Submitted to Journal of Surveying Engineering
Extended Receiver Autonomous Integrity Monitoring (eRAIM) for GNSS/INS Integration Steve Hewitson and Jiling Wang Abstract The integration of Globe Navigation Satellite System (GNSS) with Inertial Navigation System (INS) is being heavily investigated as it can deliver more robust and reliable systems than either of the individual systems. In order to ensure the integrity of navigation solutions, it is necessary to incorporate an effective quality control scheme which uses redundant information provided by both the measurement and dynamic models. As the GNSS Receiver Autonomous Integrity Monitoring (RAIM) algorithms are well developed, here they are adapted to integrated GNSS/INS systems referred as extended RAIM (eRAIM), which are derived from the least squares estimators of the state parameters in a Kalman filter (KF), to assess GNSS/INS performance for a tightly coupled scenario. In addition to the RAIM capabilities, eRAIM procedures are able to detect faults in the dynamic model and isolate them from the measurement model. The analysis includes outlier detection and identification capabilities, reliability and separability measures of integrated GNSS/INS systems. The performance of the system is also investigated with respect to diminishing satellite visibility conditions.
Subject Headings: Navigation, Integrated systems, Quality control, Monitoring Introduction The integration of GNSS with INS delivers an inherently more robust and reliable system than either of the individual systems. INS, while autonomous and immune to interference, are prone to unbounded navigation errors that exponentially increase with time. GNSS navigation on Dr. Steve Hewitson, Dr. Jinling Wang,
[email protected], School of Surveying and Spatial Information Systems, The University of New South Wales, Sydney, NSW 2052, Australia
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the other hand has exceptional long-term stability but is dependant on, and therefore limited by, signal quality and availability. In addition to improved reliability, GNSS/INS integration provides superior short and long-term accuracy, improved availability and higher data rate in comparison to either standalone GNSS or INS. The next generation of GNSS including modernized GPS and GLONASS and the new Galileo systems will significantly enhance the measurement geometry, redundancy and corresponding model (Hewitson & Wang, 2005) while ongoing improvements in INS development continually allow for better dynamic modeling at lower costs. Furthermore, it is widely accepted that the KF provides optimal estimates of the navigation parameters of a dynamic platform, assuming the state and observation models are correct. Unfortunately, such assumptions do not always hold and inevitably, significant deviations of the assumed models are not uncommon for dynamic systems. However, if the modeling assumptions are correct, the state parameter estimates are unbiased with minimum variance within the class of linear unbiased estimates. In order to take full advantage of the superior performance of integrated GNSS/INS and ensure the integrity of the navigation solution it is necessary to incorporate an effective and complementary quality control scheme which makes full use of the redundancy information provided by both the measurement and dynamic models. The most effective quality control procedures currently employed in integrated navigation systems are generally based on the predicted residuals (innovations) of the KF or a derivative of the maximum solution separation method (Brown & McBurney, 1988). Sturza (1998) presents a fault detection algorithm based on hypothesis testing in parity space. Quality control for integrated navigation systems using innovations and recursive filtering is described in Teunissen (1990). A fault detection algorithm based on the GPS solution separation method is adapted to an integrated GPS/INS system in Brenner (1995). In Diesel & King (1995), an innovation based Autonomous Integrity Monitoring Extrapolation for integrated navigation systems is presented and analyzed. Gillesen & Elema (1996) present the results of an innovations based detection, identification and adaptation (DIA) procedure with reliability analysis in an integrated navigation system. Two integrity procedures, the extrapolation and solution separation methods, are tested in Lee &
