SELECTIVE ANGLE MEASUREMENTS FOR A 3D-AOA INSTRUMENTAL VARIABLE TMA ALGORITHM

Transcription

1 SELECTIVE ANGLE MEASUREMENTS FOR A 3D-AOA INSTRUMENTAL VARIABLE TMA ALGORITHM Kutluyıl Doğançay Reza Arablouei School of Engineering, University of South Australia, Mawson Lakes, SA 595, Australia ABSTRACT The method of instrumental variables has been successfully applied to pseudolinear estimation for angle-of-arrival target motion analysis (TMA) The objective of instrumental variables is to modify the normal equations of a biased leastsquares estimator to make it asymptotically unbiased The instrumental variable (IV) matrix, used in the modified normal equations, is required to be strongly correlated with the data matrix uncorrelated with the noise in the measurement vector At small SNR, the correlation between the IV matrix the data matrix can become weak The concept of selective angle measurements (SAM) overcomes this problem by allowing some rows of the IV matrix data matrix to be identical This paper demonstrates the effectiveness of SAM for a previously proposed 3D angle-only IV TMA algorithm The performance improvement of SAM is verified by simulation examples Index Terms Selective angle measurements, 3D target motion analysis, angle-of-arrival localization, instrumental variables INTRODUCTION Tracking of radio emitters has found many applications in mobile user localization, asset localization, sensor networks target tracking in electronic warfare, to name but a few In this paper we consider the use of selective angle measurements (SAM) in 3D angle-only target motion analysis (TMA) The objective of angle-only 3D-TMA is to estimate the position, velocity possibly acceleration of a target from its azimuth elevation angle measurements collected by a moving observer (ownship) Whilst the TMA problem has been studied extensively in the 2D plane, there is little work reported on the angle-only 3D-TMA problem In [] a 3D localization algorithm was derived using an orthogonal vector estimation approach [2], [3] to estimate the location of a stationary radio emitter from angle measurements A 3D-pseudolinear estimator (3D-PLE) its weighted instrumental variable (WIV) version were also presented shown to provide much improved estimation performance close to the maximum likelihood estimator (MLE) For moving targets, [4] proposes a so-called 3D improved PLE (3D-IPLE) drawing on the MLE cost function approximation used in [5], as well as some trigonometric approximations The resulting estimator is the 2D-PLE for the 2D-projection of the 3D-TMA problem concatenated with the 3D-orthogonal vector estimator derived in [] The bias performance of the resulting estimator is improved by employing the bias compensation method in [6] the method of WIV [7] A different approach resembling the PLE in [] has been adopted in [8], resulting in a 2D-PLE for the xycomponents of the target motion parameter vector a linear least-squares solution for the z-component The bias performance of this new 3D-PLE was improved by applying bias compensation WIV only to the 2D-PLE part of the estimator The poor performance of the 3D-WIV estimator in large noise was analysed The method of SAM, originally developed for self-localization [9], was employed to improve the performance of the 3D-WIV estimator in low SNR situations In this paper we consider the application of SAM to the WIV estimator in [4] It is shown via numerical simulations that the SAM estimator can improve the performance of this WIV estimator at large noise levels thanks to its assurance of strong correlation between the IV matrix data matrix Section 2 describes the 3D-TMA problem sets out the assumptions made Section 3 summarizes the MLE for 3D- TMA An overview of the 3D-IPLE is provided in Section 4 Section 5 describes the method of SAM applied to the WIV estimator in [4] Comparative simulation examples are presented in Section 6 The paper concludes in Section 7 2 3D-TARGET MOTION ANALYSIS PROBLEM Fig depicts the 3D-TMA problem The objective of 3D- TMA is to estimate the target location p k from N noisy azimuth elevation angle measurements taken at time instants k {,,, N } The angles are related to the target location p k = [p x,k, p y,k, p z,k ] T observer location r k = [r x,k, r y,k, r z,k ] T through θ k = tan y k x k, π < θ k π (a) φ k = sin z k s k, π 2 < φ k π 2 (b) /5/$3 25 IEEE 95

