REPORT DOCUMENTATION PAGE

Transcription

1 EPOT DOCUMENTATION PAGE For Approved OMB NO The public reporting burden for this collection of inforation is estiated to average 1 hour per response, including the tie for reviewing instructions, searching existing data sources, gathering and aintaining the data needed, and copleting and reviewing the collection of inforation. Send coents regarding this burden estiate or any other aspect of this collection of inforation, including suggesstions for reducing this burden, to Washington Headquarters Services, Directorate for Inforation Operations and eports, 5 Jefferson Davis Highway, Suite 14, Arlington VA, -43. espondents should be aware that notwithstanding any other provision of law, no person shall be subject to any oenalty for failing to coply with a collection of inforation if it does not display a currently valid OMB control nuber. PLEASE DO NOT ETUN YOU FOM TO THE ABOVE ADDESS. 1. EPOT DATE (DD-MM-YYYY. EPOT TYPE Conference Proceeding 4. TITLE AND SUBTITLE Perforance Analysis of the Converted ange ate and Position Linear Kalan Filter 3. DATES COVEED (Fro - To - 5a. CONTACT NUMBE W9NF b. GANT NUMBE 6. AUTHOS S. V. Bordonaro,, P. Willett, Y. Bar-Shalo 5c. POGAM ELEMENT NUMBE 61 5d. POJECT NUMBE 5e. TASK NUMBE 5f. WOK UNIT NUMBE 7. PEFOMING OGANIZATION NAMES AND ADDESSES University of Connecticut - Storrs 438 Whitney oad Ext., Unit PEFOMING OGANIZATION EPOT NUMBE Storrs, CT SPONSOING/MONITOING AGENCY NAME(S AND ADDESS (ES U.S. Ary esearch Office P.O. Box 1 esearch Triangle Park, NC DISTIBUTION AVAILIBILITY STATEMENT Approved for public release; distribution is unliited. 14. ABSTACT In active sonar and radar applications easureents consist of range, bearing and often range rate - all nonlinear functions of the target state (usually odeled in Cartesian coordinates. The converted easureent Kalan filter (CMFK first converts the range and bearing easureents into Cartesian coordinates to allow for the use of a linear Kalan filter. The 15. SUBJECT TEMS Converted ange ate and Position 1. SPONSO/MONITO'S ACONYM(S AO. SPONSO/MONITO'S EPOT NUMBE(S 5783-CS SUPPLEMENTAY NOTES The views, opinions and/or findings contained in this report are those of the author(s and should not contrued as an official Departent of the Ary position, policy or decision, unless so designated by other docuentation. 16. SECUITY CLASSIFICATION OF: 17. LIMITATION OF a. EPOT b. ABSTACT c. THIS PAGE ABSTACT UU UU UU UU 15. NUMBE OF PAGES 19a. NAME OF ESPONSIBLE PESON Yaakov Bar-Shalo 19b. TELEPHONE NUMBE Standard For 98 (ev 8/98 Prescribed by ANSI Std. Z39.18

2 eport Title 3 Perforance Analysis of the Converted ange ate and Position Linear Kalan Filter ABSTACT In active sonar and radar applications easureents consist of range, bearing and often range rate - all nonlinear functions of the target state (usually odeled in Cartesian coordinates. The converted easureent Kalan filter (CMFK first converts the range and bearing easureents into Cartesian coordinates to allow for the use of a linear Kalan filter. The extension of the CMKF to use range rate as a linear easureent however has been liited to cases with sall bearing errors. The use of range rate as a nonlinear easureent requires the use of a nonlinear filter such as the extended Kalan filter (EKF. Due to the uncertain perforance of the EKF, various odifications have been proposed, including use of a pseudo easureent, an alternative linearization of the easureent prediction function, and sequentially processing the converted position and range rate easureents (applied to the EKF and the Unscented Kalan Filter. Coon to these approaches is that the easureent prediction function reains nonlinear. A easureent conversion fro range, bearing and range rate to Cartesian position and velocity has recently been proposed 4. This anuscript expands the evaluation of this new approach by coparing to the Sequential EKF, the Sequential Unscented Kalan Filter (UKF and the posterior Craer-ao lower bound (PCLB. The new ethod is shown to have iproved ean square error perforance and exhibits iproved constancy over the previously proposed ethods, especially in cases with poor bearing accuracy. Conference Nae: Asiloar Conf.\ on Signals, Systes and Coputers}, Asiloar, CA, Nov.\ 13 Conference Date: Noveber 14, 13

