Multi-sensor data fusion based on Information Theory. Application to GNSS positionning and integrity monitoring Nourdine Aït Tmazirte, Maan E. El Najjar, Cherif Smaili and Denis Pomorski LAGIS UMR 89 CNRS/Université-Lille Avenue Paul Langevin 59655 Villeneuve d Ascq, France Maan.El-Badaoui-el-najjar@univ-lille.fr Abstract Integrity monitoring is considered now as an important part of a vehicle navigation system. Localisation sensors faults due to systematic malfunctioning require integrity reinforcement of multi-sensors fusion method through systematic analysis and reconfiguration method in order to exclude the erroneous information from the fusion procedure. In this paper, we propose a method to detect faults of the GPS signals by using a distributed information filter with a probability test. In order to detect faults, consistency is examined through a log likelihood ratio of the information innovation of each satellite using mutual information concept. Through GPS measurements and the application of the autonomous integrity monitoring system, the current study illustrates the performance of the proposed fault detection algorithm and the pertinence of the reconfiguration of the multisensors data fusion. Keywords:data fusion, Fault detection, information theory, localisation, GNSS, RAIM. I. INTRODUCTION Autonomous navigation system requires a safety positioning system. When leading safety, positioning services not only need to provide an estimate of the vehicle location, but also uncertainty estimation. In practice, an upper bound on the positioning error, representing an integrity risk, is required to determine if a positioning system can be used for a given task. At GNSS (Global Navigation Satellite System) receiver level, standard approaches introduce a Fault Detection & Exclusion (FDE) stage to monitor the integrity of the estimation of the position. This is known as Receiver Autonomous Integrity Monitoring (RAIM) [] []. Numerous FD studies have been conducted with real time sensors measurements. When we do not dispose of a sensor model, which is always the case with localisation sensors, FDE methods are approached in a statistical manner. Classically, stochastic approaches are developed through KALMAN filters [3], Particle Filter [4] or interval analysis [5]. These procedures for FD introduce residual testing through statistical approach [6]. In this residual test scheme, generally, FD is conducted by examining the probabilistic property of the estimated state itself [5], yet a precise reference system unaffected by failures is required. The implementation of this kind of architecture requires a thresholding process in order to pick out measurements errors. This kind of process could easily detect sporadic measurements errors but hardly detect the gradually increasing error of measurements. In addition, thresholds are fixed classically with a heuristic manner. Recently, the information filter (IF), which is the informational form of the KALMAN filter (KF), has proved to be attractive for multi-sensors fusion, like in [7],[8] or [9]. The IF uses an information matrix and an information vector to represent the co-variance matrix and the state vector usually used in a KF. This difference in representation makes the IF superior to the KF concerning multiple sensor fusion, as computations are simpler and no prior information of the system state is required [] []. In addition, consistency test using log likelihood ratio (LLR), based on information matrix, permits to have a good balance between the detection probability and the false alarm probability. With the help of information filtering, the main interest of this paper is to ensure the reliability of the GNSS positioning in a navigation system. In this paper, a new GNSS integrity monitoring method is presented by applying a distributed IF and a LLR test. Specifically, the presented (FDE) scheme takes advantage of the highly LLR change between the information of each satellite and the mutual information of a set of visible satellites by a GNSS (GPS) receiver. The performance of the distributed IF is notable after reconfiguration of the data fusion procedure. This paper is organized as follows: Section briefly introduces the general principle of distributed IF. In section 3, a detailed GPS integrity monitoring algorithm that employs distributed IF and ratio test is illustrated. A result using real GPS measurement 743