2 x Observer trajectory z r k k y s k k p k Target trajectory Fig 3D-TMA geometry using azimuth elevation angles θ k, φ k Here tan is the 4-quadrant arctangent, x k = p x,k r x,k y k = p y,k r y,k z k = p z,k r z,k s k = p k r k denotes the Euclidean norm We make the following assumptions throughout the paper: The target moves with a constant velocity during the observation interval k {,,, N } The azimuth elevation angle measurements are taken at regular time instants t k = kt/(n ) T is the length of the observation interval Let p denote the initial target location vector at k = v the constant velocity vector Then the target location at time t k is p k = p + t k v (2a) = M k ξ (2b) M k = t k t k t k ξ = [ ] p v is the 6 target motion parameter vector to be estimated from noisy angle measurements Given an estimate of target motion parameters ˆξ, p k can be estimated by substituting ˆξ for ξ in (2) The azimuth elevation angle measurements are corrupted by independent zero-mean Gaussian noise: θ k = θ k + n k, n k N (, σ 2 n k ) φ k = φ k + m k, m k N (, σ 2 m k ) n k m k are independent (3) The target is observable This requires the observer to outmaneuver the target while collecting the angle measurements [ 3] 3 MAXIMUM LIKELIHOOD ESTIMATOR Under the independent additive Gaussian noise assumption, the likelihood function for the angle measurements is p( ψ ξ) = (2π) N K /2 { exp 2 ( ψ ψ(ξ)) T K ( ψ } ψ(ξ)) ψ = [ θ, θ,, θ N, φ, φ,, φ N ] T is the 2N vector of noisy angle measurements, ψ(ξ) = [θ (ξ), θ (ξ),, θ N (ξ), φ (ξ), φ (ξ),, φ N (ξ)] T is the 2N vector of azimuth elevation angles as a function of ξ with θ k (ξ) = tan y k(ξ) x k (ξ), φ k(ξ) = sin z k(ξ) s k (ξ) s k (ξ) = x k(ξ) y k (ξ) = M k ξ r k, z k (ξ) K = diag(σ 2 n,, σ 2 n N, σ 2 m,, σ 2 m N ) is the 2N 2N diagonal covariance matrix of the angle noise, K denotes the determinant of K The MLE of the target motion parameters ˆξ ML is given by (4) ˆξ ML = arg min J ML (ξ) (5) ξ R 6 J ML (ξ) is the ML cost function J ML (ξ) = 2 et (ξ)k e(ξ), e(ξ) = ψ ψ(ξ) (6) Equation (5) describes a nonlinear least squares estimator with no closed-form solution A numerical solution can be obtained by using the Gauss-Newton (GN) algorithm: ˆξ(i + ) = ˆξ(i) + (J T (i)k J(i)) J T (i)k e(ˆξ(i)), i =,, (7) Here J(i) is the 2N 6 Jacobian matrix of ψ(ξ) with respect to ξ evaluated at ξ = ˆξ(i): 96

3 J(i) = s xy,(ˆξ(i)) at (ˆξ(i))M s xy,n (ˆξ(i)) at N (ˆξ(i))M N s (ˆξ(i)) bt (ˆξ(i))M s N (ˆξ(i)) bt N (ˆξ(i))M N a k (ˆξ) is a unit vector orthogonal to the 2D-projection of the estimated 3D-range vector b k (ˆξ) is a unit vector orthogonal to the estimated 3D range vector []: sin θ k (ˆξ(i)) a k (ˆξ(i)) = cos θ k (ˆξ(i)) (9) sin φ k (ˆξ(i)) cos θ k (ˆξ(i)) b k (ˆξ(i)) = sin φ k (ˆξ(i)) sin θ k (ˆξ(i)) () cos φ k (ˆξ(i)) s xy,k (ξ) = [ ] xk (ξ) y k (ξ) is the projected range vector parameterized by ξ Assuming a priori knowledge of the noise covariance matrix K, the Cramer-Rao lower bound (CRLB) for the 3D- TMA problem is given by (8) CRLB = (J T o K J o ) () J o is the Jacobian matrix evaluated at the true target motion parameter vector 4 OVERVIEW OF 3D-IPLE In [4] a matrix equation linear in ξ is obtained from a smallnoise approximation of the MLE cost function following [5] The resulting 3D-IPLE can also be derived by rewriting (a) as sin( θ k n k ) cos( θ k n k ) = y k x k lumping the noise terms together to get [4] [sin θ k, cos θ k, ]M k ξ = [sin θ k, cos θ k, ]r k + η k (2) η k = s k cos φ k sin n k, rewriting (b) as or sin( φ k m k ) cos( φ k m k ) = z k s xy,k [sin φ k cos θ k, sin φ k sin θ k, cos φ k ]M k ξ = [sin φ k cos θ k, sin φ k sin θ k, cos φ k ]r k + ν k (3) ν k = s k (cos φ k sin φ k cos n k cos φ k sin φ k ) Stacking (2) (3) for k =,,, N gives H = [M T u,, M T N u N, Hξ = d + ω (4) M T v,, M T N v N ] T d = [r T u,, r T N u N, r T v,, r T N v N ] T ω = [η,, η N, ν,, ν N ] T u k = [sin θ k, cos θ k, ] T v k = [sin φ k cos θ k, sin φ k sin θ k, cos φ k ] T We note that (4) is made up of two linear matrix equations concatenated; one for the 2D-PLE (projection of the 3D-TMA problem into the xy-plane) the other for the orthogonal vector estimator [] The 3D-IPLE in [4] is the least-squares solution of (4): ˆξ IPLE = (H T H) H T d (5) 5 SELECTIVE ANGLE MEASUREMENTS The correlation between H ω introduces undesirable estimation bias Modifying the normal equations for the 3D- IPLE H T Hˆξ IPLE = H T d to G T Hˆξ IV = G T d gives the IV estimator: ˆξ IV = (G T H) G T d (6) This estimate is asymptotically unbiased if G T H is full-rank E{G T ω} = [5] An effective simple method for constructing the IV matrix satisfying these requirements asymptotically was proposed in [7], which comprises the following steps: Obtain an initial estimate ˆξ IPLE Estimate the angles based on this initial estimate (cf (4)) ˆθ k = θ k (ˆξ IPLE ), ˆφk = φ k (ˆξ IPLE ) Substitute the angle estimates for angle measurements in H to construct the IV matrix G: G = H θ =ˆθ,, θ N =ˆθ N, φ = ˆφ,, φ N = ˆφ N Introducing a weighting matrix ( W = K diag s xy, (ˆξ IPLE ) 2,, s xy,n (ˆξ IPLE ) 2, ( s (ˆξ IPLE ) 2 s xy, (ˆξ IPLE ) 2 σn 2 ),, ( ) sn (ˆξ IPLE ) 2 s xy,n (ˆξ IPLE ) 2 σn 2 N ) results in a weighted IV estimator (3D-IWIV) [4]: ˆξ IWIV = (G T W H) G T W d (7) 97

4 z axis (m) y axis (m) 2 5 x axis (m) Fig 2 Simulated 3D-TMA geometry 5 which is asymptotically unbiased efficient [6] The matrices G H are required to correlate well so that the product G T H is well-conditioned If the unit vectors that make up these matrices do not align well as a result of large differences between ˆθ k θ k, between ˆφ k φ k, the correlation between G H can be weakened This will increase the condition number of G T H MSE To avoid ill-conditioning of G T H, some rows of the IV matrix can be forced to be identical to the corresponding rows of H, depending on the difference between the angle estimates measurements This leads to the idea of SAM: { (ˆθ k, ˆφ (ˆθ k, k ) = ˆφ k ) if θ k ˆθ k < α k φ k ˆφ k < β k ( θ k, φ k ) otherwise (8) Here α k β k are threshold values that should be chosen proportional to σ nk σ mk, respectively The angle estimates strongly influence the threshold values The appropriate ranges for the thresholds are 2σ nk < α k < 5σ nk 2σ mk < β k < 5σ mk The matrices G H are maximally correlated if ˆθ k = θ k ˆφ k = φ k, k =,,, N, in G This amounts to replacing G with H, which results in a least-squares estimator with severe bias problems To retain some benefits of bias reduction, (8) offers a compromise solution We will refer to the 3D-IWIV estimator employing (8) as the 3D-SAM-IWIV The performance improvement of SAM is determined by the reduction in the condition number of G T W H The relation of MSE performance to condition number will be demonstrated in the next section 6 SIMULATIONS The simulated TMA geometry is depicted in Fig 2 The target moves with a constant velocity of v = [6, 3, ] T m/s starting from an initial position p = [5,, 2] T m The observer takes N = 3 angle measurements at regular time Bias norm RMSE D PLE 3D IPLE MLE D PLE 35 3D IPLE 3 MLE 25 CRLB Fig 3 Bias RMSE performance intervals (t k+ t k = 5 s) while following a three-leg constant-velocity trajectory (r = [,, 5] T m the observer velocity alternates between [25, 3, ] m/s [25, 3, ] m/s from one leg to another) The azimuth elevation angle noise is iid with stard deviation σ The MLE is implemented using the GN algorithm, initialized to the 3D-IPLE run for iterations The 3D-IWIV 3D-SAM-IWIV estimators use the 3D-IPLE to construct the IV matrix H the weighting matrix W For the 3D-SAM-IWIV estimator the threshold parameters are α = = α N = 5σ β = = β N = 5σ All bias RMSE values are estimated using 2, Monte Carlo simulation runs Fig 3 shows the simulated bias norm RMSE of the 3D-PLE in [8], 3D-IPLE, 3D-IWIV, 3D-SAM-IWIV MLE The RMSE plot also includes the CRLB (square root of trace of CRLB) For σ > 3 both the 3D-IWIV MLE exhibit a sharp increase in RMSE This is due to the loss of correlation between G H for the 3D-IWIV estimator the threshold effect for the MLE The 3D-SAM-IWIV avoids rapid deterioration of RMSE by assuring a strong correlation between G H The method of SAM causes a nonvanishing residual correlation between the IV matrix 98