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O’Laughlin (2000) with respect to the detection of the presence of a slowly growing error (SGE) for a tightly coupled GPS/INS system. Nikiforov (2002) presents fault, detection and exclusion algorithms for multi-sensor integrated navigation systems based on the KF innovations. More recently, Farrell (2005) describes a Q, R matrix decomposition based detection and exclusion technique for GPS/INS with performance comparable to the parity space approach. Hwang (2005) adapts the Novel Integrity-Optimised RAIM (NIORAIM) concept to the multiple solution separation method. In this paper, the RAIM algorithms and quality measures described in Hewitson et al. (2004) and Hewitson & Wang (2006) are adapted to an integrated GNSS/INS system. This adapted RAIM procedure is referred to as extended RAIM (eRAIM). In addition to RAIM capabilities, eRAIM procedures should also be able to detect faults in the dynamic model and isolate them from the measurement model and vice versa. The eRAIM algorithms are derived from the least squares estimators of the state parameters in a Gauss-Markov KF (Wang et al., 1997) and are used to assess GNSS/INS RAIM performance for a tightly coupled simulation scenario. Herein, the measurement model incorporates pseudorange and accumulated delta range measurements. The dynamic model is driven by the INS to provide a reference trajectory for the measurement model. Analyzes compare the outlier detection and identification capabilities and reliability and separability measures of a combined GPS/GLONASS/Galileo/INS system, GPS/GLONASS/INS system and a GPS/INS system. The performance of the GPS/INS system is also investigated with respect to diminishing satellite visibility conditions.
GNSS/INS Integration Herein, the GNSS/INS integration is performed using a tightly coupled KF scheme. The tight coupling approach, used herein, processes the raw pseudorange and delta-range GPS measurements together with the corresponding INS predicted measurements in a single filter. In this way the INS is essentially providing the reference trajectory via the dynamic model based on the strapdown INS error equations for the filter measurement update, which is based on the difference between the INS predicted measurements and the GPS measurements. The major advantage of this approach is that the measurement update can still be executed when a standalone
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GPS solution is unavailable. The model used in an extended Kalman filter (EKF) to describe the system dynamics depend on the selected error states and the error model describing these error states. Herein, the psi-angle error model is adopted to describe the nine INS navigation errors relating to position, velocity and attitude. The three gyroscope and three accelerometer bias states are modeled by first order Gauss-Markov processes and the receiver clock error is modeled by a 2-state (bias and drift) random process model (Huddle 1983; Pue, 2003). The KF state vector has 17 elements, including position, velocity, attitude errors, accelerometer and gyroscope biases and receiver clock bias and drift, plus twelve or more GNSS range bias states (Hewitson 2006). Regardless of the states and error model employed, the state evolution model of the discrete extended KF is described by:
xˆ k− = Фk,k-1 xˆ k −1 +wk
(1)
and the linearised measurement model relating the measurements to the state of the system is expressed as:
zk=Hkxk+εk
(2)
where xˆ k −1 is the mk dimension updated state parameter vector at epoch k − 1 ; xˆ k− is the mk dimension predicted state vector for epoch k made at epoch k − 1 ; Фk,k-1 is the mk× mk-1 state transition matrix from epoch k − 1 to k ; wk is the mk dimension random error vector representing the dynamic process noise at epoch k ; zk is the nk dimension measurement vector at epoch k ; Hk is the nk× mk geometry matrix relating the measurements to the state parameters and εk is the nk dimension error vector of the measurement noise at epoch k . Furthermore, a closed loop update procedure is employed where the navigation errors are used to correct the INS navigation solution. This updating scheme controls the error growth of the INS navigation solution provided the GNSS measurement update is available. The extent of control is dependant on the number of GNSS measurements available.
