IMM Estimator for Ground Target Tracking with Variable Measurement Sampling Intervals Mahendra Mallick Toyon Research Corporation 75 Aero Camino, Suite A Goleta, CA 93117-3139, USA
[email protected] Abstract – Common ground target dynamic models include the nearly constant velocity (NCV), nearly constant acceleration (NCA), and nearly constant turn (NCT) models. Most of the papers on the interacting multiple model (IMM) estimator use a constant Markov chain transition probability matrix (TPM) corresponding to a constant measurement sampling interval. However, a multi-sensor ground target tracking system usually employs ground moving target indicator radar, electro-optical, infrared, video, acoustic, and seismic sensors, for which the sampling intervals are different. Modeling such systems requires using a variable sampling interval in the IMM estimator, which in turn requires the use of a nonconstant TPM. An analytic expression for the TPM with variable sampling interval exists for two dynamic models. When the number of dynamic models is greater than two, the TPM can be numerically calculated efficiently. We present the technical approach for the IMM estimator with variable sampling intervals. Preliminary numerical results are presented for a maneuvering target with the NCV, NCA, and NCT models using 200 Monte Carlo simulations. Keywords: Ground target tracking, maneuvering target, interacting multiple model (IMM), variable Markov chain transition probability matrix.
1 Introduction The motion of a ground target usually includes multiple dynamic models such as the nearly constant velocity (NCV), nearly constant acceleration (NCA), nearly constant turn (NCT), and stop models [1]-[3], [7], [9], [12]. A ground target can move on-road or off-road [7]. It is a standard practice to use the interacting multiple model (IMM) [1]-[4], [6], [8] [14]-[15] or variable structure IMM (VS-IMM) estimator [7] to estimate the state of a maneuvering target. A multi-sensor ground target tracking system usually employs ground moving target indicator (GMTI) radar [7], [10] electro-optical (EO), infrared (IR), video, acoustic, and seismic sensors, for which the measurement sampling intervals are different. In realistic scenarios, the measurement sampling interval even for a single sensor can vary. Most of the published papers on the IMM or VS-IMM estimator [1]-[4], [6]-[8], [14]-[15] use a constant Markov chain transition probability matrix (TPM) corresponding to a constant measurement
Barbara F. La Scala Dept. of Electrical and Electronic Engineering University of Melbourne Victoria 3010, Australia
[email protected] sampling interval. Therefore, for multi-sensor ground target tracking problems, it is necessary to use a TPM in the IMM or VS-IMM which can handle variable measurement sampling intervals. In general, the TPM for a stationary or homogeneous Markov chain can be formally expressed as a matrix exponential function of the product of the transition probability rate matrix and measurement sampling interval [13]. When there are two dynamic models, the matrix exponential function can be simplified to obtain a simple analytic expression for the TPM as function of the measurement sampling interval [13], [5]. Alternatively, one can directly integrate the stochastic differential equation for the TPM and obtain the analytic expression. A simple analytic expression for the TPM as function of the measurement sampling interval does not exist at present when the number of dynamic models is greater than two. For computational purposes, one can compute the matrix exponential function efficiently and easily [11] for any number of models. In order to demonstrate the use of a TPM with the variable measurement sampling interval in the IMM estimator, we consider the motion of a ground target in two dimensions which involves the NCV, NCA, and NCT models in different parts of the trajectory. The dimensions of the target state for the NCV, NCA, and NCT models in 2D are four, six, and five respectively. The target full state is a seven dimensional consisting of 2D position, 2D velocity, 2D acceleration, and angular velocity along the Z axis. We use range, azimuth, and radial velocity measurements [10] from two GMTI sensors with variable measurement sampling intervals to estimate the target state and present numerical results to demonstrate the technical approach. We use the symbol “:=” to define a quantity. The outline of the paper is as follows. Section 2 describes the time dependent transition probability matrix using the continuous-time Markov chain associated with multiple dynamic models of a target. The expression for the TPM (Π ) as a function of the transition probability rate matrix (Λ ) and measurement sampling interval (τ ) is presented in Section 2. Section 3 summarizes the NCV and NCT models and gives details of the NCA model. Section 3.4 describes the adjusted full state, state transition matrix or time evolution function, and process noise covariance matrix. Section 4 presents the GMTI sensor measurement
model. Section 5 summarizes the steps of the IMM estimator for the problem. Finally, Sections 6 and 7 present numerical results and conclusions.
