Bias Estimation for Optical Sensor Measurements with Targets of Opportunity

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1 Bias Estimation for Optical Measurements with s of Opportunity Djedjiga Belfadel, Richard W Osborne, III and Yaakov Bar-Shalom Electrical and Computer Engineering University of Connecticut Storrs, CT, USA djedjigabelfadel@uconnedu rosborne@engruconnedu, ybs@engruconnedu Abstract This paper provides a solution for bias estimation of multiple passive sensors using common targets of opportunity The measurements provided by these sensors are assumed timecoincident (synchronous) and perfectly associated Since these sensors provide only line of sight (LOS) measurements, the formation of a single composite Cartesian measurement obtained from fusing the LOS measurements from different sensors is needed to avoid the need for nonlinear filtering The evaluation of the Cramer-Rao Lower Bound (CRLB) on the covariance of the bias estimate, ie, the quantification of the available information about the biases, combined with simulations, shows that this method is statistically efficient, even for small sample sizes Index Terms Bias estimation, bias observability, statistical efficiency, CRLB, composite measurements, maximum likelihood I ITRODUCTIO In order to carry out data fusion, registration error correction is crucial in multisensor systems This requires estimation of the sensor measurement biases Measurement bias, in target tracking problems, can result from a number of different sources Some primary sources of bias errors include: measurement biases due to the deterioration of initial sensor calibration over time; attitude errors caused by biases in the gyros of the inertial measurement units of (airborne, seaborne or spaceborne) sensors; and timing errors due to the biases in the onboard clock of each sensor platform [8] The effect of biases introduced in the process of converting sensor measurements from polar (or spherical) coordinates to Cartesian coordinates has been discussed extensively in [] together with the limit of validity of the standard transformation One of the most studied examples of converted measurement tracking involves the conversion from polar or spherical measurements to Cartesian coordinates for use in a linear Kalman filter If the conversion process is unbiased, the performance of a converted measurement Kalman filter is superior to a mixed coordinate EKF (ie target motion in Cartesian coordinates and measurements in polar coordinates) [] The approaches for conversion include the conventional conversion, the Unbiased Converted Measurement (UCM), the Modified Unbiased Converted Measurement (MUCM), and the Unscented Transform (UT) Recently, a decorrelated version of the UCM technique (DUCM) has been developed to address both conversion and estimation bias [5], [6] Another example of biased measurement conversion is the estimation of range-rate from a moving platform To measure range rate using the Doppler effect, it is necessary to nullify the impact of platform motion The conventional nullification approach suffers from a similar bias problem as the position measurement conversion [3] A novel scheme was proposed in [3] and [4] by applying the DUCM technique to own-doppler nullification to eliminate this bias Time varying bias estimation based on a nonlinear least squares formulation and the singular value decomposition using truth data was presented in [8] However, this work did not discuss the CRLB for bias estimation An approach using maximum a posteriori (MAP) data association for concurrent bias estimation and data association based on sensor-level track state estimates was proposed in [9] and extended in [1] For angle-only sensors, imperfect registration leads to LOS angle measurement biases in azimuth and elevation It is important to correct for these bias errors so that the multiple sensor