Coordinates Transformation for Target Maneuver Detection

2007 IEEE International Conference on Control and Automation Guangzhou, CHINA - May 30 to June 1, 2007 FrCP.23 Coordinates Transformation for Target Maneuver Detection A. Karsaz, N. Pariz H. Khaloozadeh, A. Khorami Department of Electrical Engineering Department of Electrical Engineering Ferdowsi University & Kh. N. Toosi University Ferdowsi University Mashhad, Iran Mashhad, Iran [email protected], [email protected], [email protected], [email protected] Abstract--Ship borne targets normally maneuver on circular II. MODELS OF UNCERTAINTY [7] paths which have lead to tracking filters on circular turns. In this The basic models to be considered in this paper are the paper, an innovation technique is presented to transform the tracking-maneuvering target problems from Polar coordinate to Bayesian and Fisher models which use in [8]. Theses models Cartesian coordinate, therefore a standard linear Kalman filter are specific cases of the state space structure-white process. can be easily applied to them. Mathematical relation between The Bayesian models are one of the most important and measurement noise covariance in polar coordinate and the common used models of uncertainty. In Bayesian models, measurement noise covariance in Cartesian coordinate for uncertainty are modeled by random variables and/or stochastic Kalman implementation is obtained in this approach via a processes with completely specified either probability theorem. distributions or completely specified first and second Keywords-Maneuvering targets, Polar to Cartesian moments. transformation, target maneuver detection. The complete definition of the Bayesian, discrete time model for linear systems is summarized now. X ( n + 1) = F (n ) X (n ) + G ( n) w( n) I. INTRODUCTION z (n ) = H ( n ) X ( n ) + v ( n ) X ( n) state T HERE exist many approaches and methods for tracking maneuvering targets [1]-[3]. For example, switching between Kalman filters of different orders, estimation of z (n ) v (n ) observation white observation uncertainty (1) acceleration as input during a maneuver and to correct the w(n) white system driving uncertainty state using batch least squares methods or recursive estimation X (0 ) initial condition for on-line implementation. Lanka [4] and Korn, Gully and ℜ( n1 ) E{v( n1 ) v T (n 2 )} =  Willsky [5] have developed an extended Kalman filter using a n1 = n 2 circular model of motion. Singer [6] modeled target  0 n1 ≠ n 2 Q( n1 ) E{w(n1 ) wT ( n 2 )} =  acceleration as a random process with known exponential n1 = n 2 autocorrelation. This model is capable of tracking a  0 n1 ≠ n 2 maneuvering target, but the performance of the estimation is E{x( 0) x T ( 0)} = ψ reduced when target moved at a constant velocity. A E{x(0)} = 0 , E{w(0)} = 0 , E{v(0)} = 0 generalized likelihood ratio (GLR) method for maneuver detection and estimation was presented by Korn, Gully and In many applications, the input disturbance, w(.) can be Willsky [5]. This algorithm proposes the use of two modeled as being completely unknown. A model where w(.) is hypotheses, null hypothesis for a target without maneuver, and completely unknown is a type of Fisher model. Of course, alternative hypothesis for a target with maneuver. When the conceptually such Fisher models have to be handled in a log likelihood ratio is over a threshold, a maneuver is detected. different fashion from Bayesian models where w(.) is viewed This system needs a bank of correlators to detect the