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eRAIM Algorithms Through adopting least squares principles for the state estimations in a KF, the same quality control algorithms, as for the single epoch snapshot approach, can be used. In such a case, the predicted state parameters can be included as measurements in the adjustment. The algorithms used herein are based on the adopted DIA procedure. For a detailed background to the DIA procedure used herein, see Wang and Chen (1994), Hewitson et al. (2004) and Hewitson & Wang (2006). The inclusion of dynamic information, in the form of the state parameter predictions of the KF, increases the redundancy of the adjustment. The improvement in redundancy has significant affects on the quality control. It has been shown that by integrating the measurements zk with the predicted values of the state parameters xˆ k− , optimal estimates for the state parameters xk can be obtained using least squares principles. The corresponding measurement model is (e.g. Wang et al, 1997): lk = Ak+ vk
(3)
and lk = [zk ; xˆ k− ]; vk = [ v z k ; v xˆ − ]; Ak = [Hk; E] k
where lk is the least squares measurement vector containing zk and xˆ k− ; Ak is the (nk + mk )× nk design matrix; v z k is the nk dimension residual vector of the measurements zk ; v xˆ − is the mk k dimension residual vector of predicted state parameters xˆ k− ; E is an mk × nk identity matrix. The corresponding stochastic model is described with the stochastic variance covariance
Cl k of the measurement vector lk, including measurement covariance matrix Rk and the predicted error covariance matrix Pk−
⎡R 0 ⎤ C lk = ⎢ k − ⎥ ⎣ 0 Pk ⎦
(4)
The optimal estimates for the state parameters xˆ k and the error covariance matrix Q xˆ are k
xˆ k = ( ATx C l−k1 Ak ) −1 AkT C l−k1lk
(5)
Q xˆ k = ( ATx C l−k1 Ak ) −1
(6)
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The filtering residuals v k , consisting of v z k and v xˆ k , can then be calculated from
⎡ v zk ⎤ ⎥ = Ak xˆ k − lk vk = ⎢ ⎢⎣ v xˆ k− ⎥⎦
(7)
From the least squares principles the cofactor matrix of the filtering residuals Qv is k
Qvˆ = C lk − AQ xˆ k AT
(8)
The most import components of the eRAIM are outlier identification, reliability and separability. They are briefly explained in the following three sub-sections. More detailed information can be found in Hewitson (2006). Outlier Identification Once a fault has been detected with a global detection algorithm such as the variance factor test, the w-test can then be used to identify the corresponding measurement, where the test statistic is (e.g. Baarda, 1968; Cross et al., 1994; Teunissen, 1998)
wi =
eiT C lk vˆ eiT C lk Qvˆ C lk ei
(9)
where ei is a unit vector in which the ith component has a value equal to one and dictates the measurement to be tested. Under the null hypothesis, wi has a standard normal distribution for a outlier ∇S i , wi has the following non-centrality.
δ i = ∇Si eiT C lk Qvˆ C lk ei
(10)
The critical value for the test is depended on the significance level α . Reliability The internal reliability is expressed as a minimal detectable bias (MDB) that specifies the lower bound for detectable outliers with a certain probability and confidence level. The MDB is
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determined, for correlated measurements with single error considered (e.g. Baarda, 1968; Cross et al., 1994), by
∇0 Si =
δ0 T i
e C lk Qvˆ C lk ei
(11)
where δ 0 is the non-centrality parameter, which depends on the detectability γ and false alarm rate (significance level α ). The external reliability is then evaluated as
∇ 0 xˆ = Q x AT C lk ei ∇ 0 Si
(12)
Separability For situations arising where an outlier is sufficiently large to cause many w-test failures, resulting in many alternatives, it is essential to ensure any two alternatives are separable so that a good measurement is not incorrectly flagged as an outlier. The probability of incorrectly flagging a good measurement as the detected outlier is dependant on the correlation coefficient of the test statistics wi and w j (e.g. Förstner, 1983; Wang and Chen 1994).
ρ ij =
eiT C lk Qvˆ C lk e j eiT C lk Qvˆ C lk ei ⋅ eTj C lk Qvˆ C lk e j
(13)
The correlation coefficient has the property ρ ij ≤ 1 , where 1 and 0 correspond to full and zero correlation between two test statistics, respectively. The greater the correlation between two test statistics, the more difficult they are to separate. The degree of correlation of measurements is dependant on the measurement redundancy and geometric strength. If the measurement redundancy is equal to 1 an outlier can be detected but cannot be identified as all the identification test statistics are fully correlated. When determining the separability, it suffices to only consider the maximum correlation coefficient ρ ij max (∀j ≠ i ) for each statistic.
