The carrier phase-based Global Positioning System (GPS) has become an essential technique for a wide range of precise positioning applications, such as kinematic positioning and vehicle navigation and guidance. However, there are several constraints on the use of this technique. Firstly, an assurance of satellite signal line-of-sight is a critical requirement, yet GPS signals can be obstructed by buildings, bridges, and even tree foliage. Under these circumstances, the GPS system is unable to continuously carry out its positioning task because of the insufficient number of tracked satellites. Furthermore, the impact of baseline-dependent GPS errors, such as orbit uncertainties, and atmospheric effects, further constrains the applicable baseline length between reference and mobile user receiver to perhaps 10–15 km. These constraints have led to the development of several network-based GPS kinematic positioning techniques, including the virtual reference station approach [1], and the area correction parameter techniques [2]. Additional limitations of using GPS are the relatively low carrier phase data output rate (e.g., typically 1–10 Hz), and the need to deploy more than one GPS antenna to derive full attitude information. Such constraints can be so restrictive that they may hinder the widespread adoption of carrier phases-based GPS techniques for many precise positioning and navigation applications.

Some of the restrictions of carrier phase-based GPS technology can be addressed by its integration with an inertial navigation system (INS). The INS is a self-contained navigation unit providing position, velocity and attitude information based on measurements by its ensemble of sensors (typically a set of accelerometers and gyroscopes). Its disadvantage is that INS navigation accuracy deteriorates rapidly with time due to the presence of sensor biases and a double-integration mechanisation algorithm. However, an appropriate integration of INS with GPS can take advantage of each technology’s strengths, delivering a high data-rate complete navigation solution with both superior short-term and long-term accuracies [3–5]. Even during a GPS signal blockage, it is still possible to carry out positioning in the INS stand-alone mode for short periods of time; the so-called INS bridging mode. The critical duration of the bridging capability varies as a function of the quality of the INS [6]. Nevertheless, the baseline length constraint cannot be overcome using the integrated GPS/INS approach as the quality of double-differenced GPS observations decreases with increasing baseline length. A solution to this problem would be to combine measurements from a number of reference receivers to model the baseline-dependent GPS errors, through applying various parameterisation techniques [7–9]. The use of multiple GPS reference receivers therefore permits the baseline lengths to be much longer than that in the single reference station scenario. As a result, all of the aforementioned constrains of precise kinematic GPS technique would be effectively addressed by augmentation of GPS/INS with multiple reference station carrier phase data.

This paper proposes a GPS/INS integration scheme with multiple GPS references for use in highly precise long-baseline kinematic positioning. This approach utilises measurements from multiple reference stations to model the GPS baseline-dependent errors and to apply them to mobile receiver observations before updating the integration filter. Hence, the applicable baseline length is extended with centimetre-level positioning accuracy. After discussing the concept of the employment of multiple reference stations with the integrated GPS/INS, some technical issues required for the implementation of the algorithm are described. This is followed by the description of measurement simulations and test results with an emphasis on the effects of employing multiple reference stations in the integrated GPS/INS data processing.

The proposed integration approach is comprised of two components, namely the stationary reference measurement processing and the mobile positioning instrumentation. The role of the multiple GPS reference stations is to generate the so-called network corrections that model the baseline-dependent errors at a mobile station, whereas the mobile platform component is an integrated GPS/INS device. As illustrated in Figure 1, the mobile component consists of INS mechanisation, GPS carrier phase processing, and the Kalman filter that estimates the INS navigation and sensor errors. Using such an integration scheme, high positioning accuracy can be maintained through the continuous calibration/estimation of the INS errors. The network correction terms substantially reduce the baseline-dependent GPS errors at the mobile station compared with the conventional GPS/INS integration approach based on a single reference station.

