Consistent Unbiased Linear Filtering with Polar Measurements

Transcription

1 Consistent Unbiased Linear Filtering Polar Measurements Dietrich Fränken Data Fusion Algorithms & Software EADS Deutschl GmbH D-8977 Ulm, Germany Abstract The problem of tracking objects moving in Cartesian space sensors delivering polar measurements has been under investigation of several researchers for quite some time now. Different proposals for using measurement conversion techniques in combination a linear Kalman filter have been made in order to reduce the range bias that shows up in the filter estimates when a Cartesian pseudo-measurement is created from the polar measurements by applying the respective conversion formulae in combination a corresponding linearized form of the measurement error covariance matrix. It turns out that the actual behavior of these different approaches strongly depends on the specific situation under consideration where observed effects range from a truly (approximate) suppression of conversion bias up to an even increased bias. In this paper, a systematic approach to analyzing the bias effects of measurement conversion in certain typical tracking situations is presented. Starting from this approach, a new measurement conversion technique is proposed that yields consistent unbiased estimates in these cases. Keywords: Object tracking, measurement conversion, Kalman filter, linear filtering, bias compensation. I. INTRODUCTION In many tracking applications, the dynamics of moving objects are modeled in Cartesian coordinates. The usage of sensors yielding polar measurements range bearing (or azimuth) plus, possibly, elevation then leads to a mixed linear/non-linear estimation problem. In order to circumvent this difficult problem, measurement conversion is frequently used. This means that the measurements as well as adequately chosen measurement error covariance matrices are converted from the original polar coordinate system into a corresponding Cartesian system. Then, the stard Kalman filter update equations are invoked using both the converted measurements the transformed measurement error covariance matrix. Performing this conversion in a straight-forward manner, one is faced the problem of a range bias in the resulting estimates. After a subtractive approach to debiasing the converted measurements has been proposed in [, it has been shown in [ that the true bias of the converted measurements really is multiplicative in nature. While the compensation factor accounting for that bias turns out to be independent of the true state, the error variance is not. As the true state is clearly not available, a conditioning of the error variance on the measurements is used by the authors in order to derive an applicable result. Because the two approaches of [ [ yield rather similar results unless one is faced very large bearing errors, only the second one will be considered in more detail from this point on. In [3, several measurement conversion approaches have been compared. In particular, the consideration of a fixed measurement scenario has lead to an alternative multiplicative debiasing method measurement-conditioned error matrix that we will investigate as well. In [4, another proposal for a conversion solely based on the measurements has been made. The authors argue that the derivation in [ suffers from some inconsistency between the conversion of measurement (conditioning on truth only) associated covariance matrix (conditioning first on the true range then on the measurements). By conditioning on the measurements only, they compute an additive conditional measurement bias an associated covariance matrix hence propose to subtract this bias from the modified converted measurement of [. But, a closer inspection shows that this approach overall again means to apply the multiplicative range compensation factor of [3 to the measurement obtained by the straight-forward conversion rule. And although the authors of [4 have noticed the better credibility of their estimates (i. e., a better consistency between estimation error estimated error) when compared those of [, the major reason for this has not been mentioned. Our presentation will fill this gap. It should be noted at this point that in all of the aforementioned papers, when conditioning on the measurements is performed, it is (explicitly or implicitly) assumed that the true state is a function of the independent quantities measurement measurement error rather than taking the measurement as a function of the true state plus an error being independent thereof. We do not want to argue about this point any further, but merely take the results from the cited publications as given there. In [5, some fundamental flaws of all converted measurement filtering techniques have been brought to attention, e. g., non-white sequences of state-dependent (pseudo-)linear measurements plus noise covariance matrices being conditioned on the state /or the measurement, thus not fulfilling the preconditions for the Kalman filter to constitute the recursive minimum mean square error estimator. Starting from this observation, the authors derive the (approximate) best linear unbiased estimator (BLUE) for the tracking problem [5, [6, [7. Although not derived as a converted measurement