3 4 Perforance Analysis of the Converted ange ate and Position Linear Kalan Filter Steven V. Bordonaro Naval Undersea Warfare Center Newport, I 841, USA Eail: steven.bordonaro@navy.il Peter Willett Dept. of Electrical and Coputer Engr., University of Connecticut Storrs, CT , USA Eail: willett@engr.uconn.edu Yaakov Bar-Shalo Dept. of Electrical and Coputer Engr., University of Connecticut Storrs, CT , USA Eail: ybs@engr.uconn.edu Abstract In active sonar and radar applications easureents consist of range, bearing and often range rate - all nonlinear functions of the target state (usually odeled in Cartesian coordinates. The converted easureent Kalan filter (CMFK first converts the range and bearing easureents into Cartesian coordinates to allow for the use of a linear Kalan filter. The extension of the CMKF to use range rate as a linear easureent however has been liited to cases with sall bearing errors. The use of range rate as a nonlinear easureent requires the use of a nonlinear filter such as the extended Kalan filter (EKF. Due to the uncertain perforance of the EKF, various odifications have been proposed, including use of a pseudo easureent, an alternative linearization of the easureent prediction function, and sequentially processing the converted position and range rate easureents (applied to the EKF and the Unscented Kalan Filter. Coon to these approaches is that the easureent prediction function reains nonlinear. A easureent conversion fro range, bearing and range rate to Cartesian position and velocity has recently been proposed 4. This anuscript expands the evaluation of this new approach by coparing to the Sequential EKF, the Sequential Unscented Kalan Filter (UKF and the posterior Craer-ao lower bound (PCLB. The new ethod is shown to have iproved ean square error perforance and exhibits iproved constancy over the previously proposed ethods, especially in cases with poor bearing accuracy. I. INTODUCTION A coon tracking scenario is one in which the easureents consists of range and bearing while the target is tracked in Cartesian coordinates (a natural choice, since target dynaics are linear in the Cartesian coordinate syste. A coon approach is to eploy an EKF to handle the fact that the easureents are a nonlinear function of the target state. An alternative approach is to first convert the easureents to Cartesian coordinates, allowing the tracking to be perfored with a linear Kalan filter. Perforance of this approach, referred to as the Converted Measureent Kalan Filter (CMKF, exceeds that of a ixed coordinate EKF if an unbiased conversion fro polar to Cartesian coordinates is used 14. Perforance is further enhanced if estiation bias is eliinated by evaluating the converted easureent error covariance using the state prediction 5. In addition to range and bearing, in any active sonar and radar applications easureents also include range rate. Previous approaches to incorporate range rate have converted range and bearing to Cartesian coordinates, but left range rate as a nonlinear function of the state. A recently proposed ethod 4, however, provides a natural extension of the CMKF to include range rate by converting range, bearing and range rate to Cartesian position and velocity. The converted easureent is then used in a linear Kalan filter. The ethod, referred to here as the converted easureent Kalan filter with range rate (CMKF, was shown in 4 to have iproved perforance over an EKF and an EKF with alternate linearization as describe in 3. In this anuscript the evaluation is expanded to include the sequential EKF using a pseudo easureent 7,, 1 and the sequential UKF using the raw range range easureent 8, 13. II. POBLEM STATEMENT Active sonar and radar systes produce easureents in polar coordinates, often with the additional easureent of range rate: z AW = r α = h(x (1 ṙ where r, α, and ṙ are the easured range, bearing and range rate; h is the easureent function, and x is the target state. The easureent error for the raw easureents is assued to be Gaussian with covariance atrix AW = σ r ρσ r σṙ σ α ρσ r σṙ σ ṙ ( where σ r,, and