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Adaptive Kalman Filtering for Vehicle Navigation Journal of Global Positioning Systems (2003) Vol. 2, No. 1 : 42-47 Adaptive Kalman Filtering for Vehicle Navigation Congwei Hu1,2, Wu Chen1,Yongqi Chen1 and Dajie Liu2 (1) Department of Land Surveying and Geo-Information, Hong Kong Polytechnic University, Hong Kong, (2)Tongji University, Shanghai, China Received: 13 June 2003 / Accepted: 20 November 2003 Abstract. Kalman filters have been widely used for cities. Then the turning points of the trajectory are navigation and system integration. One of the key extracted and compared with the map data at the same problems associated with Kalman filters is how to assign locations to determine the transformation parameters (Hu suitable statistical properties to both the dynamic and the et al, 2003). Currently Kalman filters have been widely observational models. For GPS navigation, the used in different GPS receivers. However, a conventional manoeuvre of the vehicle and the level of measurement Kalman filter is vulnerable for the determination of the noise are environmental dependent, and hardly to be turning points precisely. predicted. Therefore to assign constant noise levels for The Kalman filtering is an optimal estimation method such applications is not realistic. that has been widely applied in real-time dynamic data processing. A Kalman filter estimates the state of a In this paper, real-time adaptive algorithms are applied to dynamic system with two different models namely GPS data processing. Two different adaptive algorithms dynamic and observation models. The dynamic model are discussed in the paper. A number of tests have been describes the behaviour of state vector, while the carried out to compare the performance of the adaptive observation model establishes the relationship between algorithms with a conventional Kalman filter for vehicle measurements and the state vector. Both models are navigation. The test results demonstrate that the new associated with statistical properties to describe the adaptive algorithms are much robust to the sudden accuracy of the models. For many applications, the model changes of vehicle motion and measurement errors. statistic noise levels are given before the filtering process and will maintain unchanged during the whole recursive process. Commonly, this a priori statistical information is Key words: GPS, adaptive filtering, vehicle navigation, determined by test analysis and certain knowledge about real time positioning the observation type beforehand. If such a priori information is inadequate to represent the real statistic noise levels, Kalman estimation is not optimal and may cause to an unreliable results, sometimes even leads to filtering divergence (Mohamed and K.P. Schwarz 1999). For vehicle navigation, sudden acceleration or 1 Introduction deceleration and sudden change of the directions are impossible to predict. Therefore it is difficult to design a For some navigation applications, we need to know the system with constant noise variances that will satisfy all precise position when a vehicle turns. In map matching situations. One of the common problems with vehicle processing, for example, the turning points of a vehicle navigation using Kalman filter is so called ‘over have to be accurately determined in order to establish shooting’ problem. That is the effect that the dynamic reliable map matching (Yu et al, 2002). In most cities, model keeps position estimation along with previous the local maps are based on local datum and they have to trend while a vehicle actually turns to another direction. be transformed to WGS 84 coordinate system for the use of satellite navigation technology. One of the quick ways Adaptive filtering is trying to determine the statistic to establish the transformation parameters is to drive a car parameters of the dynamic system based on the behaviour equipped with a GPS receiver in the different parts of the of the system during data processing, and it has been paid Hu etal.: Adaptive Kalman Filtering for Vehicle Navigation 43 much attention in Kalman filtering theory (Jia and Zhu, where k denotes epoch number; X is the state vector at 1984, and Gustafsson, 2000). Different adaptive Kalman k filtering algorithms have been studied for surveying and epoch k; Φk+1,k is the state transition matrix; Γ k+1,k is navigation applications. Chen (1992) and Mohamed and vector of system disturbance; Ω is vector of dynamic Schwarz (1999) applied adaptive Kalman filters for the k integration of GPS and inertial navigation system (INS). model noise; Lk+1 is vector of observation at