2 approach, the approximate BLUE filter can be indeed written in a corresponding Kalman filter form a compensation factor just like in [, but a measurement noise covariance matrix that now does not depend on the measurement but on the predicted state the prediction error covariance [8. The possible effects induced by this dependency, e. g., in combination multiple model approaches for tracking, remain undiscussed at this point. And, the fact that other conversion techniques solely depend on the current measurement still renders them attractive for practical implementation. In [8, the behavior of different popular measurement conversion techniques in a tracking environment has been investigated. In particular, it has been shown that the technique of [, shifting away the measurement from the sensor, tends to worsen the range bias when being applied to nonmaneuvering targets that move constant velocity while the fixed-measurement technique of [3, shifting the measurement towards the sensor, effectively reduces the range bias in these cases (unless the bearing error becomes extremely large where this approach starts showing a noticeable bias towards the inside). We will continue considering cases like these in the following, thus being representative for targets zero or low maneuvering behavior. A method for analyzing the (asymptotic) behavior of estimated state (position) covariance in these cases will be derived. With this result, we will confirm the aforementioned statements about the debiasing behavior of the approaches in [ [3. We will also show that the (almost) debiasing approach in [3 yields estimates that are overly confident respect to the crossrange error thus inconsistent. Finally, a new conversion method will be derived that not only yields consistent estimates but also is able to reduce the cross-range estimation error significantly. All presentation is done in spatial dimensions. II. CONVERTED MEASUREMENT FILTERS Consider an object located in the Cartesian plane at a location z T (x, y) (rcos β,r sin β) () (true) range r bearing β plus a sensor producing measurements z T m (x m,y m )(r m cos β m,r m sin β m ) () the measured range r m r + r the measured bearing β m β + β where the range error r the bearing error β are assumed to be independent zero mean Gaussian rom variables variances σr σβ, respectively. We further assume that a predicted state (estimate) ẑ p (estimated) prediction error covariance matrix ˆP p is given. Linear filtering converted measurements now means to compute a vector z c a matrix R c in such a way that a stard Kalman filter update can be used for the state estimation error covariance (for the sake of brevity, we only consider the position part of the state here): ẑ u ẑ p + K(z c ẑ p ), ˆPu ˆP p KSK T, K ˆP p S, S ˆP p + R c. (3) The classical, straight-forward approach to converted measurement filtering simply reads [ σ z c z m, R c T r βm rmσ β Tβ T m (4) the transformation matrix [ cos βm sin β T βm m. (5) sin β m cos β m As has been shown in [, the estimate z m for the true state z is biased, that is, the abbreviations λ E[cos( β) e σ β / µ e σ β λ, (6) one has z c z m E[z c z λz. (7) Hence, it has been proposed to use, instead of (4), the (conditionally) unbiased estimate z c λ z m E[z c z z. (8) The second order error statistics of this unbiased estimator has been computed as well but, as has been mentioned in the introduction, the computation of this true error statistics requires the knowledge of the true state. Because this is not available in practice, it has been suggested in [ to rather use R c T βm [ µ(cσ m + σ r)+cr m µsσ m instead the additional abbreviations T T β m (9) c cosh(σ β), s sinh(σ β), σ m r m + σ r. () The fixed-measurement approach of [3 can be written in a similar form. Here, one has z c λz m () [ µ(cσ R c T m + σr) βm µsσm Tβ T m () σ m as defined in eq. (). All converted measurement filters shown in detail in this section (classical, [, [3) share a couple of common properties: The converted measurement is merely a scaled version of the classical variant the scaling factor does not depend on the measurement (nor on the predicted or the true state). The used measurement error covariance matrix has one of its (orthogonal) semi-axes in alignment the measurement while the respective scaling of the semi-axes depends on the measured range only. Yet, we see a (maybe) surprising subtle difference in here: In comparison the classically converted measurement, the proposal in [ forces the converted measurements to the outside, i. e., away from the sensor, while the variant in [3 shifts it just in the opposite direction, i. e., towards the sensor. The implications of this difference will become apparent later on.