ηi =
λij .
(12)
j ≠i
2 Time Dependent Transition Probability Matrix
The transition probability matrix Π(τ ) satisfies the Kolmogorov equation [13] (13) Π′(τ ) = Π(τ )Λ, Π(0) = I . The solution of the linear differential equation (13) is [13] Π(τ ) = e Λτ . (14)
Let M (t ) denote the mode or dynamic model of the target at time t. We assume that the mode of the target is described by a continuous-time Markov chain {M (t ), t ≥ 0} with r modes {ai }ir=1 [13]. Transitions occur
When r = 2, an analytic expression for Π(τ ) can be obtained [13], [5], Π(τ ) =
at random times {ζ n } . The values
ξ n = M (ζ n+ ),
ξ n ∈ {a1 , a2 ,..., ar }
(1)
1 η 2 + η1e −(η1 +η2 )τ η1 + η 2 η 2 − η 2 e −(η1 +η2 )τ
η1 − η1e −(η1 +η2 )τ . η1 + η 2 e −(η1 +η2 )τ
(15)
is specified by the underlying point process {ζ n } and the
When r > 2, it is difficult to get an analytic expression for Π(τ ) . However, we can evaluate (14) numerically using efficient matrix exponential evaluation algorithms [11]. In this paper, we use the “expm” function in Matlab for evaluating Π(τ ) for an arbitrary value of τ .
imbedded Markov chain {ξ n } [13]. Let µ j (t ) and π ij (t , t ′) denote the state (or mode) probability at time t
3 Dynamic Models
of M (t ) at the times {ζ n+ } form a discrete-state Markov sequence, known as the Markov chain embedded in the Markov process {M (t ), t ≥ 0} . A Markov chain {M (t )}
and transition probability from time t to t ′ , respectively [13], µ j (t ) := P{M (t ) = a j }, (2)
π ij (t , t ′) := P{M (t ′) = a j | M (t ) = ai }, t < t ′. (3) The mode probability and mode transition probabilities satisfy [13] π ij (t , t ′) = 1, t < t ′, (4) j i
µ (t )π ij (t , t ′) = µ j (t ′), t < t ′.
The Markov process is stationary or homogeneous if the transition probabilities depend only on the time difference τ = t ′ − t [13], π ij (τ ) = P{M (t + τ ) = a j | M (t ) = ai }, τ > 0. (6) (7)
j
Let Π(τ ) denote the r × r transition probability matrix whose
(i, j ) th element is
π ij (τ ) .
The transition
probability rates of {M (t )} are defined by [13]
λij =
∂π ij (τ ) ∂τ
= π ij′ (0 + ). τ =0
(8)
Let Λ denote the r × r transition probability rate matrix with elements {λij } [13]. The matrix equivalent of (8) is
Λ := Π′(0+ ). (9) Differentiating (7) with respect to τ and then setting τ to zero, we get λij = 0. (10) j
Define (11)
Then 1 /ηi represents the expected sojourn time in the
i th mode. From (10) we get
M k := M (t k ) at time t k is one of the r possible modes M k ∈ {a1 , a 2 ,..., a r }. (16) In this paper, we use the direct discrete-time dynamic models [1]. The dynamic models for the NCV and NCA are linear while the dynamic model for the NCT is nonlinear. A general linear discrete-time dynamic model for the target motion is given by [1]-[3], [9] x k = Fk , k −1x k −1 + k , k −1w k −1 (17) where Fk ,k −1 ,
k ,k −1 ,
and w k −1 represent the state
transition matrix, process noise gain matrix, and zeromean white process noise, respectively. The NCV and NCA models in 2D satisfy (17) and the white process noise acceleration w k −1 is a 2×1 vector
w k −1 =
wk −1, x wk −1, y
,
(18)
with the properties [1]-[3], [9]
+
ηi := −λii .
at a measurement time t k . We assume that the mode
(5)
i
For the stationary Markov chain [13] π ij (τ ) = 1, τ > 0.