The GPS baseline-dependent errors are estimated based on the pre-determined coordinates of the reference stations. It is prerequisite to correctly resolve the double differenced (DD) carrier phase ambiguities between the reference stations to generate the accurate corrections. For real-time measurements with baselines over a few tens of kilometres, several integer ambiguity resolution algorithms have been proposed [10–12]. These techniques are characterised by an attempt to form linear combinations with longer wavelengths and less noise, using the L1 and L2 carrier phase measurements. Due to the considerable magnitude of the ionospheric effect in the L1 and L2 carrier phases, the ambiguities of the wide-lane linear combinations are resolved first, and then the L1 ambiguities are fixed from the ionospheric-free measurements. This is followed by computing the L2 ambiguities from those of the resolved L1 and wide-lane combinations. The ambiguity estimation and validation procedure are applied for each step of the resolution. While the least squares ambiguity decorrelation adjustment (LAMBDA) method is used for the ambiguity estimation, the W-ratio test is applied for the validation [13,14].

The performance of positioning and navigation with the multiple GPS reference stations is largely dependent on the ability of an algorithm to separate the site-dependent errors from the DD residuals computed by subtracting the ambiguities and geometric distances from the DD measurements. Figure 2 illustrates an estimation procedure of the baseline-dependent errors based on a Kalman filtering. The 4-states model has been implemented to describe the dynamics of the baseline-dependent errors (the 1st–3rd state), and the multipath error (the 4th state). As given in Equation (1), a position, velocity, and acceleration (PVA) model is applied for the first three states, whereas the multipath (the 4th state) is modelled as a first-order Markov process [15–16]: (1) [ x ˙ L 1 ( t ) x ˙ L 2 ( t ) ] = [ F 0 0 F ] [ x L 1 ( t ) x L 2 ( t ) ] + [ w L 1 ( t ) w L 2 ( t ) ]

A detailed expression of the dynamic matrix can be given by: (2) F = [ 1 T T 2 2 0 0 1 T 0 0 0 1 0 0 0 0 e − α T ]where T is the sampling interval and α is the correction time.

The error states in Equation (1) are as follows: (3) x L 1 = [ ∇ Δ P 1 ∇ Δ V 1 ∇ Δ A 1 ∇ Δ M 1 ] T (4) x L 2 = [ ∇ Δ P 2 ∇ Δ V 2 ∇ Δ A 2 ∇ Δ M 2 ] Twhere ∇Δ_P is the DD baseline-dependent error, ∇Δ_V and ∇Δ_A are the velocity and the acceleration of the error respectively, and ∇Δ_M is the multipath.

The so-called corrections, modelled by interpolating the baseline-dependent errors at the reference stations with respect to the position of a mobile station, should be applied for mobile GPS receiver measurements to reduce these errors. Over the past decades, a number of interpolation methods have been proposed, including the linear combination model, distance-based linear interpolation method, and linear interpolation method [17]. However, the performances of these methods are equivalent according to Dai et al. [18]. To interpolate the baseline-dependent errors at three or more stations, the coefficients α and β should be computed using the following equation [9]: (5) [ α β ] = ( A T A ) − 1 A T Cwhere: (6) A = [ Δ X R 1 Δ Y R 1 Δ X R 2 Δ Y R 2 ⋮ ⋮ Δ X R ( n − 1 ) Δ Y R ( n − 1 ) ] (7) C = [ ∇ Δ P , R1 ∇ Δ P , R1 ⋮ ∇ Δ P , R ( n − 1 ) ]as well as ΔX and ΔY are the plane coordinate differences referred to the master reference station (MR), the subscripts R_i indicate the reference station R_i, and the elements of the vector C are the baseline-dependent errors.

