1. Introduction
With the rapid development of automatic driving as well as advanced vehicle control and safety systems in urban environments, the requirements in terms of navigation accuracy, continuity, and availability are increasing. Real-time kinematic (RTK) in surveying and mapping is a precise positioning technique by processing double-differenced carrier-phase measurements of Global Navigation Satellite System (GNSS) signals [
1]. The double-difference technique can remove the error sources that are highly correlated over space and time, such as ephemeris errors. Range information with remained multipath and receiver noise can achieve cm-level accuracy only after the integer ambiguities in the carrier-phase measurements have been resolved correctly. Thus, integer ambiguity resolution (AR) becomes a critical part of the RTK technique. The conventional AR techniques are classified into three categories according to the usage of receiver measurements [
2]: ambiguity resolution in the measurement domain [
3], search technique in the coordinate or position domain [
4], and search technique in the ambiguity domain [
5,
6]. The third class based on the theory of integer least-squares (ILS) is most widely used; it contains abundant techniques, such as the Fast Ambiguity Resolution Approach (FARA) [
5], the Least-Squares Ambiguity Decorrelation Adjustment (LAMBDA) [
6], the Least-Squares Ambiguity Search Technique (LSAST) [
7]. A typical AR procedure in the ambiguity domain comprises two steps. Firstly, the float solution of ambiguities and the corresponding variance–covariance matrices are obtained by real-valued least squares estimation. Secondly, the ambiguity search process is employed to fix the integer values and adjust the baseline solution. The performance of the AR procedure in the ambiguity domain depends on a precise float estimation. Thus, it is susceptible to the number of available satellites and the quality of observations.
In harsh urban environments, the surrounding obstacles may bring two representative predicaments to the reception of satellite signals: blockages and multipath. GNSS signal blockages always make the RTK techniques suffer from a poor geometry and a reduction in the number of satellites in view. However, this dilemma has improved with the development of other constellations such as BeiDou Navigation Satellite System (BDS), which has been providing Positioning, Navigation, and Timing (PNT) services in the Asia-Pacific region since December 2012 [
8]. Taking into account the use of multi-constellation systems, the number of satellites in view for users has effectively increased. Meanwhile, there is potential for the positioning accuracy and stability of RTK to be improved with the combination of GPS and BDS. In addition to solving the reduction in the total number of satellites, the ideal RTK technology also needs to deal with the frequent satellite visibility changes caused by GNSS signal blockages. For carrier-phase measurements, once the satellite signals need to be reacquired by the receiver due to changes of the availability status, the corresponding integer ambiguities have to go through a period of re-searching and fixing. In the interim, unless the fixed solution can be obtained at the time of cycle-slip occurrence, at best, only the accuracy of the float solution can be provided. For the AR techniques in the ambiguity domain, both the float solution and the success rate impose requirements on the accuracy of the pseudorange. Unfortunately, the second predicament in satellite observations under urban environments is that due to the multipath generated by the reflections on building structures, there exist large correlation errors in pseudorange measurements. Therefore, the standalone GNSS positioning methods such as RTK cannot meet the requirements under GNSS-challenged environments.
An alternative way to enhance the system performance is exploiting other sensors that are aiding. Inertial Navigation System (INS) and GNSS integrations are most widely applied because of their complementary properties. The Inertial Navigation System (INS) [
9], vision system [
10,
11], and Wi-Fi [
12] all have been take into consideration. Among the above, INS measurements are immune to external interference. They can precisely provide high-rate velocity and attitude, and give an ephemeral positioning by dead reckoning when satellite observation is absent. In turn, the accurate positioning results obtained by RTK can correct inertial errors accumulated over time. In an RTK/INS integrated navigation system, the overall accuracy still depends on the positioning result of RTK; thus, the purpose of the effective integration is to exploit inertial information to increase the search speed and success rate of the AR technique. According to the measurement model, RTK/INS integrated systems can be classified as loosely and tightly coupled structures. In the case of loosely coupled techniques, RTK and INS calculate the position and velocity results independently [
13]; thus, they fail to enhance the accuracy and availability by providing additional information for the AR procedure.
By contrast, in tightly coupled RTK/INS integrations, using inertial information as an accurate position prediction in an AR procedure can effectively improve the performance of the navigation system when the satellite observation condition is poor. Conventional tightly coupled RTK/INS algorithms generally include two stages in order to ensure that their own precision is identical to the RTK results. The first stage is to use the INS measurements to assist the AR process, and the second stage is determination of whether the AR results are valid. If AR validation failed, the pseudorange measurements are utilized to correct the state variables, including position, velocity, attitude, and the biases of inertial sensors. Otherwise, the carrier-phase measurements with fixed solutions are used as more accurate ranging information to complete the correction instead.
