An Improved Single-Point Track Initiation Using GMTI ...

1

An Improved Single-Point Track Initiation Using GMTI Measurements Mahendra Mallicka , Yaakov Bar-Shalom, b,1 T. Kirubarajanc, and Mark Morelande d a Independent Consultant, Smith River, CA, USA b ECE Dept., University of Connecticut, Storrs, CT, USA c ECE Dept., McMaster University, Hamilton, ON, Canada d ECE Dept., RMTI University, Melbourne, VIC, Australia [email protected], [email protected], [email protected], [email protected]

Abstract—A ground moving target indicator (GMTI) radar measures range, azimuth, and Doppler or range-rate of a target. The range-rate measurement, which is the key measurement of a GMTI radar, is used to differentiate a ground moving target from clutter when the radial velocity of the target exceeds the minimum detectable velocity (MDV). The range and azimuth measurements provide an accurate estimate of the target position. A number of track initiation algorithms using GMTI measurements are known at present. All track initiation algorithms set the initial velocity to zero and a large prior covariance for velocity is used based on the maximum target speed. The range-rate measurement is then used to update the target velocity or state. The rangerate measurement contains only the component of the target velocity along the radar line-of-sight (RLOS) and no component of velocity in the perpendicular direction is measured. As a result, existing track initiation algorithms produce a bias in the velocity estimate. This paper first corrects two errors in the single-point track initialization part of the paper [30] using the range-rate measurement to obtain the initial velocity estimate and associated covariance. Secondly, we present an improved track initiation algorithm based on a heading-parametrized multiple model (HPMM) method that uses the sign of the range-rate measurement, prior knowledge of target maximum speed, and MDV. The proposed algorithm reduces the bias in the initial velocity estimate and the superior performance of the algorithm is demonstrated using Monte Carlo simulations. Index Terms—Ground moving target indicator (GMTI) radar, minimum detectable velocity (MDV), track initiation using GMTI measurements, heading-parametrized multiple model (HPMM) method, GMTI filtering.

I. I NTRODUCTION The ground moving target indicator (GMTI) radar has been widely used for surveillance and tracking of ground moving targets for more than a decade [5]–[8], [14]–[16], [20], [21], [23], [30]. The minimum detectable velocity (MDV) is an important design parameter for a GMTI radar [8], [25]. If the magnitude of the target radial velocity is less than the MDV, then the radar cannot differentiate the target return from endo-clutter ground returns and a target will not be detected. Therefore, a GMTI radar can detect a target only when the radial velocity is greater than the MDV. The measurements of a GMTI radar are slant range, azimuth, and range-rate (or Doppler) [6], [14], [15], [20], [28], [30]. The number of published papers on GMTI tracking has significantly increased 1 Y.

Bar-Shalom was supported by ARO grant W911-NF-10-1-0369.

since the introduction of GMTI radar; however, many papers do not model the range-rate in the measurement model and only consider range and azimuth. Only a few papers [6], [12], [14], [15], [20], [21], [23], [30] include range-rate in the GMTI measurement model. The track initiation algorithm is a simple yet important algorithm in filtering and multitarget tracking. If a track is initiated in a statistically consistent manner, then it can prevent filter divergence, improve measurement-to-track association, reduce the number of false tracks, and help track management in a multitarget tracking scenario [3], [4]. Two types of track initiation methods, single-point (SP) and two-point difference (TPD) track initiation [2], [17], [18], [22], [26, Chapter 6], [30], exist. The SP method has advantages over the TPD in terms of estimation accuracy [17] and ease of use in multitarget tracking. To the best of our knowledge, only a few papers [12], [15], [30] address SP track initiation using range, azimuth, and range-rate measurements for the GMTI filtering problem. The range-rate is the projection of the velocity of the target relative to the sensor along the radar line-of-sight (RLOS) [30]; the present paper uses this model. In practice, the contribution of the sensor velocity is removed from range-rate to provide the measurement of radial velocity [28] as used in [6], [12], [14]– [16]. The work in [12] uses the slant range and azimuth measurements and associated variances to obtain the 2D position estimate and corresponding covariance using the method described in [3, Section 1.7.2]. The position estimate has a negligible bias. The 2D velocity estimate is set to zero and a large velocity covariance is used based on the maximum possible target speed v max . Then the range-rate measurement, treated as a function of target position and velocity, is used to update the full state using the linear minimum mean square error (LMMSE) estimator. The SP track initiation method in [15] is similar to that in [12], except that the algorithm in [11] is used to obtain unbiased Cartesian position measurements and the associated covariance. The paper [30] uses an SP track initiation algorithm that is similar to that in [15]. In this method, the velocity estimate is updated using the range-rate measurement by the LMMSE estimator, and the cross-covariance between position and velocity components is zero. Methods in [12], [15] produce a small cross-covariance between position and velocity components.

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We noticed two errors in the SP track initialization part of [30] using the range-rate measurement to obtain the initial velocity estimate and associated covariance. Section III presents corrections to this algorithm. An improved algorithm for the initial covariance is also presented, which computes the crosscovariance between the position and velocity components of the initial covariance. The track initiation algorithms in [12], [15], [30] contain a bias in the initial velocity estimate, since this is set to zero and then the range-rate measurement is used to update the target velocity or state. We present an improved track initiation algorithm in Section IV based on a heading-parametrized multiple model (HPMM) method, which uses the sign of the range-rate measurement, the prior knowledge of target maximum speed, and the MDV. The new algorithm reduces the bias in the velocity estimate and the superior performance of the algorithm is demonstrated using Monte Carlo simulations. For clarity, we use an italics for scalar quantities and boldface for vectors and matrices in our notation convention. A lower or upper case Roman letter represents a name (e.g. “p” for “platform,” “RMS” for “root mean square,” etc.). We use “:=” to define a quantity and A  denotes the transpose of the vector or matrix A. The n−dimensional identity matrix and m × n null matrix are denoted by I n and 0m×n , respectively. The outline of the paper is as follows. We explain the GMTI filtering problem in Sections II. Sections III and IV present corrections to the SP track initiation algorithm [30] and the HPMM based track initiation algorithm, respectively. Based on the suggestion of a reviewer [20], we present the details of the GMTI track initiation and filtering algorithms in Section V for comparative evaluation. This approach uses the affine coordinate transformation in [27] and we call this approach the Affine Coordinate Transformation Method. Finally, Sections VII and VIII present numerical results using four GMTI tracking examples and conclusions. Appendix A presents some useful results involving the Gaussian distribution and Appendix B presents detailed equations for the converted unbiased Cartesian measurements using the slant range and azimuth measurements. II. T HE GMTI F ILTERING P ROBLEM In this Section, we define the states of the target and GMTI platform and present the dynamic and measurement models for GMTI filtering. In a stand-off GMTI groundtarget tracking scenario, the GMTI platform travels at a certain altitude to collect measurements on ground-moving targets. The trajectories of the target and GMTI platform in a typical scenario are shown in Fig. 1. In certain cases, the RLOS can be obscured by the terrain. For simplicity, we use a flat-Earth approximation and assume that a target moves with the nearly constant velocity (NCV) motion [2] in the XY plane. The nearly constant acceleration (NCA) and coordinated turn (CT) models [2], [16] are also used for ground target tracking for maneuvering targets. We consider two types of motions for the GMTI platform. In order to be consistent with the scenario in [30], we assume that the platform moves with constant velocity (CV) in the

XY plane. In another scenario, the GMTI platform moves at a constant altitude above the XY plane with CV. We assume that the position and velocity of the GMTI platform are precisely known and terrain obscuration is absent. Let x(k) denote the target state at time t k , expressed in a tracker coordinate frame (T frame) x(k) = [ξ(k)

˙ η(k) ξ(k)

 , η(k)] ˙

(1)

where ξ and η are the Cartesian X and Y position coordinates and ξ˙ and η˙ are the corresponding velocity components. When the platform moves in the XY plane, the platform state x p (k) at time tk in the T frame is expressed by xp (k) = [ξp (k) ηp (k)

ξ˙p (k) η˙ p (k)] ,

(2)

where ξp and ηp are the Cartesian X and Y position coordinates and ξ˙p and η˙ p are the corresponding velocity components of the platform. Let p(k) and p p (k) denote the target and platform position vectors, respectively, at time t k , p(k) = [ξ(k) η(k)] ,

(3) 

pp (k) = [ξp (k) ηp (k)] .

(4)

Y

Target nominal trajectory

X 600 km/h 100 km

Sensor trajectory

Fig. 1.

Target and GMTI sensor trajectories.