data is presented in section 4, followed by the conclusion in section 5. II. A. Information filter INFORMATION THEORY AND FILTERING An Extended KF can easily be expressed in another form, called Nonlinear Information Filter (NIF) or inverse covariance filter []. Instead of working with the estimation of states x(i j) and the variance P(i j), the IF deals with the information state vector y(i j) and information matrix Y(i j), where: y i j = P i j. x i j () Y(i j) = P (i j) () The information matrix is closely associated with the Fisher information. The physical mean of the Fisher information is the surface of a bounding region containing probability mass. It measures the compactness of a density function. x k = F k. x k + w k (3) Where x(k) is the state vector, F(k) the state transition matrix and w(k) the process noise. The observation is also modeled: z k = H k. x k + v(k) (4) Where z(k) is the observation vector, H k the observation matrix and v(k) is a white noise. The update stage of the NIF is written : Where y k k = y k k + i k (5) Y k k = Y k k + I k (6) i k = H T k. R k. z(k) (7) i(k) is a vector containing the contribution in term of information from an observation z(k). I k = H T k. R k. H T k (8) I(k) is the associated information matrix. The couple (i k, I(k)) represents the Information Contribution (IC) of the observation. In multiple sensor problems: z i k = H i k. x k + v i k i = N (9) the estimate does not represent a simple linear combination of contributions from individual sensors : N x k k x k k + i= W i k. z i k () H i k. x k k (with W i (k) independent gain matrices). The innovation generated from each sensor is correlated. Indeed, they share common information through the prediction x(k k ). In information form, estimates can be constructed from linear combinations of observation information. y k k = y k k + N i= i i k () It is because information terms i i (k) from each sensor are assumed to be uncorrelated. Now it is straightforward to evaluate the contribution of each sensor (unlike for a KF). Each sensor node simply generates the information terms i i (k). These are summed at the update step of the NIF to produce a global information estimate. B. Mutual Information To evaluate the IC of each sensor, a well known concept of Information theory can be used: the Mutual Information (MI). The MI I(x, z) is an a priori measure of the information which will be gained by x with a set of observations z. I(x, z) is defined in [] as: P x, z I x, z = E ln P x. P z = E ln P x z = E ln P x P z x P z In term of entropies, the MI is written: () I x, z = H x + H z H x, z = H x H x z (3) With H(x), H(z) the respective entropies of x and z. H x z and H z x the conditional entropies. 744
III. FAULTY DETECTION AND EXCLUSION : APPLICATION TO GPS INTEGRITY MONITORING GNSS integrity monitoring or Receiver Autonomous Integrity Monitoring (RAIM) is a technology developed to control the integrity of global navigation satellite system signals in a receiver system. During a long time, integrity was exclusively used in aviation or marine navigation. The integrity represents the ability to associate to a position a reliable indication of trust. With the development of different safety-critical urban applications, the notion of integrity became an important field of research, in particular in urban canyon, where satellites signals are subject to hostile environment. Usually, RAIM algorithms use a comparison of the result of a main equation system and subsystems results [4], we propose to use the information theory to quantify the information contribution of each satellite (Figure ). In other words, we propose to detect and to exclude faulty satellite upstream, meaning before the equation system resolution. As well detailed in [3], the process model depends on the dynamical characteristics of the vehicle. The system is assumed to evolve according to the equation: x k + = A. x k + w k + (4) Where the state vector is composed of the eight following variables: x k = X k, X k, Y k, Y k, Z k, Z k, c t k, c t k (5) [X, Y, Z] representing the receiver position,[x, Y, Z ]the velocity in Earth Centered Earth Fixed (ECEF) frame, t the clock range and t the clock drift. The state transition matrix is given by: A = T T T T (6) Where T is the sample period (in our case second). The measurement noise w. is assumed to be time invariant: w. = T ς x cς g T, Tς x, T ς y, Tς y, T ς z, Tς z, cς f T +, cς g T (7) Where ς x, ς y, ς z represent the process noise variances related to the states x, y and z, ς f and ς g are the variances related to clock offset and velocity, and c the speed of light. Figure. Satelllite Information contribution for GNSS Monitoring Making the assumption that w~n, Q means that w is assumed to be a white Gaussian noise with covariance Q : A. State space representaion for GNSS Positionning GNSS positioning with pseudo-range is a Time of Arrival method [5]. Pseudo-ranges are the distances between visible satellites and the receiver plus the unknown difference between the receiver clock and the GNSS time. Thus, GNSS positioning is a four dimensional problem: the 3D coordinates (x i, y i, z i ) of the user and the clock offset dt i are unknown. Q = ας x γς x ας x γς x ας x γς x γς f + ας g βς g βς g γς g (8) 745