There are two different implementations of the conventional tightly coupled RTK/INS algorithms. Both of them use an extended Kalman filter (EKF) to correct state variables and solve integer ambiguity estimation in the ambiguity domain. The first kind of tightly coupled structure is realized by fusing raw pseudorange and carrier-phase measurements as well as INS navigation solutions through an EKF; then, the float ambiguities can be estimated as filter states [
14]. Subsequently, based on the information of the filter covariance matrix, a search process such as LAMBDA is performed in the ambiguity domain. The resulting fixed solution is then used to correct the remaining state variables. This implementation exploits the ambiguity constraints between adjacent epochs, making it practicable for consumer-grade receivers to achieve the desired accuracy in open sky conditions. However, adding float ambiguities to filter states also has one obvious drawback. Since the ambiguities at the previous moment affect both current state estimation and the accuracy of the carrier-phase measurement as ranging information, this algorithm is very sensitive to cycle slips in the receiver’s phase-locked loop (PLL). If the ambiguity estimation for a certain satellite is biased, then this biased estimation will affect the entire filtering process until the corresponding satellite becomes unavailable. Furthermore, in the case of frequent changes in satellite visibility, ambiguities at each GNSS epoch must be researched and fixed; thus, the benefits brought by the invariance constraint of the ambiguities will no longer exist.
In order to free the ambiguities from filter states, the second kind of tightly coupled implementation uses the Least-Squares (LS) estimation to obtain the float solution of ambiguities instead of the EKF, and its next integer search and correction steps are the same as the implementation mentioned above [
15]. Compared with the AR procedure in GNSS standalone systems, not only are the pseudorange and carrier-phase measurements utilized as observables, but also the position vectors derived from the INS solution are taken as additional observables to solve this LS problem. This inertial-aided AR algorithm is expected to obtain more precise estimation and reduce the search space under INS motion constraints. Moreover, it only uses the observations of the current epoch, so the second kind of implementation is not affected by the intermittent GNSS reception. However, since both implementations search integer solutions in the ambiguity domain, the success rate of AR is unavoidably dependent on the quality of pseudorange measurements. When the pseudorange measurements have large errors due to multipath, the accuracy of the above two kinds of tight integration techniques will be degraded.
Therefore, in order to obtain high precision positioning results continuously in urban environments where satellites frequently change and large pseudorange errors exist, this paper proposes a new implementation of the tightly coupled RTK/INS integration. The new tightly-coupled algorithm proposed in this paper still contains the two stages: INS-aided AR and a correction step. However, in order to avoid the influence of large pseudorange errors in float ambiguity estimation, we perform the AR process in the Position Domain. Thus, the proposed algorithm will be abbreviated as “ARPD-RTK/INS” in the following sections. In our INS-aided AR process, the INS solution is used to provide a more accurate initial value of location and a more efficient search volume due to the use of motion constraints, when compared with the standalone-GNSS AR techniques in the position domain [
4,
12]. To get rid of the influence of the pseudorange error, we determine to use the posterior probability density function (PDF) to represent the weights of candidate points. Taking into account the integer property of the carrier-cycle ambiguity, the posterior PDF is only updated by the double-differenced carrier-phase measurements of GPS and BDS, which does not involve pseudorange measurements. Furthermore, we expect that a suboptimal integer solution can be obtained in the AR process, which is able to provide more precise ranging information than the pseudorange for the ensuing correction step, even if it is not necessarily the fixed integer solution. Thus, the ultimate solution is obtained by the weighted average of candidate ambiguity vectors instead of finding the fixed integer solution with the maximum probability.
This paper is organized as follows.
Section 2 describes the architecture of conventional RTK/INS integrations and the transition and observation models on which to base our positioning process.
Section 3 illustrates the flow chart of the new ARPD-RTK/INS algorithm, and explains the steps to solve the integer ambiguities in the position domain.
Section 4 presents the experimental results of the proposed algorithm, and evaluates the performance by comparing with other algorithms on both real and simulated datasets.
Section 5 summarizes our conclusions and future research directions.
2. The Theoretical Basis of Tightly-Coupled INS/RTK Integrations
2.1. Definition of State Vectors and Motion Model
In the RTK/INS integrated navigation systems, in order to clearly illustrate the positioning procedure, we first define the state vectors as well as the motion and observation models. Categorized according to different uses, there are three types of state vectors: position vector, inertial parameter vector, and ambiguity vector.
where
k is the discrete-time index;
is the position vector at time
k, which only contains three-dimensional position information; and
is the inertial parameter vector, which contains velocity, Euler angles, accelerometer bias, and gyroscope bias. The inertial parameter vector only exists when the navigation system integrates with the inertial navigation system.
is the ambiguity vector, and
and
represent the single-differenced integer ambiguities of satellite
s at
L1 and
L2 frequencies separately, obtained by the difference between rover
r and base station
b. is the total number of observable satellites at time
k.