3 III. ANALYSIS OF ASYMPTOTIC BEHAVIOR As has been mentioned above, we now analyze the asymptotic behavior of the Kalman filter estimate that results from an application of any technique based on scaled converted measurements. As we have stated before, we will consider the case out process noise only, for the sake of simplicity, we for now assume a stationary target at true position z as well as a Kalman filter yielding position estimates ẑ only. Finally, we set β out loss of generality due to the rotational symmetry of the estimation problem. This means that the range direction coincides the x-axis while the cross-range direction is aligned the y-axis. It is well-known that the Kalman filter performs an information fusion of predicted state estimation error covariance the measurement the corresponding measurement error covariance. Thus, taking n independent (unscaled) converted measurements z k assumed corresponding measurement error variances R k denoting the filter estimate the estimated error covariance by ẑ ˆP, respectively, eqs. (3) become equivalent to ˆP I, I I k, ẑ Pi I i, i I k R k, i k, i k I k z k. (3) Now, we are interested in the asymptotic filter output, i. e., the estimates that result from a couple of successive filter updates. With ī : E [i k E [i nī (4) Ī : E [I k E [I nī, (5) all expected values here in the following are to be understood as conditioned on the true state z like in eq. (7) we are able to compute the asymptotic expected state estimate z. Due to Slutzky s theorem [9, one has z E [ẑ E [ I i ( ) ( E ) I k i k (6) n n E [I/n E [i/n Ī ī for sufficiently large n. Likewise, we can compute the asymptotic expected covariance estimate P E [ ˆP E [ I E [I nī (7) as delivered by the filter. In the following, we will assume equality for the given approximate values. Finally, we note that the covariance estimate delivered by the filter differs from the actual covariance of the state estimate. The latter can be computed following a similar line of arguments as above defining k i k I k z i k I k Ī ī (8) E [ k T j E [ k (9) j k E [ k E [ T j. () One finds (ẑ z)(ẑ z) T (I i Ī ī)(i i Ī ī) T I (i IĪ ī)(i IĪ ī) T I T ( n )( n ) T I k k I T I k T k + k T j I T j j k () thus (again exploiting a variant of Slutzky s theorem) the approximate expected covariance of the state estimate E [ (ẑ z)(ẑ z) T nī E [ k T k Ī T. () We now evaluate the results just found by exploiting the specific form of converted measurement z k assumed measurement error variance R k. With bearing errors β k measured ranges r k as well as the nominal measurement error variances C k in range respectively in cross-range direction, we have [ rk z k T βk (3) Thus, there holds R k T βk [ R k C k I k T βk D rk T T β k D rk i k T βk D rk [ rk Eq. (6) immediately implies [ ϱ E [i k ī λ T T β k. (4) [ ϱ k [ cos( βk ) sin( β k ) From eq. (5), we get [ Ixx I I k xy (5) (6) ϱ k r k. (7) I xy ϱ E [ϱ k. (8) I yy I xx cos ( β k )+ ( I xy I yy R k sin ( β k ), ) cos( β k ) sin( β k ), sin ( β k )+ cos ( β k ) (9) (3)

4 hence find as well as E [I k Ī λ [ A B A cosh(σ β ) R + sinh(σ β ) C, B sinh(σ β ) R + cosh(σ β ) C [ R E [ C E (3) (3) (33) where we have made frequent use of the general relationship E [cos(m β k ) λ (m ) (34) holding for all integer values m. The asymptotic expected state estimate thus becomes z E [ẑ Ī ī [ ϱ A. (35) λ The expected variance of the state estimate can be computed in a similar way. Starting from [ rk cos( β k k ) ϱ A λ I xx r k sin( β k ) ϱ A λ I xy [ ( ϱk cos( β k ) λ δ k cos( β k ) ) λ σ (36) k ϱ k sin( β k ) λ δ k sin( β k ) the abbreviations δ k ϱ [ A σ k ϱ [ A +, (37) one obtains, after somewhat lengthy computations, E [ [ k T X k Y X λ ( cosh(σβ)e [ ϱ k λ cosh(σβ)e [ϱ k δ k + λ4 4 cosh(4σ β)e [ δk ) + λ E [δ kσ k E [ϱ k σ k + 4λ E [ σk, Y λ ( sinh(σ β)e [ ϱ k λ sinh(σ β)e [ϱ k δ k (38) (39) + λ4 4 sinh(4σ β)e [ δk ). Honoring eq. (), we have thus found the expected variances of the state estimate in range in cross-range direction to be σrange X nλ 4 A 4 σcross Y nλ 4 B 4, (4) respectively. A closed-form evaluation of the various expected values occurring in the different formulae up to now is, if at all possible, a tedious job. But, all these expressions depend on the range only ( not on the bearing) in many applications like, e. g., long-range radar tracking, one may safely assume to have σ r r. Under this assumption, one may approximate ϱ r R (4) as well as E [ ϱ r + σ k r R 4, E [ δk δ, E [ σk σ, E [ϱ k δ k ϱδ, E [ϱ k σ k ϱσ, E [δ k σ k δσ δ ϱ [ A R C σ ϱ [ A R + C (4) (43) where R C are approximated by the respective entries C k of R k r k being replaced by the true range r. In order to wrap up the analysis part, we now honor the possible occurrence of a scaling factor in the measurement conversion, i. e., instead of z k the value αz k is used