σṙ are the standard deviations of the range, bearing and range rate easureent noise. The correlation coefficient for the correlation between the range and range rate easureent noise is ρ 1. Since target otion is linear in Cartesian coordinates, state estiation is best perfored in this coordinate syste. The Kalan filter for a nearly constant velocity target otion assuption is described in. Defining the state as x = x y ẋ ẏ, using Hk and P k to represent the easureent prediction atrix the state estiate s covariance atrix, the Kalan gain and covariance update steps are Proc. of the Asiloar Conference on Signals, Systes, and Coputers Nov. 13

4 5 S k+1 = k+1 + H k+1 P k+1 k H k+1 (3 W k+1 = P k+1 k H k+1s 1 k+1 (4 P k+1 k+1 = P k+1 k W k+1 S k+1 W k+1 (5 ˆx k+1 k+1 = ˆx k+1 k + W k+1 zk h k+1 k (6 III. APPOACHES Two effective techniques that predate the CMKF are the sequential EKF using a pseudo easureent and the sequential UKF. Coon to these two approaches is that range and bearing are first converted to Cartesian position. This leaves the range rate as the only easureent coponent that is a nonlinear function of the target state. Various approaches have been developed for the conversion of range and bearing to Cartesian position. In 14, it was shown that the conventional conversion fro polar to Cartesian coordinates introduces a bias in the expected value of the converted easureent. Various reedies to this bias have been proposed 14, 17, 15, 16, 6 and applied to tracking with range rate in 7, 13,, 1. The range rate easureent is a nonlinear function of the target state xẋ + yẏ ṙ = x + y + w ṙ (7 where wṙ is the range rate easureent noise. The sequential EKF and sequential UKF differ in their approach to handle this nonlinear easureent. The sequential EKF attepts to reduce the nonlinearity between the state and the easureent by replacing the range rate easureent, ṙ, with a pseudo easureent consisting of r ṙ. η pseudo = r ṙ = h pseudo η (x+w η = xẋ+yẏ ρσ r σṙ +w η (8 where ρσ r σṙ is a debiasing ter. The use of this pseudo easureent was proposed in 9 and applied to the second order EKF in 7,, 1. According to 13, the pseudo easureent has disadvantages when the range and range rate easureent noises are not statistically independent (as is the case for certain wavefors 1. For these cases, use of the UKF has been proposed 8, 13 to handle the strong nonlinearities with the use of range rate instead of the range/range rate product. In this case η is siply the range rate easureent, ṙ, η raw = ṙ = h raw xẋ + yẏ η (x + wṙ = x + y + w ṙ (9 The sequential EKF and UKF process the position easureents first, followed by processing of pseudo easureent (8 for the EKF 7,, 1 or the raw range range easureent 8, 13 for the UKF. In order to process these easureents sequentially, the range rate based easureent, or η raw, ust first be decorrelated fro the position coponents of the easureent. This is achieved as follows. η pseudo The covariance atrix of the converted easureent error can be partitioned into the position and pseudo range rate blocks 7 pp CONV = pη ηp ηη (1 where and Let pp xx = xy yx yy, ηp = xη yη ( L = ηp ( pp 1 = L 1 L B = Ix L 1 (1 (13 By pre-ultiplying B on both sides of the easureent conversion equation, a new easureent prediction function can be obtained in which the position easureent is unodified, and the pseudo range rate is replaced with a decorrelated pseudo range rate, ε ε L 1 x + L y + η (14 with the corresponding easureent prediction function h pseudo ε = L 1 x + L y + xẋ + yẏ (15 for the pseudo range rate approach, and h raw ε = L 1 x + L y + xẋ + yẏ x + y (16 for the raw range rate approach. For the second order EKF with decorrelated pseudo range rate easureents, (3 (6 are processed for the converted position portion of the easureents. The state estiate, ˆx p = ˆx ŷ ˆẋ ˆẏ, and state covariance, P p k+1 k+1, updated using the position easureent only, are subsequently processed using the decorrelated pseudo range rate easureent in a second order EKF 7. Siilarly, in the sequential UKF, (3 (6 are processed for the converted position portion of the easureents. The state estiate, ˆx p, and state covariance estiate, P p k+1 k+1, updated using the