Simultaneously, the motion equation is as follows:
where the function
denotes the transition model,
is the system noise whose distribution is independent of time. The transition model depends on the INS measurements. Using a strap-down inertial system, we need to transform the specific force vector and angular rate measured in the body frame into the Earth-centered Earth-fixed (ECEF) frame. Then, the kinematic equations are as follows [
17]:
where
denotes the rotational transformation from the body frame to the ECEF frame, and
and
denote the Earth relative velocity and position in the ECEF frame, respectively. The equations above give their time derivatives. The vector
is the specific force that can be obtained from INS measurements directly.
is the local gravity vector related to the current position
.
is the skew-symmetric matrix of the Earth rotation vector, and
is the skew-symmetric matrix of the angular rate
. It is noteworthy that the specific force and angular rate measurements are accompanied by noise. The simplified measurement models of the accelerometer and gyroscope are:
where
and
are the output vectors of inertial sensors;
and
represent the random noise; and
and
respectively represent the accelerometer and gyroscope biases that are contained in the state vector. The biases are modeled as constant vectors and corrected by the GNSS solutions.
is the identity matrix, and
and
denote the matrices of the scale factor and cross-coupling errors for accelerometer and gyroscope, respectively.
2.2. Observation Model
Involving the new GNSS observations, this subsection discusses the definition of the observation model. At each GNSS epoch, a new observation
is given by:
The function presents the observation model. is the measurement noise whose distribution is independent of not only time but also the system noise. In the positioning process, the observation vector usually contains a dual-frequency carrier phase as well as a code pseudorange, and eliminates the noise sources that are highly correlated over space by the double-differencing method, as given by Equation (11).
Here, represents the double-differenced carrier-phase vector at L1 frequency with a selected reference satellite t. Analogously, is the double-differenced pseudorange vector, and and denote double-differenced integer ambiguities and distances, respectively. , , and are the corresponding double-differenced carrier phase, pseudorange, and integer ambiguities at L2 frequency, respectively. ε is the measurement noise of the relevant observation.
Taking into account the clock bias between GPS and BDS, the reference satellite t of each constellation needs to be determined separately. Satellites at lower elevation angles are more likely to be blocked by surrounding buildings or interfered by larger atmospheric errors. To alleviate the hand-over problem, we choose the satellite with the highest elevation angle as a reference satellite.
It should be noted that the conventional tightly coupled RTK/INS algorithms often use both carrier-phase and pseudorange measurements as the observation vector when there is no fixed solution. However, the new algorithm proposed in this paper, in order to avoid the negative effects caused by large pseudorange errors, only uses the first two items of Equation (11), i.e., double-differenced carrier phases, to update state vectors.
2.3. Conventional Tightly Coupled RTK/INS Integrations with AR in the Ambiguity Domain
In this subsection, we introduce two conventional RTK/INS integrated techniques, both of which complete the AR in the ambiguity domain. As we mentioned above, the typical tightly coupled RTK/INS algorithms generally include two stages: INS-aided AR, and the correction process. The total accuracy is guaranteed by the ability to successfully find a fixed solution. According to the different way that INS participates in the AR process, the existing tightly coupled algorithm is divided into two categories [
10,
11], as mentioned in the introduction
Section 1. The first implementation is depicted in
Figure 1. It uses INS measurements to help estimate float ambiguities through an EKF, and then searches for integer solutions in the ambiguity domain by LAMBDA. Compared with the ambiguity covariance matrix obtained only from the double-differenced carrier phase and pseudo-range observations, this tightly coupled algorithm takes into account a priori motion information, so the covariance matrix obtained by EKF has a more efficient search space. The filter states include position, velocity, attitude, biases of inertial sensors, and float ambiguities. Therefore, if no change in the satellite tracking status is detected, the integer solutions will be used as the initial values of filter states at the next moment, as shown by the red arrow in
Figure 1. However, if there is an undetected cycle slip, the incorrect initial value can cause the subsequent filtering process to fail. At the same time, the accuracy of this tightly-coupled algorithm is affected by the quality of pseudorange measurements, which can cause a bias in the ambiguity estimation of a continuous tracking satellite.
The second kind of implementation is illustrated in
Figure 2. Float ambiguities are obtained by least squares estimation. The next step is to search for integer solutions in the ambiguity domain and correct the state variables using the fixed carrier-phase observations. If no valid integer solution is found, pseudorange observations will be used to correct the state variables by an EKF instead. For the float ambiguity estimation process, there are not only double-differenced carrier-phase and pseudorange observations as constraints, but also the positioning result of the INS solution as additional observables, as shown by the green arrow in
Figure 2. The inertial navigation assistance makes it practicable to complete the AR process when there are fewer than four satellites in a single constellation system. In contrast to the first kind of implementation, this algorithm only utilizes the current epoch observations in the AR process, so it is not affected by frequent changes in satellite visibility. From the second flowchart, it also can be clearly concluded that the quality of the pseudorange observation directly affects the success rate when searching for a fixed solution in the ambiguity domain. To avoid this undesirable influence, our new ARPD-RTK/INS algorithm searches for integer ambiguities in the position domain.