measurements and/or tracks can be referenced to a common tracking coordinate system (frame) If uncorrected, registration error can lead to large tracking errors and potentially to the formation of multiple tracks (ghosts) on the same target In this paper, bias estimation is investigated when only targets of opportunity are available We assume the sensor locations are known and we estimate their orientation biases Estimation of sensor biases only was discussed in [7] The estimation of range and location biases for active sensors was presented in [11] First, we discuss the bias estimation for synchronous biased sensors Then we evaluate the corresponding Cramer-Rao lower bound (CRLB) on the covariance of the bias estimates, which is the quantification of the available information on the sensor biases Section II presents the problem formulation and solution in detail Section III describes the simulations performed and gives the results Finally, Section IV gives the conclusions II PROBLEM FORMULATIO The fundamental frame of reference used in this paper is a 3D Cartesian Common Coordinate System (CCS) defined by the orthogonal set of unit vectors (e x, e y, e z ) In a multisensor scenario, sensor platforms will typically have a

2 sensor reference frame associated with it (measurement frame of the sensor) defined by the orthogonal set of unit vectors (e ξs, e ηs, e ζs ) The origin of the measurement frame of the sensor is a translation of the CCS origin, and its axes are rotated with respect to the CCS axes The rotation between these frames can be described by a set of Euler angles We will refer to these angles φ s +φ s, ρ s +ρ s, ψ s +ψ s of sensor s, as roll, pitch and yaw respectively [8], where φ s is the nominal roll angle, φ s is the roll bias, etc Each angle defines a rotation about a prescribed axis, in order to align the sensor frame axes with the CCS axes The xyz rotation sequence is chosen, which is accomplished by first rotating about the x axis by φ s, then rotating about the y axis by ρ s, and finally rotating about the z axis by ψ s The rotations sequence can be expressed by the matrices T s (ψ s, ρ s, φ s ) =T z (ψ s )T y (ρ s )T x (φ s ) cos ψ s sin ψ s = sin ψ s cos ψ s 1 cos ρ s sin ρ s 1 sin ρ s cos ρ s 1 cos φ s sin φ s (1) sin φ s cos φ s Assume there are S synchronized passive sensors, with known position in the CCS, ξ s = [ξ s, η s, ζ s ], s = 1,,, S, and t targets, located at x i = [x i, y i, z i ], i = 1,,, t, in the same CCS With the previous convention, the operations needed to transform the position of a given target i expressed in the CCS coordinate into a position expressed in a sensor s coordinate system (x is = [x is, y is, z is ] ) can be expressed as x is = T (ω s )(x i ξ s ) i = 1,,, t, s = 1,,, S () where ω s = [φ s, ρ s, ψs] is the nominal orientation of sensor s and T (ω s ) is the appropriate rotation matrix and the translation (x i ξ s ) is the difference between the vector position of the target i and the vector position of the sensor s, both expressed in the CCS As shown in Figure 1, the azimuth angle α is is the angle in the sensor xz plane between the sensor z axis and the line of sight to the target, while the elevation angle ɛ is is the angle between the line of sight to the target and its projection onto the xz plane, that is [ αis ɛ is ] = ( ) tan 1 x is ( zis tan 1 y is x is +zis ) (3) The model for the biased noise-free LOS measurements is then [ ] [ ] α b is g1 (x = i, ξ s, ω s, b s ) = g(x g (x i, ξ s, ω s, b s ) i, ξ s, ω s, b s ) (4) ɛ b is y Fig 1 Optical sensor coordinate system with the origin in the center of the focal plane where g 1 and g denote the sensor Cartesian coordinatesto-azimuth/elevation angle mapping that can be found by inserting equations () and (3) into (4) The bias vector of sensor s is b s = [φ s, ρ s, ψ s ] (5) and θ is = [x is, y is, z is, ψ s, ρ s, φ s ] (6) denotes