maneuver as a random vector with known covariance matrix Q(.). For onset time. some applications the Fisher modeling of w(.), can be viewed In this Paper, we presented a new approach to transform as the limiting Bayesian case, where. measurement noise covariance matrix from Polar coordinate to Cartesian coordinate. III. FILTERING OF THE BAYESIAN MODELS The desired form of the filtering solution is a difference 1-4244-0818-0/07/$20.00 ©2007 IEEE 2978 equation (recursive relationship) expressing Xˆ ( N + 1 N ) in IV. TRANSFORMATION BETWEEN POLAR COORDINATE AND CARTESIAN COORDINATE MEASUREMENT COVARIANCE MATRIX terms of Xˆ ( N N ) on z(N+1). The measurement noise in the maneuvering model is related to The logic, which yields the desired equation, can be the measurement noise in Cartesian coordinates by Roecker summarized in the following steps: and McGillem [10]. Tracking with a polar model of motion is 1. Assume that Xˆ ( N + 1 N + 1) is to be calculate just Xˆ ( N N ) not easily implemented by using a Kalman filter in Cartesian and z(N+1). coordinate because the process model is nonlinear, and the 2. Use the one-step prediction logic to change the problem to covariance matrix of each radar obtained in polar coordinate. calculate Xˆ ( N + 1 N + 1) from Xˆ ( N + 1 N ) and z(N+1). Therefore for Kalman filter implementation in Cartesian 3. Solve a Fisher estimation problem where Xˆ ( N + 1 N + 1) and coordinate, transformation between polar coordinate and Cartesian coordinate measurement noise covariance matrix is z(N+1) are considered on an unknown vector X(N+1). necessary. In the Bayesian model, stochastic probabilistic models are The transformation between Cartesian coordinates and polar used for all the uncertainties. Thus x(0), v(n) and w(n) are coordinates is given by modeled as zero mean uncertainty random variables. The matrix H(n), F(n) and G(n) in Eq. (1) assumed to be R = x2 + y2 (4) known function of time n. The problem to be considered in y θ = tan −1 ( ) (5) how to use the observation up to time n2 , z(1),...,z( n2 ), to x X = R cos θ (6) estimate the state X (n1 ) at some time n1. Y = R sin θ (7) The solution of the problem filtering, after some manipulation In polar coordinates the state is radius and angle of leads us to the Kalman filter with equations: Xˆ ( N +1 | N ) = F ( N ) Xˆ ( N | N ) + K ( N +1)[z( N +1) target ([ R θ ]) . − H ( N + 1)F ( N ) Xˆ ( N | N )] A. Theorem K ( N + 1) = ∑( N + 1 | N ) H T ( N + 1)ℜ−1 ( N +1) ∑( N +1 | N + 1)) = ∑( N + 1 | N ) − If the states R and θ in polar coordinates are two independent measurements with white noise zero mean ∑( N +1 N ) H T ( N +1)[ℜ( N + 1) + H ( N +1) variables, then the states in Cartesian coordinates are two ∑( N +1 | N ) H ( N +1)T ]−1 H ( N + 1) ∑( N + 1) (2) independent measurements with white noise zero mean ∑( N +1 | N ) = F ( N ) ∑( N | N )F T ( N ) variables. In addition the measurement noise vector in Cartesian coordinates is related to the measurement noise + G( N )Q( N )GT ( N ) ∑(0 | 0) = 0, Xˆ (0 | 0) = 0 vector in polar coordinates by the following equation, δ x2  cos 2 θ 0 R02 sin 2 θ 0  δ R2   2 =  2  2  δ y   sin θ 0 R0 cos θ 0  δ θ  2 2 (8) K (N ) is the Kalman gain and notation Xˆ ( N + 1 | N ) denotes the prediction at the ( K + 1) th sample point given the measurement where δ R and δ θ are too small. up to and including the K th whilst Xˆ ( N | N ) denotes the B. Proof th Define, estimation at the N sample point given the measurement up to and including the N th . ∑( N | N ) is the error