a Doppler measurement, which is equal to the time derivative of the carrier measurement. The subscript i denotes the frequency of each measurement.

Doppler measurements have been typically used to detect a cycle slip for single-frequency measurement. A Doppler is an apparent frequency change in a wave and is caused by the relative motion between a satellite and a user. An integrated Doppler frequency shift over some time period is equal to a carrier measurement change such that [13]

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \Delta {\varphi }_i={\varphi }_i\left(k\right)-{\varphi }_i(k-1)={\int }_{k-1}^k{f}_{d}dt,\end{array}\end{array}$$

(2)

where k is the current epoch, and $f_d$ is an instantaneous Doppler frequency between a receiver and a satellite. If the time interval between epochs is short, (2) can be expressed as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\int }_{k-1}^k{f}_{d}dt\approx \frac{d_i\left(k\right)+d_i(k-1)}{2}·\Delta t+{\epsilon }_{\Delta {\varphi }_i},\end{array}\end{array}$$

(3)

where $d_i$ is a coarse Doppler shift measured by a receiver, which is immune to a cycle slip. $\Delta t$ represents the time interval between successive epochs. ${\epsilon }_{\Delta {\varphi }_i}$ includes nuisance errors like carrier phase measurements and noise. The shorter the time interval between successive epochs, the smaller the error caused by the Doppler measurement [9].

If there is no cycle slip between two epochs, the difference of carrier phase measurement should be equal to the integrated Doppler measurements. However, these two values will have a significant difference when there is a cycle slip because the integer ambiguity in the carrier phase measurement at the two epochs is no longer the same. Note that the Doppler measurements, d, are not impacted by a carrier phase cycle slip.

Therefore, a cycle slip can be inferred from the following

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill T_{MDACSD}& =\Delta {\varphi }_i^s-\left\{\frac{d_i^s\left(k\right)+d_i^s(k-1)}{2}·\Delta t\right\},\hfill \end{array}\end{array}$$

(4)

where the superscript s refers to the satellite from which the signal is transmitted.

The DACSD method has the advantage of being able to detect cycle slips even in a single-frequency environment. Additionally, unlike the cycle slip detection method that detects through a differential method between frequencies or receivers, we can identify a specific frequency or measurement at which a cycle slip occurs. However, the noise level of the test statistics is high, making it difficult to detect a small cycle slip.

2.2. Proposed Single-Frequency Cycle Slip Detection for Small UAV Platforms

In this section, we propose a time-difference carrier phase (TDCP) combination method that better detects a cycle slip of single-frequency carrier measurements with a lower noise level than Doppler measurements. The TDCP combination method uses the differential value of the carrier phase measurement and can be expressed as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\lambda }_i·\frac{{\varphi }_i^s\left(k\right)-{\varphi }_i^s(k-1)}{\Delta t}& \approx {\lambda }_i{\dot{\varphi }}_i^s\hfill \\ \hfill & =\left({\mathbf{v}}^s-\mathbf{v}\right)·{\mathbf{1}}^s+\dot{b}+{\epsilon }_{{\dot{\varphi }}_i^s},\hfill \end{array}\end{array}$$

(5)

where ${\mathbf{v}}^s$ is the satellite velocity vector, $\mathbf{v}$ is the user velocity vector, and $\dot{b}$ is change rate in a receiver clock. The rates of ionospheric and tropospheric delay are assumed to be negligible. ${\epsilon }_{{\dot{\varphi }}_i^s}$ is the combined noise term during measurement interval, and ${\mathbf{1}}^s$ is a line-of-sight vector of the satellite s. (5) can be rearranged as below

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\lambda }_i·\frac{{\varphi }_i^s\left(k\right)-{\varphi }_i^s(k-1)}{\Delta t}-{\mathbf{v}}^s·{\mathbf{1}}^s=-\mathbf{v}·{\mathbf{1}}^s+\dot{b}+{\epsilon }_{{\dot{\varphi }}_i^s},\end{array}\end{array}$$

(6)

Here, the left-hand side of (6) represents a value that we can calculate or obtain through measurements. The receiver velocity vector and receiver time drift value on the right-hand side are unknown and need to be estimated with a set of dual frequency measurements. When (6) is expressed in a matrix form with two frequencies, it can be represented as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathbf{Y}}_{\mathrm{dual}}& =\mathbf{GX}+{\epsilon }_{{\dot{\varphi }}^s},\hfill \end{array}\end{array}$$