The dynamic models for the target and sensor are described, respectively, by [2] x(k) = F(k − 1)x(k − 1) + w(k − 1),

(5)

xp (k) = F(k − 1)xp (k − 1),

(6)

where F(k−1) and w(k−1) are the state transition matrix and zero-mean white Gaussian process noise with covariance matrix Q(k − 1), respectively, for the time interval [t k−1 , tk ] [2]. The state transition matrix and process noise with covariance matrix (using the discretized continuous-time NCV model) are given by [2]   I2 Δk I2 , Δk := tk − tk−1 , (7) F(k − 1) = 02×2 I2   I2 Δ3k /3 I2 Δ2k /2 Q(k − 1) = q , (8) I2 Δ2k /2 I2 Δk

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where q is the power spectral density of the continuous-time process noise [2]. The details of the GMTI measurement model are presented in [15]. The azimuth angle in [15] follows the standard convention (e.g., [28]) and is measured from the local North (Y axis) in the clockwise direction. To be consistent with the convention in [30], the azimuth angle is measured from the local East (X axis) in the counter-clockwise direction. Secondly, the radial velocity is used as the measurement model in [15], [28], whereas [30] and the current paper use the rangerate as the measurement model. The range vector at time t k is given by r(k) = p(k) − pp (k).

(9)

Let u be the unit vector along the range vector r or radar line-of-sight (RLOS) u = r/r = r/r.

(10)

Let v(k) and vp (k) be the target and platform velocities at time tk , respectively. The measurement models for range, azimuth, and range-rate are described by rz (k) = r(x(k), xp (k)) + nr (k), θz (k) = θ(x(k), xp (k)) + nθ (k),

(11) (12)

r˙z (k) = r(x(k), ˙ xp (k)) + nr˙ (k),

(13)

where the measurement noises n r (k), nθ (k), and nr˙ (k) are assumed to be mutually uncorrelated zero-mean Gaussian with variances, σr2 , σθ2 , and σr2˙ , respectively. Note that we use nr (k), nθ (k), nr˙ (k) as measurement noises while wr (k), wθ (k), wr˙ (k) are used as measurement noises in [30]. We assume that the measurement errors are uncorrelated at all times and are uncorrelated with the process noise at all times. The measurement functions for range, azimuth, and range-rate are given by r(k) = [(p(k) − pp (k)) (p(k) − pp (k))]1/2 , −1

θ(k) = tan

(η(k) − ηp (k), ξ(k) − ξp (k)),

r(k) ˙ = u(k) (v(k) − vp (k)),

(14) (15) (16)

where the azimuth in (15) lies in [0, 2π). At each measurement time t k , the range and azimuth measurements are converted to unbiased Cartesian position measurements (ξz (k), ηz (k)) [11], [30] with associated position covariance R c (k). Let z(k) and R(k) denote the position measurement with range-rate measurement and associated covariance at time t k . Then z(k) = [ξz (k) ηz (k) r˙z (k)] ,   Rc (k) 02×1 R(k) = . 01×2 σr2˙

(17) (18)

The measurement model is nonlinear due to the range-rate. The nonlinear measurement model can be written as z(k) = h(x(k), xp (k)) + n(k),

(19)

where n(k) is a zero-mean measurement noise with covariance R(k). If the GMTI platform moves in 3D with a CV motion,

then we can define three dimensional position vector and velocity of the GMTI platform at time t k in T frame by ζp (k)] ,

(20)

η˙ p (k) ζ˙p (k)] .

(21)

pp (k) = [ξp (k) ηp (k) vp (k) = [ξ˙p (k)

The dynamic model in (6) holds with the state transition matrix   I3 Δk I3 F(k − 1) = . (22) 03×3 I3 The definitions for the range vector r in (9) and unit vector u along the RLOS in (10) still hold if we use p(k) = [ξ(k) η(k) ˙ v(k) = [ξ(k)

0] ,

(23)



η(k) ˙ 0] .

(24)

We use a simpler form of F(k − 1) in our simulations. III. C ORRECTIONS TO THE SP T RACK I NITIATION A LGORITHM [30] The range-rate measurement model at time t 0 can be written as [30, (47) and (48)]. r˙z (0) = Hv (v(0) − vp (0)) + nr˙ (0), Hv = [cos θ(0)



sin θ(0)] = u (0),

(25) (26)

where θ(0) is the azimuth angle at time t 0 and v(0) ∼ ¯ v ) is the prior velocity of the target with N (0, P ¯ v = diag(σs2 , σs2 ). P

(27)

Here σs represents the prior speed standard deviation in each direction. The velocity estimate at time t 0 based on the rangerate measurement r˙ z (0) is given by [30, (49)] as ¯ v H + σ 2 )−1 r˙z (0) + vp (0) ¯ v H (Hv P ˆ (0|0) = P v v v   r˙  σs2 cos θz (0) ξ˙p (0) , (0) + r ˙ = 2 z η˙ p (0) σs + σr2˙ sin θz (0)

(28)

where vp (0) is the platform velocity at t 0 . The correct expresˆ (0|0) is sion for v ¯ v H + σ 2 )−1 (r˙z (0) + Hv vp (0)) ¯ v H (Hv P ˆ (0|0) = P v v v r˙  2 σs cos θz (0) (r˙z (0) + Hv vp (0)). = 2 σs + σr2˙ sin θz (0)

(29)

Next, we provide the justification for (29). The velocity update equation using the range-rate measurement is [2] ˆ (0|0) = Wv (0)νr˙ (0), v

(30)

where Wv (0) and νr˙ (0) are the linear minimum mean square error (LMMSE) or Kalman gain and innovation for the rangerate at time t0 , respectively. The range-rate innovation is given by [2] (31) νr˙ (0) = r˙z (0) − rˆ˙z (0), where rˆ˙z (0) is the predicted range-rate at time t 0 . The Kalman gain is given by [2] ¯ v H /Sr˙ , Wv (0) = P v

(32)

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Using the expression for ν r˙ (0) from (35) in (30), we get

where the innovation variance for range-rate is ¯ v H + Rr˙ = σ 2 + σ 2 . Sr˙ = Hv P v s r˙

Since the prior mean of the velocity at time t 0 is zero, the predicted range-rate at time t 0 is given by rˆ˙z (0) = −Hv vp (0).

(34)

Using (34) in (31) gives νr˙ (0) = r˙z (0) + Hv vp (0).

Pv =

σs2





(36)

Thus, (32) and (36) yield (29). The first line on the right hand side of [30, (50)] is correct. However, the final expression for P v in the second line of [30, (50)] contains typographical errors. The final expression for Pv in [30, (50)] is

(35)

−(σs2 /Sr˙ ) cos θz (0) sin θz (0) (1 − σs2 /Sr˙ ) cos θz (0) −(σs2 /Sr˙ ) cos θz (0) sin θz (0) (1 − σs2 /Sr˙ ) sin θz (0)

The (1,2) and (2,1) elements of P v in (37) are correct. How-

Pv = σs2

ˆ (0|0) = Wv (0)(r˙z (0) + Hv vp (0)). v

(33)



¯ (0) = [ξz (0) ηz (0) 0 0] , x   Rc (0) 02×2 ¯ P(0) = , 02×2 σs2 I2

(39) (40)

where Rc (0) is the 2 × 2 covariance matrix for unbiased

.

(37)

ever, the diagonal elements of P v in (37) contain typographical errors. The correct expressions for P v is

1 − (σs2 /Sr˙ ) cos2 θz (0) −(σs2 /Sr˙ ) cos θz (0) sin θz (0) 2 1 − (σs2 /Sr˙ ) sin2 θz (0) −(σs /Sr˙ ) cos θz (0) sin θz (0)

ˆ (0|0) = [ξz (0) ηz (0)] denote the initial position Let p estimate using the first converted unbiased Cartesian measurements. The associated position covariance is R c (0). Let ˆ (0|0) and v ˆ (0|0) Method 1 refer to the algorithm which uses p ˆ (0|0), where for the position and velocity components of x ˆ (0|0) is given by (28) and the associated velocity covariance v is given by (37). Method 1 represents the incorrect algorithm in [30] and ignores the cross-covariance between the position and velocity components of the state estimate. We use Method 2 to represent the algorithm in which the error in Method 1 [30] is corrected. The cross-covariance between the position and velocity components is still zero in Method 2. When the platform velocity is high, as in typical GMTI tracking scenarios, the cross-covariance in the initial covariance is not negligible. A better approach is to calculate the Jacobian of the range-rate measurement function with respect to position and velocity and then update the state and covariance obtained from converted Cartesian measurements [11], [30]. The Jacobian H r˙ (0) of the range-rate measurement function with respect to position and velocity is presented in [30, (21)–(22)] and below in (47). The detailed steps of the improved covariance calculation are presented next. We refer to this method as Method 3. ¯ ¯ (0) and P(0) Let x denote the state estimate and covariance based on converted Cartesian position measurements [11], [30] and prior speed variance, respectively. Then,



 .

