Recursive Filtering for Target Tracing with Range and Direction Cosine Measurements Zhansheng Duan Yu Liu X. Rong Li Department of Electrical Engineering University of New Orleans New Orleans, LA 748, U.S.A. {zduan,lyu,xli}@uno.edu Abstract Due to the nonlinear relationship between Cartesian coordinates and range-direction-cosine coordinates, target tracing of state described in Cartesian coordinates with range and direction cosine measurements is a nonlinear filtering problem. Measurement conversion based Kalman filter available for this type of problem has some serious drawbacs. Depending on whether measurement of the third direction cosine is directly available, two recursive linear minimum mean-squared error () filters for target tracing with range and direction cosine measurements are developed in this paper. Illustrative numerical examples show that in terms of credibility and accuracy, the proposed filters should be preferred. Keywords:, target tracing, range, direction cosine, nonlinear filtering. Introduction In tracing applications, target dynamics are usually modeled in Cartesian coordinates, while the measurements are directly available in the original sensor coordinates. Due to the nonlinear relationship between Cartesian coordinates and sensor coordinates, tracing in Cartesian coordinates using measurements available in sensor coordinates is in essence a nonlinear filtering problem. Existing nonlinear filtering algorithms, e.g., extended Kalman filter (EKF) [], unscented filtering () [], the second-order Stirling interpolation based filter (DD) [3], and Gaussian filter [4], for general nonlinear systems can be applied to this problem. But since the nonlinear relationship between the Cartesian coordinates and sensor coordinates is explicitly nown, specifically designed nonlinear filters may behave better. In a trac-while-scan surveillance system, the radar can provide measurements, e.g., range, bearing, elevation and range rate, in polar or spherical coordinates [5]. Specifically designed target tracing algorithms for this type of radar abound. One popular idea is to convert the measurement model in polar or spherical coordinates to Cartesian Research supported in part by NSFC grant 666, ARO through grant W9NF-8--49, ONR-DEPSCoR through grant N4-9-- 69, and NAVO through Contract # N636-9-P-3S. coordinates so that the converted measurement model taes a pseudo linear form with respect to the target motion state described in Cartesian coordinates. In this way, it is hoped that the Kalman filter can be applied. Unfortunately, the converted measurement errors are state dependent and have biases. Numerous ways have been proposed to debias the converted measurements [5]. For instance, [6, 7, 8] proposed to use additive debiasing, and [9, ] proposed to use multiplicative debiasing. [] proposed the recursive filter with respect to the converted measurements. [, 3] extended debiased conversion to the case when range rate measurements are also available. For the scan-while-trac surveillance system, such as the phased array radar, the measurements are usually provided in RUV coordinates [5, 4, 5, 6, 7, 8, 9]. That is, what is available is range and two direction cosine measurements. Specifically designed target tracing algorithms for this type of radar are also available. For example, by using the firstorder and second-order Taylor series expansions, [8, ] developed Kalman filters for this type of problem based on debiased measurement conversion. As pointed out in [5, ], the Kalman filter based on debaised conversion of measurement model from polar or spherical coordinates to Cartesian coordinates