Simulation Studies Single frequency pseudorange and accumulated delta-range GNSS measurements and delta-velocity and delta-theta INS measurements were generated using a modified version of the GPSoft® Satellite Navigation, Inertial Navigation System and Navigation System Integration and
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KF Toolboxes (2007 Braasch). The 3D reference trajectory in Figure 1, along with a simple straight and constant velocity, was used to simulate the GNSS and INS measurements of an airplane. The measurement update rate was 1 Hz. Figure 1 3D Simulation Reference Trajectory The pseudorange and accumulated delta range measurements are generated with standard deviations of 1 m and 0.1 m, respectively, considering measurement noise and modeling error etc. Accumulated delta-range measurements are taken 50 milliseconds prior to the filter update time as well as at the filter update time to derive the delta-range observables. The delta-range measurements were then used to estimate the receiver’s velocity and provide the dynamic information in KF for the state predictions. The satellite coordinates were calculated from the Keplerian elements of the nominal GPS, GLONASS and Galileo constellations (see Hewitson & Wang, 2006) and all simulations were conducted with a 5-degree masking angle. The INS was simulated with an output rate of 200 Hz. The simulated errors included initial alignment and velocity errors (0.1 milli-rad and 0.02m/s horizontally and zero vertically), and accelerometer and gyroscope biases (50 micro-g and 0.01deg/hour). The error growth characteristics of the simulated INS can be seen in Figure 2.
Figure 2 Horizontal position error of simulated INS. In order to assess the performance of tightly coupled GNSS/INS systems with respect to outlier detection and identification several scenarios were considered. First, the low dynamic 3-Dimensional trajectory (Figure 1) was used to investigate the eRAIM performance of GPS/INS, GPS/GLONASS/INS and GPS/GLONASS/Galileo/INS systems. The eRAIM performance was then also assessed with respect to SGE and instantaneous biases under simulated low visibility conditions using a simple straight and constant velocity trajectory. All the integrity solutions used in the following analyzes were derived from the least squares principles and the critical value used for the w-test was 3.2905 for a false alarm rate ( α ) of 0.1%. The internal reliabilities (MDBs) were determined with a detectability (γ ) of 80%, as recommended by Cross
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et al. (1994). 3-Dimensional trajectory Analysis The simulated GNSS and INS measurements for the 3D trajectory in Figure 1 were used to determine the horizontal position errors, w-statistics, internal reliabilities and maximum correlation coefficients (MCCs) associated with GPS/INS, GPS/GLONASS/INS and GPS/GLONASS/Galileo/ INS systems. The measurements were simulated for a helicopter with a speed of approximately 4.7 m/s after accelerating from rest at approximately 0.12 m/s2. The centripetal acceleration was approximately 0.34 m/s2 for all turns. Line plots of the horizontal position errors and statistics and histograms of the internal reliabilities and MCC were then generated to analyze the performances and distribution of values over the whole trajectory. The histograms were separated with respect to the pseudorange and delta range measurements and the predicted states for each estimation scenario. It should be noted that the correlation coefficient results, however, were not isolated to type of measurement being analyzed, i.e. the results for the pseudorange measurements include the correlations with the delta range and predicted state statistics as well as the statistics corresponding to the other pseudoranges. Integrated GPS/INS Figure 3 shows the horizontal position errors for the GPS/INS scenario where 7-8 satellites were visible for the entire trajectory. Figure 3 GPS/INS EKF horizontal position error for 3D trajectory. Figure 4 shows the w-statistics for the measurements, predicted states and the variance factor of the measurement updates. It is clear that no false alarms have occurred. Though there are some small failures for the w-test it is usual practice to ignore these if the variance factor test has passed. No biases have been induced and these w-test failures are due to the simulated measurement noise. Figure 4 EKF w-test statistics for GPS/INS measurements (top), predicted states (middle) and unit variance statistics (bottom) for 3D trajectory. Figure 5 gives the distribution of the MDBs for the pseudorange and delta-range measurements and the predicted states. The MDBs corresponding to the pseudorange measurements range from about 4.39 m to approximately 4.66 m with the majority of the values between 4.4 m and 4.5 m. The Delta-
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range MDBs spread over values from 1.26 m/s to about 1.306 m/s with most of the values between 1.26 m/s and 12.603 m/s. The MDB values for the predicted states fall between 0.5 and 4.6. It should be noted that the predicted states include the receiver clock bias and drift errors as well as the position and velocity error states and that the INS does not provide any additional observable for the clock states.