the augmented parameter vector for target i and sensor s For a given target, each sensor provides the noisy LOS measurements where z is = g(x i, ξ s, ω s, b s ) + w is (7) x z w is = [w α is, w ɛ is] (8) The measurement noises wis α and wɛ is are zero-mean, white with corresponding standard deviations σs α, σs ɛ and are assumed mutually independent The problem is to estimate the bias vector for all sensors and obtain composite position measurements in Cartesian coordinates from LOS measurements that will then be used in a tracking filter We shall obtain the maximum likelihood (ML) estimate of the parameter vector θ = [x 1,, x t, b 1,, b S ] (9) which contains all the elements of θ is, i = 1,, t, s = 1,, S, rearranged and transformed into CCS, by maximizing the likelihood function where t S Λ(θ) = p (z is θ) (1) i=1 s=1 p (z is θ) = πr s 1/ ( exp 1 ) [z is h is (θ)] Rs 1 [z is h is (θ)] and (11) h is (θ) = g(x i, ξ s, ω s, b s ) (1)

3 The ML estimate (MLE) is then ˆθ ML = arg max Λ(θ) (13) θ In order to find the MLE, one has to solve a nonlinear least squares problem for the exponent in (11) This will be done using a numerical search via the Iterated Least Squares (ILS) technique [1] A Conditions for bias estimability First condition for bias estimability: For a given target we have measurements from each sensor (the two LOS angles of the target point) We assume that each sensor sees all the targets at a common time 1 Therefore the number of measurements of t targets seen by S sensors is t S Given that the dimensions of the position and bias vectors of each target are 3, and knowing that the number of measurements has to be at least equal to the number of parameters to be estimated (target positions and biases), we must have t S 3( t + S ) (14) For example, to estimate the biases of 3 sensors (9 bias components) we need 3 targets (9 position components), ie, the search is in an 18-dimensional space For sensors (6 bias components) we need at least 6 targets (18 position components) to have complete observability (estimability) In this case a search in a 4-dimensional space is needed Second condition of bias estimability: This is the invertibility of the Fisher Information matrix [1], to be discussed later B Iterated Least Squares Given the estimate ˆθ j after j iterations, the ILS estimate after the (j + 1)th iteration will be ˆθ j+1 = ˆθ j + [ (H j ) R 1 H j] 1 (H j ) R 1 [z h( ˆθ j )] (15) where z = [z 11,, z is,, z t S ] (16) h( ˆθ j ) = [h 11 ( ˆθ j ),, h is ( ˆθ j ),, h t S ( ˆθ j ) ] (17) R 1 R R = R S where R s is the covariance matrix of sensor s, and H j = h ( θ j) θ θ= ˆθj (18) (19) is the Jacobian matrix of the vector consisting of the stacked measurement functions (17) wrt (9) evaluated at the ILS 1 This can also be the same target at different times, as long as the sensors are synchronized estimate from the previous iteration j In this case, the Jacobian matrix is, with the iteration index omitted for conciseness, H = [ H 11 H 1 H t1 H 1 H t S ] where H is = x 1 y 1 z 1 x t y t z t ψ 1 ρ 1 φ 1 ψ S ρ S φ S g is x 1 g is y 1 g is z 1 g is x t g is y t g is z t g is ψ 1 g is ρ 1 g is φ 1 g is ψ S g is ρ S g is φ S () (1) The appropriate partial derivatives are given in the Appendix C Initialialization In order to perform the numerical search via ILS, an initial estimate ˆθ is required Assuming that the biases are null, the LOS measurements from the first and the second sensor α i1, α i and ɛ i1 can be used to solve for each initial Cartesian target position, in the CCS, as x i = ξ ξ 1 + ζ 1 tan α i1 ζ tan α i tan α i1 tan α i () yi = tan α i1 (ξ + tan α i (ζ 1 ζ )) ξ 1 tan α i (3) tan α i1 tan α i zi (ξ 1 ξ ) cos α i + (ζ ζ 1 ) sin α i =η 1 + tan ɛ i1 sin (α i1 α i ) (4) D Cramer-Rao Lower Bound In order to evaluate the efficiency of the estimator, the CRLB must be calculated The CRLB provides a lower bound on the covariance matrix of an unbiased estimator as [1] E{(θ ˆθ)(θ ˆθ) } J 1 (5) where J is the Fisher Information Matrix (FIM), θ is the true parameter vector to be estimated, and ˆθ is the estimate The FIM is J = E { [ θ ln Λ(θ)] [ θ ln Λ(θ)] } θ=θtrue (6) where the gradient of the log-likelihood function is λ(θ) = ln Λ(θ) (7)