position easureent only, are subsequently processed by the decorrelated range rate easureent using the second order unscented transfor 13. Siga points are generated using the state and covariance estiate that has been updated by the position estiates and then passed through the nonlinear function, h raw ε, to provide the tie and easureent update. IV. NEW CONVETED MEASUEMENT APPOACH WITH ANGE ATE MEASUEMENTS The approach used in the recently introduced converted easureent Kalan filter with range rate (CMKF is to convert the raw easureent of range, bearing and range rate into a easureent of position and velocity in Cartesian coordinates. The raw easureent is converted in a anner that

5 6 is unbiased and consistent, and that allows for and describes the correlation between the range and range rate easureent errors. The converted easureent error covariance estiate is evaluated at the prediction (as opposed to the easureent to avoid estiation bias 5. In order to develop this conversion, consider the inclusion of a non-inforative cross range rate easureent, ċ. The conversion function to Cartesian is therefore, z C = x y ẋ ẏ = D (α where D is the direction cosine atrix, r ṙ ċ cos α sin α sin α cos α cos α sin α sin α cos α (17 (18 Cross range rate is non-inforative because it is not truly easured, but a priori knowledge about the distribution of expected cross range rates (based on knowledge of possible target speeds can be used in calculating the converted easureent error covariance as describe in IV-B. A. Estiation of the Mean The conversion (17 has ultiplicative bias of a factor of e σα/. An unbiased version of the easureent conversion (17 can be developed as an extension of the Unbiased Converted Measureent for position 15 z U = x y ẋ ẏ = eσ α / D (α B. Estiation of the Covariance r ṙ ċ (19 For convenience, the converted easureent error covariance, C, will be developed in a coordinate syste along the line of sight (LOS to the target,, and then converted to Cartesian coordinates, i.e. C = D (α D (α ( The calculation of the coponents of C requires the true target velocity and position. Since these are not available in practice, the evaluation is perfored at the predicted target state, ˆx k+1 k. First the predicted target state and covariance are rotated into the estiate s LOS coordinate syste: ˆx = D (α t ˆx k+1 k (1 P = D (α t P k+1 k D (α t ( where the predicted target bearing is α t = tan 1 ( ˆx k+1 k ˆx 1 k+1 k (3 and ˆx n is the nth coponent of ˆx. The individual coponents of evaluated at the prediction are as follows, = 1 ( 1 + P + σ r β + ( 1 δ + (4 = (5 13 = 1 ˆx ρσ r σṙ β+ 1 ˆx3 13 δ+ (6 ( = 1 ( + σr β 1 δ (7 3 = 1 ˆx4 14 β 1 ˆx4 14 δ (8 ( 33 = 3 ( 33 + σṙ β σċ β ( 3 ( 33 δ δ (9 where β ± = 1 1 ± e σ α e σ αt e σ α δ ± = 1 1 ± e σ αt and σα t is the approxiate bearing variance of the predicted track estiate based on a linearization of P, σα t = P (3 1 ˆx n is the nth coponent of ˆx and P n is the (n eleent of P. The value of σċ used in (9 is set based on an a priori estiate of the standard deviation of target cross range rate. Since the cross range rate easureent, ċ, is noninforative, the reaining coponents of the easureent noise covariance in the LOS coordinate syste, (e.g. 44, 34 are infinite. It is therefore useful to deal with the inverse of ( 1 1 = 1:3,1:3 (31 Since the inverse of the direction cosine atrix, D (α, is its transpose, the easureent noise covariance for (19, C, is C 1 = D (α t 1 D (α t (3 The inconsistency in the use of σċ, and the rational, requires explanation. The value in (9 influences the Kalan gain for the ṙ easureent and an a prior estiate of σċ is used to iprove the consistency of the tracker. For the coponents of that influence the Kalan gain for ċ, σċ is set to infinity (or, equivalently, the coponents of 1 set to zero. This is to ensure that the tracker is not biased towards the value substituted for ċ, since it is not truly easured. The consistency of the conversion ethod was exained using the Noralized Error Squared (NES and shown to be consistent in 4. Note that although the converted easureent has diension 4, the expected NES, eq. ( is 3, since velocity errors along the cross range are ultiplied by zero.