has the following serious drawbacs. First, the converted measurement errors are state dependent; second, their covariances are estimated conditioned on the measurement or state; third, the converted measurement error sequence is not white any more. However, in the assumptions of the Kalman filter, the measurement noise is independent of the state, its covariance is unconditional, and it is white. These drawbacs cannot be overcome in the existing debiased conversion based methods [6, 7, 8, 9, ]. Since the debiased conversion based Kalman filters in [8, ] follow the same idea, they are not free of these drawbacs either. In this paper, we want to extend the wor in [] to target tracing with range and direction cosine measurements. That is, we want to develop a recursive filter for target tracing in the RUV coordinates. It is shown that the main difficulty to fulfill this goal stems from the lac of direct measurement of the third direction cosine. For the
case when the third direction cosine is measured, we propose a neat recursive filter with range and direction cosine measurements; otherwise, by constructing a pseudo measurement of the third direction cosine, we also propose a slightly more complicated recursive filter with range and two directly measured direction cosine measurements. Illustrative examples are provided to compare the proposed filters with some existing algorithms in terms of credibility and estimation accuracy. The paper is organized as follows. Sec. formulates the problem. Sec. 3 presents a recursive filter when the third direction cosine measurement is directly available. Sec. 4 presents a recursive filter when the third direction cosine measurement is not directly available. Sec. 5 provides some illustrative examples. Sec. 6 gives conclusions. Problem formulation To simplify discussion, consider only the following linear target dynamics described in Cartesian coordinates X = F X + G W X = [ x ẋ y ẏ z ż ] is the target state vector at time, W is a zero-mean white noise sequence with covariance Q, E[X ] = X, cov(x ) = P. A radar located at the origin of the Cartesian coordinates measures range r m, two direction cosines um and vm of the target as r = r m = r + n r u m = u + n u () v m = v + n v () x + y + z, u = x /r, v = y /r n r, nu and nv are all zero-mean white noise sequences with variances σr, σ u and σ v, respectively; also, nr, nu and n v are independent of each other, and independent of X and W. The range and two direction cosine measurements can be transformed into x m and ym in the Cartesian coordinates as x m = r m u m = (r + n r )(u + n u ) = x + r n u + u n r + nr nu (3) y m = r m v m = (r + n r )(v + n v ) = y + r n v + v n r + nr nv (4) In this paper, we want to estimate the state as best as we can in the sense. Remar: Note that in the above formulation, we only need to now the first two moments of the measurement noise, while in [, 6, 7, 8, 9, ], distribution of the measurement noise must be nown. For example, it was assumed therein that measurement noise is Gaussian distributed. 3 Recursive filter with measured w m From the appendix of [], it is easy to verify that the following two lemmas hold for estimation. Lemma For a scalar-valued γ, if γ is uncorrelated with x and z, then E [γx z] = E[γ]E [x z] Lemma For estimation error covariance, cov( x) = E[(x ˆx)(x ˆx) ] = E[xx ] E[ˆxˆx ] cov( x, ỹ) = E[(x ˆx)(y ŷ) ] = E[xy ] E[ˆxŷ ] ˆx = E [x z], x = x ˆx ŷ = E [y z], ỹ = y ŷ First, let us consider the case in which the third direction cosine measurement is also available, w m = w + n w w = z /r and n w is a zero-mean white noise sequence with variance σw. Also, n w