as the (presumeably bias-corrected) converted measurement in the Kalman filter updates (3). When doing so, the expected estimate as given in eq. (35) has to be scaled by α while the variances of the state estimate as given in eq. (4) have to be scaled by α. In view of the further proceeding, we finally introduce the scaled values X A 4 R 4 X Ỹ A 4 R 4 Y. (44) Overall, we thus get expected variances of the state estimate according to σrange α X R 4 A 4 nλ 4 A 4 σcross α Ỹ R 4 A 4 nλ 4 B 4 (45) while the filter, due to eqs. (7) (3), delivers the expected estimates ˆσ range nλ A ˆσ cross nλ B. (46) Herein, X Ỹ can be written as mixed second order polynomials in /R /C, X X rr [R + X rc R C + X cc [C, Ỹ Y rr [R + Y rc R C + Y (47) cc [C, coefficients X rr, X rc, X cc as well as Y rr, Y rc, Y cc depending on r, σr, σβ. Fig. displays several performance parameters that can be obtained by applying different measurement conversion techniques to a typical long-range scenario. In addition to the three approaches mentioned before (classical, [, [3), a new conversion technique is evaluated that will be derived in the next section. Unlike this technique, the three other approaches produce range biases depending on the bearing accuracy σ β. For long-range radar applications, bearing errors are expected to be small to moderate (up to a few degrees). We see that here the classical uncompensated conversion leads to

5 µ r 6 4 Range bias No debiasing - Unbiased conv. [ Fixed meas. [3 New, q Stard deviation in range direction σ range RMSE range σ cross RMSE RMSE in range direction Stard deviation RMSE in cross-range direction Overall RMSE σ β / deg Figure. Expected performance parameters for measurement conversion as a function of the bearing accuracy σ β. From top to bottom, the plots show (a) the range bias µ r E [ˆr r, (b) the stard deviation in range direction σ range σrange E[ (ˆr r), (c) the corresponding root mean square error RMSE range µ r + σ range, (d) the stard deviation in cross-range direction σ cross, (e) the overall RMSE RMSE RMSE range +σcross after n updates for a target at range r 4 km using a sensor range accuracy σ r 5m. True behavior is shown in solid, estimated output as delivered by the filter in dashed lines. estimates that are indeed biased in the range direction. But, we note that the estimate is not closer to the sensor as eq. (7) λ< might suggest, but further away (by about a factor of λ ). With the analysis of this section, we have been able to specifiy the cause for this apparent discrepancy that has been mentioned (although neither analyzed nor quantified) in literature before, e. g., in [5: The bias in eq. (7) as derived in [ is µ r / km σ range / km RMSE range / km σ cross / km RMSE / km y/km Fixed meas.: True Filter: σ range m, σ cross 8 m 4.4 Filter output (analysis) Filter output ( runs) 4. True error (analysis) True error ( runs) x/km Figure. Expected value of estimated variance as delivered by the fixedmeasurement (FM) filter of eqs. () () plus actual expected variance of the state estimate after n updates for a target at range r 4 km bearing β 9 using a sensor range accuracy σ r 5m bearing accuracy σ β.75. Shown are the 9 % confidence ellipses according to eqs. (46) (45) as well as the corresponding results from M Monte Carlo runs. Note the different scaling of the axes. that of the converted measurements alone, the influence of the measurement-dependent covariance matrix on the successive Kalman filter updates is not taken into account. The plots also show that the approach suggested in [ even worsens (approximately doubles) the bias. In contrast to that, the fixedmeasurement approach of [3 shows virtually no bias for up to moderate bearing errors. This is due to the triangulation effect that occurs when fusing measurements shifted towards the sensor in combination measurement covariances where the assumed cross-range variance is much larger than the range variance, see [8 for an illustrative example. For small moderate errors, the actual stard deviations of the range estimates yield comparable results for all of the investigated methods. But, the corresponding variances as reported by the filters show some differences. While the fixed-measurement approach yields variance estimates close to its true behavior, the new approach proposed in this paper tends to overestimate its error. Honoring the fact that one is not interested in the stard deviation of the filter estimate from its mean, but in the root mean square error (RMSE), i. e., in the mean squared deviation of the filter estimate from truth, we note that both the classical approach as well as the approach in [ are, respect to the range estimate, inconsistent in the sense that they yield variance estimates being smaller than the actual error variances. The same is true for all