covariance R = Ro + ∆R matrix and ∑ ( N + 1 | N ) is the error covariance matrix of the (9) θ = θ o + ∆θ where, ∆R and ∆θ are two zero mean independent random one-step prediction. The measurement covariance matrix ℜ is white noise variables as bellow: usually introduced in polar coordinate, but we need it in ∆R : N (0, δ R ) Cartesian coordinate for Kalman filter implementation at Eq. (2). ∆θ : N (0, δ θ ) δ 2 0  ℜ= x 2 δ R : Standard diversion of target radius measurement  0 δ y  (3) δθ : Standard diversion of target angle measurement δ x : Standard diversion of target trajectory measurement at X axis Ro : Target radius without noise θo : Target angle without noise δ y : Standard diversion of target trajectory measurement at Y axis Suppose that x and y in Cartesian coordinates have been Maneuvering Targets are difficult to track with Kalman filter perturbed by a noise as bellow since the target model of tracking filter might not fit the real X = X o + ∆X target trajectory [9]. (10) Y = Y 0 + ∆Y 2979 where the states X o and Yo define the target trajectory in A target maneuvering tracking algorithm which turns in two- Cartesian coordinates without noise. The relation equation dimensional space is simulated. Several values of between variables ∆X , ∆Y and ∆R , ∆θ is desired. measurement noise covariance elements, range measurement By Euler equation; and target azimuth measurement was used. It is assumed that the target moves in a plane, which is the X o + jYo = Ro e jθ o = Ro cos θ o + jRo sin θ o (11) two-dimensional case, such as a ship. At first example the target is traveling at zero acceleration X + jY = Re jθ u (t ) 2*1 in X and Y direction equal a x (t ) = 0 m/s 2 , = R cos θ + jR sin θ a y (t) = 0 m/s 2 . In this simulation, the sampling time is = ( R o + ∆R ) cos(θ o + ∆θ ) + j ( R o + ∆R) sin(θ o + ∆θ ) (12) = ( R o + ∆R )(cos θ o cos ∆θ − sin θ o sin ∆θ ) T=0.015 second and the matrices of Q is 0.1. A Monte Carlo simulation of 10000 runs was performed on these paths + j ( R o + ∆R )(sin θ o cos ∆θ + cos θ sin ∆θ ) ( R 0 = 5000 (m) and θ 0 = 30 (deg) with constant velocity). As By using Eq. (8) we have, X o + ∆X + j (Yo + ∆Y ) = R o cos θ o cos ∆θ − Ro sin θ o sin ∆θ Fig.1 and Fig.2 shows, measurement noise covariance + ∆R cos θ o cos ∆θ − ∆R sin θ o sin ∆θ elements generated for R and θ axis are both Gaussian (13) random variable. Fig. 3 shows actual and noisy Trajectory + j( Ro sin θ o cos ∆θ + Ro cos θ o sin ∆θ + ∆R sin θ o cos ∆θ + ∆R cos θ o sin ∆θ ) (measurement trajectory) in polar coordinate with constant Suppose we know that ∆R and ∆θ both have small values. velocity. Fig. 3 is only apparent on finer scale (600 time index so we have point from 10000). cos ∆θ ≈ 1 Histogram R, σ =100 (m) R sin ∆θ ≈ ∆θ 1800 ∆θ∆R ≈ 0 1600 By substitution (11) in (13) and using above relation we have, 1400 X o + jY o + ∆ X + j ∆ Y = R o cos θ o − R o ∆ θ sin θ o + ∆ R cos θ o − ∆ R ∆ θ sin θ o 1200 + j( R o sin θ o + R o ∆ θ cos θ o 1000 + ∆ R sin θ o + ∆ R ∆ θ cos θ o ) (14) 800 = R o cos θ o + j R o sin θ o 600 − R o ∆ θ sin θ o + ∆ R cos θ o 400 + j( R o ∆ θ cos θ o + ∆ R sin θ o ) 200 Therefore, 0 ∆X = ∆R cos θ o − Ro ∆θ sin θ o (15) 1.15 1.16 1.17 1.18 1.19 1.2 1.21 1.22 1.23 1.24 1.25 Distance (m) 4 x 10 ∆Y = Ro ∆θ cos θ o + ∆R sin θ o (16) Fig. 1. Histogram of state R in polar coordinate with a Gaussian As mentioned before ∆R and ∆θ are two zero mean measurement noise ( δ R = 100 m ) independent random white noise variables, then X , Y are zero mean white noise with following variances : var(∆X ) = cos 2 θ 0 var(∆R ) + R02 sin 2 θ 0 var( ∆θ ) 2000 Histogram θ, σ =2 (deg) θ var(∆Y ) = sin 2 θ 0 var(∆R ) + R02 cos 2 θ 0 var(∆θ ) δ x2  cos 2 θ 0 R02 sin 2 θ 0  δ R2  1800  2 =  2   1600 δ y   sin θ 0 R02 cos 2 θ 0  δ θ2  1400 δ 2 0 1200 ℜ= x  1000  0 δ y2  800 600 Because, Ro , cosθ o and sinθ o are constant at each time. The 400 values of δ R , δ θ can obtained from radar catalogs and have 200 known values. 