epoch k+1; Wang et al (1997) applied a simplified adaptive algorithm B represent design matrix for observation; V is in kinematic GPS positioning. Chen et al (1999) uses k+1 adaptive filters to estimate the velocity of permanent GPS observation noise; and δkj is the δ -function of stations. Kronecker. In this paper, we investigate the performance of two Kalman filtering estimation can be expressed as: different adaptive Kalman filters for vehicle navigation using GPS, one based on the fading memory and one Prediction: based on the variance estimation. Both algorithms make ˆ ˆ use of the predicted residuals. The fading memory X k+1/k = Φk+1,k X k/k (3) approach tries to estimate a scale factor to increase the predicted variance components of the state vector. The T T QX(k+1/k) = Φk+1,k QX(k/k)Φk+1,k + Γ k+1,k QΩk Γ k+1,k (4) variance estimation method, on the other hand, directly calculates the variance factor for the dynamic model. Updating: Both algorithms closely examine whether there are ˆ ˆ ˆ divergence in the filtering process. If there is no X k+1/k+1 = X k+1/k + K k+1(Lk+1 − Bk+1 X k+1/k ) (5) divergence, the conventional Kalman filtering is used. Otherwise, the adaptive algorithms are applied. Two QX(k+1/k+1 ) = (I − K k+1Bk+1 )QX(k+1/k) (6) examples of vehicle navigation with DGPS are given in the paper. It demonstrates that the positioning accuracy K = Q BT (B Q BT + Q )−1 with the adaptive approaches is significantly better than k+1 X (k+1/ k ) k+1 k+1 X (k+1/ k ) k +! L(k+1) the conventional Kalman filtering, especially when the (7) vehicle turns around the corners. ˆ where X k+1/ k is the predicted state vector; QX (k+1/ k ) is the variance matrix for Xˆ ; K is the gain matrix; 2 The Adaptive Kalman Filtering Algorithms k+1/ k k+1 ˆ X k+1/ k+1 is the estimation of filtering; and QX(k+1/k+1 ) is Considering a general linear dynamic system: its variance matrix. X k+1 = Φk+1,k X k + Γ k+1,k Ωk (1) The adaptive filtering with fading memory algorithm Lk+1 = Bk+1 X k+1 + Vk+1 (2) with The Kalman filtering estimation at epoch k can be considered as ‘weighted’ adjustment between the new E(Ωk ) = 0 , measurements (observation model) and the predicted state vector based on the dynamic model and all previous cov(Ωk ,Ω j ) = QΩk δkj , measurements. If too much ‘weight’ were put to the dynamic model, the estimation would ignore the E(V ) = 0 , information from measurements and causes the k divergence of the filtering process. The idea of fading cov(V ,V ) = Q δ , memory is simple. By applying a factor S>1 to the k j Lk kj predicted covariance matrix to deliberately increase the variance of the predicted state vector (Eq. 8), more cov(Ωk ,V j ) = 0 , ‘weight’ will be given to the measurements. ' T T ˆ Q X(k+1/k) = S(Φ Q Φ + Γ Q Γ ) E(X 0 ) = X 0/ 0 , k+1,k X(k/k) k+1,k k+1,k Ωk k+1,k (8) Q = var(X ) . X 0 / 0 0 The main difference between different fading memory algorithms is on how to calculate the scale factor S. One approach is to assign the scale factor as a constant, S =1 44 Journal of Global Positioning Systems 〜1.4. When S =1, it becomes the conventional Kalman 1 (V T V ) filtering. Obviously there are some drawbacks with a m k+1/k k+1/k constant factor. For example, as the filtering proceeds, the S ≥ k+1 precision of the filtering will decrease because the effects 1 N −1 1 (V T V ) of old data will become less and less. The best way is to N ∑ m k−N + j/k−N k−N + j/k−N use a variant scale factor that will be determined base on j=0 k−N + j the dynamic and observation model accuracy. In this (16) paper, an algorithm is derived based on the size of the predicted residuals, which represent the difference where mi is the number of measurements at epoch i. between the measurements and the predicted state vector. When S≤1，it indicates that the filtering is in a steady The predicted residual vector is expressed as: state processing. When S>1, it indicates that the filtering ˆ may be in an unstable state or in a state of divergence. In Vk+1/k = Lk+1 − Bk+1 (X k+1/k ) (9) practice, in order to avoid false alarm, a threshold S0>1 is For a linear dynamic system, we have (Jia and Zhu, 1984) selected. When S>S0, the adaptive algorithm is applied. Otherwise S =1 and the conventional Kalman filtering T E(Vk+1/kVk+1/k ) algorithm is used. T (10) = Bk+1QX(k+1/k)Bk+1 + QL(k+1 ) The adaptive filter with variance component Let us introduce a scale factor S≥1 to the predicted estimation covariance matrix T This approach tries to estimate the variance factor of the Q& X (k+1/ k ) = S(Φk+1,k QX (k / k )Qk+1,k (11) dynamic model directly using the predicted residuals.
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