Figure 5 Histogram of internal reliabilities along the 3D trajectory for pseudorange (top) delta-range (middle) measurements and predicted states (bottom) for GPS/INS. The MCC distributions associated with the pseudorange and delta-range measurements and predicted states are given in Figure 6. Here, the correlation coefficients corresponding to the pseudorange statistics approximately range between 0.3 and 0.62. The delta-range statistics range from about 0.35 to 0.65. The correlation coefficients corresponding to the predicted states lie between the values 0.5 and 0.9. There are a large proportion of the predicted states with corresponding correlation coefficients between 0.8 and 0.9. This is due to the weighting of the dynamic model with respect to the clock drift.
Figure 6 Histogram of MCCs along the 3D trajectory corresponding to pseudorange (top) delta-range (middle) measurements and predicted states (bottom) for GPS/INS. Integrated GPS/GLONASS/INS The horizontal position error for the GPS/GLONASS/INS system is given in Figure 7. We can see that the estimation accuracy is noticeably greater than for the GPS/INS system (see Figure 3). In this case there were 15 to 16 satellites visible for the entire trajectory.
Figure 7 GPS/GLONASS/INS EKF horizontal position error for 3D trajectory. In Figure 8 we can see an outlier has occurred at 7 minutes with a failure of the variance factor test due to measurement noise. The MDBs of the GPS/GLONASS/INS system (see Figure 9) are significantly lower and more stable overall with respect to the pseudoranges, delta-ranges and predicted states than the GPS/INS system (Figure 5). The lower limits of the delta-range and predicted state MDBs however, appear to have been reached at the values of approximately 1.26 m and 0.5 respectively.
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Figure 8 EKF w-test statistics for GPS/GLONASS/INS measurements (top) and predicted states (middle) and unit variance statistics (bottom) for 3D trajectory.
Figure 9 Histogram of internal reliabilities along the 3D trajectory for GPS/GLONASS/INS. Notable improvements are also achieved over the GPS/INS case with respect to the pseudoranges, delta-ranges and predicted measurement statistic correlations. Figure 0 reveals that the pseudorange correlations are reduced by approximately 0.1 for both the lower and higher limits, the deltarange lower limit is reduced by about 0.05 with upper limit reduced by over 0.1 and the lower limit for the predicted states is also lowered by over 0.1 with a huge reduction of greater than 0.25 of the upper limit. Figure 10 Histogram of MCCs along the 3D trajectory corresponding to pseudorange (top) delta-range (middle) measurements and predicted states (bottom) for GPS/GLONASS/INS. Integrated GPS/GLONASS/Galileo/INS There were 23 to 24 visible satellites for the GPS/GLONASS/Galileo/INS scenario and the corresponding horizontal position error is given in Figure 11. A slight improvement in the accuracy over the GPS/GLONASS/INS system has been achieved. No false alarms were raised in this scenario (see Figure 92). Again the lower limits of the delta-range and predicted state MDBs (see Figure 103) seem to have been reached. The lower limit for the pseudorange is approximately 1.26 m with the delta-range MDB lower limit now slightly lower at 0.4 for the GPS/GLONASS/Galileo/INS case. There is also some overall improvement with respect to the predicted state MDBs with the weight of the distribution centred slightly more towards the lower limit.
