Traditionally, the state vector of navigation filter is defined as [11]: (1) X = [ δ R e δ V e δ ψ e δ ω b δ A b δ T ] Twhere δR^e is position error, δV^e is velocity error, δψ^e is attitude error, δω^b is gyroscope bias error, δA^b is accelerometer bias error, δT = [δb δd]^T is the clock error vector, the superscript e and b represent ECEF frame and body frame respectively.

The corresponding system model of navigation filter is: (2) [ δ R ˙ e δ V ˙ e δ ψ ˙ e δ ω ˙ b δ A ˙ b δ T ] = [ 0 I 0 0 0 0 G - Ω ˜ e Ω ˜ e - 2 Ω ˜ e A ˜ e 0 C b e 0 0 0 - Ω ˜ e - C b e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Λ ] [ δ R e δ V e δ ψ e δ ω b δ A b δ T ] + [ w 1 w 2 w 3 w 4 w 5 w 6 ] where A ˜ e = [ 0 f z - f y - f z 0 f x f y - f x 0 ] , Ω ˜ e = [ 0 ω e 0 - ω e 0 0 0 0 0 ] , Λ = [ 0 1 0 0 ] , G = μ r 3 [ - 1 + 3 x 2 r 2 3 x y r 2 3 x z r 2 3 x y r 2 - 1 + 3 y 2 r 2 3 y z r 2 3 x z r 2 3 y z r 2 - 1 + 3 z 2 r 2 ]where [x y z] is the vehicle's current position in ECEF-frame, r = x 2 + y 2 + z 2 [ f x f y f z ], is the specific force vector in the body frame, C b e is the body-to-Earth-frame coordinate transformation matrix, and ω_e is the Earth's angular rate [20].

For the integrated navigation filter, the error components of pseudo range, pseudo range rate and pseudo range acceleration are used as measurements which are proportional to corresponding estimated states of prefilter [11]. Since the emphasis was on the prefilter model, the measurement equation was deliberately omitted here, for more details please refer to [11,21].

In federated ultra-tight GPS/INS integration, the prefilters are responsible for implementing code and carrier tracking, and also providing measurement information and corresponding measurement noise matrices for the integrated navigation filter [9–11]. For specific appliactions different prefilter models have been investigated [14,15]. A most commonly used prefilter model will be introduced here, the performance of which will be compared with that of a simplified prefilter model with an adaptive Kalman filter. The system model for the prefilter is written as follows [14]: (3) [ A ˙ δ τ ˙ δ ϕ ˙ δ f ˙ δ a ˙ ] = [ 0 0 0 0 0 0 0 0 λ L 1 2 π λ C A L 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 ] [ A δ τ δ ϕ δ f δ a ] + [ 1 0 0 0 0 0 1 λ L 1 2 π λ C A 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 ] [ w A w τ w ϕ w f w a ]where A is the normalized signal amplitude, δØ is the carrier phase tracking error, δf is the carrier frequency error, δa is the carrier frequency rate error, δτ is the code phase error, λ_L1 ≈ 0.19 m is the GPS L1 carrier wavelength, λ C A L 1 ≈ 293 m is the GPS CA code chip length, w_A is the process noise for the normalized signal amplitude, w_τ represents the code/carrier divergence, w_Ø represents the carrier phase noise due to the clock bias, w_f represents the carrier frequency noise due to the clock drift, and w_a represents the carrier phase acceleration noise due to the receiver dynamics.

The outputs of normalized early-minus-late envelope code discriminator and two-quadrant arctangent carrier discriminator are used as measurements, and the measurement equation is written as follows [14,15]: (4) Z = [ δ τ δ ϕ ] = [ 0 1 0 0 0 0 0 1 0 0 ] [ A δ τ δ ϕ δ f δ a ] = [ I E 2 + Q E 2 - I L 2 + Q L 2 I E 2 + Q E 2 + I L 2 + Q L 2 tan - 1 ( Q P / I P ) ] + [ ν 1 ν 2 ]where v₁ and v₂ are the output noises of code and carrier discriminators respectively.