methods (apart from the new one derived next) when looking at the cross-range variance. In other words, none of the measurement conversion methods proposed in literature so far is able to yield estimates where the estimated cross-range variance as delivered by the filter is a reliable measure of the actual quality of the state estimate being reported. In addition to that, we see that the new method yields the smallest crossrange variance among the cidates. We also note that, for the chosen setting, the cross-range variance for the approach in [ is smaller than that of the fixed-measurement approach in [3 that, because that error is dominant in this case, among those two approaches one can expect the former one to yield a

6 smaller overall estimation error despite its range bias. But, the last statement may not hold in general, especially if bearing errors become larger /or if the number n of updates is increased because variances decrease according to /n while the range bias stays the same may thus, after all, become dominant. Examples will be given later on. For increasing bearing errors, we note that the fixedmeasurement approach can be expected to show a noticeable bias towards the inside in agreement what has been mentioned in the introduction. IV. CONSISTENT UNBIASED CONVERSION In view of eqs. (35) (4), a truly debiasing scaling factor is given by α λr A λ [cosh(σ β) + sinh(σ β) R C (48) which, for typical long-range applications R C moderate bearing errors σ β, approximately equals λ. Ifα is chosen according to eq. (48), eq. (45) simplifies to X σrange nλ A 4 σcross Ỹ nλ B 4 (49) a conversion yielding exactly consistent estimates in both range in cross-range direction (having ˆσ range σrange ˆσ cross σcross) must obey X A Ỹ B. (5) One could now try solve the homogeneous system of quadratic equations in /R /C that results from combing eqs. (3), (47), (5). This leads, apart from the trivial solution /R /C after elimination of C, to a cubic equation in /R exactly one real solution (at least for practically relevant parameter values). Although this solution happens to be positive, it causes extremely large variances in range direction (becoming even larger than those in cross-range) rendering this approach useless. Instead, we consider unbiased conversions yielding exactly consistent estimates in cross-range direction only (thus having Ỹ B ). From Y rr [R + Y rc R C + Y cc [C sinh(σ β ) R + cosh(σ β ) C, (5) one can easily compute R as a function of C or vice versa (multiple solutions are discussed shortly later on). Likewise, one readily determines the values R C if one sets the terms in eq. (5) equal to a nominal value /σ, i. e., if one requires a specific cross-range variance via σcross Ỹ nλ 4 B 4 ˆσ cross nλ B σ nλ. (5) For reference purposes, we will consider a value σ obtained from σcross according to eq. (49) interim values R R C C as given by the fixed measurement approach in eq. (), i. e., R µ(c(r + σ r )+σ r), C µs(r + σ r ), (53) σ cross C /C σ range C vs. R Stard deviation in range direction vs. R Stard deviation in crossrange direction vs. R 5 5 True error Filter output FM true error FM filter output Ref. true error Ref. filter output R /R Figure 3. Expected performance of a filter yielding asymptotically unbiased estimates consistent cross-range variance for a setting as in Fig.. From top to bottom, the plots show, as functions of R, (a) the values C according to eq. (5), (b) the expected variance of the range estimate the corresponding filter output, (c) the expected variance of the cross-range estimate the corresponding filter output. Variances are normalized to the values R C of eq. (53), respectively, plots start R Rref. denote the values R C resulting from eqs. (5) (5) given this σ by Rref C ref, respectively. Clearly, this is only one possible choice for R, we can construct a whole family of new consistent unbiased conversion techniques by first selecting R qrref some q (not too large) before computing C according to eq. (5) finally determining the debiasing range correction factor from eq. (48). Fig. 3 shows how C as well as the variances of the filter estimates the expected variance estimates depend on the particular choice. Fig. 4 confirms some of these findings. Further case studies suggest that the following (yet to be proven) statements hold for a large range of practically interesting parameter sets r, σr, σβ : The value R ref is a certain lower bound for suitable values R. For values not smaller than that, eq. (5) yields exactly one positive solution for C. By construction, the choice R R ref yields an actual cross-range variance being identical to that of the fixedmeasurement approach of [3. A choice (moderately) larger than that leads to a reduced yet still consistent cross-range variance thus to a smaller overall RMSE.

