Gaussian process regression approach for bridging GPS outages in integrated navigation systems M.M. Atia, A. Noureldin and M. Korenberg A Kalman filter (KF) enhanced by the Gaussian process regression (GPR) technique is suggested to bridge GPS-outages in navigation solutions where inertial navigation systems (INS) and GPS are integrated. A KF utilises linearised dynamic models. If a low-cost MEMS-based INS with complex stochastic nonlinearity is considered, performance degrades significantly during short periods of GPSoutages owing to linearised models. Proposed is a novel usage of GPR as a nonlinear INS-errors predictor. During GPS availability, the correct vehicle state, sensor measurements, and INS output deviations from GPS are collected. During GPS-outages, GPR is applied to this data set to predict INS deviations enabling the KF to estimate all INS errors. The proposed technique was tested on real road experiments showing significant improvements during long GPS-outages.
Introduction: An inertial navigation system (INS) has good short-term accuracy while GPS has consistent long-term accuracy. Thus, the integration improves the overall INS/GPS integrated solution [1]. A Kalman fitter (KF) [2] is an optimal linear state estimation technique that requires a linear dynamic model. If the dynamic model is not linear, the KF is extended to work on a linearised model. In INS/ GPS integration, the INS errors dynamic model is linearised and used to predict INS errors. When GPS measurements are available, the KF performs an update step to accurately estimate INS errors and correct the INS output. In denied GPS environments, the KF depends only on the linearised dynamic model to predict INS errors. Thus, and owing to the inaccuracy of this linearised model, performance degrades significantly even in short periods of GPS-outages. Low-cost MEMS-based INS with high levels of nonlinearity cause further degradation of performance even with careful calibration and systematic errors compensation. These errors eventually result in large position drifts during even short periods of GPS-outages. Owing to its flexible non-parametric nature and computational simplicity, Gaussian processes (GP) models are widely used to solve machine learning problems such as predictions and model selection procedures such as nonlinear optimisation problems [3– 5]. For example, in [3], the model of robotic inverse dynamics was learned using GPR. In addition, in [4], the wireless propagation modelling inside complex indoor environments was adequately modelled by GPR. Being nonparametric means that a model that can correctly fit the data is not required [3– 5]. In addition, by using nonlinear mapping from input space to feature space, highly nonlinear regression models can be learned [4]. Moreover, training data points are not required to be continuous or equally spaced [4, 5] like the case in [6]. Furthermore, using GP, uncertainness in the data (process and observations) can be adequately handled. Methodology: A. A low-cost 3D RISS platform consists of one MEMS grade vertically aligned gyroscope, two horizontal accelerometers, and a vehicle odometer [7]. The state of the vehicle is determined by the vector {Ak , rk , pk , fk , lk , hk , Vek , Vnk , Vuk}, where wk is the latitude of the vehicle, lk is its longitude, and hk is its altitude, Vek , Vnk , Vuk are the East, North, and up velocity, respectively, pk is the pitch angle, rk is the roll angle, and Ak is the azimuth angle. The INS error state vector xk ¼ {dAk , drk , dpk , dfk , dlk , dhk , dVek , dVnk , dVuk , daod , dfx , dfy , dwz} where daod , dfx , dfy , dwz are stochastic errors of the odometerderived acceleration, transversal accelerometer, forward accelerometer, and the gyroscope, respectively. The nonlinear INS error state dynamic model is generally given by xk ¼ f (xk21 , uk21 , wk21) where uk are the sensors readings, and wk is the process noise. The mathematical details of this dynamic model can be found in [7]. The measurement model involves velocity, position, and azimuth updates and is given as zk ¼ h(xk , vk) where zk is GPS velocity, position, and azimuth, and vk is the measurement noise. In the KF, both dynamic and measurement models are linearised using Taylor series expansion. The KF performs the prediction step according to this linearised dynamic model. When GPS observations zk are available, the KF performs the update step using the INS velocity, position, and azimuth deviations from GPS measurements to estimate the complete INS error state vector xk and, hence, correct the INS output.