0 34 36 38 40 42 44 46 48 50 52 54 Target Azimuth (deg) V. SIMULATION RESULTS Fig. 2. Histogram of state θ in polar coordinate with a Gaussian As an evaluation of this transformation technique, a Monte measurement noise ( δ θ = 2 o ) Carlo simulation was performed and various paths with circular and straight trajectories were implemented. 2980 As proof by Theorem A, and using Eq. (6), (7) for evaluate the Fig. 6 shows the simulation result when target has a constant X and Y directly from noisy R and θ (show in Fig.4, Fig.5), velocity and maneuver detection is done by using linear Gaussian distribution can see for both X and Y too. Kalman filter (Eq. 2). Target Range Target Azimuth 5500 32 31 θ, state (deg) R, state (m) 5000 30 29 4500 28 0 200 400 600 0 200 400 600 time index T time index T Target Range measurement Error Target Azimuth measurement Error 400 4 R, state Error(m) θ, state Error(m) 200 2 0 0 -200 -2 -400 -4 0 200 400 600 0 200 400 600 time index T time index T Figure 3: Actual and noisy Trajectory in polar coordinate with constant velocity and their Gaussian measurement errors Histogram of X, Histogram of Y, 1800 1800 1600 1600 1400 1400 1200 1200 1000 1000 800 800 600 600 400 400 200 200 0 7000 7500 8000 8500 9000 9500 10000 0 Target Distance (m) 7000 7500 8000 8500 9000 9500 10000 Target Distance (m) Fig.4. Histogram of X in Cartesian coordinate evaluated Fig.5. Histogram of Y in Cartesian coordinate evaluated by Eq. (6) by Eq. (7) 2981  Target Trajectory [8] H. Khaloozadeh and A. Karsaz, "A Modified Algorithm based on Mixed 3600 Bayesian-Fisher Model for Target Maneuver Detection", IEEE SPCOM, Actual Trajectory 2004, pp 7868-7872. Estimated Trajectory [9] T. Kawase, H. Tsurunosono, "Two-Stage Kalman Estimator Using 3400 Advanced Circular Prediction for Maneuvering Target Tracking", IEEE, 1998. 3200 [10] James a. Roecker and Clare D. McGillem, "Target Tracking in Maneuver Centered Coordinates" IEEE, 1988. Y (m) 3000 BIOGRAPHIES 2800 Ali Karsaz received the B.S. degree in Electrical Engineering from the Amirkabir University of 2600 Technology, Tehran, Iran, in 1999. He Obtained his M.Sc. degree in Electrical Engineering in 2004 at 2400 Ferdowsi University of Mashhad. He is currently as 4000 4500 5000 5500 6000 6500 a Ph.D. student in Control Engineering at the X (m) Ferdowsi University of Mashhad, Mashhad, Iran. Fig. 6. Actual and Estimation Trajectory in Cartesian coordinate with His interest area is Artificial Neural Networks, Stochastic and constant velocity using linear Kalman filter (Eq. 2). nonlinear Modeling and Estimation, System Identification, Time Series Analysis and prediction, Inertial Navigation Systems. By using Eq. (4), (5) trajectory of target in polar coordinate are obtained (Fig .7). Nasser Pariz with honors from the Ferdowsi University, Mashhad, Iran, in 1988, 1991, and 1999, respectively. He is the Director of Electrical Fig. 8, shows the actual and estimation of target parameter in Engineering Department, Ferdowsi University. His R and θ direction in the present of target maneuvering. research interests