(38)

converted Cartesian position measurements [11], [30]. Let ˆ (0|0) and P(0|0) denote the state estimate and covariance x based on the range, azimuth, and range-rate measurements at time t0 . Then, ˆ (0|0) = x ¯ (0) + Wr˙ (0)(r˙z (0) − rˆ˙z (0)), x

(41)

¯ P(0|0) = P(0) − Sr˙ (0)Wr˙ (0)Wr˙ (0) ,

(42)

where the Kalman gain and innovation variance for range-rate are given by  ¯ Wr˙ (0) = P(0)H r˙ (0) /Sr˙ (0),

(43)

¯ r˙ (0) + σ 2 . Sr˙ (0) = Hr˙ (0)PH r˙

(44)

It can be shown that 1 ∂ r˙ = (v − vp ) (I − uu ), ∂p r

(45)

∂ r˙ (46) = u . ∂v The above two equations are valid in 2D and 3D at any time t. Then ∂ r˙ 1 Hr˙ = (47) = [ (v − vp ) (I − uu ) u ]. ∂x r Simplification of (47) yields [30, (21)] at time t k Hr˙ (k) =[sin θ(k)l(k)/r(k), − cos θ(k)l(k)/r(k), cos θ(k), sin θ(k)],

(48)

where ˙ ˙ η˙p (k)) cos θ(k)]/r(k). l(k) := [(ξ(k)− ξ˙p (k)) sin θ(k)−(η(k)− (49)

5

We can compute H r˙ (0), r(0), sin θ(0), cos θ(0), and l(0) using ¯ (0) and sensor state at time t0 . It is more efficient to compute x Hr˙ (0) using (47) at time t 0 , since this is valid in 2D and 3D. IV. H EADING -PARAMETRIZED M ULTIPLE M ODEL M ETHOD The radial velocity of the target [19], in general, is given by the inner product of the target velocity v and the unit vector u along the RLOS vr = v · u = v cos φ,

0  φ < 2π,

(50)

where φ is the angle measured from the RLOS to the target velocity in the counter-clockwise direction, as shown in Fig. 2. We note that the radial velocity is the same for φ and 2π −φ (or −φ) (see Fig. 2). The velocity v 1 in Fig. 2 has the same magnitude as v (v1  = v) but makes an angle 2π − φ with the RLOS. Therefore, v and v 1 have the same radial velocity. The radial velocity is positive or negative when the target velocity lies in the forward-looking or backward-looking direction, respectively, relative to the RLOS. If the magnitude of the radial velocity is less than the MDV, then a target will not be detected [25]. Thus, for a target to be detected, the radial velocity must satisfy |v r | ≥ MDV.

I0

Target

v

I

r

ru

v v u

Sensor

2S

Fig. 2.

I

v1

I0

v1

,

v1 u

v cos I ,

1

cos (MDV / v ).

φ0 = cos−1 (MDV/v), for 0  φ0 < π/2.

(53)

We note that as the speed increases, the angle φ 0 increases. If the MDV is not used, then φ 0 = π/2. For typical values of the MDV, the angle φ0 is close to π/2. The Gaussian distribution, which is commonly used in filtering, has an infinite support. The interval [0, v max ] for target speed is finite. In order to use a Gaussian distribution for target speed in an approximate manner, we assume that the speed has a Gaussian distribution with mean s¯ = v max /2 and the interval [0, v max ] contains a probability of 0.99935. √ This leads to the speed standard deviation σ s = vmax / 12. We use this interpretation in all cases in the paper when we use a Gaussian distribution for a finite interval. We divide the left and right sides of the RLOS into a number of heading intervals to be used in the HPMM method. Thus, there are two models corresponding to the left and right sides of the RLOS. Subsection IV-A describes the specification of heading intervals and calculation of the mean and standard deviation of the speed and heading in each interval. Computation of the mean and covariance of the Cartesian velocity in an interval is described in Subsection IV-B. Subsection IV-C presents computation of the initial state and covariance for each model. The initial velocity estimates for these two models are different. We run two extended Kalman filters (EKFs) concurrently for these two models and compute the model probability for each model. Subsections IV-D and IV-E describe the details of the EKF and computation of model probabilities, respectively. A. Specification of Heading Intervals

Target radial velocity and MDV.

Given the first range-rate measurement r˙ z (0) at time t0 , an estimate of the radial velocity is calculated by ˆ  (0)vp (0), vˆr (0) = r˙z (0) + u

Additionally, we incorporate the MDV in our model. Suppose the target velocity is in the forward-looking direction as shown in Fig. 2. For a given target speed v, let φ 0 be the maximum value of φ in the interval [0, π/2) for the target to be detected. Then,

(51)

1 ˆ (0) = [ξz (0) − ξp (0) ηz (0) − ηp (0)] . u (52) r The sign of vˆr˙ (0) provides useful information regarding the direction of target velocity relative to the RLOS. When the prior velocity is set to zero in existing GMTI track initiation algorithms before processing the range-rate measurement, this information is discarded. We use the sign of vˆr˙ (0) in selecting the direction of target velocity in the forward-looking or backward-looking direction. If the sign of vˆ r˙ (0) is positive or negative, then we infer that the target velocity is in the forward-looking or backward-looking direction, respectively, relative to the RLOS. Suppose the sign of vˆr˙ (0) is positive as shown in Fig. 2. Then we infer that the true velocity is either to the left or right of the RLOS (see Fig. 2) in the forward-looking direction.

We have assumed the platform motion to be deterministic and precisely known. In real-world scenarios accurate estimates of platform position and velocity are obtained from the on-board inertial navigation system (INS) and the global positioning system (GPS). Thus, an accurate estimate of the radial velocity using (51) can be obtained. Since we do not know the speed of the target, we shall use the maximum target speed vmax to compute φ0 . If the radial velocity is negative, a similar method can be used to compute φ 0 . Let α denote the azimuth angle (heading) of the initial target velocity. For the target to be detected, α must satisfy θ(0) − φ0 < α < θ(0) + φ0 .

(54)

The headings θ(0) − α and θ(0) + α, where 0 < α ≤ φ 0 yield the same range-rate. However, they lie on opposite sides of the RLOS and represent different velocities. We use a HPMM method, similar to that used for range-parametrized multiple model approach in [1], [9], [24] for bearing-only filtering. We divide this interval into n α uniform intervals and choose the mid-point of each interval as the mean for a candidate target

6

heading. The ith mean heading is given by α ¯ i = θ(0) − φ0 + (2i − 1)Δα/2,

i = 1, 2, ..., nα ,

(55)

where α = 2φ0 /nα . The variance for the heading in each interval is σα2 = (α)2 /12. We choose nα to be an even number and partition the intervals to two sets of intervals, each containing n α /2 intervals. The first set contains the first nα /2 intervals and the second set contains the last nα /2 intervals. The mean speed and heading for the ith interval are (¯ s, α ¯ i ). These two sets of intervals lie on opposite sides of the RLOS. Fig. 3 shows the sensor-target geometry and the construction of six heading intervals with three heading intervals on opposite sides of the RLOS. As an example, the mean heading α ¯ 2 for the second interval is shown in Fig. 3.

6th

5th 4th

T (0) 3rd

Sensor

2S

2nd

I0

D2

I0

B. Mean and Covariance of Cartesian Velocity in an Interval We follow the approach in [18] to compute the mean and covariance of the Cartesian components of velocity in each interval. Let a = [s α] denote the vector containing the polar ¯ = [¯ coordinates of velocity with mean a s α ¯ ] . Assuming Gaussian distribution for speed and heading, one obtains Pa =

diag(σs2 , σα2 ).

(56)

Let v = [ξ˙ η] ˙  denote the vector containing the Cartesian ¯˙  . Then, ¯ = [ξ¯˙ η] coordinates of velocity with mean v v = [ξ˙ η] ˙  = c(a), c(a) = [s cos α

(62) (63)

(65)

ˆ where νr,i ˙ (0) and r˙z,i (0) are the range-rate innovation and predicted range-rate for the ith model,

Fig. 3. Specification of heading intervals and mean heading for an interval. In this example, nα = 6.

where

¯ i (0) = [ξz (0) ηz (0) v ¯i ] , x   Rc (0) 02×2 P¯i (0) = . 02×2 Pvi

 ¯ i (0) − Sr,i Pi (0|0) = P ˙ (0)Wr,i ˙ (0)Wr,i ˙ (0) ,

1st

¯, Pa ), a ∼ N (a; a

C. Computation of Initial State Estimate and Covariance for Each Model ¯ i (0) and P¯i (0) denote the state estimate and covariance Let x based on converted Cartesian position measurements [11], [30] and prior velocity and associated covariance, for the ith interval, respectively. Then,

ˆ i (0|0) and Pi (0|0) denote the state estimate and covariLet x ance based on the range, azimuth, and range-rate measureˆ i (0|0) ments at time t0 for the ith model, respectively. Then x and Pi (0|0) are determined using equations similar to (41)– (49): ˆ i (0|0) = x ¯ i (0) + Wr,i x (64) ˙ (0)νr,i ˙ (0),

Target

I0

For completeness, Appendix A presents the derivation of ˙ based on the expressions for E{ ξ˙2 }, E{η˙ 2 }, and E{ξ˙η} ¯ i and Pvi denote the method in Appendix 1C in [18]. Let v mean Cartesian velocity and associated covariance in the ith ¯ i and Pvi using (60) and (61). interval. We can calculate v

s sin α] .