4 Scenario 1 Scenario Y axis(m) Y axis(m) Fig Scenario 1 Fig 3 Scenario t S θ λ(θ) = H isr 1 is (z is h is (θ)) (8) i=1 s=1 Scenario 3 which, when plugged into (6), gives J = t S H is i=1 s=1 ( ) R 1 s His θ=θtrue = H ( R 1) H θ=θtrue (9) III SIMULATIOS We simulated three optical sensors at various locations observing a target at three (unknown) points in time (which is equivalent to viewing three different targets at unknown locations) Five scenarios of three sensors are examined for a set of target locations They are shown in Figures 6 Each scenario is such that each target position can be observed by all sensors As discussed in the previous section, the three sensor biases were roll, pitch and yaw angle offsets The biases for each sensor were set to 3 = 5mrad We made 1 Monte Carlo runs for each scenario In order to establish a baseline for evaluating the performance of our algorithm, we also ran the simulations without bias and with bias but without bias estimation The horizontal and vertical field-of-view of each sensor is assumed to be 6 The measurement noise standard deviation σ s (identical across sensors for both azimuth and elevation measurements) was assumed to be 34 mrad A Description of the Scenarios The sensors are assumed to provide LOS angle measurements We denote by ξ 1, ξ, ξ 3 the 3D Cartesian sensor positions, and x 1, x, x 3 the 3D Cartesian target positions (all in CCS) The three target positions are the same for all the scenarios, and they were chosen from a trajectory of a ballistic Y axis(m) target as follows (m) Fig 4 Scenario 3 x 1 = ( 86,, 68 ) x = ( 359,, 815 ) x 3 = (413,, 6451 ) Table I summarizes the sensor positions (in m) for the 5 scenarios considered B Statistical efficiency of the estimates In order to test for the statistical efficiency of the estimate (of the 18 dimensional vector (9)), the normalized estimation error squared (EES) [1] is used, with the CRLB as the covariance matrix The sample average EES over 1 Monte Carlo runs is given in Table II for all the scenarios The EES is calculated using the FIM evaluated at both the true bias values and target positions, as well as at the estimated biases and target positions According to the CRLB, the FIM has to be evaluated at the true parameter Since this is not

5 TABLE I SESOR POSITIOS (M) FOR THE SCEARIOS COSIDERED First Second Third Scenario ξ η ζ ξ η ζ ξ η ζ Scenario EES θ=θils EES θ=θt 185 EES Y axis(m) Scenarios Fig 5 Scenario 4 Fig 7 Sample average EES over 1 Monte Carlo runs for all 5 scenarios Scenario EES of ψ 1 EES of ρ 1 EES of φ 1 EES of ψ Bias EES EES of ρ EES of φ EES of ψ 3 EES of ρ 3 EES of φ Y axis(m) Scenarios Fig 8 Sample average bias EES (CRLB evaluated at the estimate), for each of the 9 biases, over 1 Monte Carlo runs for all 5 scenarios Fig 6 Scenario 5 available in practice, however, it is useful to evaluate the FIM also at the estimated parameter, the only one available in real world implementations [1], [13] The results are practically identical regardless of which values are chosen for evaluation of the FIM The 95% probability region for the 1 sample average EES of the 18 dimensional parameter vector is [1684, 1919] For all 5 scenarios the estimate is found to be within this interval and is therefore considered a statistically efficient estimate Figure 7 shows the EES for all scenarios, Figure 8 shows the individual bias component EES for all scenarios, The 95% probability region for the 1 sample average single component EES is [74, 19] For all 5 scenarios the estimates are found to be within this interval The RMS position errors for the 3 targets are summarized in Table III In this table, the first estimation scheme was established as a baseline using bias-free LOS measurements to