6 7 Unfortunately, C 1 is not invertible, so C is not available for use directly in the Kalan filter gain calculation (3. It can however be used in the inforation for of the Kalan filter, described in V-A. V. APPLICATION TO TACKING A. Inforation For of the Kalan Filter The inforation for of the Kalan filter propagates the inverse of the state covariance and uses the inverse of the easureent error covariance. This allows the use of 1 C (3 directly, in place of C, which is unavailable. The calculation of the Kalan gain in the Kalan filter (3 (4 is replaced with 1 W k+1 = P 1 k+1 k + H 1 k+1 H H 1 k+1 (33 and the state covariance update (5 is replaced with P k+1 k+1 = P 1 k+1 k + H 1 k+1 H 1 (34 B. Converted Measureent Kalan Filter with ange ate With the use of the easureent conversion function (19, each coponent of the state (for the nearly constant velocity odel is observed directly. When applied to the inforation for Kalan filter, one has H as the identity atrix. The converted easureent is therefore linear with respect to the target state, eliinating the need for the extended (or unscented Kalan filter. The Converted Measureent Kalan Filter with ange ate (CMKF is ipleented as follows: 1 Convert the raw easureents of range, bearing and range rate to Cartesian position and velocity with (19, using ċ =, and use the result, z = z U, in (6. Use the inforation for of the Kalan filter (33 (34 with 1 = C 1 fro (3. 3 Set the easureent prediction atrix, H = I 4 4, in (33 (34. 4 Let h = ˆx in (6. A. Perforance Coparisons VI. EVALUATION The perforance of the proposed Converted Measureent Kalan Filter with ange ate (CMKF has been evaluated with respect to the current state-of-art techniques. The CMKF was copared to 1 CMKF using range and bearing easureents only (POS Sequential EKF using pseudo range rate (i.e. range, range rate product as described in III (SEKF 3 Sequential UKF using range-rate as described in III (SUKF 4 Posterior Craer-ao lower bound, as defined in 18, for range and bearing easureents only (PCLBPOS 5 Posterior Craer-ao lower bound, as defined in 18, for range, bearing and range-rate easureents (PCLBPOS To allow for direct coparison, all of the existing trackers were ipleented with the conversion of range and bearing to Cartesian coordinates using the ethod described in 6, 1. The conclusions hold for other conversion ethods. Measures of perforance include ean square error (MSE for the target position and velocity estiates. An additional easure of perforance is the tracker consistency based on the Average Noralized Estiation Error Squared (. The of a consistent estiator is close to 1. Tracker consistency is iportant not only for analysis of results, but also for easureent to track association in ultitarget tracking scenarios in clutter. The tracking scenario is set up as follows: 1 True target range, r, norally distributed with ean 4, and standard deviation of 3. True target bearing uniforly distributed fro π to π 3 True target heading uniforly distributed fro π to π 4 True target speed χ distributed, scaled by 1/s 5 Sensor range accuracy, σ r = 3 6 Sensor bearing accuracy, = 1,.5, 5, 8 and Sensor range rate accuracy, σṙ =.1/s 8 Correlation between range and range rate errors, ρ =.. 9 σċ in (9 set to 1 /s Fig. 1 shows the results of 5, Monte Carlo runs of the trackers. In each of the test cases, the proposed CMKF achieved perforance that was equal to or better than the existing ethods. For sall angle error, the perforance of the CMKF atched the existing ethods (results overlap on plots, and all trackers were fairly consistent. As the angle accuracy degraded, the CMKF perforance was better in ters of MSE, and considerably better in ters of consistency. Even for severely degraded angle accuracy, = 16, the CMKF aintained consistency. VII. CONCLUSION When tracking with easureents of range, bearing and range rate, various filtering techniques have been proposed. For cases with poor bearing accuracy, the valid approaches have been liited to nonlinear filters, such as the EKF and UKF. This work has shown that a linear Kalan filter can be eployed to handle this estiation proble by using a new converted easureent technique and the inforation for of the Kalan filter. The conversion process avoids conversion bias by using an unbiased converted easureent, and precludes estiation bias by evaluating the converted easureent error covariance at the prediction. Siulations show that this linear Kalan filter has iproved MSE perforance over all the state of the art trackers, including the sequential EKF using pseudo range rate and the sequential UKF. This new approach is recoended for consideration in active radar or sonar systes that include easureents of range rate.