(57) (58)

The distribution of c(a) is given by [18] ¯ , Pv ). c(a) ∼ N (c(a); v Using the results from Appendix 1C in [18], we get ¯˙  = s¯ exp(−σα2 /2)[cos α ¯ = [ξ¯˙ η] v ¯ sin α ¯ ] . The covariance P v can be written as [18]   ¯ ¯ E{ξ˙2 } − ξ˙2 E{ξ˙η} ˙ − ξ˙η¯˙ . Pv = ¯ E{ξ˙η} ˙ − ξ˙η¯˙ E{η˙ 2 } − η¯˙2

ˆ νr,i ˙ (0) = r˙z (0) − r˙z,i (0),

(66)

vi − x˙ p (0)), rˆ˙z,i (0) = Hv¯ i (¯

(67)

¯ ˙ (0) /Sr,i Wr,i ˙ (0) = Pi Hr,i ˙ (0),

(68)

¯ ˙ (0) + σr2˙ , Sr,i ˙ (0) = Hr,i ˙ (0)Pi Hr,i

(69)

Hr,i ˙ (0) = [sin θ(0)li (0)/r(0)

− cos θ(0)li (0)/r(0) cos θ(0)

sin θ(0)],

(70)

¯ li (0) = [(ξ˙i (0)−ξ˙p (0)) sin θ(0)−(η¯˙i (0)−η˙p (0)) cos θ(0)]/r(0). (71) ¯i in (67) is responsible for reducing We note that the term v the bias in the velocity estimate. Let p(r˙z (0)|i) be the likelihood that the velocity lies in the ith interval given the range-rate measurement r˙ z (0). Assuming a Gaussian distribution for the range-rate innovation, one has p(r˙z (0)|i) = N (νr,i ˙ (0); 0, Sr,i ˙ (0)),

(72)

and the normalized weight w i for the ith model is nα /2

(59) wi = p(r˙z (0)|i)/



p(r˙z (0)|j).

(73)

j=1

(60)

The initial state estimate and associated covariance for each set is calculated as nα /2

(61)

ˆ (0|0) = x

 j=1

ˆ j (0|0), wj x

(74)

7

P(0|0) = nα /2



ˆ (0|0))(ˆ ˆ (0|0)) ]. wj [Pj (0|0) + (ˆ xj (0|0) − x xj (0|0) − x

j=1

(75) The initial state and covariance associated with the first and second sets of intervals are calculated using (74) and (75), respectively. These two estimators for the first and second sets of intervals are referred to as Methods 4a and 4b, respectively. D. Extended Kalman Filtering Using the initial state estimate and associated covariance in (74) and (75), respectively, we perform filtering using an EKF [2], [15] for Methods 4a and 4b. The EKF uses the measurement z(k) and associated covariance R(k) in (17) and (18), respectively. The Jacobian matrix representing the derivative of the measurement function h with respect to the state vector x is given by ⎤ ⎡ 1 0 0 0 1 0 0 ⎦. (76) H=⎣ 0 H H H Hr,ξ ˙ r,η ˙ r, ˙ η˙ r, ˙ ξ˙ E. Model Probabilities of Methods 4a and 4b It is not possible to select Method 4a or Method 4b based on the first measurement, since both methods give the same rangerate estimate. We process additional measurements to resolve the ambiguity and calculate the model probabilities of these two methods. Let Z(k) denote the collection of measurements up to time tk , Z(k) := {z(0), z(1), ..., z(k)}.

(77)

Let P (1|Z(k)) and P (2|Z(k)) denote the probabilities of the Methods 4a and 4b, respectively, after processing Z(k). Using the Bayes’ rule, we have P (j|Z(k)) ∝ p(z(k)|j, Z(k − 1))P (j|Z(k − 1)),

j = 1, 2, (78) where p(z(k)|j, Z(k − 1)) is the likelihood of the mode j and is given by p(z(k)|j, Z(k − 1)) = N (ν j (k); 0, Sj (k)),

j = 1, 2, (79)

and where ν j (k) and Sj (k) are the innovation and innovation covariance, respectively, for the jth mode. We set the initial probabilities to half after processing z(0), P (1|z(0)) = P (2|z(0)) = 0.5.

RT m =



(80)

V. GMTI T RACK I NITIATION AND F ILTERING USING THE A FFINE C OORDINATE T RANSFORMATION M ETHOD Based on the suggestion of a reviewer, we present the algorithms for track initiation and filtering using the GMTI measurement model described in Subsection 6.3.2.2 of [20]. This approach uses the affine coordinate transformation in [27]. We refer to this method as Method 5. The authors assume that the range, azimuth, possibly elevation, and range-rate measurements are available from a GMTI radar. We emphasize that measurements for a GMTI radar are slant range (or range), azimuth, and range-rate (or alternatively radial velocity) and the elevation is not a GMTI radar measurement [13], [28]. Measurement standard deviations for slant range, cross-range, and radial velocity are specified in [28]. Estimated geodetic coordinates of the target location on the surface of the Earth are calculated using the slant range, azimuth, terrain elevation data, and geoid undulation by a GMTI geolocation algorithm as described in [13]. Thus, the accuracy of the estimated geodetic coordinates depend on the accuracy of the terrain elevation data and accuracy of the geolocation algorithm. Therefore, it is preferable to use the original GMTI measurements. Then one could use accurate elevation data and geolocation algorithm for track initiation and track filter. In order to be consistent with the approach used in [30], we use the 2D scenario in this paper, in which the target and the GMTI sensor move in the XY -plane as shown in Fig. 1. In [12], [15] a flat-Earth approximation is used and the target moves in the XY plane with the NCV motion, whereas the GMTI sensor moves in a plane parallel to the XY plane at a fixed altitude with constant velocity. Terrain elevation and geoid undulation data are used in a track-oriented multiple hypothesis tracking (MHT) algorithm in [14] to process outof-sequence measurements (OOSMs) from two GMTI sensors in realistic multitarget ground tracking scenarios. In this paper, the elevation angle of the range vector is zero for scenarios 1-3. Therefore, we can use the approach in [20] using measured range and azimuth. The algorithm in [20] first converts the measured range and azimuth to Cartesian coordinates of the range vector in the tracker coordinate frame (T frame) using the standard conversion described in Section 1.7.2 of [3] to obtain rT m (k) = [rz (k) cos θz (k)

rz (k) sin θz (k)] .

(81)

The converted Cartesian measurements in (81) are slightly biased with a maximum bias of 0.05 m for the current scenario T [3]. The associated covariance R T m (k) for rm (k) is given in [3] as

σr2 cos2 θz (k) + rz2 (k)σθ2 sin2 θz (k) (σr2 − rz2 (k)σθ2 ) sin θz (k) cos θz (k)

(σr2 − rz2 (k)σθ2 ) sin θz (k) cos θz (k) σr2 sin2 θz (k) + rz2 (k)σθ2 cos2 θz (k)

 .

(82)

We define a sensor coordinate frame (S frame) whose origin is at the sensor location and whose X axis is along the range

8

vector rT m (k) and the Y axis is orthogonal to the X axis along the direction of increasing azimuth angle. Thus, the X, Y axes of the S frame are rotated by an angle θ z (k) about the Z axis of the T frame. The 2D rotational transformation from the T frame to the S frame is an elementary rotation about the Z axis of the T frame [29] and is given by   cos θz (k) sin θz (k) T3 (θz (k)) = . (83) − sin θz (k) cos θz (k)

We can also use (92) if A is a cross-covariance matrix between a position vector and velocity. Since T 3 (θz (k)) is an orthogonal matrix, the transformations of a S and AS to corresponding quantities in the T frame are obtained by the following inverse transformations:

We note that the S frame is not a fixed frame relative to the T frame. The orientation of the S frame changes with time due to motion of the target and sensor. Thus, the measured range vector in the S frame is given by

Let xS (k) and xSp (k) denote the target and platform states, respectively, in the S frame at time t k . Then xS (k) and xSp (k) are given by

rSm (k) = T3 (θz (k))rT m (k) = [rz (k)

0] ,

(84)

and as expected the components of r Sm (k) along the X and Y axes of the S frame are r z (k) and zero, respectively. The covariance matrix for r Sm (k) is given by  RSm (k) = T3 (θz (k))RT m (k)T3 (θz (k)).

