17-03-2014, 08:52 PM
Abstract—The problem of localization and tracking of GMTs
(ground moving targets) is investigated based on
measurements of TDOA (time-difference of arrival) and DOA
(direction of arrival) for which the measurement noises are
assumed to be independent and identically distributed (i.i.d.).
The problem of the constrained linear MMSE (minimum
mean-squared error) estimation is formulated by employing
the pseudo-measurement model from the existing literature
that imposes a quadratic constraint on the state vector
associated with the GMT dynamics. Randomization of the state
vector for the GMT process suggests to replace the hard
constraint by its expectation. We first derive a solution to a
similar quadratically constrained MMSE estimation problem.
The constrained Kalman filtering is then developed for those
estimation problems involving quadratic constraints,
applicable to localization and tracking of GMTs based on
TDOA and DOA measurements. Moreover the constrained
Kalman filter admits a simple recursive solution with
complexity comparable to that of the conventional Kalman
filter. A simulation example is used to illustrate our proposed
constrained Kalman filter in localization and tracking of
GMTs.
Keywords-Kalman filter, TDOA, DOA
I. INTRODUCTION
Localization and tracking of GMTs using unmanned
aerial vehicles (UAVs) play an important role in combat
intelligence, surveillance, and reconnaissance (ISR) such as
ground moving target identification (GMTI) and moving
surface target engagement (MSTE). We assume that the
UAVs are equipped with passive sensors, and the noisy
TDOA and DOA data can be estimated or measured
sequentially. The objective is localization and tracking of the
threat GMTs of interest.
Passive source localization based on TDOA or DOA has
received considerable attention for the past a few decades.
For instance localization based on TDOA measurements has
been studied in [6], [7], [8], [9], [10], [18], [20], [23] while
the target motion analysis (TMA) has been investigated in
[2], [3], [4], [11], [15], [16], [17], [22]. Many different
algorithms have been developed and investigated for source
localization or tracking. More recent results include the twostep
LS algorithm for localization based on TDOA and
FDOA (frequency-difference of arrival) [13], posterior
Cram´er-Rao bound for bearing-only tracking [5], and
various techniques of tracking maneuvering targets [14].
Interested readers are referred to these research papers and
also references therein for various proposed solution
approaches. It is interesting to observe from the existing
literature that the TDOA data are exploited exclusively for
localization only whereas the DOA data are often associated
with tracking by assuming constant target velocity in TMA.
In this paper we consider the same passive source
localization and tracking problem based on both TDOA and
DOA measurements. It is well known that in tracking, the
target motion is classically described by a diffusion model
[14] or state-space model driven by some white noise
process. Using the quasi-linear measurement equations from
[6], [15] on TDOA and DOA respectively, localization and
tracking of GMT will be formulated into a sequential
estimation problem. Different from the existing solution
methods, the nonlinear term in the quasi-linear measurement
equations will be treated as a constraint which is in fact a
quadratic constraint. We propose a constrained Kalman filter
to tackle the localization and tracking problem based on
sequential measurements of TDOA and DOA where the
nonlinear term in the quasi-linear measurement equation is
taken as one of the state variables to be estimated.
Randomization of the state vector for the GMT process
suggests to replace the hard quadratic constraint by its
expectation. We will develop linear MMSE estimation
subject to the soft quadratic constraint, and then propose the
constrained Kalman filter to localize and track the GMT. It
will be shown that our proposed constrained Kalman filter is
a constrained linear MMSE estimator.
The organization of the paper is as follows. The problem
of target localization and tracking is formulated in Section 2
where measurements of TDOA and DOA are assumed. The
solution and algorithm are derived in Section 3 for
localization and tracking of the GMT based on sequential
measurement data, including the constrained Kalman filter.
Simulation studies are presented in Section 4 to illustrate the
effectiveness of the proposed constrained Kalman filter in
target localization and tracking. The paper is concluded in
Section 5. The notations are standard and will be made clear
as we proceed.