The mobile platform within the coverage of the reference network can apply the following 2-D linear model to interpolate the baseline-dependent errors: (8) ∇ Δ NC = α Δ X MR + β Δ Y MR

The integration of GPS with INS has been implemented using a tightly-coupled Kalman filtering technique, which utilises a single filter to process all the data in the DD measurements domain. The integrated processing procedure employed in this study is presented in Figure 1. The error state vector for the filter includes the parameters of the navigation solution and the INS sensor errors. The psi-angle model [e.g., Equations (9–11)] is used to describe the behaviour of INS navigation errors [19]. (9) δ r ˙ n = − Ω en n δ r n + δ ν n (10) δ ν ˙ n = − ( Ω en n + 2 Ω ie n ) δ ν n + δ ∇ n − Ω ψ f n (11) ψ ˙ n = − Ω in n ψ n + δ ɛ nwhere δr, δν, δψ, and δψ are the position, velocity, and attitude error vector respectively, Ω is the skew-system form of the frame rotation rate vector, δ∇ is the accelerometer error vector, δε is the gyro drift vector, and the superscripts and subscripts indicate coordinate frames (e.g., inertial, ECEF, and navigation). The biases and scale factors of the INS sensors are modelled by a random bias and the first-order Gauss-Markov process respectively. Hence, a total of 21 error states are considered in this research. For a more detailed discussion, see, for example, Grejer-Brezinska et al. [20], Da [21], and Lee [22].

In this GPS/INS integration, the DD GPS carrier phases are used to update the Kalman filter to estimate the navigation and INS sensor errors. The DD measurements are formed by differencing between two single differences (SD) across two different satellites at each epoch. In the case of a medium baseline up to 100 km, the measurements can be mathematically represented as: (12) λ ⋅ Δ ∇ ϕ = Δ ∇ ρ + Δ ∇ d ρ − Δ ∇ d i + Δ ∇ d r ︸ the distance − dependent errors + λ ⋅ Δ ∇ N + Δ ∇ ɛwhere ∇ denotes differencing between two satellites, so that Δ∇ represents a double-differencing, λ is the wave length of the carrier phase, ρ is the true range or geometric range, dρ is the satellite orbit uncertainties, di is the ionospheric effect, dr is the tropospheric delay, N is the integer-ambiguity, and ε is the noise.

By applying the network correction, Equation (12) can be re-written as follows: (13) λ ⋅ Δ ∇ ϕ = Δ ∇ ρ + Δ ∇ N + Δ ∇ ɛ

However, Equation (13) contains an integer ambiguity term, which should be correctly resolved to ensure centimetre-level positioning accuracy. Here, an INS-aided ambiguity resolution scheme utilising the INS-predicted position in the float ambiguity estimation has been implemented. This approach enhances the performance of the ambiguity resolution by improving the precision of the DD range estimation. More details on this approach can be found in Lee et al. [23]. Equation (13) with the ambiguity resolved is applied for updating the Kalman filter to estimate its error states, which are fed back to correct the inertial solution and sensor measurements. However, it is important to note that the term Δ∇ε in Equation (13) contains not only the noise, but also the un-modelled baseline-dependent errors, which means that the positioning accuracy of the GPS/INS integration is determined by the quality of the corrections modelled by the multiple reference stations.

Three sets of simulated GPS and INS measurements have been processed in this section to test the performance of the algorithms implemented and to evaluate the achievable accuracy of the position and attitude parameter estimation.

An integration of GPS/INS with multiple-reference stations for long-baseline kinematic positioning has been proposed in this paper, with the objective of improving the accuracy of the position estimation in the case of baseline lengths up to about 100 km. The algorithms concerned with implementing the proposed scheme are addressed, which include the AR between reference stations, the estimation of the baseline-dependent errors, the network correction modelling, and the integrated GPS/INS processing with an emphasis on the Kalman filter design. Three GPS and INS simulated measurement sets on mobile platforms were processed to characterise not only the performance of the algorithms implemented, but also the achievable accuracy of the position and attitude estimation. The results show that the accuracy of the position component was significantly improved to be a few centimetres through modelling the baseline-dependent errors based on the multiple reference stations. Because the accuracy of the attitude estimation based on INS depends primarily on the quality of the gyro sensors, marginal improvement was observed in the component. In conclusion, more research on quality control algorithms for the integration and further tests with real data sets will be carried out in the near future.