(85)

(86)

Remark 1: It is mentioned on page 214 in [20] that the covariance R in (6.31) is given by the independent measurement variances in spherical coordinate; R should be the covariance of converted Cartesian coordinates. For the 2D scenario, R in (6.31) of [20] is R T m (k) in (82). Remark 2: In a multisensor–multitarget ground target tracking system, a topographic coordinate frame (TCF) [13], [14] is defined, whose origin has the geodetic longitude λ 0 , geodetic latitude φ0 , and geodetic height h 0 . The X, Y, and Z axes are chosen along the local east, north, and upward directions, respectively. This TCF represents the T frame in which the target state, dynamic model, and measurement models for all targets and sensors are specified. Let zS (k) and RS (k) denote the range vector measurement in the S frame with the range-rate measurement and associated covariance at time t k . Then, zS (k) = [(rSm (k)) r˙z (k)] ,  S  Rm (k) 02×1 S . R (k) = 01×2 σr2˙

(87) (88)

Using (84) in (87) and (86) in (88), we get S

z (k) = [rz (k) 0 S

R (k) =



r˙z (k)] ,

diag(σr2 , rz2 (k)σθ2 , σr2˙ ).

A = T3 (θz (k))AS T3 (θz (k)).

(94)

xS (k) = [(pS (k))

(vS (k)) ] ,

(95)

xSp (k) = [(pSp (k))

(vpS (k)) ] ,

(96)

zS (k) = HS (xS (k) − xSp (k)) + nS (k),

(97)

where nS (k) is a zero-mean measurement noise with covariance RS (k) and (98) HS = [I3 03×1 ]. A. Filter Initialization ¯ ¯ (0) and P(0) Let x denote the state estimate and covariance based on the converted Cartesian position measurements using standard conversion [3] and prior speed variance, respectively. Then, T T ¯ (0) = [ξp (0) + rm,x x (0) ηp (0) + rm,y (0) 0  T  Rm (0) 02×2 ¯ P(0) = , 02×2 σs2 I2

0] ,

(99) (100)

where RT m (0) is the 2 × 2 covariance matrix for converted Cartesian position measurements in (82). The initial state estimate is updated in the S frame using the range-rate mea¯ S (0), which can be computed surement. Therefore, we need x ¯ S (0) is given using (99) and (91). Using the result in (90), P by ¯ S (0) = diag(σ 2 , r2 (0)σ 2 , σ 2 , σ 2 ). P (101) r z θ s s

(89)

It follows from (89), (97), and (98) that the measurement model for the range-rate in the S frame is

(90)

r˙z (0) = ξ˙S (0) − ξ˙pS (0) + nr˙ (0).

Thus, the measurement vector z S (k) and associated covariance matrix RS (k) are easily specified in the S frame. Let a denote a 2 × 1 position vector or velocity in the T frame. Then the corresponding vector a S in the S frame is given by (91) aS = T3 (θz (k))a. Similarly, let A denote a 2×2 covariance matrix for a position vector or velocity in the T frame. Then the corresponding covariance matrix A S in the S frame is given by AS = T3 (θz (k))AT3 (θz (k)).

(93)

where the 2 × 1 position vector and velocity for the target and sensor in the S frame are determined by (91). The measurement model in the S frame is

It can be shown that RSm (k) = diag(σr2 , rz2 (k)σθ2 ).

a = T3 (θz (k))aS ,

(92)

(102)

The state estimate and covariance are updated using equations similar to (41) and (42) in the S frame

where

¯ S (0) + WrS˙ (0)(r˙z (0) − rˆ˙z (0)), ˆ S (0|0) = x x

(103)

¯ S (0) − Sr˙ (0)WS (0)WS (0) , PS (0|0) = P r˙ r˙

(104)

rˆ˙z (0)) = −ξ˙pS (0),

(105)

¯ S (0)HS (0) /Sr˙ (0), WrS˙ (0) = P r˙

(106)

HSr˙ (0) = [0

0 1

0],

(107)

9

and the expression for S r˙ (0) in (44) also holds here. The initial ˆ (0|0) and covariance matrix P(0|0) in the T state estimate x frame are calculated by the inverse of the position and velocity ˆ S (0|0) and PS (0|0) and the cross-covariance components of x S term in P (0|0) using the transformations in (93)–(94).

ˆ (k−1|k−1) and covariance matrix Given the state estimate x P(k − 1|k − 1) in the T frame at time t k−1 , the predicted state ˆ (k|k − 1) and associated covariance P(k|k − 1) are estimate x determined in the T frame by [2] ˆ (k|k − 1) = F(k − 1)ˆ x x(k − 1|k − 1),

(108)

P(k|k − 1) = F(k − 1)P(k − 1|k − 1)F (k − 1) + Q(k − 1). (109) We calculate the rotational transformation matrix T 3 (θz (k)) using the azimuth measurement θ z (k) and transform the preˆ S (k|k − 1) dicted state estimate and covariance to obtain x S and P (k|k − 1) in the current S frame using (91) and (92), respectively. The updated state estimate and covariance are calculated by [2] ˆ S (k|k) = x ˆ S (k|k − 1) + WS (k)(zS (k) − ˆ x zS (k)), S

S

S

S



P (k|k) = P (k|k − 1) − W (k)S (k)W (k) , where

vr = v ρ/r,

(116)



B. Kalman Filtering

S

Given the first range-rate measurement r˙ z (0) at time t0 , an estimate of the radial velocity is given by (51). Assuming that the target moves in the XY plane, the radial velocity for the 2D or 3D scenario is given by

(110) (111)

⎡

⎤ ξˆS (k|k − 1) − ξpS (k) ⎢ ⎥ zˆS (k) = ⎣ ηˆS (k|k − 1) − ηpS (k) ⎦ , ˆ ξ˙S (k|k − 1) − ξ˙S (k)

(112)

p

WS (k) = PS (k|k − 1)HS (k) SS (k)

−1

,

(113)

SS (k) = HS (k)PS (k|k − 1)HS (k) + RS (k).

(114)

ˆ (k|k) and covariance matrix P(k|k) in the The state estimate x T frame are calculated using the inverse of transformations. VI. GMTI T RACK I NITIATION AND F ILTERING USING 3D S ENSOR T RAJECTORY In a stand-off GMTI ground-target tracking scenario, the GMTI platform travels at a certain altitude to collect measurements on ground-moving targets. In some cases, the RLOS is obscured by the terrain. We use a flat-Earth approximation and assume that a target moves with NCV motion in the XY plane. The GMTI platform is assumed to move at a constant altitude above the XY plane with constant velocity. We assume that the position and velocity of the GMTI platform are precisely known and terrain obscuration is absent. Let ρz (k) denote the estimate of the ground range corresponding to the range measurement r z (k) at time tk . Then,

ρz (k) = rz2 (k) − ζp2 (k). (115) Using the range and azimuth measurements at time t k and the corresponding variances, we calculate the converted unbiased Cartesian measurements (ξz (k), ηz (k)) and associated covariance Rc (k) [11], [15], using the approach presented in the Appendix B.

where ρ := [ξ−ξp η−ηp ] represents the ground range vector at any observation time. The ground range vector reduces to the range vector for the 2D scenario. We observe from (116) that a positive or negative value of the radial velocity indicates that the velocity of the target lies in the forward or backward direction of ground range vector. As defined in subsection IV-D, let z(k) and R(k) denote the position measurement with the range-rate measurement and associated covariance, respectively, at time t k . Then (17)–(19) and (76) are also valid for z(k), R(k), the measurement model, and the Jacobian matrix H, respectively, for the 3D scenario. We find that results for Methods 2, 3, and 5 are nearly the same. Therefore, we consider Methods 3, 4a, and 4b for the 3D scenario. A. Filter Initialization Using the range, azimuth, and range-rate measurements at time t0 and associated measurement error variances, we compute the unbiased Cartesian measurements (ξ z (0), ηz (0)) and the associated covariance matrix R c (0) by the algorithm ˆ (0|0) presented in Appendix B. The initial state estimate x and associated covariance matrix P(0|0) for Method 3 are calculated using (39)-(49). Given the first range-rate measurement r˙ z (0), an estimate vˆr (0) of the radial velocity is calculated by (51). The sign of the estimated radial velocity is used to construct the heading intervals in the forward or backward direction of the estimated ˆ (0) = [ξz (0) − ξp (0) ηz (0) − ηp (0)] , ground range vector ρ as described in Subsection IV-A. The initial state estimate and covariance matrix for each of the two models are determined using the approach described in Subsections IV-B – IV-C. B. Extended Kalman Filtering We run two concurrent EKFs using the approach described in subsection IV-D for each EKF and calculate the model probabilities for the two EKFs using the algorithm in subsection IV-E. VII. N UMERICAL S IMULATION R ESULTS We consider four scenarios, Scenarios 1–4, involving standoff GMTI ground-target tracking, where the platform speed (180.556 m/s) is much higher than the target speed. Usually, the GMTI platform travels at a given altitude. In order to be consistent with the equations in [30], we have set the altitude of the platform trajectory to zero in Scenarios 1–3. The GMTI platform moves with constant velocity in the XY plane in Scenarios 1–3. The scenario 4 is similar to the Scenario 3 except that the GMTI platform travels with a constant velocity at a fixed altitude of 10 km. A single target moves in the XY plane with NCV motion [2] in all scenarios. The speed of the