Let x k := x(t k ) denote the continuous-valued target state
E{w k −1} = 0 2×1 , E{w k −1w ′l −1} = δ kl
2 σ wx
0
0
2 σ wy
. (19)
In (19), we have used the assumption that the X and Y components of the process noise are uncorrelated. The units of (σ wx ,σ wy ) for the NCV and NCA are ( m / s 2 , m / s 2 ) and ( m / s 3 , m / s 3 ) respectively. The form
of the process noise gain matrix
k ,k −1 is
different for the
NCV and NCA models. The process noise covariance matrix Q k ,k −1 is given by
Q k ,k −1 := E{ =
k ,k −1w k ,k −1w ′k ,k −1 2 σ wx
0
0
2 σ wy
k ,k −1
′k ,k −1}
3.3 Nearly constant turn model (20)
′k ,k −1.
Next, we present the definition of the state, state transition matrix or nonlinear time evolution function, process noise gain, and process noise covariance matrix for the NCV, NCA, and NCT models in Sections 3.1, 3.2, and 3.3, respectively. Section 3.4 presents the above quantities for the full model.
The state for the NCT model consists of 2D position, velocity, and angular velocity components and is defined by [1], [9], [12] x = [x y x y ω ] ′. (26) k
k
k
The state for the NCA model consists of 2D position, velocity, and acceleration components and is defined by x k := [xk y k xk y k xk y k ]′ . (22) The state transition matrix, process noise gain, and process noise covariance matrix are given, respectively, by 0 ∆k 1 0 0 1 0 0
0 0 0 0
0 0
0 ∆k 0 1 0 0
∆2k
/2 0 2 ∆k / 2 0 ∆k 0 , ∆k 0 1 0
(23)
0 1
∆3k
/6 0 ∆3k / 6 0 ∆2k / 2 0 , k , k −1 := 2 ∆k / 2 0 ∆k 0 0 ∆k
k
w k −1
wk −1, x = wk −1, y . wk −1,α
(24)
Qk,k−1 = 2 ∆6 / 36 σwx k
2 ∆5 /12 2 ∆4 / 6 σwx σwx 0 0 0 k k 2 6 2 5 2 σwy∆k / 36 σwy∆k /12 σwy∆4k / 6 0 0 0 2 5 2 4 2 3 σwx∆k /12 σwx∆k / 4 σwx∆k / 2 0 0 0 2 5 2 4 2 3 . σwy σwy σwy 0 ∆k /12 0 ∆k / 4 0 ∆k / 2 2 ∆4 / 6 2 ∆3 / 2 2 ∆2 σwx σwx σwx 0 0 0 k k k 2 4 2 3 2 0 σwy∆k / 6 0 σwy∆k / 2 0 σwy∆2k
(25) We emphasize that the NCA model presented in (23)-(25) is different from the piecewise constant Wiener process acceleration model [1].
(28)
The components of w k −1 represent the X and Y components of acceleration and the Z component of angular velocity, respectively. We assume that the components of w k −1 are uncorrelated, i.e.
E{w k −1} = 0 3×1 , E{w k −1w ′l −1}
3.2 Nearly constant acceleration model
1 0 0 Fk , k −1 := 0
k
where f is the nonlinear time evolution function and w k −1 is a 3 ×1 zero-mean white process noise
3.1 Nearly constant velocity model The state for the NCV model consists of 2D position and velocity components and is defined by [1], [9] x k := [xk y k xk y k ]′ . (21) Expressions for the state transition matrix, process noise gain, and process noise covariance matrix are presented in [1], [9].
k
The nonlinear discrete-time NCT model is [1], [9], [12] x k = f ( x k −1 , k − 1) + k , k −1w k −1 (27)
2 2 = δ kl diag{σ wx ,σ wy ,σ w2 α }.