7 y/km y/km New, q : True Filter: σ range 38 m, σ cross m Filter output (analysis) Filter output ( runs) True error (analysis) True error ( runs) x/km New, q : True Filter: σ range 53 m, σ cross m x/km Figure 4. Same representation as in Fig., but now for the new filter R qrref q (top) / q (bottom). For R Rref, the filter yields an estimate for the range variance being larger than the actual one. This undesirable (yet tolerable) effect grows increasing values of R. One has Rref >R as well as Cref >C. It should be noted here that the derivations so far make frequent use of the true range r. Clearly, this is not available in practice. But, as before, one merely can substitute it by the measured range r m (still assuming σr r ) use the above formulae unalteredly up to this detail for filtering purposes (as has been done for the Monte Carlo runs in Fig. 4). V. TRACKING RESULTS Up to now, we have considered a stationary target only. In order to investigate how the results of the previous sections translate into the problem of tracking a moving target, we have simulated targets performing undisturbed (no process noise) linear motions in the (x, y)-plane constant velocity. Fig. 5 shows the obtained results also lists relevant parameters, where initial range r bearing β as well as speed v heading h are corresponding nominal values. The actual initial position velocity deviated romly from these nominal values Cartesian stard deviations (for both spatial directions each) σ p km σ v m/s, respectively. Estimates were obtained zero process noise Kalman filters having states position velocity in two spatial dimensions assumed constant velocity. The position parts of the filters were initialized based on the first measurement the minimum mean square error estimator for the conversion problem (for details, see [8), the velocity parts zero velocity large variances ([ m/s ). We used the classical measurement conversion (black), the conversion according to [ (red) the one proposed in [3 (orange), the newly developed unbiased conversion technique R qrref q (green), plus finally, for comparison, the simplified BLUE filter of [6, [7 (blue, yielding virtually the same results as the original BLUE filter of [5 here). The figure shows the RMSE the logarithm of the average normalized estimation error squared, ANEES 4M M µ [ (ẑ z) T ˆP (ẑ z), (54) µ frequently used for judging the estimators consistency (values of the ANEES larger than indicate that the filter is overly confident about its estimation quality). Like in [8 (but in none of the earlier publications), it also displays the mean error in range direction, i. e., the error in the direction given by the line of sight of the true target position respect to the sensor (i. e., the quantity to be eliminated by debiasing approaches). Clearly, this error the variances in range in cross-range direction (not shown separately) together induce the RMSE. The results demonstrate the superior performance of the new method among the measurement conversion techniques. After an initial transient, it is out range bias, its smaller cross-range variance leads to an overall reduced RMSE. The occurrence of a noticeable bias towards the inside for the fixed-measurement approach of [ in case of larger bearing errors is confirmed. Which one of the two methods in [ in [3 yields a smaller RMSE in fact depends on the situation. We note that, for larger bearing error, the approach of [ produces an RMSE that is larger than that of [3, after a higher number of updates, even larger than that obtained using the classical approach. The range bias of the method [ is also the reason for the observation in [5 where, for slower targets much closer to the sensor, as is shown in the bottom plot, it has been observed that the method in [ lacks credibility (the range bias of 3 m here significantly contributes to the overall RMSE of m). The better credibility of this approach when compared the classical approach as reported in [ look at the results for the ANEES obtained those two methods in any of the long-range plots suggesting this conclusion is not due to a reduced bias, but to a much better estimate of the cross-range variance. Finally, we note that the proposed new conversion technique yields a performance comparable to that of the approximate BLUEfilter in some cases while it, due to overestimating the range variance, results in smaller values for the ANEES throughout. VI. CONCLUSION A systematic approach to the analysis of measurement conversion techniques for linear filters running on polar measurements has been presented. We have derived closed form formulae that allow for the asymptotic (approximate) computation of a possible range bias as well as of variances of state estimates estimated variances as reported by such filters for both the range the cross-range direction. Based on these results, we have proposed a new conversion technique that, unlike other methods known from literature, yields truly unbiased state estimates for all possible bearing error variances consistent estimates for the cross-range variance.