10

target for Scenarios 1–4 are 11.029 m/s (low speed), 33.526 m/s (high speed), 29.056 m/s (medium speed), and 29.056 m/s (medium speed), respectively. The target velocity relative to the RLOS is in the backward looking direction in Scenarios 1–2, whereas it is in the forward looking direction in Scenarios 3–4, as shown in Fig. 4. The angle φ defined in Fig. 2 at time t 0 in Scenarios 1–2 and 3–4 is 138.1◦ and 43.8◦ , respectively. We use 2 m/s for MDV in all scenarios yielding a value of 86.8 ◦ for φ0 defined in Fig. 2. Thus, we observe that the effect of including the MDV in selecting the feasible angles {α} for target velocity relative to the RLOS is small. We used nα = 10 in all scenarios and hence five heading-intervals are used for Methods 4a or 4b. 0.6

v

0.4

RLOS

0.2

0.4

0.4

Alternate target velocity

0.6 0.8

0.2

v1

u

v1

Target velocity 0.6

u

0

1 0.8

0.2

RLOS Alternate target velocity

0

v Target velocity

-0.2

-1 -1

-0.5

0

0.5

Scenarios 1 and 2

1

Scenarios 3 and 4

-0.4

and NIS [2], [15] defined, respectively, by NEES(k) = 1/(4M )× M  i=1

(ˆ xi (k|k) − xi (k))P−1 xi (k|k) − xi (k)), i (k|k)(ˆ

(119)

NIS(k) = 1/(3M )× M  i=1

(ˆzi (k|k) − zi (k))S−1 zi (k|k) − zi (k)), i (k)(ˆ

(120)

where Pi (k|k) is the filter covariance in the ith Monte Carlo run. The NEES can be calculated at any time t k , k = 0, 1, ... and the NIS is calculated at time t k , k > 0. Their ideal values are unity. We present results using the six methods, Methods 1, 2, 3, 4a, 4b, and 5. Methods 1-3, 4a-4b, and 5 are explained in Sections III, IV, and V, respectively. Let P 1 (0|0), P2 (0|0), P3 (0|0), P4a (0|0), and P4b (0|0), and P5 (0|0), denote the average initial covariances calculated from 1000 Monte Carlo runs for Methods 1, 2, 3, 4a, 4b, and 5, respectively. We note that P2 (0|0) ≈ P3 (0|0). Therefore, we have not presented results for P2 (0|0).

-0.6

Fig. 4.

Radar line-of-sight (RLOS) and target velocity

A. Numerical Results for Scenario 1

The trajectories of the target and GMTI platform in Scenario 1 are shown in Fig. 1. The parameters for the target and sensor for Scenarios 1–3, are presented √ in Table I. The prior speed standard deviation σ s = vmax / 3 is calculated using a maximum target speed v max of 80 mph (= 35.762 m/s). We performed 1000 Monte Carlo simulations for each scenario. In order to assess the measures of performance of various methods, we calculate the sample bias, sample MSE matrix (MSEM), normalized estimation error squared (NEES), and normalized innovation squared (NIS) [2], [15]. Let x i (k) and ˆ i (k|k) denote the true state and state estimate, respectively, x at time tk for a given estimator in the ith Monte Carlo run. Then the sample bias in the state estimate at time t k for the estimator is given by b(k) = (1/M )

M 

(ˆ xi (k|k) − xi (k)),

(117)

i=1

where M is the number of Monte Carlo runs. Let Σ(k|k) denote the sample MSEM for an estimator at time t k , defined by Σ(k|k) = (1/M )

M  i=1

(ˆ xi (k|k) − xi (k))(ˆ xi (k|k) − xi (k)) .

(118) Let zi (k), zˆi (k), and Si (k) denote the measurement, predicted measurement, and associated innovation covariance in the ith Monte Carlo run, respectively. The consistency of the filter covariance and innovation covariance at time t k calculated by the six methods can be rigorously compared using the NEES

P1 (0|0), P3 (0|0), P4a (0|0), P4b (0|0), and P5 (0|0) have the following numerical values: ⎡ ⎤ 877.124 2661.519 0 0 ⎢ 2661.519 9215.572 ⎥ 0 0 ⎥, P1 (0|0) = ⎢ ⎣ 0 0 34.407 114.422 ⎦ 0 0 114.422 392.887 (121) ⎡ ⎤ 877.064 2661.314 −4.843 1.414 ⎢ 2661.314 9214.870 −16.587 4.843 ⎥ ⎥ P3 (0|0) = ⎢ ⎣ −4.843 −16.587 34.436 114.413 ⎦ . 1.414 4.843 114.413 392.890 (122) ⎡ ⎤ 877.371 2662.366 −5.974 −0.993 ⎢ 2662.366 9218.474 −20.462 −3.402 ⎥ ⎥. P4a (0|0) = ⎢ ⎣ −5.974 −20.462 8.532 25.143 ⎦ −0.993 −3.402 25.143 85.463 (123) ⎡ ⎤ 877.283 2662.063 −3.935 3.437 ⎢ 2662.063 9217.438 −13.476 11.771 ⎥ ⎥. P4b (0|0) = ⎢ ⎣ −3.935 −13.476 7.995 24.321 ⎦ 3.437 11.771 24.321 86.011 (124) ⎡ ⎤ 877.111 2661.525 0 0 ⎢ 2661.525 9215.580 ⎥ 0 0 ⎥, P5 (0|0) = ⎢ ⎣ 0 0 34.407 114.422 ⎦ 0 0 114.422 392.887 (125) Comparing the results in (121) and (122), we observe that the 2 × 2 covariances for the position and velocity components of the state are nearly the same; only the cross-covariances between them are different.

11

TABLE I TARGET AND SENSOR PARAMETERS . Variable Target initial mean position (ξ(0), η(0)) (m) ˙ Target initial mean velocity (ξ(0), η(0)) ˙ (m/s) Target initial mean speed (m/s) Power spectral density of the process noise q (m2 /s3 ) Target prior speed standard deviation σs (m/s) Sensor initial position pp (0) (m) Sensor initial velocity vp (0) (m/s) Measurement standard deviations σr (m), σθ (rad), σr˙ (m/s)

Scenario 1 (117.491, 201.326) (9.948, 4.762) 11.029 0.5 20.6 (96074.945, -27813.756) (46.731, 174.403) (10, 0.001, 1)

The sample MSEMs based on 1000 Monte Carlo runs for the six methods have values as follows: ⎡ ⎤ 871.993 2612.546 −11.554 −37.220 ⎢ 2612.546 9014.191 −30.355 −94.953 ⎥ ⎥ Σ1 (0|0) = ⎢ ⎣ −11.554 −30.355 2358.777 8070.266 ⎦ , −37.220 −94.953 8070.266 27623.859 (126) ⎡ ⎤ 871.993 2612.546 −4.998 2.934 ⎢ 2612.546 9014.191 −17.706 8.725 ⎥ ⎥, Σ2 (0|0) = ⎢ ⎣ −4.998 −17.706 5.024 14.025 ⎦ 2.934 8.725 14.025 50.146 (127) ⎡ ⎤ 871.844 2612.019 −5.184 2.237 ⎢ 2612.019 9012.326 −18.342 6.339 ⎥ ⎥. Σ3 (0|0) = ⎢ ⎣ −5.184 −18.342 5.026 14.029 ⎦ 2.237 6.339 14.029 50.144 (128) ⎡ ⎤ 931.299 2775.002 −10.890 −16.766 ⎢ 2775.002 9443.058 −35.003 −61.599 ⎥ ⎥ Σ4a (0|0) = ⎢ ⎣ −10.890 −35.003 33.821 111.103 ⎦ . −16.766 −61.599 111.103 378.172 (129) ⎡ ⎤ 931.090 2774.319 −2.689 7.215 ⎢ 2774.319 9440.826 −7.193 24.437 ⎥ ⎥. Σ4b (0|0) = ⎢ ⎣ −2.689 −7.193 3.087 7.515 ⎦ 7.215 24.437 7.515 29.015 (130) ⎡ ⎤ 930.470 2772.230 −5.908 −2.874 ⎢ 2772.230 9433.729 −18.418 −10.802 ⎥ ⎥. Σ5 (0|0) = ⎢ ⎣ −5.908 −18.418 5.190 14.221 ⎦ −2.874 −10.802 14.221 50.022 (131) Results in the lower right 2×2 block in (126) corresponding to the velocity component of the state show that velocity estimate given by [30, (49)] has large estimation error. In contrast, the results in (127) and (128) show that velocity estimation using Method 2 or 3 has much lower estimation errors. The MSE matrices Σ2 (0|0) and Σ3 (0|0) are nearly the same. Comparison of P2 (0|0) with Σ2 (0|0) and P3 (0|0) with Σ3 (0|0) shows that the covariance calculated by Method 3 is closer to the MSE matrix than that for Method 2. Table II presents the root mean square (RMS) error, sample bias for position and velocity estimates, standard deviations for velocity estimates, and NEES at time t 0 from 1000 Monte Carlo runs for the six track initiation algorithms. The number of degrees of freedom for NEES and NIS are 4000 and 3000, respectively. Therefore, 99% probability intervals for NEES and NIS are [0.943, 1.058] and [0.934, 1.067], respectively.