(29)
Expressions for f, process noise gain, and process noise covariance matrix are presented in [1], [9], and [12].
3.4 Full model The NCV, NCA, and NCT dynamic models use the state vector x k := [xk y k xk y k xk y k ω k ] ′ (30) where the time evolution of the state is given by x k = Fkj,k −1x k −1 + Γkj,k −1w kj −1 , j = 1,2, (31) j j j x k = f (x k −1 , k − 1) + k , k −1w k , k −1 , j = 3, (32) for j = 1, 2, 3 corresponding to the NCV, NCA, and NCT models, respectively. Note that the NCT model is nonlinear as the state transition matrix includes elements of the state at the previous sampling time. Secondly, derivatives of any order are non-zero for the NCT model. Therefore, when the X and Y components of acceleration are included in the state vector, corresponding rows cannot be set to zero. We can show that x(t + ∆) cos ω∆ − sin ω∆ x(t ) = . (33) y (t + ∆ ) sin ω∆ cos ω∆ y (t ) We combine the time evolutions of the position, velocity, and angular velocity with the time evolution of the acceleration from (33) to get the time evolution for the state for the NCT model. In this framework, the state transition matrices for the NCV and NCA models and the nonlinear time evolution for the NCT models are
1 0 ∆k
Fk1, k −1
0 1 0 0 = 0 0
0 1 0
∆k 0 1
0 0 0 0 0 0
0 0 0
0 0 0
1 0 ∆k
Fk2, k −1
0
and Q3k,k −1 =
0 0 0 0 0 0 0 0 0 ,
(34)
0
∆2k / 2
0
2 3 ∆k σ wx
0
0 1 0 0
∆k 0 1 0
0 ∆k 0 1
∆2k / 2 0 ∆k 0
0 0 0 0
0 0
0 0
0 0
1 0
0 0 0 , 0
1−c s k, k − 1 − k, k −1 ωk−1 ωk−1 1−c s
(35)
ωk−1
c k, k − 1 0 0 s 0 0
0
0
0
−s
0
0
0 , (36)
c
0
0
0
k, k − 1
k, k − 1
k, k − 1
0 0
0
0
0 0
0
0
0 0
0
0
2 2 ∆k σ wx
0
0 0
0
0 0
2 3 ∆k / 2 σ wy
2 2 ∆k σ wy
0
0 0
0
0 0 0 0
0 0
0 0
0 0
0 0
0 0
0 0 0 2 2 0 0 σ wα ∆k
4 GMTI Measurement Model
[
c −s 0 k, k −1 k, k − 1 s c 0 k, k − 1 k, k − 1 0 0 1
∆ k = t k − t k −1 ,
(37)
(39)
Similarly, the process noise covariance matrices are Q1k , k −1 = 0
0
σ k2α 0
0 0 .
(47)
σ k2r
Dropping the subscript k, the GMTI measurement model is described by [10] hr (x, s) = r = [( x − s x ) 2 + ( y − s y ) 2 + s z2 ]1 / 2 , (48) h α (x, s) = α =
2 3 0 ∆k / 2 0 0 0 0 σ wx 2 4 2 3 0 σ wy ∆ k / 4 σ wy ∆ k / 2 0 0 0
/2
0
0
(38)
ck , k −1 = cos(ωk −1∆ k ).
