In recent literature a derivative free Kalman filter (DFKF) a method that propagates mean and covariance using non-linear transformation is frequently used. In this paper i) factorised version of EKF (UD Extended Kalman Filter or UDEKF) and ii)DFKF are studied and evaluated using various sets of simulated data of the non-linear systems. Sensitivity study of DFKF with respect to tuning parameters used in creation of sigma points and the associated weight is carried out. DFKF is more accurate and easier to implement. A data fusion scheme is involved. It is observed that fusion enhances the estimation accuracy of the state of non-linear plant. Application of DFKF to non linear parameter estimation problem is also demonstrated

Kashyap, SK

Raol, JR

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Evaluation of Derivative Free Kalman Filter and Fusion in Non-Linear Estimation

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EVALUATION OF DERIVATIVE FREE KALMAN FILTER AND FUSION IN NON-LINEAR ESTIMATION S. K. Kashyap J.R. Raol Scientist, Flight Mechanics and Control Division National Aerospace Laboratories Bangalore-560 01 7 Email: sudesh@css.nal.res.in Email: jrraol@css.nal.res.in Head, Flight Mechunics and Control Division National Aerospace Laboratories Bungalore-560 01 7 Abstract In recent Iireranrre CI derivarivefree Kalman .filter ( D F U )  a merhod thar propagates mean and covariance using non- linear lransformaiioiz is f reqvent~~ used. In this paper i) facrorized versiun uf EKF (VD &tended Kalman Filter or UDEKF) and ii) DFKF are studied and evaluated using various sets of‘ simulared data of the non-linear sysslems. Seusirivif?. stzui~: of DFKF with respect to tuning parameters used iri creafion qfsigmapoints and the associated weighrs is carried out. DFKF is more acnrrate and earier to implement. A dara .fusion scheme is evoh,ed and presented bared on DFKF for similar sensors. Its performance is evuluured It is observed thatfiision enhances the estimarion accurac); of the slate of noii-linear plant. .4ppIication of DFKF to non-linear prrrumeler eslimation problem is also demonstrated. Keywords: Non-liuear systems. Targel tracking, Kalmon filieriiig. Deriwtive f;ee trarrsformatio~i and Kalman filter, Data fusion. Paramefer estima/Ion 1. Introduction 1.1. Derivative Free Transformation and Kalman Filter (DFKF) Fig. 1 shows pictoiial representation of DFT [l]. Consider propagation of a random variablexof dimension L ( L  = 2 )  through a non-linear fimction y = f ( x ) .  Assume that mean and coveriance of sigma points, shown by black dots in left side of fig. I ,  for random variable are 7 and P, respectively. These sigma points and their associated weights are deterministically created by the following equations: 1-4244-0038-4 2006 IEEC CCEWCCGEI, ouawa, M~~ 2006 Here, ‘in’ and ‘c’ stand for mean and covariatlce respectively. The scaling parameters used for the creation Of sigma points and their associated weights are: a that detelmines the spread of sigma points around : , f l  is used to incorporate any prior knowledge about distribution of:, A = a  ( L  + K) - L and K is the secondaly tuning parameter. The sigma points created are propagated through the non-linear function ( yi = f ( X , ) .  where, i = 0. ..., 2L ) resulting in transformed sigma points (black dots in right side of fig. 1). The mean and coval-iance of transformed points are formulated as: 2 Subsequently the sigma-point computations for basic state estimation are given in details. A data fusion scheme for similar sensors using DFKF to extract more information about an entity of interest is then evolved based on the DFKF. Various aspects like: DF-DFKF initializatioo, augmented State and its error co\rariance, sigma-points computation, state and covariance propagation, and state and covariance update will be discussed and presented in details in the final pqer. 664 2. Results and Discussions 2.1. Kinematics Consistency The performance of the filters for kinematics consistency using realistic longitudinal short period and lateral-directional data generated from a sut-degree-of freedom simulation of an aircraft is evaluated. The basic kinematics required in state estimation is then presented by state and observation Models. The measurement noise with SNR of 10 is added only to the observables and no noise is added to the rates and accelerations during the data generation. The results shown in fig. 2 are generated for 25 Monte Carlo simulations. Fig. 2 shows the comparison of true, measured and estimated observables. From the plots it is clear that wherever (e.g. between 0-5 seconds or around 10 second) the non-linearity in measurement data is more severe, the performance of UDEKP is degraded as compared to DFKF. 2.2. Data Fusion Consider the vehicle reentry problem shown in fig. 3. It is assumed that a vehicle entering the atmosphere at high altitlide and at high speed is tracked by two ground-stationed sensors with different accuracies. It is assumed that sensors are placed nearby. The