Figure 81 GPS/GLONASS/Galileo/INS EKF horizontal position errors for 3D trajectory.
Figure 92 EKF w-test statistics for GPS/GLONASS/Galileo/INS measurements (top) and predicted states (middle) and unit variance statistics (bottom) for 3D trajectory.
Figure 103 Histogram of internal reliabilities along the 3D trajectory for pseudorange (top) delta-range (middle) measurements and predicted states (bottom) for GPS/GLONASS/Galileo/INS.
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Obvious improvements in the correlation coefficients corresponding to the pseudoranges, deltaranges and predicted states of the GPS/GLONASS/Galileo/INS system are shown in Figure 4. Both the lower and upper limits of the correlations corresponding to the pseudoranges have been lowered by approximately 0.05 and 0.1, respectively. The delta-range test statistic correlation coefficients have greatly stabilized and appear to have reached a lower bound limit of just over 0.3. The lower limit of the predicted state correlation coefficients have also improved by about 0.5
Figure 14 Histogram of MCCs along the 3D trajectory corresponding to pseudorange (top) delta-range (middle) measurements and predicted states (bottom) for GPS/GLONASS/Galileo/INS.
Outlier Detection and Identification during Low Satellite Visibility A simple straight trajectory was used to assess the performance of GNSS/INS integration under low satellite visibility conditions with respect to outlier detection and identification. The measurements were simulated for an airplane moving along the trajectory with a constant speed of 200 m/s. Initially, a 6m instantaneous bias was induced in the same satellite at exactly 4.5, 5.5, 6.5 and 7.5 minutes corresponding to satellite visibilities of 4, 3, 2 and 1 respectively. The variance factors, w-statistics, internal reliabilities and correlation coefficients were then examined. Further, outlier identification analyzes were then made with respect to the presence of a SGE. Instantaneous Bias Detection and Identification Figure 115 shows the degrading satellite visibility and the corresponding horizontal position error. The results of the statistical tests, namely the variance factor and w-tests, used for detection and identification of the induced biases are given in Figure 126. From the variance factor results it is clear that the outliers corresponding to 4, 2 and 1 visible satellites were correctly detected. Though it is not obvious, the bias induced corresponding to 3 visible satellites was also detected with a test statistic of 3.2748 and threshold of 3.0912. The satellite corresponding to the induced bias was correctly identified in the 4, 3 and 2 visible satellites scenarios, however the pseudorange in the 1 visible satellite case was fully correlated (see Figure 137) with, and inseparable from, the predicted position and clock bias states. For this reason the predicted position and clock bias states were also identified. Furthermore, Figure 137 Also shows that the 6m biases were above the MDB threshold for the duration of the simulation.
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Figure 115 GPS/INS EKF horizontal position error under low satellite visibility conditions with instantaneous biases induced.
Figure 126 EKF w-test statistics for GPS/INS measurements (top) and predicted states (middle) and unit variance statistics (bottom) under low satellite visibility conditions with instantaneous biases induced.