are power and control systems. received the B.Sc., M.Sc., and Ph.D. degrees in electrical engineering Hamid Khaloozadeh, received the B.S. degree in VI. CONCLUSIONS control engineering from Sharif University of In this paper, a new transformation is presented to transform technology (Tehran, Iran), in 1990, the M.S. degree in control engineering from K.N.Toosi university Polar coordinate to Cartesian coordinate tracking target of technology Tehran, Iran), in 1993, and the Ph.D. problems. Linear Kalman filter in Cartesian coordinate can degree in control engineering from Tarbiat easily implemented by using this technique. Simulation results Moddaress university (Tehran, Iran), in 1998. Since 1998 to 2004 he was a faculty member at Ferdowsi University show a high performance of the proposed innovation of Mashhad. He is currently an assistant Professor teaches in the technique and effectiveness of this scheme in tracking Department of Electrical Engineering in K.N.Toosi University of maneuvering targets. Technology (Tehran, Iran). His interest area is Digital Control, Nonlinear Modeling, System Identification, Optimal Control, Adaptive Control, Stochastic Estimation and Control, Neural Networks and Time Series Analysis. REFERENCES Ali Khorami was born in Tehran in Iran, on July 9, 1975. He studied at the Ferdowsi University of [1] A. N. Shiryayev, "On optimal methods in quickest detection problems", Mashhad/Iran. He received B.S. degree in Electrical Theory of Probability and Its Application, 8, 1, pp. 22-46. 1963. Engineering from this University. Obtained his [2] L. Pelkowitz, and S. C. Schwartz, "Asymptotically Optimum Sample M.Sc. in Electrical Engineering at Ferdowsi Size for Quickest Detection", IEEE Transactions on Aerospace and university of Mashhad. Electronic Systems, AES-23, pp. 263-271, 1987. [3] T. C. Wang , and P. K. Varshney, "A Tracking algorithm for maneuvering targets", IEEE Transactions on Aerospace and Electronics Systems, Vol. 29, No. 3, pp. 910-924, 1993. [4] O. Lanka, "Circle Maneuver Classification for Maneuvering Radar Targets Tracking", Tesla Electronics, Vol. 17, pp. 10-17, March 1984. [5] J. Korn, S. W. Gully, and A. S. Willsky, "Application of the generalized likelihood ratio algorithm to maneuver detection and estimation", in Proc. 1982 American Control Conference, pp. 792-798. [6] ]R. A. Singer, "Estimating optimal tracking filter performance for manned maneuvering targets", IEEE Trans Aerospace and Electronic Systems, AES- 6, pp. 473-483. 1970. [7] F. C. Schweppe, Uncertain Dynamic Systems, Prentice-Hall. 1973. 2982  Target Range Target Azimuth 8000 30.6 30.4 θ, state (deg) R, state (m) 7000 30.2 6000 30 5000 29.8 0 5000 10000 0 5000 10000 time index T time index T Target Range Estimation Error Target Azimuth Estimation Error 20 0.6 R, state Error(m) θ, state Error(m) 10 0.4 0 0.2 -10 0 -20 -0.2 0 1000 2000 3000 0 5000 10000 time index T time index T Figure 7: Actual and Estimation Trajectory in polar coordinate with constant velocity and their errors 4 x 10 Target Range Target Azimuth 2 45 40 θ, state (deg) R, state (m) 1.5 35 1 30 0.5 25 0 5000 10000 0 5000 10000 time index T time index T Target Range Estimation Error Actual and measurement Trajectory in polar coordinat 20 0.2 R, state Error(m) θ, state Error(m) 10 0.1 0 0 -10 -0.1 -20 -0.2 0 1000 2000 3000 0 5000 10000 time index T time index T Figure 8: Actual and Estimation Trajectory in polar coordinate with maneuvering target and their errors a (t ) = ax2 (t ) + a y2 (t ) = 0.1 m / s 2 2983