IEEE - Aerospace and Electronic Systems - October 2021 - 21

Hennessy et al.
searching in Doppler-rate and Doppler-acceleration [13].
This is effective for the detection of space objects because
although the velocity is significant, and the rate at which
the geometry changes is significant, the trajectory itself is
stable and predictable. This is a common approach used
by other space surveillance radars, particularly incorporating
Doppler-rate, or chirp, adjustments to match radial
acceleration [22],[23].
However, previous results with the MWA searching
through these Doppler-rates were all based on a priori
information to determine the beamforming. Despite using
truth information for the beamforming, these detections
still required a search in delay, Doppler, Doppler-rate, and
Doppler-acceleration to form detections. Naively extending
these methods for uncued searching would be problematic,
with the four-dimensional search space needing
to be extended significantly to search in directional and
directional rate parameters.
Instead of forming a matched filter from a significant
number of measurement search parameters, it is possible
to limit the search space to realistic trajectories. As
detailed in earlier work [24], this Orbit Determination
before Detect (ODBD) approach can be achieved by
searching through hypothesized target position and velocity
and then deriving the measurement parameters. In this
way, the search space is limited in dimension, and the
measurement space can be completely unconstrained.
This type of approach is well suited to space surveillance
as orbital trajectories have very defined and predictable
motions, and are generally not manoeuvring. In fact,
the simplest form of an orbit is a two-body, or Keplerian,
orbit, which is completely determined by only six parameters.
This treats the Earth and the object of interest as point
masses, and assumes that gravity is the only force, with
the acceleration experienced given by the object's position
r and the standard gravitational parameter for Earth m:
r€ ¼
m
jrj3 rr;
(6)
From this, given a Cartesian position r and velocity _r,
an object's trajectory can be determined for all time. With
a Keplerian system, orbits form conics with one focal
point being at the (gravitational) center of the Earth; however,
in this work, we are only interested in elliptical
orbits. It should be noted that the orbits are only elliptical
in the Earth-centered inertial (ECI) coordinate frame. The
ECI coordinate system has an origin at the center of the
Earth; however, the coordinate system does not rotate
with the Earth. A six-tuple position and velocity will sufficiently
define an orbit for all time.
Because there is no closed-form solution to Kepler's
equation, numerical methods mustbeusedtocalculate
the parameters of interest over the CPI. We use Taylor
series approximations for the measurement parameters
OCTOBER 2021
of interest-Doppler and its subsequent rates as well as
the directions and their subsequent rates. Essentially,
this determines the motion-induced polynomial phase
signal coefficients from the position, velocity, (6), and
its instantaneous derivatives. Equation (3) can be
rewritten as
fD ¼
1
rrx r_ rx
þ
rrx
rtx r_ tx
rtx
(7)
with expressions for f_D, f€D, and any number of subsequent
derivatives of the Doppler frequency easily determined
in a similar manner. Likewise, the expression for
the topocentric right-ascension and declination, a and d,
respectively, are derived from the ECI coordinate frame.
These are the directional parameters used in place of azimuth
and elevation, and are determined from the slantrange
vector from the target to the receiver rrx ¼
r rrx ¼½rrxX
; rrxY ; rrxzT
a ¼ tan1 rrxY
rrxX
d ¼ tan1
B
!
rrxZ
@q
2 þ rrxY
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rrxX
1
C
A:
(8)
(9)
Although (6) is specific to Keplerian orbital trajectories,
any motion model could be applied to determine the
measurement parameters; taking
matching linear velocity.
CONSTRAINED SEARCH SPACE
The work in the previous section limits the search space
for uncued detection to the six parameters required to
define the orbit, the initial position, and velocity. This
enables searches to be limited to only the sets of measurement
parameters corresponding to orbits of realistic
Earth-captured Keplerian orbits. This is still a significant
search space. Not only is it six dimensional, a
given Cartesian position will still leave three slack variables
in the velocity vector. However, for a given position,
it is possible to further restrict the extent of
potential orbital velocities.
A method to significantly constrain this search is to
limit the search to solely match circular orbits. This
ensures that any resulting orbit will be realistic, it drastically
limits the potential range of resulting velocities, and
most importantly, it will not be overly restrictive. The vast
majority of objects in an Earth-captured orbit are in circular
or near-circular orbits. This is especially true away
from the equator.
IEEE A&E SYSTEMS MAGAZINE
21
€r ¼ 0 would result in
IEEE - Aerospace and Electronic Systems - October 2021

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