Scenario 2 (100.000, 200.00) (29.035, 16.763) 33.526 0.5 20.6 (96074.945, -27813.756) (46.731, 174.403) (10, 0.001, 1)

Scenario 3 (100.000, 200.00) (25.163, 14.528) 29.056 0.5 20.6 (-27813.756, -96074.945) (174.403, -467.312) (10, 0.001, 1)

Results in Table II show that all the methods have nearly the same RMS position error (102 m) and the position bias for all the methods is small since range and azimuth measurements are available. The RMS velocity errors for Methods 2, 3, and 5 are nearly the same (7.4 m/s) where as Method 4b has a lower RMS velocity error of 5.7 m/s. Method 4a for Scenario 1 corresponds to the wrong direction of velocity and has a larger RMS velocity error of 20.3 m/s. Method 1 has a very large RMS velocity error of 173.2 m/s and large bias errors of 48.5 m/s and 166.2 m/s along the X and Y directions, respectively, due to the error in [30, (49)]. The velocity estimates for Methods 2, 3, and 5 have nearly the same bias errors of about -2.1 m/s and -7.1 m/s along the X and Y directions, respectively, since the prior initial velocity is set to zero. Method 4b, which corresponds to the correct direction of velocity, has velocity estimation bias errors of 1.5 m/s and 5.3 m/s along the X and Y directions, respectively. Low values of the standard deviations for velocity errors for all methods indicate that the errors show a small variation about the bias. We shall see in scenarios 2 and 3 that the bias errors in velocity estimates increase as the speed increases. We observe from Table II that the NEES is outside the 99% probability intervals. This is expected, since the covariances are calculated based on only one GMTI measurement vector at time t0 . The model probabilities for Methods 4a and 4b from the first Monte Carlo run are presented in Fig. 5. This trend is observed in all Monte Carlo runs. We observe that the correct model in Method 4b is selected after about six measurements. Therefore, one can discard the model with lower probability after about six scans in a multitarget tracking application. RMS position and velocity errors for Methods 2, 3, and 5 are nearly the same. Therefore, we present the RMS position and velocity errors using Methods 3, 4a, and 4b in Figs. 6 and 7 from 1000 Monte Carlo runs. The values for Method 1 are much higher and are not included in these plots. The RMS position and velocity errors in Figs. 6 and 7 show that the Method 4b corresponding to the correct direction of velocity performs the best. The NEES and NIS using Methods 3, 4a, and 4b are presented in Figs. 8 and 9 from 1000 Monte Carlo runs. The NEES using Methods 3 and 4b lie inside the 99% probability bounds after six and nine measurements. The NIS using Methods 3 and 4b lie inside the 99% probability bounds at all measurement times. Since the initial velocity estimate for Method 4a is incorrect for the current scenario, the NEES

12

TABLE II S CENARIO 1 (TARGET SPEED = 11.029 M / S ): RMS E RRORS , B IASES , AND NEES. Variable RMS position error (m) RMS velocity error (m/s) Sample position bias (m) Sample velocity bias (m/s) Std. of velocity error (m/s) NEES

Method 1 101.8 173.2 (0.5, 2.0) (48.5, 166.2) (1.0, 0.3) 18.371

Method 2 101.8 7.4 (0.5, 2.0) (-2.1, -7.1) (1.0, 0.3) 0.820

Method 3 101.8 7.4 (0.6, 2.3) (-2.1, -7.1) (1.0, 0.3) 0.816

1

RMS Vel. Error (m/s)

18

0.7 0.6 0.5 0.4 0.3

16 14 12 10 8

0.2

6

0.1

4 0

5

10

15

20

25

Method 3 Method 4a Method 4b

2

30

0

5

10

Time Index

Fig. 5.

Fig. 7.

25

30

2 Method 3 Method 4a Method 4b

100

90

1.6

80

1.4

70

1.2

60

1

50

0.8

0

5

10

15

20

25

Method 3 Method 4a Method 4b 99% Lower bound 99% Upper bound

1.8

NEES

RMS Pos. Error (m)

20

Scenario 1: RMS velocity error using 1000 Monte Carlo runs.

110

30

0

Time Index

Fig. 6.

15

Time Index

Scenario 1: Model probabilities for Methods 4a and 4b.

40

Method 5 101.8 7.4 (0.6, 2.0) (-2.1, -7.1) (1.0, -0.3) 0.820

20

Method 4a Method 4b

0.8

0

Method 4b 101.8 5.7 (0.7, 2.8) (1.5, 5.3) (0.9, 0.8) 0.873

22

0.9

Model Probability

Method 4a 101.9 20.3 (0.8, 3.0) (-5.7, -19.4) (1.1, 0.2) 1.906

Scenario 1: RMS position error using 1000 Monte Carlo runs.

and NIS lie within the 99% probability bounds after 12 measurement times. B. Numerical Results for Scenario 2 The target-sensor geometry in Scenario 2 is the same as that in Scenario 1 (see Fig. 1). The target speed (33.526 m/s) in Scenario 2 is about three times higher than that in Scenario 1. Table III presents numerical results for the Scenario 2. We have not included the results from Method 1 in Table III since the method contains errors. We observe from Table III that the RMS velocity error (24.2 m/s) of existing methods (Methods 2, 3, and 5) is much higher than that (13.0 m/s) of Method 4b. Similarly, the velocity bias (1.0, 3.7) m/s for Method 4b is much smaller compared with that (-6.8.0, -23.2) m/s for Methods 2, 3, and 5.

5

10

15

20

25

30

Time Index

Fig. 8.

Scenario 1: NEES using 1000 Monte Carlo runs.

C. Numerical Results for Scenarios 3 and 4 The target-sensor geometry in Scenario 3 is different from that in Scenario 1 and is shown in Fig. 4. Table IV presents numerical results for Scenario 3. Method 4a and 4b correspond to the correct and incorrect directions of velocity for this scenario. Results in Table IV show that the velocity estimation accuracy of Method 4a is significantly higher than that of Method 2, 3, or 5. The plots for model probabilities, RMS position error, RMS velocity error, NEES, and NIS for Scenario 3, are presented in Figs. 10, 11, 12, 13, and 14, respectively. In Scenario 3 the GMTI platform travels in the XY plane with constant velocity, whereas in Scenario 4, the GMTI platform travels in a plane parallel to the XY plane with constant velocity at a fixed altitude of 10 km. The results for Scenarios 3 and 4 are nearly the same. Therefore, we have not

13

TABLE III S CENARIO 2 (TARGET SPEED = 33.526 M / S ) : RMS E RRORS , B IASES , AND NEES. Variable RMS position error (m) RMS velocity error (m/s) Position bias (m) Velocity bias (m/s) NEES

Method 2 101.8 24.2 (0.5, 2.0) (-6.8,-23.2) 1.132

Method 3 101.8 24.2 (0.8, 3.0) (-6.8, -23.2) 1.129

Method 4a 101.9 35.5 (1.1, 3.7) (-10.0, -34.0) 5.504

Method 4b 101.9 13.0 (1.0, 3.7) (-3.7,-12.4) 1.419

Method 5 101.8 24.2 (0.6,2.0) (-6.8,-23.2) 1.132

TABLE IV S CENARIO 3 (TARGET SPEED = 29.056 M / S ): RMS E RRORS , B IASES , AND NEES. Variable RMS position error (m) RMS velocity error (m/s) Position bias (m) Velocity bias (m/s) NEES

Method 2 102.1 20.1 (2.0, -0.5) (-19.3, 5.5) 1.027

Method 3 102.1 20.1 (1.2, -0.3) (-19.3, 5.5) 1.023

Method 4a 102.0 8.7 (0.6,-0.1) (-8.3, 2.3) 1.050

1.15

Method 4b 102.0 31.6 (0.4,-0.04) (-30.4, 8.7) 4.289

Method 5 102.1 20.1 (2.0, -0.5) (-19.3, 5.5) 1.027

120 Method 3 Method 4a Method 4b 99% Lower bound 99% Upper bound

RMS Pos. Error (m)

1.1

Method 3 Method 4a Method 4b

110

NIS

1.05

1

100 90 80 70 60

0.95 50 0.9

0

5

10

15

20

25

40

30

0

5

10

Time Index

Fig. 9.