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B. Gaussian process regression: A Gaussian process defines probability distributions over functions. The objective is to generate a nonlinear regression model that makes predictions for all possible inputs based on all observed data. The observed data set D {(xi , yi)|i ¼ 1, 2, . . ., n} consists of n multi-dimension data points drawn from the noisy process yi ¼ f(xi + 1), where 1 is additive Gaussian noise with zero mean and variance s2n. For input vectors arranged in a matrix X and corresponding output vectors arranged in a matrix Y, GPR can provide a data-driven nonlinear mapping from X to Y and predictions for any unseen inputs X∗ . This nonlinear regression is based on the covariance kernel function [3]. The covariance kernel function determines the covariance between any two output vectors and can take many forms (see [3] for examples). The most popular kernel function used is 2 the exponential function given by Cov(yi , yj ) = k(x i , xj ) + sn d(i − j) where k(xi , xj ) = s2f exp − 12 (xi − xj )T M (xi − xj ) . The function Cov(yi , yj ) calculates the covariance between output vectors in terms of input vectors and some hyper-parameters M and sf. M is a matrix whose diagonal elements are set to a length-scale parameter ℓ such that M ¼diag(ℓ). The predicted output for unseen input vector x∗ is as a Gaussian probability distribution whose mean and variance mx∗ , sx∗ are given by: mx∗ = k(x∗ , X )(K + s2n I)−1 Y , s2x∗ = k(x∗ , x∗ ) − k(x∗ , X )T × (K + s2n I)−1 k(x∗ , X ), where K is the covariance matrix calculated by the kernel function for all given output vectors in Y. The hyper-parameters s2f , ℓ are learned from data by maximising the log marginal likelihood of the observations {X, Y} conditioned on the hyper-parameters [3]. C. Input/output data for the EKF/GPR: The data gathered in good GPS availability should not include positions because their values are unbounded. Thus, only the current sensors’ values and corrected vehicle velocity and attitude (INS output corrected by the KF) will be gathered as inputs and the INS velocity and azimuth deviations from GPS will be used as the target outputs. Learning the hyper-parameters is performed on a large data set previously collected for this purpose. When there is a GPS outage, the GPR model is used to predict the INS velocity and azimuth deviations. The predicted INS deviations are fed to the KF as virtual updates with noise nk ¼ s2x∗ . The KF then performs a virtual update step to estimate all INS errors and, hence, correct INS output. Experimental results: The augmented EKF/GPR technique was examined in post-processing offline mode with real-road test experiments inside a land vehicle on two different trajectories at the City of Kingston (Ontario, Canada). The gyroscope and accelerometers used are from a Crossbow MEMS grade IMU300CC-100. The odometer readings were logged in through the vehicle OBD II interface. The reference solution was provided by a NovAtel G2 Pro-Pack SPAN unit, which is a high end tactical grade INS integrated with GPS. For each trajectory, a total of six artificial GPS-outages periods with seven different starting points and 11 GPR different data set sizes were simulated separately. Thus we implemented a total of 924 separate artificial GPS-outage simulations (runs). For each run we calculated the 2D position root mean square error (RMSE) in metres during GPS-outages. This large number of runs is necessary to analyse and verify the proposed method. Data set size effect: Figs. 1a and b show the improvement rate over all runs and RMSE percentage reduction (%RMSE) against the GPR data set size, respectively. These Figures show strong correlation between the results of the two different trajectories. In addition, they show that improvement rate and RMSE reduction increases with the data set size. 100 90 80 70 60 50 40 30 20 10 0
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GPS outage period effect: Fig. 2a shows the relation between the GPS outage period and RMSE improvement. It can be seen that with longer GPS outages, the %RMSE reduction increases.
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Conclusion: In this research, an enhanced KF augmented with a nonlinear Gaussian process regression technique to bridge GPSoutages was introduced. Based on a large number of experiments, an improvement in position RMSE was achieved for 80% of the experimental runs. Moreover, the proposed bridging technique showed an average position %RMSE reduction of 50% over this large number of experimental runs. This ratio increases with a long GPS-outage period. # The Institution of Engineering and Technology 2011 8 November 2010 doi: 10.1049/el.2010.7164 One or more of the Figures in this Letter are available in colour online.
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M.M. Atia, A. Noureldin and M. Korenberg (Department of Electrical and Computer Engineering, Queens University, 99 University Ave, Kingston, ON K7L 3N6, Canada)
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Fig. 2 GPS-outage period effect and performance in 240 s outage
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References Overall average performance: We determined that 80% of the runs showed an improvement in RMSE with %RMSE reduction of 50%. In addition, %RMSE reduction significantly increases during long GPSoutages. Fig. 3 shows two examples of RMSE values during 240 s GPS-outages with and without the GPR bridging technique. Fig. 2b shows the performance for part of the vehicle trajectory during 240 s GPS outage. RMSE values during 4 min GPS outage with different outage start times
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1 Grewal, M.S., Weill, L.R., and Andrews, A.P.: ‘Global Positioning Systems, Inertial Navigation, and Integration’ (John Wiley and Sons, 2001) 2 Yang, Y., Niu, X., and El-Sheimy, N.: ‘Real-time MEMS based INS/GPS integrated navigation system for land vehicle navigation application’, Institute of Navigation, National Technical Meeting 2006, NTM, 2006 3 Rasmussen, C.E., and Williams, C.: ‘Gaussian processes for machine learning’ (MIT Press, 2006) 4 Duvallet, F., and Tews, A.D.: ‘WiFi position estimation in industrial environments using Gaussian processes’. IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, Nice, France, September 2008 5 Stein, M.L.: ‘Statistical interpolation of spatial data: some theory for Kriging’, Springer Ser. Stat., 1999 6 Shen, Z., Georgy, J., Korenberg, M.J., and Noureldin, A.: ‘FOS-based modelling of reduced inertial sensor system errors for 2D vehicular navigation’, Electron. Lett.’, 2010, 46, (4), pp. 298–299 7 Georgy, J., Noureldin, A., Korenberg, M., and Bayoumi, M.: ‘Low cost 3D navigation solution for RISS/GPS integration using mixture particle filter’, IEEE Trans. Veh. Technol., 2010, 59, (2), pp. 599–615
Fig. 3 2D position RMSE a Trajectory 1 b Trajectory 2
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