(7)

where ${\mathbf{Y}}_{\mathrm{dual}}=$ $\left[\begin{array}{c}{\lambda }_i·\frac{{\varphi }_1^1\left(k\right)-{\varphi }_1^1(k-1)}{\Delta t}-{\mathbf{v}}^1·{\mathbf{1}}^1\\ ⋮\\ {\lambda }_i·\frac{{\varphi }_2^m\left(k\right)-{\varphi }_2^m(k-1)}{\Delta t}-{\mathbf{v}}^m·{\mathbf{1}}^m\end{array}\right]$, G = $\left[\begin{array}{cc}-{\mathbf{1}}^1& 1\\ ⋮& ⋮\\ -{\mathbf{1}}^m& 1\end{array}\right]$, and X = $\left[\begin{array}{c}\mathbf{v}\\ \dot{b}\end{array}\right]$. Here, X is estimated by pseudo-inverse using matrix Y and matrix G

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \widehat{\mathbf{X}}& ={\left({\mathbf{G}}^T\mathbf{G}\right)}^{-1}{\mathbf{G}}^T{\mathbf{Y}}_{\mathrm{dual}}.\hfill \end{array}\end{array}$$

(8)

In this method, the measurements used for the user velocity and receiver clock error rate estimation should be free of any cycle slips. Therefore, we only used m dual-frequency measurements that have passed the dual-frequency carrier phase (DFCP) combination, which is often referred as a geometry-free method [14]. This is because the data that have passed through DFCP are highly likely to be free of cycle slips. Then, the $\widehat{\mathbf{X}}$ is applied for the cycle slip detection of s single-frequency carrier phase measurement as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill T_{TDCP}={\lambda }_i·\frac{{\varphi }_i^s\left(k\right)-{\varphi }_i^s(k-1)}{\Delta t}-{\mathbf{v}}^s·{\mathbf{1}}^s-\left[\begin{array}{cc}-{\mathbf{1}}^s& 1\end{array}\right]·\widehat{\mathbf{X}}.\end{array}\end{array}$$

(9)

When there are no cycle slips, $T_{TDCP}$ includes noise in carrier phase as well as estimated errors in $\widehat{\mathbf{X}}$. An experimental distribution of $T_{TDCP}$ is addressed in later sections.

The TDCP combination method has the advantage of being able to detect a small cycle slip even in single-frequency measurements. In contrast to the DFCP method, it is possible to specify the measurement in which cycle slip has occurred. Compared to DACSD, the noise level of the proposed TCDP is reduced by using the differential value of the carrier measurement. For this reason, the TDCP approach detects a smaller cycle slip and its false alarm is relatively lower than the one of DACSD.

3. Results

We collected data via flight tests and set a threshold for each cycle slip method with the flight test data. The flight tests were conducted at Hongik University’s 4th Industrial Revolution Campus located in Hwaseong, South Korea. During flight, dual-frequency measurements of GPS, Galileo, and Beidou constellations were collected at 1Hz rate. Each flight took approximately 10 min to complete, and a total of 45 flight tests were conducted. Our testbed of a small UAS was based on a S550 frame shown in Figure 1a. The receiver and antenna onboard the testbed were Ublox ZED-F9P and Trimble AV17, respectively. The speed of the UAV ranged from 1 to 6 m/s.

Figure 2 shows the test statistics of DACSD and TDCP based on the flight test data. To establish a criterion for determining the occurrence of actual cycle slips, we adopted the results of the DFCP method by using dual-frequency carrier phase measurements only.

Table 1. Performance comparison of MDACSD and DFCP in the aspects of false alarm and missed detection of flight data.

Method	False Alarm Rate	Missed Detection Rate	Data
DFCP	0%	0%	738,612
DACSD	22.4913%	0.058669%	738,612
TDCP	0%	0.025206%	738,612

lists the number of false alarms and missed detection of cycle slips resulted from the DACSD and TDCP methods. There were no false alarms in the TDCP combination. On the other hand, in the case of DACSD, it can be observed that false alarms have occurred in many cases.

Table 2. Performance comparison of DACSD and DFCP in the aspects of missed detection through injected cycle slips.

Dataset	Injected Cycle Slip of 1 Wavelength		Injected Cycle Slip of 2 Wavelength		No. of Simulation
	TDCP	DACSD	TDCP	DACSD
1	952	0	952	291	959
2	711	0	711	302	711
3	1190	0	1190	288	1190
4	1080	0	1080	318	1080

shows the number of false alarms and missed detection of simulated cycle slips injected to flight test data. For the smaller cycle slip of one wavelength, TDCP performs significantly better than DACSD.

4. Conclusions

We introduced a novel cycle slip detection method for single-frequency measurements using mixed dual- and single-frequency measurements. This method could result in more precise cycle slip test statistics compared to previous Doppler-based approaches. Our test results based on flight test data confirmed that the proposed TDCP approach provides lower false alarms and missed detection rates than Dopper-based approaches. Therefore, using this approach is advantageous in the RTK process as it allows a tighter cycle slip check on the set of single-frequency measurements obtained along with dual-frequency measurements.