is statistically independent of nr, nu and n v. Correspondingly in the Cartesian coordinates, we have Denote z m = rm wm = (r + n r )(w + n w ) = z + r n w + w n r + n r n w (5) Z = [ x m y m z m ] Then by the two lemmas above, for the case in which all three direction cosine measurements u m, vm and wm are directly available, the following theorem holds. Theorem ( filter with measured w m). If the third direction cosine measurement w m is also available, given ˆX = E [X Z ], P = MSE( ˆX ), the estimate of X is: Prediction: ˆX = F ˆX P = F P F + G Q G Update: ˆX = ˆX + C S Z P = P C S C
Z = Z Ẑ Ẑ = [ ˆX (), ˆX (3), ˆX (5)] C = [P (:, ), P (:, 3), P (:, 5)] S = [S (i, j)] 3 i,j= S (, ) = P (, ) + σ u E[r ] + σ r E[u ] + σ r σ u S (, ) = S (, ) = P (, 3) + σ re[u v ] S (, 3) = S (3, ) = P (, 5) + σ re[u w ] S (, ) = P (3, 3) + σ ve[r ] + σ re[v ] + σ rσ v S (, 3) = S (3, ) = P (3, 5) + σ r E[v w ] S (3, 3) = P (5, 5) + σ w E[r ] + σ r E[w ] + σ r σ w Proof: Given ˆX and P, it follows easily from the property of estimation that ˆX = E [X Z ] = F ˆX P = MSE( ˆX ) = F P F + G Q G The estimator E [X Z ] always has the following quasi-recursive form [] ˆX = E [X Z ] = E [X Z, Z ] Z = ˆX + C S P = MSE( ˆX ) = P C S C From Lemma and the property of estimation, we have Ẑ = E [Z Z ] = [E [x m Z ], E [y m Z ], E [z m Z ]] E [x m Z ] = E [x + r n u + u n r + nr nu Z ] and similarly, = E [x Z ] = ˆx = ˆX () E [y m Z ] = ŷ = ˆX (3) E [z m Z ] = ẑ = ˆX (5) From Lemma, we have C = cov( X, Z ) = E[X Z ] E[ ˆX Ẑ ] x x m x y m x z m ẋ x m ẋ y m ẋ z m = E y x m y y m y z m ẏ x m ẏ y m ẏ z m z x m z y m z z m ż x m ż y m ż z m ˆx ˆx ˆx ŷ ˆx ẑ ẋ ˆx ẋ ŷ ẋ ẑ E ŷ ˆx ŷ ŷ ŷ ẑ ẏ ˆx ẏ ŷ ẏ ẑ ẑ ˆx ẑ ŷ ẑ ẑ ż ˆx ż ŷ ż ẑ E[x x m ] = E[x (x + r n u + u n r + n r n u )] = E[x x ] and similarly, E[x y m ] = E[x y ], E[x z m ] = E[x z ] E[ẋ x m ] = E[ẋ x ], E[ẋ y m ] = E[ẋ y ], E[ẋ z m ] = E[ẋ z ] E[y x m ] = E[y x ], E[y y m ] = E[y y ], E[y z m ] = E[y z ] E[ẏ x m ] = E[ẏ x ], E[ẏ y m ] = E[ẏ y ], E[ẏ z m ] = E[ẏ z ] E[z x m ] = E[z x ], E[z y m ] = E[z y ], E[z z m ] = E[z z ] E[ż x m ] = E[ż x ], E[ż y m ] = E[ż y ], E[ż z m ] = E[ż z ] Thus C = [P (:, ), P (:, 3), P (:, 5)] Furthermore, from Lemma, we have S = cov( Z ) = E[Z Z ] E[Ẑ Ẑ ] = E xm xm x m ym x m zm y mxm y mym y mzm z mxm z mym z mzm E ˆx ˆx ˆx ŷ ˆx ẑ ŷ ˆx ŷ ŷ ŷ ẑ ẑ ˆx ẑ ŷ ẑ ẑ E[x m xm ] = E[(x + r n u + u n r + n r n u )( )] = E[x x ] + E[r (nu ) ] + E[u (nr ) ] + E[(n r ) (n u ) ] = E[x x ] + σ u E[r ] + σ r E[u ] + σ r σ u and similarly, E[x m ym ] = E[ym xm ] = E[x y ] + σ r E[u v ] E[x m z m ] = E[z m x m ] = E[x z ] + σ re[u w ] E[y m y m ] = E[y y ] + σ ve[r ] + σ re[v ] + σ rσ v E[y m z m ] = E[z m y m ] = E[y z ] + σ re[v w ] E[z m zm ] = E[z z ] + σ w E[r ] + σ r E[w ] + σ r σ w
Thus S (, ) = P (, ) + σ ue[r ] + σ re[u ] + σ rσ u S (, ) = S (, ) = P (, 3) + σ re[u v ] S (, 3) = S (3, ) = P (, 5) + σ re[u w ] S (, ) = P (3, 3) + σ v E[r ] + σ r E[v ] + σ r σ v S (, 3) = S (3, ) = P (3, 5) + σ r E[v w ] S (3, 3) = P (5, 5) + σ w E[r ] + σ r E[w ] + σ r σ w Remar: Since r, u, v and w are all nonlinear functions of X, the expectations E[r ], E[u ], E[v ], E[w ], E[u v ], E[u w ] and E[v w ] in Theorem can be approximated by unscented transformation (UT). Only the UT for E[r ] is shown below for illustration, and the others can be done similarly. E[r ] = E[x + y + z ] 6 α (j) ((X (j) ()) + (X (j) (3)) + (X (j) (5)) ) X () α () = j= 6 = ˆX, X (±i) = ˆX ± [((6 + κ)p ) ]i κ 6 + κ, α(±i) =, i =,,, 6 (6 + κ) [A ] i is the i-th column of the Cholesy decomposition of square matrix A. The design parameter κ provides an extra degree of freedom to fine tune the higher order