2 3 ∆k σ wx
Rk =
σ kr2 0
sk , k −1 = sin (ω k −1∆ k ),
2 4 ∆k / 4 σ wx
]
v k := [vkr vkα vkr ] ′. (45) We assume the sensor position s k as error-free. We assume that v k is a zero-mean independent Gaussian noise with diagonal covariance R k , v k ~ N (0, R k ), (46)
where ∆ k is the measurement sampling interval, and
.
where s k ∈ ℜ 3 and v k ∈ ℜ 3 are sensor position and measurement noise at time k, respectively: ′ s k := skx sky s kz , (44)
0
k, k −1
0
/2
(α ) , and range-rate (r ) of a target. The GMTI measurement model at time k is described by [10] z k = h(x k , s k ) + v k , (43)
0
ωk−1
0 0
A GMTI radar sensor measures the range (r ) , azimuth
0 0
0
k, k − 1
0 1
2 3 0 ∆k / 2 0 0 0 σ wx 2 4 2 3 0 σ wy∆k / 4 σ wy∆k / 2 0 0
(42)
f 3(xk−1, k −1) = 1 0
2 4 ∆k / 4 σ wx
0
0 0 0 0 0 0 0 0 0
1 0 0 0
0 0 = 0 0
(41)
0 0 0
0 2 3 σ wy ∆k
2 2 ∆k σ wx
0
0
2 2 σ wy ∆k
/2
tan −1 ( x − s x , y − s y ),
0 0 0 (40 0 0 0 ,
0 0
0 0
0 0
0 0
0 0 0 0 0 0
0
0
0
0
0 0 0
) 2 6 σwx ∆k / 36
0
2 5 σwx ∆k /12
0
2 4 σwx ∆k / 6
0
2 6 σwy ∆k / 36
0
2 5 σwy ∆k /12
0
2 5 σwx ∆k /12
0
2 4 σwx ∆k / 4
0
2 3 σwx ∆k / 2
0
2 5 σwy ∆k /12
0
2 4 σwy ∆k / 4
0
2 4 σwx ∆k / 6
0
2 3 σwx ∆k / 2
0
2 2 σwx ∆k
0
0
0
2 4 σwy ∆k / 6
0
2 3 σwx ∆k / 2
0
2 2 σwy ∆k
0
0
0
0
0
0
0
0
0
2 4 σwy ∆k / 6 0
0
tan ( x − s x , y − s y ) + 2π , if tan −1 ( x − s x , y − s y ) < 0,
(49)
hr (x, s) = r =
Qk2,k −1 = 0
if tan −1 ( x − s x , y − s y ) > 0 ,
−1
0
2 3 σwy ∆k / 2 0 ,
(x − sx )x + ( y − s y ) y [( x − s x ) 2 + ( y − s y ) 2 + s z2 ]1/ 2
.
(50)
5 IMM Estimator In this Section we give a brief overview of the IMM estimator. For full details see [1]-[4], [6], [8], [14], [15]. The Markov chain transition probabilities for r models and the time interval ∆ k are denoted by {π ij (∆ k )} . These values are calculated by evaluating the matrix exponential function in (14) with τ = ∆ k . Let µ kj denote the probability of the mode j at time t k . The IMM estimator is then described by the following steps.
g(ωk−1, ∆k ) = Step 1. Calculate mixing probabilities 1 µ ki|−j1|k −1 = π ij (∆ k ) µ ki −1 , cj
(51)
ωk−1 xˆk−1|k−1
r
π ij (∆ k ) µ ki −1.
cj =
xˆk−1|k−1
(52)
ωk−1
i =1
sk,k−1∆k −
(53)
=
(54)
µ ki|−j1|k −1[ Pki −1|k −1 + (xˆ ik −1|k −1 − xˆ 0k ,−j1|k −1 ) (xˆ ik −1|k −1 − xˆ 0k ,−j1|k −1 )′],
xˆ kj|k −1 = Fkj,k −1xˆ 0k ,−j1|k −1 , j = 1,2,
(55)
for a linear dynamic model and xˆ kj |k −1 = f 3 (xˆ 0k ,−j1|k −1 ),
for a nonlinear dynamic model. covariance Pkj|k −1 is given by
(56)
The predicted mode
′ Pkj|k −1 = Fkj,k −1Pkj−,01|k −1 Fkj,k −1 + Q kj −1 , j = 1,2,
(
)
(57)
for a linear dynamic model and by
(59)
. x =xˆ k0 ,−31|k −1
The gradient matrix F 3 for the NCT model is [1] k, k − 1 ωk−1
1−c 0 1
Fk3, k −1 =
k, k − 1
ωk−1
1−c − k, k −1
0
ωk−1
s
k, k −1
ωk−1
0
g (ωk−1, ∆ ) , 3 k g (ωk−1, ∆ )
0
0
0 0
s
c
0
0
k, k − 1
0 0
0
0
c
0 0
0
0
s
0 0
0
0
k
g (ωk−1, ∆ )
−s
k, k − 1
1
0
c
k, k − 1
g (ωk−1, ∆ )
0
0 0
k, k − 1
k, k − 1
k, k − 1
0
2
k
4
k
−s
k, k − 1
g (ωk−1, ∆ )
c
k, k − 1
g (ωk−1, ∆ )
0
1
5
ωk−1
6
.