6 GDOP GDOP Tar1 Sens1 GDOP Tar1 Sens GDOP Tar1 Sens3 GDOP Tar Sens1 GDOP Tar Sens GDOP Tar Sens3 GDOP Tar3 Sens1 GDOP Tar3 Sens GDOP Tar3 Sens3 TABLE II SAMPLE AVERAGE EES OVER 1 MOTE CARLO RUS FOR 5 SCEARIOS (BASED O 18 1 = 18 DEGREES OF FREEDOM CHI-SQUARE DISTRIBUTIO) Scenarios EES θ=θt EES θ=ˆθils TABLE III SAMPLE AVERAGE POSITIO RMSE (M) FOR THE 3 TARGETS, OVER 1 MOTE CARLO RUS, FOR THE 3 ESTIMATIO SCHEMES Scenarios Fig 9 GDOPs for the 5 scenarios considered First Second Third Scheme RMSE RMSE RMSE estimate the target positions For the second scheme, we used biased LOS measurements but we only estimated target positions In the last scheme, we used biased LOS measurements and we simultaneously estimated the target positions and sensor biases Bias estimation yields significantly improved target RMS position errors in the presence of biases Each component of θ should also be individually consistent with its corresponding σ CRLB (the square root of the corresponding diagonal element of the inverse of FIM) In this case, the sample average bias RMSE over 1 Monte Carlo runs should be within 15% of its corresponding bias standard deviation from the CRLB (σ CRLB ) with 975% probability Table IV demonstrates the consistency of the individual bias estimates To confirm that the bias estimates are unbiased, the average bias error b, from Table V, over 1 Monte Carlo runs confirms that b is less then (which it should be with 95% probability), ie, these estimates are unbiased In order to examine the statistical efficiency for a variety of target-sensor geometries, the sensors locations were varied from one scenario to another in order to vary the Geometric Dilution of Precision (GDOP), defined as GDOP = RMSE r σα + σɛ (3) where RMSE is the RMS position error for a target location (in the absence of biases), r is the range to the target, and σ α and σ ɛ are the azimuth and elevation measurement error standard deviations, respectively Figure 9 shows the various GDOP levels in the 9 target-sensor combinations for which statistical efficiency was confirmed IV COCLUSIOS In this paper, we presented an algorithm that uses targets of opportunity for estimation of measurement biases The first step was formulating a general bias model for synchronized optical sensors The association of measurements is assumed to be perfect Based on this, we used a ML approach that led to a nonlinear least-squares estimation scheme for concurrent estimation of the 3D Cartesian positions of the targets and the angle measurement biases of the sensors The bias estimates, obtained via ILS, were shown to be unbiased and statistically efficient APPEDIX The appropriate partial derivatives of (1) are z k g is g is g is z k g is g is g is = + + (31) = + + (3) = + + (33) z k z k z k = + + (34) = + + (35) = + + (36) = g is + g is + g is (37) = g is + g is + g is (38) = g is + g is + g is (39) z k z k z k = g is + g is + g is (4) = g is + g is + g is (41) = g is + g is + g is (4)

7 TABLE IV SAMPLE AVERAGE BIAS RMSE OVER 1 MOTE CARLO RUS AD THE CORRESPODIG BIAS STADARD DEVIATIO FROM THE CRLB (σ CRLB ), FOR ALL COFIGURATIOS (MRAD) First Second Third Scenario ψ ρ φ ψ ρ φ ψ ρ φ RMSE σ CRLB RMSE σ CRLB RMSE σ CRLB RMSE σ CRLB RMSE σ CRLB TABLE V SAMPLE AVERAGE BIAS ERROR b OVER =1 MOTE CARLO RUS FOR ALL COFIGURATIOS (MRAD) (TO COFIRM THAT THE BIAS ESTIMATES ARE UBIASED) First Second Third Scenario ψ ρ φ ψ ρ φ ψ ρ φ 1 b b b b b Given that () can be written as x is x is = y is = T s (x i ξ s ) z is T s11 T s1 T s13 = T s1 T s T s3 T s31 T s3 T s33 therefore and x i ξ s y i η s z i ζ s (43) x is = T s11 (x i ξ s ) + T s1 (y i η s ) + T s13 (44) y is = T s1 (x i ξ s ) + T s (y i η s ) + T s3 (45) z is = T s31 (x i ξ s ) + T s3 (y i η s ) + T s33 (46) = T s11, = T s1, = T s31, = T s1, = T s, = T s3, = T s13 = T s3 (47) = T s33 = T s 11 (x i ξ s ) + T s 1 (y i η s ) + T s 13 (48) = T s 11 (x i ξ s ) + T s 1 (y i η s ) + T s 13 (49) = T s 11 (x i ξ s ) + T s 1 (y i η s ) + T s 13 (5) = T s 1 (x i ξ s ) + T s (y i η s ) + T s 3 (51) = T s 1 (x i ξ s ) + T s (y i η s ) + T s 3 (5) = T s 11 (x i ξ s ) + T s (y i η s ) + T s 3 (53) = T s 31 (x i ξ s ) + T s 3 (y i η s ) + T s 33 (54) = T s 31 (x i ξ s ) + T s 3 (y i η s ) + T s 33 (55)