As shown in Equation (3), the code phase error δτ is proportional to the carrier frequency error δf, and in order to decrease the state space dimension the “carrier aided code tracking process” can be implemented “outside” the prefilter. In addition, as shown in Equation (4), if the normalized discriminators were used the measurements would contain no information of estimated signal amplitude, and the observability of which would be a substantial problem [16]. In some cases, the baseband I/Q information were used as measurements directly [14,15] including the signal amplitude; however, a non-linear filter algorithm is required to implement code and carrier tracking [7,14,15]. Considering the above analysis, the code phase error and signal amplitude will be excluded from the state space of prefilter in the following discussion.

For the jth (j = 1, 2, …, N) channel, the system model of the simplified pre-filter is defined as follows (discrete form): (5) X k + 1 j = [ δ ϕ k + 1 j δ f k + 1 j δ a k + 1 j ] = [ 1 T 0.5 T 2 0 1 T 0 0 1 ] [ δ ϕ k j δ f k j δ a k j ] + [ w ϕ w f w a ]where δ ∅ k j (rad) is the carrier phase tracking error at time instant k, δ f k j (rad/s) is the carrier frequency error at time instant k, and δ a k j (rad/s²) is the carrier frequency rate error at time instant k.

The measurement is the output of two-quadrant arctangent carrier discriminator, and the corresponding measurement model is: (6) Z k j = tan - 1 ( Q p / I p ) = [ 1 0 0 ] X k j + ν 2

Since the navigation solution accuracy is insufficient for carrier phase tracking [14], the carrier phase is modified by channel filter directly, as shown in Figure 3. The carrier NCO control information is provided as follows: (7) { φ ˜ j , k + 1 - = rem ( φ ˜ j , k + + ( f I F + Δ f ˜ j , k ) ⋅ T , 2 π ) φ ˜ j , k + 1 + = rem ( φ ˜ j , k + 1 - X k j ( 1 ) + X k j ( 2 ) ⋅ T + 0.5 ⋅ X k j ( 3 ) ⋅ T 2 , 2 π )where f_IF represents the centre frequency of down-converted intermediate frequency(IF) COMPASS signal, Δf̃_j,k represents the carrier Doppler frequency, φ ˜ j , k + and φ ˜ j , k + 1 + are the carrier phases after prefilter update, φ ˜ j , k - and φ ˜ j , k + 1 - are the carrier phases before prefilter update, X k j represents the updated state of prefilter, the subscript k represents the current time instant, rem represents remainder after division operation, and T is the prefilter update period.

In Equation (7), the carrier Doppler frequency is obtained from corrected INS information and COMPASS ephemeris as follows: (8) Δ f ˜ j , k = f B 3 c 〈 LOS j , k , V j , k sat - V k usr 〉where f_B3 is the carrier frequency [1,2], c is light velocity, LOS_j,k is the unit line-of-sight vector from user to satellite j, V j , k sat and V k usr represent the satellite and user velocity in ECEF frame respectively, and <,> represents vector dot operation.

Since the code tracking related parameter has been excluded from the state space of this simplified prefilter model, the code tracking is controlled by the carrier tracking process. The code NCO control information is provided as: (9) { f j , k + 1 code = f C A B 3 - Δ f ˜ j , k f C A B 3 f B 3 Δ τ j , k = f code , k j f sample n j , k = [ l code Δ τ j , k ] τ ˜ j , k + 1 = τ ˜ j , k + n j , k ⋅ Δ τ j , k - l codewhere f C A B 3 is the code frequency [1,2], f_sample is the sampling frequency, l_code is the code length [1,2], and [A] represents the nearest integer greater than or equal to A.

Equation (8) shows that the carrier Doppler frequency is obtained from INS information directly and Equation (9) shows that code phase is controlled by code frequency and propagated forward sequentially. As the code frequency is obtained from carrier frequency Doppler, the feedback from traditional prefilter to code NCO computation is omitted as shown in Figure 3 with a dashed arrow.