measurements from either of sensors are m terms of range and bearing. The strong nonlinearities present in vehicle dynamic are due to the different types of forces acting on i t  The most dominant force is aerodynamic drag as a function of vehicle speed and altitude. Gravitational force accelerates the vehicle towards the center of Earth. In initial phase of flight, vehicle has almost ballistic trajectory hut as density of the atmosphere increases, drag effects become more important and the vehicle rapidly decelerates until its motion is almost vertical. The initial state of vehicle i s  equal to [6500.4,349.14,-1.8093,4.7967,0.6932]. The data is simulated for total number of N ~ 1450 scans. The vehicle is continuously tracked by two sensors in proximity at (n, = 6375 Km,y, = 0 Km).  The rate at which measurements arrive is at a frequency of 5Hz i.e. sampling intervalT 4 . 2  seconds. It is assumed that first sensor gives good bearing information but has noisy range measurement and vice-versa for second sensor (thought this may not be true in general, it is assumed here for the sake of performance evaluation of the algorithm). The standard deviations of range and bearing noises used in simulation are: Sensor 1: ui,= IKm, uIo= 0.05 deg, Sensor 2: 02,= 0.22 Km, uZo= 1 deg. The results are generated for 25 Monte Carlo simulations and performances of DF-DFKF and two DFKF (i.e. for sensor 1 & sensor 2 respectively) are compared. It is clear from fig. 4 that fused state, ils compared to estimated state from other two methods, is close to tnie state. It is clear that data fusion increases the estimation accuracy that would not have been possible using single sensor measurements. 2.3. Parameter Estimation DFKF algorithm is applied to perfonn estimation of non- dimensional longitudinal parameters using simulated short period data of an aircraft. The data is simulated with a sampling time of 0.03 second by giving a doublet input to the elevator. Random process noise (zero mean and Gaussian) with standard deviation of 0.001 is added to certain states. The noisy measurements with SNR-I0 are generated. For estimating the parameters using DFKF. they are modeled as augmented states in the state model. In this case there are 4 states and 11 parameters to be estimated using 7 observables. The initial states and parameters for the DFKF are assumed to be 10% off from their true values. The initial estimation covariance matrix is chosen to reflect this uncertainty. The estimated values of the parameters are compared with the tNe values of the derivatives in Table 1. The estimates are fairly close to the true values. The convergence of the pitching moment rclated derivatives: Cmm, C,,, , ,C,,,4, C ,,,, ‘i is shown in Fig. 5.  It is clear that even in the presence of noise in the data, the parameters converge close to their true values. However. some deviation is observed for CAo estimate. 0 -  3. Conclusions The performances of UDEKI: and DFKF are compared for applications l i e  kinematic consistency checking using realistic longitudinal short period and lateral-directional data of an aircraft. It is observed that DFKF performs better than UDEKF and hence can he used for many nun-linear filtering and control applications. Also, a sensitivity study of DFKF was camed out. A data fusion scheme for similar sensors is proposed and its performance evaluated. Application of DFKF is also illustrated for parameter estimation. References Simon J. lulier, Jeffrey K. Uhlmann, A New Extension ofthe Kalman Filter to Nonlinear Systems, Proceeding qf AeroSenre, I I” I?ifernorioiiol Symposium Aerospace/De/ense Sensing, Simziluriim and Controls, 1997.pp.lS2-193. Simon J. Julier, Jeffrey K. Uhlmann, Unscented Filtering and Nonlinear Estimation, Proceeding ofthe IEEE, Vol.92, No. 3, Mbrch 2004,pp. 401-422. Raol I.R., Girija G., Jatinder Singh, Modelling and Parameter Estimation of Dynamic Systems, IEE Contml Engineering Series Book, Vol. No. 65, IEE, London, August 2004. Table I Estimated parameters of an aircraft ....... ~ .. ... ~. ......... ... ....... Figure 1. Derivative free transformation UDEKF -- DFW --True I - - -Measured 167 I, ' I  DFKF ?#easured 0 5 1 0 1 5 2 0 2 5  0.01 0 5 10 15 20 25 1 0,14,~.- . . - D e 0.12 .s 0 1  o 5 10 15 in 25 Time (wc) TJ c 0.1 i I 0.05' I 0 5 10 15 20 25 0 5 10 15 20 25 -7 40201- 4010 4000 i I 5 10 15 20 25 Tm (re=) Figure 2. Comparison of true, measured, and estimated obsewation data ~~ ~~ . ~ . .  .6400 6450 6500 xl (Km) Figure 3. True positions of re-entered vehicle tn Earth atmosphere 0 100 IS0 200 110 50 -0.1 -i. ~ Time (.ac, Figure 4. Comparison of true, estimated and fused state xg % 0 2 4 6  8 2 4 6 8  -1 8 Time (sec) Figure 5. Parameter convergence for an aircraft-pitching moment derivatives 

Evaluation of derivative free Kalman filter and fusion in non-linear estimation

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Evaluation of derivative free Kalman filter and fusion in non-linear estimation

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