(ground moving targets) is investigated based on
measurements of TDOA (time-difference of arrival) and DOA
(direction of arrival) for which the measurement noises are
assumed to be independent and identically distributed (i.i.d.).
The problem of the constrained linear MMSE (minimum
mean-squared error) estimation is formulated by employing
the pseudo-measurement model from the existing literature
that imposes a quadratic constraint on the state vector
associated with the GMT dynamics. Randomization of the state
vector for the GMT process suggests to replace the hard
constraint by its expectation. We first derive a solution to a
similar quadratically constrained MMSE estimation problem.
The constrained Kalman filtering is then developed for those
estimation problems involving quadratic constraints,
applicable to localization and tracking of GMTs based on
TDOA and DOA measurements. Moreover the constrained
Kalman filter admits a simple recursive solution with
complexity comparable to that of the conventional Kalman
filter. A simulation example is used to illustrate our proposed
constrained Kalman filter in localization and tracking of
GMTs.
Keywords-Kalman filter, TDOA, DOA
I. INTRODUCTION
Localization and tracking of GMTs using unmanned
aerial vehicles (UAVs) play an important role in combat
intelligence, surveillance, and reconnaissance (ISR) such as
ground moving target identification (GMTI) and moving
surface target engagement (MSTE). We assume that the
UAVs are equipped with passive sensors, and the noisy
TDOA and DOA data can be estimated or measured
sequentially. The objective is localization and tracking of the
threat GMTs of interest.
Passive source localization based on TDOA or DOA has
received considerable attention for the past a few decades.
For instance localization based on TDOA measurements has
been studied in [6], [7], [8], [9], [10], [18], [20], [23] while
the target motion analysis (TMA) has been investigated in
[2], [3], [4], [11], [15], [16], [17], [22]. Many different
algorithms have been developed and investigated for source
localization or tracking. More recent results include the twostep
LS algorithm for localization based on TDOA and
FDOA (frequency-difference of arrival) [13], posterior
Cram´er-Rao bound for bearing-only tracking [5], and
various techniques of tracking maneuvering targets [14].
Interested readers are referred to these research papers and
also references therein for various proposed solution
approaches. It is interesting to observe from the existing
literature that the TDOA data are exploited exclusively for
localization only whereas the DOA data are often associated
with tracking by assuming constant target velocity in TMA.
In this paper we consider the same passive source
localization and tracking problem based on both TDOA and
DOA measurements. It is well known that in tracking, the
target motion is classically described by a diffusion model
[14] or state-space model driven by some white noise
process. Using the quasi-linear measurement equations from
[6], [15] on TDOA and DOA respectively, localization and
tracking of GMT will be formulated into a sequential
estimation problem. Different from the existing solution
methods, the nonlinear term in the quasi-linear measurement
equations will be treated as a constraint which is in fact a
quadratic constraint. We propose a constrained Kalman filter
to tackle the localization and tracking problem based on
sequential measurements of TDOA and DOA where the
nonlinear term in the quasi-linear measurement equation is
taken as one of the state variables to be estimated.
Randomization of the state vector for the GMT process
suggests to replace the hard quadratic constraint by its
expectation. We will develop linear MMSE estimation
subject to the soft quadratic constraint, and then propose the
constrained Kalman filter to localize and track the GMT. It
will be shown that our proposed constrained Kalman filter is
a constrained linear MMSE estimator.
The organization of the paper is as follows. The problem
of target localization and tracking is formulated in Section 2
where measurements of TDOA and DOA are assumed. The
solution and algorithm are derived in Section 3 for
localization and tracking of the GMT based on sequential
measurement data, including the constrained Kalman filter.
Simulation studies are presented in Section 4 to illustrate the
effectiveness of the proposed constrained Kalman filter in
target localization and tracking. The paper is concluded in
Section 5. The notations are standard and will be made clear
as we proceed.