With α = c T 3 3 β = c T (9) () γ = T () The observation [x i, y i, z i ] of each satellite i, is calculated thanks to broadcasted ephemeris. The atmospheric errors are S S also modeled δρ i Iono, δρ i Tropo. The pseudo-range can be modeled as follow: R(. ) = ς ς j B. Mutual Information for FDE ς n (8) As proved in [4] the entropy, for a multivariate Gaussian distribution (which is assumed to be the case in Eq. 4 for our study), goes as the log-determinant of the covariance. Precisely, the differential entropy of a d-dimensional random vector X drawn from the Gaussian N(μ, ε) is: R s i = x i x S + y i y S + z i z S + c. dt i dt s S S + δρ i Iono + δρ i Tropo () Being non-linear, the observation model is linearised around a predicted state [X pr ] to obtain an observation matrix: H X = N X. ln N X. dx = d ln(π) + d. + ln ε (9) H = x x j x n y y j y n z z j z n (3) From Eq.3 & Eq. 9, the MI in our case is defined: I X, Z = d ln π P(X) + d. + ln ( d + ln P(X Z) ) It becomes a basic Log Likelihood Ratio : + d. ln π (3) With And j x = R j s x = (xs s x pr ) R pr (4) y j = R j s y = (ys y pr ) R pr s (5) j z = R j s z = (zs s z pr ) R pr (6) s R pr = x S x pr + y S y pr + z S z pr (7) Each satellite noise is assumed to be uncorrelated with all others. So, the noise described as follow v~n, R is simply represented by its diagonal time-invariant matrix R: I X, Z = ln P(X) P(X Z) (3) A Global Information Mutual Information (GOMI) can be defined. P(X) is the prediction covariance and P(X Z) is the updated covariance. The GOMI becomes, in information form: I X, Z = ln Y(k k) Y(k k ) I X, Z = Y k k + i= ln I i(k) Y(k k ) N (3) (33) We also define the Partial Observation Mutual Information (POMI) which will take in account only one information term of sensor i. I i X, Z = ln Y k k + I i(k) Y(k k ) (34) The GOMI will prove to be an adapted residual for fault detection, whereas the POMI will isolate the faulty observation. 746
C. Principle of the FDE method The structure of the proposed GNSS integrity monitoring method is illustrated in figure which correspond to a classical prediction/correction filtering using IF but including a FDE stage. With n measurements, n Information Contribution (IC IC n ) (Eq. 7 & Eq.8) are computed. A GOMI (Eq. 33) is then computed using the n IC. The GOMI permits to have an autothresholding stage in order to detect the inconsistency of one or several satellites. In case of fault detection, a POMI (Eq. 34) is computed for each satellite. These different POMIs are used to isolate the faulty satellites measurements in order to be excluded from the correction step of the NIF algorithm. IV. EXPERIMENTAL RESULTS In order to test the performance of the RAIM developed approach, real data acquisition has been carried out with CyCab vehicle produced by Robosoft (www.robosoft.fr/) with several embedded sensors. In this work, measurements of GPS RTK Thales Sagitta system and open GPS Septentrio Polarxe@ (Figure 4) are used. Figure. Classical GNSS Integrity Monitoring schema In the proposed FDE approach, described in the following flowchart, in figure 3, only five measurements are needed for the integrity monitoring. Figure 4. Experimental vehicle. The data acquisition has been carried out around LORIA Lab. During these experiments we remarked, as shown in figure 5, that when the positionning system uses all visible satellite without a FDE stage, one satellite introduces a bias in the positionning process. It seems that this satellite was in a bad geometric configuration in respect to the building in front (Bat C in the figure 4, 5 and 6). To compare, the test trajectory reference with centimetric accuracy (GPS RTK) is plotted in green in figure 6. This trajectory is about eighty meters length. Figure 3. Information Theory based proposed method 747
Figure 5. Trajectory after fusion without FDE Figure 7. Global Observation Mutual Information for Detection Figure 6. Reference trajectory given by RTK GPS The fault exclusion step is realised using the POMI computed for each satellite. In fact, the POMI permits to identify the inconsistency of the faulty satellite like discribed in figure 8. One can see the surges of POMI of satellite 6 represented in blue. In figure 7 and 8, the results of the detection and exclusion process are presented for the set of visible satellites during the test trajectory. GOMI concept, introduced in section helps to auto-threshold information contribution of all satellites. After few steps of initialization, the contribution of each satellite goes to stabilize around a value. Indeed, a satellite, if still visible, should not bring a contribution different to the information gives by the steps before. Figure 7, one can easily see the different surges of information contribution. That because a pseudo-range is generally overestimated when the corresponding satellite waves is subject to multi-path. Thus, the IC of the faulty satellite is also overvalued. This IC becomes inconsistent, and need to be excluded. In this test, we detected only one satellite real errors, but the proposed approach can be generalized for multiple faulty satellites. Figure 8. Partials Observation Mutual Information for Exclusion In figure 9, is shown the performance of the fusion with FDE after exclusion of satellite 6 from the fusion procedure. We note that the trajectory in figure 9 is close to the GPS RTK trajectory showed in figure 6. 748