Comparing Equations (3) and (5), it can be observed that the code/carrier divergence has been excluded from process noises in the simplified prefilter model resulting in degradation in the carrier tracking process [11,18]. Therefore, an adaptive Kalman filter was incorporated to compensate for the code/carrier divergence. Multi-model-based and innovation-based adaptive estimations are most commonly used adaptive Kalman filtering algorithms [19]. Since the simplified prefilter model is already determined an innovation-based adaptive estimation was adopted here. In innovation-based adaptive estimation, the process and measurement noise covariance matrices are adapted using innovation sequences; however, since the code/carrier divergence is a process noise the adaptation emphasis is on Q k j in this manuscript. The initial value of Q k j is calculated as follows: (10) Q ^ 0 j = [ 2 π 2 f B 3 2 ⋅ h 0 0 0 0 8 π 4 f B 3 2 ⋅ h - 2 0 0 0 ( 4 π 2 f B 3 2 / c 2 ) ⋅ d R 2 / d t 2 ]where dR²/dt² is the vehicle's light of sight acceleration, and h₀ and h₋₂ represent the white component and random walk of the oscillator frequency noise [22].

The state correction sequence is used to adapt the process noise covariance matrix Q k j which is computed as follows [19]: (11) Q ^ k j = 1 N ∑ m = m 0 k Δ X m j ( Δ X m j ) T + P k / k - Φ P k - 1 / k - 1 Φ Twhere Δ X k j is the state correction sequence which is computed as: (12) Δ X m j = Δ X m / m j - Δ X m / m - 1 j

While a standard Kalman filter, shown in Equation (13), is used to estimate the states of a traditional prefilter model, an adaptive Kalman filter is used for the simplified prefilter model: (13) { X k | k - 1 = Φ X k - 1 | k - 1 P k | k - 1 = Φ P k - 1 | k - 1 Φ T + Q ^ k - 1 K k = P k | k - 1 H T ( H P k | k - 1 H T + R ) - 1 X k | k = X k | k - 1 + K k ( Z k - H X k | k - 1 ) P k / k = ( I - K k H ) P k / k - 1

Associated matrix multiplication operations in above Kalman filter algorithms are implemented to compare the computational complexities of both filter models. Table 1 shows the number of multiplication operations in detail.

In Table 1, N is the window length as used in Equation (11). For example, if N = 5 was considered, a 1 - 128 + 45 503 ≈ 65.6 % reduction in total number of multiplications is achieved.

The COMPASS B3 frequency signal considered in this paper is BPSK modulated as GPS L1 frequency signal [1,2], the system model of navigation filter is almost in accordance with Equation (2) by just replacing λ_L1 with λ_B3 and λ C A L 1 with λ C A B 3. However, as in [11,23], only the delta pseudorange and delta pseudorange residuals are chosen as measurements to the navigation filter. For the jth channel, the relationship between measurements and estimated states of pre-filter is as follows: (14) [ Δ δ ρ k j Δ δ ρ ˙ k j ] = [ λ B 3 ⋅ ( δ ϕ k j - δ ϕ k - 1 j ) 2 π λ B 3 ⋅ ( δ f k j - δ f k - 1 j ) 2 π ] = [ λ B 3 ⋅ ( X k j ( 1 ) - X k - 1 j ( 1 ) ) 2 π λ B 3 ⋅ ( X k j ( 2 ) - X k - 1 j ( 2 ) ) 2 π ]where λ_B3 is the COMPASS B3 carrier wavelength, X k j and X k - 1 j are the estimated state of prefilter at time instant “k” and “k − 1”.

The corresponding measurement equation of integrated navigation filter is as follows (only a single channel is list): (15) [ Δ δ ρ k j Δ δ ρ ˙ k j ] = [ - Δ u T - T u pre T T 2 2 u pre T A ˜ e 0 T 2 2 u pre T C b e 0 T 0 - Δ u T - T u pre T A ˜ e 0 - T u pre T C b e 0 0 ] [ δ R e δ V e δ ψ e δ ω b δ A b δ b δ d ] + [ ν Δ δ ρ k j ν Δ δ ρ ˙ k j ]where u_pre is the unit LOS vector from receiver to satellite in previous filter update period, u is the unit LOS vector from receiver to satellite in current filter update period, Δu is the difference of above two LOS vectors.