(61)
k
k−1|k−1 k,k−1
k
∂h(x, s) ∂x
(63) (64)
. x= xˆ kj |k −1
Step 5. Mode Likelihood For each mode j = 1,2,..., r , calculate likelihood λkj
the mode
λkj = p(z k | xˆ kj|k −1 , Pkj|k −1 ) ~ N ( kj ;0, S kj ).
(65)
Step 6. State and Covariance Update For each mode j = 1,2,..., r calculate the updated state and covariance using (66) xˆ kj|k = xˆ kj|k −1 + K kj kj , Pkj|k = Pkj|k −1 − K kj S kj (K kj ) ′,
(67)
where the gain is given by (68)
Step 7. Mode Probability Update For each mode j = 1,2,..., r update the mode probabilities using 1 (69) µ kj = λkj c j , c where r
λkj c j .
c=
(70)
j =1
Step 8. Combined State Estimate and Covariance The combined state estimate and associated covariance are then given by r
k
µ ki xˆ ik |k ,
xˆ k |k =
(71)
i =1
k
r
(60) where
sk,k−1
K kj = Pkj|k −1[H kj ( xˆ kj|k −1 )]′ (S kj ) −1 .
3
∂f (x) ∂x
ck,k−1∆k −
k−1|k−1 k,k−1
H kj (xˆ kj|k −1 ) :=
for nonlinear dynamic model where Fk3,k −1 (xˆ 0k ,−31|k −1 ) :=
ωk−1
ωk−1
where
)
(58)
yˆk−1|k−1
1−ck,k−1
S kj := H kj (xˆ kj|k −1 )Pkj|k −1[H kj ( xˆ kj|k −1 )]′ + R kj
′ Pk3|k −1 = Fk3,k −1 ( xˆ 0k ,−31|k −1 ) Pk3−,01|k −1 Fk3,k −1 ( xˆ 0k ,−31|k −1 ) + Q 3k −1 .
(
sk,k−1∆k −
Step 4. Innovation and Innovation Covariance The innovations { kj } and innovation covariances {S kj } for all three models are given, respectively, by j ˆj ˆj (62) k := z k − z k | k −1 = z k − h( x k | k −1 )
xˆ kj|k −1 using
k, k − 1
ωk−1
+
k−1|k−1 k,k−1
i =1
where the state space model is given in Section 3.4. Step 3. Prediction For each mode j = 1,2,..., r calculate the predicted state
s
1− ck,k−1
ωk−1
k−1|k−1 k,k−1
µ ki|−j1|k −1xˆ ik −1|k −1 , i =1
1 0
ωk−1
yˆk−1|k−1
−(xˆk−1|k−1sk,k−1 + yˆk−1|k−1ck,k−1)∆k (xˆ c −y s )∆
r
xˆ 0k ,−j1|k −1 =
r
−
−(xˆk−1|k−1sk,k−1 + yˆk−1|k−1ck,k−1)∆k (xˆ c − yˆ s )∆
Step 2. Mixing Suppose we have xˆ ik −1|k −1 and Pki −1|k −1 , i = 1, 2,…, r. We calculate the mixed initial state and covariance for each mode filter, j = 1, 2,..., r by
Pk0−, 1j|k −1
sk,k−1
ck,k−1∆k −
µ ki [Pki |k + (xˆ ik |k − xˆ k |k )(xˆ ik|k − xˆ k |k )′].