7 8 MSE Position MSE Position MSE Position MSE Position = 1 deg =.5 deg = 5 deg. 3 x 14 = 8 deg. 5 x 14 MSE Velocity (/s MSE Velocity (/s MSE Velocity (/s MSE Velocity (/s = 1 deg. 1 5 =.5 deg = 5 deg = 8 deg = 1 deg =.5 deg = 5 deg = 8 deg. PCLBPOS PCLBPOS SEKF SUKF POS CMKF Fig. 1. MSE and for the position only based tracker (POS, sequential EKF using pseudo range rate (SEKF, sequential UKF using rangerate (SUKF, and the new Converted Measureent Kalan Filter with ange ate (CMKF. The PCLB for position only easureents and position easureents with range rate is also shown. ACKNOWLEDGMENT Steven Bordonaro was supported by ON In-house Laboratory Independent esearch (ILI. Peter Willett was supported by the Office of Naval esearch under contract N Yaakov Bar Shalo was supported under AO W9NF and ON N EFEENCES 1 Y. Bar-Shalo, Negative correlation and optial tracking with doppler easureents, IEEE Transactions on Aerospace and Electronic Systes, vol. 37, no. 3, pp. 17, Jul 1. Y. Bar-Shalo, X.. Li, and T. Kirubarajan, Estiation with Applications to Tracking and Navigation. New York & Boston: John Wiley and Sons, 1. 3 D. Bizup and D. Brown, The over-extended Kalan filter - don t use it! in Proceedings of the Sixth International Conference of Inforation Fusion, vol. 1, 3, pp S. Bordonaro, P. Willett, and Y. Bar-Shalo, Consistent linear tracker with position and range rate easureents, in Signals, Systes and Coputers (ASILOMA, 1 Conference ecord of the Forty Sixth Asiloar Conference on, 1, pp , Decorrelated, unbiased converted easureent kalan filter, IEEE Trans. on Aerospace and Electronic Systes., Accepted. 6 Z. Duan, C. Han, and X.. Li, Coents on unbiased converted easureents for tracking, IEEE Trans. on Aerospace and Electronic Systes., vol. 4(4, pp , 4. 7, Sequential nonlinear tracking filter with range-rate easureents in spherical coordinates, in Proceedings of the 7th International Conference on Inforation Fusion, vol. 1. Citeseer, 4, pp Z. Duan, X.. Li, C. Han, and H. Zhu, Sequential Unscented Kalan filter for radar target tracking with range rate easureents, in Inforation Fusion, 5 8th International Conference on, vol. 1. IEEE, 5, pp. 8 pp. 9 A. Farina and F. Studer, adar data processing, vol. I: Introduction and tracking, vol. II: Advanced topics and applications, Fitzgerald, Effects of range-doppler coupling on chirp radar tracking accuracy, Aerospace and Electronic Systes, IEEE Transactions on, no. 4, pp , L. Gao, J. Lan, J. Jiang, and X. Fan, Perforance analysis of sequential filter based on unbiased converted easureents with doppler, in Electronics and Optoelectronics (ICEOE, International Conference on, vol. 1. IEEE,, pp. V L. Jiao, Q. Pan, Y. Liang, and F. Yang, A nonlinear tracking algorith with range-rate easureents based on unbiased easureent conversion, in Inforation Fusion (FUSION, 1 15th International Conference on. IEEE, 1, pp M. Lei and C. Han, Sequential nonlinear tracking using UKF and raw range-rate easureents, IEEE Transactions on Aerospace and Electronic Systes, vol. 43, no. 1, pp. 39 5, January D. Lerro and Y. Bar-Shalo, Tracking with debiased consistent converted easureents versus EKF, IEEE Trans. on Aerospace and Electronic Systes., vol. 9(3, pp. 5 1, M. Longbin, S. Xiaoquan, Z. Yiyu, and Y. Bar-Shalo, Unbiased converted easureents for tracking, IEEE Trans. on Aerospace and Electronic Systes., vol. 34(3, pp , M. Miller and O. Druond, Coparison of ethodologies for itigating coordinate transforation bias in target tracking, in Proceedings of SPIE Conference on Signal and Data Processing of Sall Targets, vol. 448,, pp P. Suchoski, Explicit expressions for debiased statistics of 3D converted easureents, Aerospace and Electronic Systes, IEEE Transactions on, vol. 35, no. 1, pp , P. Tichavsky, C. Muravchik, and A. Nehorai, Posterior Craer-ao bounds for discrete-tie nonlinear filtering, IEEE Transactions on Signal Processing, vol. 46, no. 5, pp , May 1998.