The variances of measurement noises can be obtained from estimation error covariance matrices of pre-filters as follows [11]: (16) R k j = [ V a r ( ν Δ δ ρ k j ) 0 0 V a r ( ν Δ δ ρ ˙ k j ) ] = [ ( λ B 3 2 π ) 2 ( P k j ( 3 , 3 ) - P k - 1 j ( 3 , 3 ) - 2 T ⋅ P k - 1 j ( 2 , 3 ) - T 2 ⋅ P k - 1 j ( 1 , 3 ) ) 0 0 ( λ B 3 2 π ) 2 ( P k j ( 2 , 2 ) - P k - 1 j ( 2 , 2 ) - 2 T ⋅ P k - 1 j ( 1 , 2 ) ) ]where P k j and P k - 1 j are the estimation error covariance matrix of jth prefilter at time instant “k” and “k − 1”.

With the simplified prefilter model and adaptive Kalman filter, the federated COMPASS/INS integration implementation process is shown in Figure 4, where Φ represents the system matrix of jth prefilter defined in Equation (5), R prefilter j represents the measurement noise variance of jth prefilter, Z k nav - filter represents the measurements of integrated navigation filter at current instant k, and R k nav - filter represents the measurement noise covariance of integrated navigation filter at current instant k. The flow diagram consists of the following steps:

COMPASS IF data was sampled through a hardware sampling system, which will be discussed in “Test description” in detail.

For the simulation tests, a complex GNSS/INS signal hardware simulator was used to generate the COMPASS radio frequency(RF) signal and INS's accelerometers and gyroscopes data (INS data for short). A hardware sampling system was constructed to sample and store the digitized COMPASS IF signal and INS data. Data collection process for the simulation case is shown in Figure 5.

The data collection system consists of complex GNSS/INS signal hardware simulator, COMPASS B3 RF module, FCFR-PCIe9801 data sampling card [24], RCK-I-ET224-MC electronic disk [25] and FS725 rubidium clock [26]. The function of each component is as follows:

GNSS/INS hardware simulator provides synchronized COMPASS B3 frequency RF signal and INS data; the vehicle scenario and signal strength can be configured by users for their corresponding applications.

The ultra-tight COMPASS/INS integration algorithm was implemented in MATLAB, and the parameters defined in baseband signal processing part are listed in Table 2.

In Table 2, h₀ and h₋₂ are quantitative description of oscillator biases of COMPASS B3 RF module and FS725 rubidium clock. With the complex GNSS/INS signal hardware simulator, a reference trajectory with known dynamics was generated with 10 mg accelerometer bias errors and 200 deg/h gyroscope bias errors. The reference position and velocity in ECEF frame are shown in Figure 6(a), where “star” represents the initial position while ‘square’ represents the end. The COMPASS signal strength was kept at −130 dBm (C/N₀ ≈ 35 dB − Hz). The satellites 01, 02, 03, 04, 07 and 08 were visible during the simulation test, and the sky plot of the satellites is shown in Figure 6(b).

A field test was conducted to collect real COMPASS B3 frequency IF data and INS data. A COMPASS B3 frequency antenna and an INS were used to replace the complex GNSS/INS signal hardware simulator in simulation case. The corresponding data collection process in the field is shown in the Figure 9. The INS used in this case had 5 mg accelerometer bias errors and 150 deg/h gyroscope bias errors.

The antenna was located on the roof of an office building to guarantee a strong COMPASS signal (C/N₀ ≈ 45 dB − Hz), and the INS was collocated with the antenna. A high precision GPS receiver was used to provide the truth reference of the antenna which are −2,207,210.269, 5,171,488.332 and 3,000,859.525 m in the ECEF frame. The satellites 01, 03, 04, 05, 06, 08, 09 and 10 were visible during the field test, and the sky plot of the satellites is shown in Figure 10.