Figure 137 EKF w-test statistic correlation coefficients of bias induced pseudorange with respect to other pseudoranges, predicted states and clock bias (top) and pseudorange MDBs for GPS/INS under low satellite visibility conditions with instantaneous biases induced (bottom). Slow Growth Error Detection and Identification Slow growth errors, or slowly increasing ramp errors, are less common that instantaneous biases but far more difficult to detect. The difficulty in detecting SGE arises from the fact that effect of the error is similar in magnitude to the dynamics of a moving system, whereas instantaneous biases are inconsistent and imply unrealistic vehicle dynamics. In order to show a major integrity issue regarding the detection and identification of SGE for integrated GNSS/INS systems under low satellite visibility conditions, a 0.2m/s error was induced in the only satellite visible for the duration of the simulation. The SGE was induced after 250 seconds when one satellite was visible. The satellite visibility and horizontal position error for this simulation is shown in Figure 148. Figure 148 GPS/INS EKF horizontal position error in East (middle) and North (bottom) components under low satellite visibility (top) conditions with SGE induced. In Figure 19, it can be seen that the error is not detected until 160 seconds after it is induced and when 4 satellites are visible. At this time the SGE is equivalent to a 32m bias. The MDBs for the pseudoranges in this simulation are comparable to those given in Figure 137 for the same visibilities. This suggests that a minimum of 4 range measurements are required to detect an SGE in an integrated GNSS/INS system. With less than for satellites visible an SGE is indistinguishable from normal system dynamics.
Figure 19 EKF w-test statistics for GPS/INS measurements (top), predicted states (middle) and unit variance statistics (bottom) under low satellite visibility condition with SGE induced.
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Conclusions Investigations into the performance of RAIM as adapted to integrated GNSS/INS systems, named as eRAIM, have been carried out considering different GNSS constellations in this paper. Fault scenarios include the detection and identification capabilities of the integrated systems with respect to instantaneous biases and SGE. It been shown that for integrated GNSS/INS systems, instantaneous biases can be detected and identified when there are two or more visible satellites. The detected instantaneous biases may not be isolated when there is only one satellite visible. It is also shown that that the additional redundancy provided by an integrated GPS/Galileo/GLONASS/INS significantly enhances RAIM performance with respect to GPS/INS or GPS/GLONASS/INS systems, even for full sky visibility conditions above a 5-degree mask angle. In addition, a major integrity issue regarding the detection of SGE of an integrated GNSS/INS system has been identified. It is shown that the SGE cannot be distinguished from the normal system dynamics and is therefore an undetectable error for an integrated GNSS/INS system when less than four satellites are available. Acknowledgement This work has been supported by the Australian Research Council research grant on Robust Positioning as well as University of New South Wales Goldstar research grant on GNSS Integrity Monitoring. The authors would like to thank Dr Jianguo Jack Wang for the valuable discussions and his assistance in preparing the revised version for this paper.
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List of Figures Figure 1 3D Simulation Reference Trajectory Figure 2 Horizontal position error of simulated INS. Figure 3 GPS/INS EKF horizontal position error for 3D trajectory. Figure 4 EKF w-test statistics for GPS/INS measurements (top), predicted states (middle) and unit variance statistics (bottom) for 3D trajectory. Figure 5 Histogram of internal reliabilities along the 3D trajectory for pseudorange (top) delta-range (middle) measurements and predicted states (bottom) for GPS/INS. Figure 6 Histogram of MCCs along the 3D trajectory corresponding to pseudorange (top) delta-range (middle) measurements and predicted states (bottom) for GPS/INS. Figure 7 GPS/GLONASS/INS EKF horizontal position error for 3D trajectory. Figure 8 EKF w-test statistics for GPS/GLONASS/INS measurements (top) and predicted states (middle) and unit variance statistics (bottom) for 3D trajectory. Figure 9 Histogram of internal reliabilities along the 3D trajectory for GPS/GLONASS/INS. Figure 10 Histogram of MCCs along the 3D trajectory corresponding to pseudorange (top) delta-range (middle) measurements and predicted states (bottom) for GPS/GLONASS/INS. Figure 11 GPS/GLONASS/Galileo/INS EKF horizontal position errors for 3D trajectory. Figure 12 EKF w-test statistics for GPS/GLONASS/Galileo/INS measurements (top) and predicted states (middle) and unit variance statistics (bottom) for 3D trajectory. Figure 13 Histogram of internal reliabilities along the 3D trajectory for pseudorange (top) delta-range (middle) measurements and predicted states (bottom) for GPS/GLONASS/Galileo/INS. Figure 14 Histogram of MCCs along the 3D trajectory corresponding to pseudorange (top) delta-range (middle) measurements and predicted states (bottom) for GPS/GLONASS/Galileo/INS.