5 Condition Number 5 x Fig 4 Averaged condition number of G T W H angle noise, which explains the slight degradation in the bias norm as the angle noise is increased We also observe that the 3D-PLE [8] outperforms the 3D-IPLE [4] by a large margin In Fig 4 the averaged condition numbers of G T W H for the 3D-IWIV 3D-SAM-IWIV estimators are plotted against noise stard deviation As the angle noise increases, the condition number of the 3D-IWIV significantly exceeds that of the 3D-SAM-IWIV, which demonstrates the effectiveness of SAM in maintaining a strong correlation between G H This also explains the improved RMSE performance of the 3D-SAM-IWIV compared with 3D-IWIV (see Fig 3) 7 CONCLUSION The method of SAM has been applied to the 3D-IWIV proposed in [4] The loss of correlation between the IV matrix G the data matrix H was shown to impact the RMSE performance adversely The effectiveness of SAM in combatting this loss was demonstrated by simulation examples It was observed that both the RMSE performance condition number of G T W H are significantly improved by employing SAM in the IV matrix REFERENCES [] K Doğançay G Ibal, Instrumental variable estimator for 3D bearings-only emitter localization, in Proc 2nd Int Conf on Intelligent Sensors, Sensor Networks Information Processing, ISSNIP 25, Melbourne, Australia, December 25, pp [2] D Koks, Passive geolocation for multiple receivers with no initial state estimate, Tech Rep DSTO RR- 222, Defence Science & Technology Organisation, Edinburgh, SA, Australia, November 2 [3] K Doğançay, Bearings-only target localization using total least squares, Signal Processing, vol 85, no 9, pp 695 7, September 25 [4] L Badriasl K Doğançay, Three-dimensional target motion analysis using azimuth/elevation angles, IEEE Trans on Aerospace Electronic Systems, vol 5, no 4, pp , October 24 [5] M Gavish A J Weiss, Performance analysis of bearing-only target location algorithms, IEEE Trans on Aerospace Electronic Systems, vol 28, no 3, pp , 992 [6] K Doğançay, Bias compensation for the bearings-only pseudolinear target track estimator, IEEE Trans on Signal Processing, vol 54, no, pp 59 68, January 26 [7] K Doğançay, Passive emitter localization using weighted instrumental variables, Signal Processing, vol 84, no 3, pp , March 24 [8] K Doğançay, 3D pseudolinear target motion analysis from angle measurements, IEEE Trans on Signal Processing, vol 63, no 6, pp 57 58, March 25 [9] K Doğançay, Self-localization from lmark bearings using pseudolinear estimation techniques, IEEE Trans on Aerospace Electronic Systems, vol 5, no 3, pp , July 24 [] S C Nardone V J Aidala, Observability criteria for bearings-only target motion analysis, IEEE Trans on Aerospace Electronic Systems, vol 7, pp 62 66, March 98 [] S E Hammel V J Aidala, Observability requirements for three-dimensional tracking via angle measurements, IEEE Trans on Aerospace Electronic Systems, vol AES-2, no 2, pp 2 27, March 985 [2] J M Passerieux D Van Cappel, Optimal observer maneuver for bearings-only tracking, IEEE Trans on Aerospace Electronic Systems, vol 34, no 3, pp , July 998 [3] A Farina, Target tracking with bearings-only measurements, Signal Processing, vol 78, pp 6 78, 999 [4] A G Lindgren K F Gong, Position velocity estimation via bearing observations, IEEE Trans on Aerospace Electronic Systems, vol 4, no 4, pp , July 978 [5] L Ljung, System Identification: Theory for the User, Prentice Hall, Upper Saddle River, NJ, 2 edition, 999 [6] K Doğançay, On the efficiency of a bearings-only instrumental variable estimator for target motion analysis, Signal Processing, vol 85, no 3, pp 48 49, March 25 99