8 = T s 31 (x i ξ s ) + T s 3 (y i η s ) + T s 33 z is (56) = zis + (57) x is = (58) = x is x is + (59) z is g is x is y = is (x is + zis )(x is + y is + z is ) (6) g is x = is + zis x is + y is + (61) z is g is z is y is = (x is + y is + z is )( x is + z is ) (6) T s11 = sin ψ k cos ρ k (63) T s1 = sin ψ k sin ρ k sin φ k cos ψ k cos φ k (64) T s13 = sin ψ k sin ρ k cos φ k + cos ψ k sin φ k (65) T s1 = cos ψ k cos ρ k (66) T s = cos ψ k sin ρ k sin φ k sin ψ k cos φ k (67) T s3 = cos ψ k sin ρ k cos φ k + sin ψ k sin φ k (68) T s31 = (69) T s3 = (7) T s33 = (71) T s11 = cos ψ k sin ρ k (7) T s1 = cos ψ k cos ρ k sin φ k (73) T s13 = cos ψ k cos ρ k cos φ k (74) T s1 = sin ψ k sin φ k (75) T s = sin ψ k cos ρ k sin φ k (76) T s3 = sin ψ k cos ρ k cos φ k (77) T s31 = cos φ k (78) T s3 = sin ρ k sin φ k (79) T s33 = sin ρ k cos φ k (8) T s11 = (81) T s1 = cos ψ k sin ρ k cos φ k + sin ψ k sin φ k (8) T s13 = cos ψ k sin ρ k sin φ k + sin ψ k cos φ k (83) T s1 = (84) T s = sin ψ k sin ρ k cos φ k cos ψ k sin φ k (85) T s3 = sin ψ k sin ρ k sin φ k cos ψ k cos φ k (86) T s31 = (87) T s3 = cos ψ k cos φ k (88) T s33 = cos ρ k sin φ k (89) REFERECES [1] Y Bar-Shalom, X-R Li, and T Kirubarajan, Estimation with Applications to Tracking and avigation: Theory, Algorithms and Software J Wiley and Sons, 1 [] Y Bar-Shalom, P K Willett, and X Tian, Tracking and Data Fusion YBS Publishing, 11 [3] S V Bordonaro, P Willett, and Y Bar-Shalom, Tracking with Converted Position and Doppler Measurements, in Proc SPIE Conf Signal and Data Processing of Small s, #8137-1, San Diego, CA, Aug 11 [4] S V Bordonaro, P Willett, and Y Bar-Shalom, Consistent Linear Tracker with Position and Range Rate Measurements, in Proc Asilomar Conf, Asilomar, CA, ov 1 [5] S V Bordonaro, P Willett, and Y Bar-Shalom, Unbiased Tracking with Converted Measurements, in Proc 1 IEEE Radar Conference, Atlanta, GA, May 1 [6] S V Bordonaro, P Willett, and Y Bar-Shalom, Unbiased Tracking with Converted Measurements, in Proc 51st IEEE Conf on Decision and Control, Maui, HI, Dec 1 [7] D F Crouse, R Osborne, III, K Pattipati, P Willett, and Y Bar-Shalom, D Location Estimation of Angle-Only Arrays Using s of Opportunity, in Proc 13th International Conference on Information Fusion, Edinburgh, Scotland, July 1 [8] B D Kragel, S Danford, S M Herman, and A B Poore, Bias Estimation Using s of Opportunity, Proc SPIE Conf on Signal and Data Processing of Small s, #6699, Aug 7 [9] B D Kragel, S Danford, S M Herman, and A B Poore, Joint MAP Bias Estimation and Data Association: Algorithms, Proc SPIE Conf on Signal and Data Processing of Small s, #6699-1E, Aug 7 [1] B D Kragel, S Danford, and A B Poore, Concurrent MAP Data Association and Absolute Bias Estimation with an Arbitrary umber of s, Proc SPIE Conf on Signal and Data Processing of Small s, #6969-5, May 8 [11] X Lin, F Y Bar-Shalom, and T Kirubarajan, Exact Multisensor Dynamic Bias Estimation with Local Tracks, IEEE Tans on Aerospace and Electronic Systems, vol 4, no, pp , April 4 [1] R W Osborne, III, and Y Bar-Shalom, Statistical Efficiency of Composite Position Measurements from Passive s, in Proc SPIE Conf on Signal Proc, Fusion, and Recognition, #85-7, Orlando, FL, April 11 [13] R W Osborne, III, and Y Bar-Shalom, Statistical Efficiency of Composite Position Measurements from Passive s, IEEE Tans on Aerospace and Electronic Systems, To appear 13