Pk |k =
(72)
i =1
6 Numerical Simulation and Results The target is assumed to move in the XY plane near the origin of the Cartesian coordinate frame and the truth trajectory consists of the NCV, NCA, and NCT models. Two GMTI sensors collect the range, azimuth, and radial velocity measurements. The sensor geometry for the
scenario is shown in Figure 1. The ground range, height and speed of each sensor are 100 km, 10 km above the XY plane, and 166.7 m/s, respectively. For each sensor, the standard deviations of the GMTI range, azimuth, and radial velocity measurements are 10 meters, 0.001 radian, and 1 m/s, respectively. There are seven different segments in the truth trajectory. Table 1 presents the dynamic model, time interval, sensor index, and measurement sampling interval for each segment of the trajectory. Table 2 presents the process noise parameters associated with the dynamic model in each segment. The angular velocity and acceleration (or deceleration) for the NCT and NCA models are also presented in Table 2. Sensor 2 Trajectory Sensor 2 Height 10 Km
YT Sensor 2 Ground Range 100 Km
7/NCV
---
XT
(1e-8, 1e-8) m/s2 1e-10 rad/s2 (1e-8, 1e-8) m/s2
We performed 200 Monte Carlo simulations where 200 truth trajectories and measurement sets were generated. Figure 2 shows a sample truth trajectory and GMTI report locations from two sensors. True dynamic models associated with different parts of the truth trajectory are also shown in Fig. 2. We used 200, 20, and 180 seconds for the expected sojourn times of the NCV, NCA, and NCT models, respectively. These values determine the diagonal elements of the transition probability rate matrix Λ according to (11). The off-diagonal elements of Λ were chosen to produce meaningful results. Table 3 presents the elements of Λ . True and IMM estimated position, velocity, acceleration, angular velocity, and mode probabilities are shown in Figures 3-9.
Sensor 1 Speed 600 Km/Hr
Sensor 1 Ground Range 100 Km
sdv White noise acceleration sdv White noise angular acceleration sdv White noise acceleration sdv
Table 3. Transition Probability Rate Matrix -0.0050 0.0040 0.0010 0.0250 -0.0500 0.0250 0.0017 0.0039 -0.0056
60 deg -30 deg
6/NCT
Angular velocity: 0.0087 rad/s
Truth Trajectory and Report Locations with 99% Prob. Error Ellipses 6000
Sensor 1 Trajectory
5000
Sensor 1 Height 10 Km
4000
Figure 1. Geometry of sensor 1 and sensor 2 trajectories.
NCV
NCV Y (m)
Sensor 2 Speed 600 Km/Hr
0.3 m/s2
3000
NCA
NCT
NCA
2000
Table 1. Characteristics of Target Trajectory Segment/ Time Sensor Dynamic Interval Measurement Sensor Model (s) Sampling Index Interval (s) 1/NCV 2/NCT 3/NCA 4/NCV 5/NCA 6/NCT 7/NCV
0-145 145-305 305-325 325-385 385-435 435-635 635-735
1 2,1 2 1 2 1 2
5 8,8 1 3 2.5 4 5
Table 2. Parameters of Target Truth Trajectory Segment / Dynamic Model 1/NCV
2/NCT
3/NCA
Kinematic Parameter --Angular velocity: -0.0145 rad/s
4/NCV
Acceleration : 0.6 m/s2 ---
5/NCA
Deceleration :
Process Noise Parameter Value White noise acceleration sdv White noise acceleration sdv White noise angular acceleration sdv White noise acceleration rate sdv White noise acceleration sdv White noise acceleration rate
(1e-8, 1e-8) m/s2 (1e-8, 1e-8) m/s2 1e-8 rad/s2 (1e-12, 1e12) m/s3 (1e-10, 1e10) m/s2 (1e-8, 1e-8) m/s3
1000
NCT
NCV
0 -3000
-2000
-1000
0
1000
2000
3000
4000
X (m)
Figure 2. A sample true target trajectory with GMTI report locations from two sensors. The 99% probability error ellipses are drawn about the report locations indicating the converted position accuracy. We note in Figures 3-5 that the position and velocity components of the state are estimated with reasonable accuracy. However, Figures 6-8 show that the estimator has difficulty in estimating the acceleration and angular velocity components of the state in certain time durations. It is interesting to note in Figures 6-7 that the estimator detects the maximum and minimum locations in some cases well, but fails in some other cases. Analyzing Figure 9, we note that the mode probabilities for the first two NCV segments are reasonable but not for the last NCV segment. The mode probabilities for the NCT segments have non-negligible contributions from the NCV mode.