introduced by each satellite observation compared to the expected information. Figure 9. Trajectory after fusion with FDE Finally, the figure shows performance of the proposed approach with faulty satellite exclusion in blue. This figure shows the position computation in the ECEF frame for each axis (X,Y,Z). In red, are the position computation on ECEF frame without faulty satellite exclusion. Figure. Exclusion of a faulty satellite in GNSS positioning V. CONCLUSION AND FUTURES WORKS This paper proposes a reliable multi sensor data fusion approach with integrity monitoring. This approach integrates Fault Detection stage by using distributed NIF and LLR test using mutual information. The proposed method makes it possible to detect fault without heuristic thresholing step. It permits an automatic reconfiguration of the fusion procedure in order to exclude the faulty measure. The proposed approach is applied for a GPS integrity monitoring. Measurements were pseudoranges from the GPS satellites. The basic concept is to use the Information Filter in a distributed architecture. The likelihood function is established and examined by integrating state estimate from distributed Information Filters. Thus, the LLR test is used in fault detection, which compares the information The evaluation of FDE is conducted through simulation using real GPS measurements data. In the presented test trajectory, the measured data from GPS contain the well known wave s multipath error illustrated with an abnormal bias. Based on real data, results demonstrate performance of the proposed approach in detecting GPS measurements faults and fusion method reconfiguration. In future work, our objective is to integrate dead reckoning sensors in order to develop a tightly coupled data fusion method. These sensors are known to introduce gradually increasing errors (drift). The LLR is well adapted to detect this kind of errors. REFERENCES Erreur! Source du renvoi introuvable. [] R. Grover Brown, Gerald Y. Chin, GPS RAIM: Calculation of Threshold and Protection Radius Using Chi-Square Methods-A Geometric Approach, in. Global Positioning System: Inst. Navigat., vol. V, pp. 55 79, 997 [] T. Walter, P. Enge. Weighted RAIM for Precision Approach. Proceedings of the ION-GPS 995, Palm Springs, CA, 995. [3] B. W. Parkinson, and P. Axelrad, Autonomous GPS integrity monitoring using the pseudorange residual, Navig., J. Inst. Navig., vol. 35, no., pp. 55 74, 988. [4] J. Ahn & al., GPS Integrity Monitoring Method Using Auxiliary Nonlinear Filters with Log Likelihood Ratio Test Approach, Journal of Electrical Engineering & Technology Vol. 6, No. 4, pp. 563~57,. [5] V. Drevelle and Ph. Bonnifait, Global Positioning in Urban Areas with 3- D Maps Proceedings of the IEEE Intelligent Vehicles Symposium, Baden-Baden, Allemagne, pp. 764-769, [6] Y. C. Lee, Analysis or range and position comparison methods as a means to provide GPS integrity in the user receiver, in Proceedings of the Annual Meeting of the Institute of Navigation, Seattle, WA, Jun. 986, pp. 4, 986. [7] B. Grocholsky, H. Durrant-Whyte, and P. Gibbens, An informationtheoretic approach to decentralized control of multiple autonomous flight vehicles, Proc. SPIE: Sensor Fusion and Decentralized Control in Robotic Systems III, vol. 496, pp. 348 359, Oct.. [8] L. Wang, Q. Zhang, H. Zhu, and L. Shen, Adaptive consensus fusion estimation for msn with communication delays and switching network topologies. Decision and Control (CDC), 49th IEEE Conference on, pages 87 9,. [9] G. Liu, F. Wörgötter, and I. Markeli c, Nonlinear estimation using central difference information filter. In IEEE Workshop on Statistical Signal Processing, pages 593 596,. [] S. Grime and H. F. Durrant-Whyte. Data Fusion in Decentralized Sensor Networks. Control Engineering Practice, :849-863, 994. [] D.-J. Lee, Nonlinear estimation and multiple sensor fusion using unscented information filtering. Signal Processing Letters, IEEE, 5:86 864, 8. [] T.M. Cover and J.A. Thomas, Elements of information theory. John Wiley & Sons, New York, NY, Inc., Chapter 9 Theroem 9.4.3, 99. [3] S. Cooper and H. F. Durrant-Whyte. A Kalman filter model for GPS navigation of land vehicles. In: IEEE/RSJ/GI International Conference on Intelligent Robots and Systems, p. 57 63, 994. [4] T.M. Cover and J.A. Thomas, Elements of information theory. John Wiley & Sons, New York, NY, Inc., Chapter Definition.3, 99. 749