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Figure 15 GPS/INS EKF horizontal position error under low satellite visibility conditions with instantaneous biases induced. Figure 15 EKF w-test statistics for GPS/INS measurements (top) and predicted states (middle) and unit variance statistics (bottom) under low satellite visibility conditions with instantaneous biases induced. Figure 16 EKF w-test statistic correlation coefficients of bias induced pseudorange with respect to other pseudoranges, predicted states and clock bias (top) and pseudorange MDBs for GPS/INS under low satellite visibility conditions with instantaneous biases induced (bottom). Figure 17 GPS/INS EKF horizontal position error in East (middle) and North (bottom) components under low satellite visibility (top) conditions with SGE induced.
Figure 18 EKF w-test statistics for GPS/INS measurements (top), predicted states (middle) and unit variance statistics (bottom) under low satellite visibility condition with SGE induced.
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Figure 1 3D Simulation Reference Trajectory
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Figure 2 Horizontal position error of simulated INS.
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Figure 3 GPS/INS EKF horizontal position error for 3D trajectory.
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Figure 19 EKF w-test statistics for GPS/INS measurements (top), predicted states (middle) and unit variance statistics (bottom) for 3D trajectory.
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Figure 5 Histogram of internal reliabilities along the 3D trajectory for pseudorange (top) delta-range (middle) measurements and predicted states (bottom) for GPS/INS.
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Figure 6 Histogram of MCCs along the 3D trajectory corresponding to pseudorange (top) delta-range (middle) measurements and predicted states (bottom) for GPS/INS.
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Figure 7 GPS/GLONASS/INS EKF horizontal position error for 3D trajectory.
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Figure 8 EKF w-test statistics for GPS/GLONASS/INS measurements (top) and predicted states (middle) and unit variance statistics (bottom) for 3D trajectory.
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Figure 9 Histogram of internal reliabilities along the 3D trajectory for GPS/GLONASS/INS.
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Figure 10 Histogram of MCCs along the 3D trajectory corresponding to pseudorange (top) delta-range (middle) measurements and predicted states (bottom) for GPS/GLONASS/INS.
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Figure 11 GPS/GLONASS/Galileo/INS EKF horizontal position errors for 3D trajectory.
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Figure 12 EKF w-test statistics for GPS/GLONASS/Galileo/INS measurements (top) and predicted states (middle) and unit variance statistics (bottom) for 3D trajectory.
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Figure 13 Histogram of internal reliabilities along the 3D trajectory for pseudorange (top) delta-range (middle) measurements and predicted states (bottom) for GPS/GLONASS/Galileo/INS.
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Figure 14 Histogram of MCCs along the 3D trajectory corresponding to pseudorange (top) delta-range (middle) measurements and predicted states (bottom) for GPS/GLONASS/Galileo/INS.
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Figure 15 GPS/INS EKF horizontal position error under low satellite visibility conditions with instantaneous biases induced.
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Figure 16 EKF w-test statistics for GPS/INS measurements (top) and predicted states (middle) and unit variance statistics (bottom) under low satellite visibility conditions with instantaneous biases induced.
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Figure 17 EKF w-test statistic correlation coefficients of bias induced pseudorange with respect to other pseudoranges, predicted states and clock bias (top) and pseudorange MDBs for GPS/INS under low satellite visibility conditions with instantaneous biases induced (bottom).
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Figure 18 GPS/INS EKF horizontal position error in East (middle) and North (bottom) components under low satellite visibility (top) conditions with SGE induced.
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Figure 19 EKF w-test statistics for GPS/INS measurements (top), predicted states (middle) and unit variance statistics (bottom) under low satellite visibility condition with SGE induced.
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