moments of the approximation and κ can be any number (positive or negative) provided that 6 + κ. 4 Recursive filter without measured w m As described in the problem formulation part, w m is usually not measured in the current radar. This maes the corresponding estimation problem much harder. One popular way is to construct a pseudo measurement by w m = (u m ) (v m) (6) = (u + n u ) (v + n v ) Then the range and direction cosine measurements can be transformed to x m, ym and zm in the Cartesian coordinates, as in (3), (4) and (5), with w m replaced by (6) so that z m = rm wm = (r + n r ) (u + n u ) (v + n v ) = r (u + n u ) (v + n v ) + n r (u + n u ) (v + n v ) Denote Z = [ x m y m z m ] Then for the case without measured w m, the following theorem holds. Theorem ( filter without measured w m ). If the third direction cosine measurement w m is not available, given ˆX = E [X Z ], P = MSE( ˆX ), the estimator of X is: Prediction: Same as in filter with measured w m. Update: Same as in filter with measured w m except that Ẑ = [ ˆX (), ˆX (3), ẑ m ] C = [P (:, ), P (:, 3), C ( 3) ] S ( :, 3) = S (3, :) = S ( 3) ẑ m = E [r (u + n u ) (v + n v ) Z ] C ( 3) = E[(X ˆX )(z m ẑm )] S ( 3) = E[(Z Ẑ )(z m ẑm )] Proof: From Lemma and the property of estimation, we have Ẑ (3) = E [z m Z ] = ẑ m = E [r (u + n u ) (v + n v ) + n r (u + n u ) (v + n v ) Z ] = E [r (u + n u ) (v + n v ) Z ] From Lemma, we have C ( :, 3) = cov[ X, z m ] = C ( 3) = E[(X ˆX )(z m ẑm )] S ( :, 3) = S (3, :) = cov[ Z, z m ] = S ( 3) = E[(Z Ẑ )(z m ẑm )] Remar: In Theorem, E[r ], E[u ], E[v ] and E[u v ] can be approximated in the same way as in Theorem. Remar: Since w m is not directly measured in this case, ẑ m does not have a nice form as in Theorem. It can be approximated by some moment matching techniques [], e.g., UT, as ẑ m = E [r (u + n u ) (v + n v ) Z ] 9 j= 9 β(j) z (j)
z (j) r (j) = u (j) v (j) β () = = r (j) (u (j) (Y (j) = Y (j) ()/r(j) = Y (j) + Y(j) (8)) (v (j) + Y (j) (9)) ()) + (Y (j) (3)) + (Y (j) (5)) (3)/r(j), j = ±9, ±8,, κ 9 + κ, β(±i) = (9 + κ), Y() = Y Y (±i) = Y ± [((9 + κ)c ) ]i, i =,,, 9 Y = [ ˆX,,, ], C = diag(p, σ r, σ u, σ v ) And correspondingly, C ( 3) 9 β (j) (Y j ( : 6) ˆX )(z (j) ẑ m ) j= 9 S ( 3) 9 j= 9 βj (Z (j) (j) Ẑ )(z ẑ m ) Z (j) x (j) y (j) = [ x (j) = Y (j) y (j) () + r(j) z (j) ] Y(j) + Y (j) (7)Y(j) (8) = Y (j) (3) + r(j) + Y (j) (7)Y(j) (9) Y(j) (8) + u(j) (9) + v(j) Y(j) Y(j) (7) (7) Remar: Other moment matching techniques [], e.g., DD [3], can also be used to replace UT. 5 Illustrative examples In this section, we compare performance of the proposed recursive filter with measured w m (w), the proposed recursive filter without measured w m (),, converted measurement Kalman filter with CM (CMKF) [] and converted measurement Kalman filter with CM () [] in terms of estimation accuracy and credibility through numerical examples. Except w, all the other filters are assumed to have no access to the measured w m. Consider the following three-dimensional discrete-time constant velocity (CV) motion model [3] of a target X = F X + G W X = [ x ẋ y ẏ z ż ], X N( X, P ) F = diag(f, F, F), G = diag(g, G, G) [ ] [ ] T T F =, G = /, T = s T W N([ ], diag((q x ), (q y ), (q z ) )) The estimation accuracy measures used are root meansquared (RMS) position and velocity errors. The filter credibility measures used are average normalized estimation error squared (ANEES) and noncredibility index (NCI) [4]. All results below are averaged over 5 Monte Carlo runs. All filters are initialized with and ˆX = X, P = P X = [ 33m 6m/s 33m 6m/s 33m 6m/s ] P = diag( 4 m, m /s, 4 m, m /s, 4 m, m /s ) σ u = σ v = σ w = 3, q x = q y = qz =.5m/s Figs. through 4 show comparison results for the case with σ r = m (case ). In this case, the range measurement accuracy is the poorest among all three cases. From the simulation results, it can be seen that in terms of both RMS position and velocity errors, there is no significant difference among CMKF,, and. LMM- SEw has the smallest RMS position error. In terms of RMS velocity error, w outperforms all the others during steady state but is worse than all the others during transient. It terms of filter credibility, all filters are credible. RMS position error(m) 9 8 7 6 5 4 3 CMKF w Figure : RMS position error comparison for case. Note that, CMKF, and essentially overlap with each other except that CMKF is slightly worse than the other three. Figs. 5 through 8 show comparison results for the case with σ r = m (case ). In this case, the range measurement accuracy is relatively better. From the simulation results, it can be seen that in terms of estimation accuracy, CMKF is the worst except it beats w in velocity slightly during transient. Our beats clearly but beats only slightly. In terms of RMS velocity error, there is no significant difference among,
RMS velocity error(m/s).8.6.4...8.6.4 CMKF w NCI.6.4..8.6 w..4. Figure : RMS velocity error comparison for case. Note that, CMKF, and essentially overlap with each other. ANEES.5.4.3.. w RMS position error(m) 9 8 7 6 5 4 3 Figure 4: NCI comparison for case CMKF w.9 Figure 3: ANEES comparison for case Figure 5: RMS position error comparison for case. Note that and essentially overlap with each other except that is slightly better than. and. Our w beats all the others significantly in RMS position error. In terms of filter credibility, CMKF is far from credible. (The ANEES and NCI of CMKF are not shown because they are much larger than those of the other filters.) is barely credible and all the others are credible. Figs. 9 through show comparison results for the case with σ r = m (case 3). In this case, the range measurement accuracy is the best. From the simulation results, it can be seen that in terms of estimation accuracy, the difference between CMKF and all the others is increased a lot when compared with case. Over all three cases, it can be seen that in terms of both estimation accuracy and filter credibility, w beats all the others. Considering both this performance improvement and simplicity of the recursive filter with measured w m, it is highly desirable to have wm in the same way (signal processing mechanism) as u m and vm in () and () instead of using the pseudo measurement in (6) in the phased array radar. Although our has very close estima- tion accuracy to those of and, its filter credibility is much better. 6 Conclusions For a scan-while-trac surveillance system, such as a phased array radar, the available measurements are usually range and direction cosine. To trac the state of a moving target described in the Cartesian coordinates, one existing approach is to convert the range and direction cosine measurement model to Cartesian coordinates so that the converted measurement model taes a pseudo linear form with respect to the state, but it has some serious drawbacs. The main difficulty to develop a recursive filter for this type of problem stems from the lac of direct measurement of the third direction cosine. For the case when the third direction cosine can be directly measured, we have proposed a neat recursive filter; otherwise, by constructing a pseudo measurement of the third direction cosine, we have also proposed a slightly more complicated recursive filter.
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