8 .5 Position error. Error in range direction Normalized estimation error squared.5 RMSE xy /km RMSE xy /km µ r /m µ r /km No debiasing Unbiased conv. [ Fixed meas. [3 - New, q Simplified BLUE [6, log (ANEES) log (ANEES) RMSE xy /m t/s µ r /m t/s log (ANEES) t/s Figure 5. Top middle: Simulation results for a long-range scenario moderate (top) large (middle) bearing errors. Nominal values are r 4 km, β 9, v 5 m/s, h 75 errors σ r 5m, σ β.75 (top) / σ β 5 (middle) sample interval T sfor M 5 runs. Bottom: Corresponding results for a medium-range scenario r 4km, β 45, v 8m/s, h 45 errors σ r 5m σ β.5. Still, one carefully has to take into account the assumptions made when wishing to apply the results of this paper. We have considered medium- to long-range radar applications where, for typical sensors, the cross-range variance dominates the range variance the range measurement error is much smaller than the actual range. And being more restrictive, we have considered models out process noise only. It is clear that the results cannot be applied to highly maneuvering targets if modeled via large process noise (in the limiting case, filter estimates must follow the measurements then the approach of [, that has lead to an increased bias in the nonmaneuvering case, will indeed be debiasing). A solution to this problem cannot be given at this point. Further research will also include the generalization of the presented results to the case of 3 spatial dimensions. REFERENCES [ D. Lerro Y. Bar-Shalom, Tracking debiased converted measurements versus EKF, IEEE Trans. Aerospace Electronic Systems, vol. 9, no. 4, pp. 5 34, July 993. [ M. Longbin, S. Xiaoquan, Z. Yiyu, S. Z. Kang, Y. Bar-Shalom, Unbiased converted measurements for tracking, IEEE Trans. Aerospace Electronic Systems, vol. 34, no. 3, pp. 3 7, July 998. [3 M. D. Miller O. E. Drummond, Comparison of methodologies for mitigating coordinate transformation bias in target tracking, in Proc. SPIE Conf. Signal Data Processing of Small Targets, vol. 448, July, pp [4 Z. Duan, C. Han, X.-R. Li, Comments on Unbiased converted measurements for tracking, IEEE Trans. Aerospace Electronic Systems, vol. 4, no. 4, pp , October 4. [5 Z. Zaho, X.-R. Li, V. P. Jilkov, Y. Zhu, Optimal linear unbiased filtering polar measurements for target tracking, in Proc. 5th Int. Conf. Information Fusion (FUSION ), July, pp [6 Z. Zaho, X.-R. Li, V. P. Jilkov, Best linear unbiased filtering for target tracking spherical measurements, in Proc. SPIE Conf. Signal Data Processing of Small Targets 3, vol. 54, July 3, pp [7, Best linear unbiased filtering nonlinear measurements for target tracking, IEEE Trans. Aerospace Electronic Systems, vol. 4, no. 4, pp , July 4. [8 D. Fränken, Some results on linear unbiased filtering polar measurements, in Proc. IEEE Int. Conf. Multisensor Fusion Integration for Intelligent Systems (MFI 6), September 6, pp [9 T. S. Ferguson, A Course in Large Sample Theory. London: Chapman Hall, 996. [ Y. Bar-Shalom X.-R. Li, Estimation Tracking: Principles, Techniques, Software. Boston, London: Artech House Publishers, 993. [ X.-R. Li V. P. Jilkov, A survey of maneuvering target tracking Part III: Measurement models, in Proc. SPIE Conf. Signal Data Processing of Small Targets, vol. 4473, July, pp