15

20

Scenario 1: NIS using 1000 Monte Carlo runs.

Fig. 11.

30

Scenario 3: RMS position error using 1000 Monte Carlo runs.

35

presented plots for Method 4.

Method 3 Method 4a Method 4b

RMS Vel. Error (m/s)

30 1 Method 4a Method 4b

0.9 0.8

Model Probability

25

Time Index

0.7 0.6

25

20

15

10

0.5 5

0.4 0.3

0

0.2

0

5

10

15

20

25

30

Time Index

0.1 0

Fig. 12. 0

5

10

15

20

25

Scenario 3: RMS velocity error using 1000 Monte Carlo runs.

30

Time Index

Fig. 10.

Scenario 3: Model probabilities for Methods 4a and 4b.

D. Variation of Velocity RMS Error and Bias with Speed We summarize the RMS velocity error and velocity bias for Scenarios 1, 3, and 2 in Table V. In the second and third rows of Table V, the first quantity represents the RMS velocity error and the next two quantities inside the parenthesis represent

the velocity bias along the X and Y axes, respectively. We observe from Table V that our proposed method (Method 4a or 4b) significantly reduces the RMS velocity error and bias compared with existing Methods 2, 3, and 5. VIII. C ONCLUSIONS In this paper we have provided corrections to two errors in the single-point track initialization part of the paper [30].

14

TABLE V S UMMARY OF V ELOCITY RMS E RRORS AND B IAS IN M / S . Target Speed (m/s) Methods 2,3,5 Method 4a or 4b

Scenario 1 11.0 7.4, (-2.1, -7.1) 5.7, (1.5, 5.3) (4b)

Method 3 Method 4a Method 4b 99% Lower bound 99% Upper bound

1.6

Consider a Gaussian random variable x with mean μ and variance σ 2 . Then we have  exp(ikx)N (x; μ, σ 2 )dx = exp(−k 2 σ 2 /2) exp(ikμ),

NEES

1.5 1.4 1.3 1.2

where i =

1.1

√ −1. The Euler identity [10] gives

1 0.9

exp iθ = cos θ + i sin θ. 0

5

10

15

20

25

exp(ikx) = cos(kx) + i sin(kx).

Method 3 Method 4a Method 4b 99% Lower bound 99% Upper bound

1.4

1.3

NIS

(134)

Using (134) in (132) and equating real and imaginary parts on the left and right hand sides, we get  cos(kx)N (x; μ, σ 2 )dx = exp(−k 2 σ 2 /2) cos(kμ), (135)

1.5



1.2

1.1

sin(kx)N (x; μ, σ 2 )dx = exp(−k 2 σ 2 /2) sin(kμ). (136)

For our problem, we need evaluation of the integral with respect to cos2 x, sin2 x and cos x sin x. The following trigonometric identities are used in (135) and (136)

1

0

5

10

15

20

25

30

Time Index

Fig. 14.

(133)

When θ = kx, where k is an integer (133) yields

Scenario 3: NEES using 1000 Monte Carlo runs.

0.9

(132)

30

Time Index

Fig. 13.

Scenario 2 33.5 24.2, (-6.8, -23.2) 13.6, (-3.7, -12.4) (4b)

A PPENDIX A: U SEFUL R ESULTS INVOLVING G AUSSIAN D ISTRIBUTION

1.8 1.7

Scenario 3 29.1 20.1, (-19.3, 5.5) 8.7, (-8.3, 2.3) (4a)

cos2 x = (1 + cos 2x)/2,

Scenario 3: NIS using 1000 Monte Carlo runs.

Secondly, we presented an improved track initiation algorithm based on a heading-parametrized multiple model method which uses the sign of the range-rate measurement, the prior knowledge of target maximum speed, and the MDV. We have presented results for Scenarios 1–3, where the target and the GMTI platform move in the XY plane. In Scenario 4, the GMTI platform moves at a fixed altitude above the XY plane. We have analyzed low-speed, medium-speed, and high-speed scenarios and our results show that the velocity bias error of the existing track initiation algorithm increases with the target speed. The track initiation and GMTI filtering algorithms based on the approach presented in [20] are shown to produce results similar to the existing baseline algorithm (Method 3). We have demonstrated that the new algorithm significantly reduces the bias in the initial velocity estimate and its superior performance is shown using Monte Carlo simulations.

sin2 x = (1 − cos 2x)/2, (137)

sin x cos x = sin 2x/2.

(138)

We use the following notation convention for the moments of cosine and sine functions of Gaussian random variables as used in [18]:  mj,k (μ, σ 2 ) = cosj (x) sink (x)N (x; μ, σ 2 ) dx. (139) Then, we get m1,0 (μ, σ 2 ) = cos μ exp(−σ 2 /2)

(140)

m0,1 (μ, σ 2 ) = sin μ exp(−σ 2 /2),

(141)

m2,0 (μ, σ 2 ) = [1 + exp(−2σ 2 ) cos 2μ]/2,

(142)

m0,2 (μ, σ 2 ) = [1 − exp(−2σ 2 ) cos 2μ]/2,

(143)

m1,1 (μ, σ 2 ) = exp(−2σ 2 ) sin 2μ/2.

(144)

15

A PPENDIX B: C ONVERTED U NBIASED C ARTESIAN M EASUREMENTS For simplicity in notation, we omit the time index k. The converted unbiased Cartesian measurements (ξ z , ηz ) and the associated covariance R c are described by [11], [15] ξz = ξp + λ−1 θ ρz cos θz , ηz = ηp +

λ−1 θ ρz

(145)

sin θz ,

(146)

λθ = exp(−σθ2 /2), Rc,11 =

(λ−2 θ

−

2)ρ2z

2

cos θz +

(ρ2z

(147) +

λθ

cos 2θz )/2, (148) 2 2 2 2  − 2)ρ sin θ + (ρ + σ )(1 − λ cos 2θ Rc,22 = (λ−2 z z )/2, z z ρ θ θ (149) 2 2 2  −2)ρ +(ρ +σ )λ ] sin 2θ /2, (150) Rc,12 = Rc,21 = [(λ−2 z z z ρ θ θ λθ = exp(−2σθ2 ),

+

σρ2 )(1

σρ = (rz /ρz )σr .

(151)

ACKNOWLEDGMENT The authors thank B. Sindhu and Ravindra Dhuli of Vellore Institute of Technology for proving useful comments and suggestions and for help in preparing the revised version of the paper. R EFERENCES [1] S. Arulampalam and B. Ristic, “Comparison of the Particle Filter with Range-Parameterised and Modified Polar EKFs for Angle-Only Tracking,” Proc. of SPIE, vol. 4048, 2000. [2] Y. Bar-Shalom, X. Li and T. Kirubarajan, Estimation with Applications to Tracking and Navigation, Wiley, New York, 2001. [3] Y. Bar-Shalom, P. Willett, and X. Tian, Tracking and Data Fusion: A Handbook of Algorithms, YBS Publishing, 2011. [4] S. Blackman and R. Popoli, Design and Analysis of Modern Tracking Systems, Artech House, 1999. [5] J. N. Entzminger, Jr., C. A. Fowler, and W. J. Kenneally, “JointSTARS and GMTI, Past, Present, and Future,” IEEE Transactions on Aerospace and Electronic Systems, vol. 35, no. 2, pp. 748–761, April 1999. [6] T. Kirubarajan, Y. Bar-Shalom, K. R. Pattipati, and I. Kadar, “Ground Target Tracking with Variable Structure IMM Estimator,” IEEE Transactions on Aerospace and Electronic Systems, vol. 36, no. 1, pp. 26–46, January 2000. [7] R. Klemm, Space-time Adaptive Processing, The Institute of Electrical Engineers, London, UK, 1998. [8] W. Koch, Tracking and Sensor Data Fusion: Methodological Framework and Selected Applications, Springer, Heidelberg, 2014 [9] T. R. Kronhamn, “Bearings-only Target Motion Analysis Based on a Multihypothesis Kalman Filter and Adaptive Ownship Motion Control,” IEE Proc.-Radar, Sonar Navig., vol. 145, no. 4, pp. 247–252, August 1998. [10] S. Lipschutz, M. Spiegel, and J. Liu, Schaum’s Outline of Mathematical Handbook of Formulas and Tables, 4th Edition, McGraw-Hill, 2012. [11] M. Longbin, S. Xiaoquan, Z. Yiyu, S. Z. Kang, and Y. Bar-Shalom, “Unbiased Converted Measurements for Tracking,” IEEE Transactions on Aerospace and Electronic Systems, vol. 34, no. 2, pp. 1023–1027, July 1998.

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