Truth and Estimated Trajectories
True and Estimated Acceleration-Y 0.3
Truth IMM
5500
Truth IMM
0.25
5000 0.2
4000
0.15 AcclnY (m/s 2)
4500
Y (m)
3500 3000 2500
0.1 0.05
2000
0
1500
-0.05
1000
-0.1
500 -3000
-2000
-1000
0 X (m)
1000
2000
-0.15
3000
Figure 3. Target truth trajectory and IMM estimated trajectory from the first Monte Carlo run.
0
100
200
300
400 Time (s)
500
600
700
800
Figure 7. True and estimated Y component of target acceleration from the first Monte Carlo run.
True and Estimated Velocity-X 25 0.15
15
0.1
10
0.05 Omega (rad/s)
VelX (m/s)
True and Estimated Angular Velocities
Truth IMM
20
5 0
Truth IMM
0
-0.05
-5 -0.1
-10 -0.15
-15
0
100
200
300
400 Time (s)
500
600
700
800 -0.2
Figure 4. True and estimated X component of target velocity from the first Monte Carlo run. True and Estimated Velocity-Y 12
0
100
200
300
400 Time (s)
500
600
700
800
Figure 8. True and estimated angular velocity of the target from the first Monte Carlo run.
10
Mode Probabilities 1 0.9
Truth IMM
0.8
6
0.7 4
Mode Probability
VelY (m/s)
8
2
0
NCV NCA NCT
0.6 0.5 0.4 0.3
-2
0
100
200
300
400 Time (s)
500
600
700
800
Figure 5. True and estimated Y component of target velocity from the first Monte Carlo run.
0.2 0.1 0
0
100
200
300
400 500 Time (s)
600
700
800
900
True and Estimated Acceleration-X 0.6
Figure 9. Mode probabilities from the first Monte Carlo run.
Truth IMM
0.5
AcclnX (m/s2)
0.4
Figure 10 presents the traces of the average mean square error matrix (MSEM) and IMM calculated covariance from 200 Monte Carlo runs. The filter calculated covariance is close to the MSEM except a few cases.
0.3 0.2 0.1 0 -0.1 -0.2
0
100
200
300
400 Time (s)
500
600
700
800
Figure 6. True and estimated X component of target acceleration from first Monte Carlo run.
4
7
Trace(MSEM) and Trace(Cov-IMM)
x 10
Tr(MSEM) Tr(Cov-IMM)
6
5
Trace
4
3
2
1
0
0
100
200
300
400 Time (s)
500
600
700
800
Figure 10. Trace of the average MSEM and IMM calculated covariance from 200 Monte Carlo runs.
7 Conclusions We have described the algorithm for the IMM estimator for handling the transition probability matrix with variable measurement sampling intervals. This arises in practical multi-sensor tracking applications. Any number of dynamic models can be included in this framework. We described details of the algorithm by considering a maneuvering target with the NCV, NCA, and NCT models. We have presented preliminary results using 200 Monte Carlo simulations. Our future work will perform detailed analysis using extensive data sets to obtained improve results.
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