--I
N89-15951
Three-Axis Attitude Determination via Kalman Filtering
of
Magnetometer Data
by
Francois
Martel?,
Parimal
K.
Pal*,
and
Mark
L.
Psi&**
Abstract
A three-axis Magnetometer/Kalman Filter attitude determination system for a spacecraft in low-altitude
Earth
orbit is
developed, analyzed, and simulation tested. The motivation for developing this system
is
to achieve light weight and low cost for
an attitude determination system.
The extended Kalman
filter
estimates the attitude, attitude rates, and constant disturbance torques. Accuracy near that
of
the
International
Geomagnetic Reference Field model
is
achieved. Covariance computation and simulation testing demonstrate the
filter's
accuracy.
One test case, a gravity-gradient stabilized
spacecraft
with a pitch momentum wheel and a magnetically-anchored
damper,
is
a real satellite
on
which
this
attitude determination system
will
be
used.
This
work
is
similar
to
that
of
Heyler
[5].
The application
to
a nadir pointing satellite and the estimation of disturbance
torques represent the significant extensions contributed by
this
paper. Beyond
its
usefulness purely for attitude determination, this
system could
be
used
as
a part of a low-cost three-axis attitude stabilization system.
t
Vice President, Spacecraft Instruments Div., Ithaco Inc.
Attitude Control Analyst, Ithaco Inc.
**
Assistant Professor, Mechanical and Aerospace Engineering. Cornel1 University
Paper No.
17
for the Flight Mechanicsfistimation Theory Symposium, NASNGoddard Space Flight Center, Greenbelt
Maryland, May
10
&
11,1988.
344
Three-Axis Attitude Determination via Kalman Filtering
of
Magnetometer Data
by
FranFois Martel, Parimal
K.
Pal,
and
Mark
L.
Psiaki
1
Introduction
1.1
Objective
The objective of this work has been
to
develop a low-
cost system for estimation of 3-axis spacecraft attitude
information based solely
on
3-axis magnetometer
measurements from one satellite orbit. Such a system will be
useful
for
missions that operate in an inclined, low-Earth
orbit and require only coarse attitude information. It can also
serve
as
the sensor part of a low-cost 3-axis closed-loop
attitude control system,
or
as a back-up attitude estimator.
A single 3-axis magnetometer measurement can give
only 2-axes worth of attitude information and
no
attitude rate
or
disturbance torque information. Therefore, this attitude
determination system must use a sequence of magnetometer
measurements. It processes these measurements recursively
in a Kalman filter. This paper, then, describes the design,
development, analysis, and simulation testing of
a
Kalman
filter and reports its expected performance.
A
follow-on,
post-launch paper
is
planned
to
report actual performance.
1.2
Backgroundhior
Work
Kalman
filters have been widely applied
to
the problem
of spacecraft attitude determination [l-71. Everything from
star sensors 12.31
to
sun
sensors [4], gyroscopes [2], and
magnetometers [4,5] have been used for filter inputs, and
accuracies
as
fine
as
2
arc sec. are possible [3].
Very few attitude determination systems have
attempted to use
only
magnetometer data to estimate attitude.
Perhaps
this
is because of the low accuracy of the
measurements; even with perfect magnetometer
measurements, inaccuracy of the knowledge of the
Earth's
magnetic field may introduce errors of
0.4O
per axis.
Perhaps such systems are rare because of the complexity of
computing the
Earth's
magnetic field from spherical
harmonic models [6].
In
at least one case the benefits (low
cost and low weight) have outweighed the costs and such a
system has been developed. Heyler reports the use of such a
system
on
the NOVA program
[5].
That system was able
to
estimate spin
axis
attitude with a
2O
accuracy
as
well
as
spin
rate. These estimates were based
on
one eighth of an orbit's
worth of magnetometer readings.
The Khan filter reported in this paper uses
50
to 300
magnetometer samples distributed evenly over an orbit to
estimate 3-axis attitude, attitude rate, and disturbance torques
for a gravity-gradient-stabilized spacecraft. It is similar to the
filter described by Heyler in that 3-axis information is
derived purely from magnetometer measurement time
histories. It differs from Heyler's filter in two respects; it
estimates the attitude and rates for a different type of
spacecraft, and it estimates disturbance torques. Also
presented is
a
detailed accounting of the
various
contributions to estimation error, including the effects of
spacecraft dynamic modeling error.
13
Outline
of
Approach
The remainder of
this
paper contains descriptions of
the dynamic model of the spacecraft under consideration, the
filter design, and the filter evaluation criteria and procedures.
It concludes with the results of the filter evaluation. The
spacecraft description discusses the type of spacecraft
for
which
this
filter will work and presents notation and
equations necessary
to
the remaining sections. The filter
design section presents the overall filter structure and
two
different gain selection techniques. The section on evaluation
methodology describes the filter accuracy and stability
performance criteria and the tools that were used to gauge
these properties. The results of the accuracy and stability
evaluations are presented in the final section, which includes
examples of simulation time histories
as
well as numerical
measures of performance.
345
2
Spacecrapt
Dynamic
Model
2.1
MModOrbit
Characteristics
The Kalman filter
discussed
in
this
work
is
applicable
to
nadir pointing
Earth
satellites operating at low altitudes in
inclined orbits. The inclination and low altitude of the orbit
are necessary
to
the proper functioning of the filter. The orbit
must stay close enough to the Earth, within about
4
Earth
radii
[6],
so
that a spherical harmonic approximation of the
Earth's
magnetic field gives a reliable attitude reference.
Some inclination of the orbit
is
necessary
to
make the attitude
of all three
axes
sufficiently observable. Pitch information in
a l-orbit magnetometer time
history
gets poor for low
inclinations, although theoretically. there
is
still
some pitch
information even in equatorial orbits; the Earth's magnetic
poles
do
not coincide with its rotational poles.
This
study
considers spacecraft in nearly circular orbits at
1.1
to
1.2
Earth radii. Filter analysis and testing has been done for the
inclinations
43O
and
57'.
2.2
Spacecraft
Attitude
Dynamics
Model
The generic spacecraft (S/C) under consideration
is
a
gravity gradient stabilized spacecraft. One model
also
has
a
pitch momentum wheel for passive yaw stiffening and a
magnetically anchored damper for passive libration damping.
The following equations of motion model the spacecraft
attitude dynamics for purposes of filter state propagation:
h,
=
0
(3)
where
a
is
the
S/C's
inertial angular velocity vector,
I,
is the moment and product of inertia matrix,
n
is the
total
extemal vector torque acting on the S/C,
\
is the
constant vector angular momentum of the pitch wheel,
q
is a
quaternion that represents the orientation of the S/C-fixed
coordinate system with respect
to
an Earth-fmed coordinate
system,
.(uscE
is
the S/C's Earth-relative angular velocity,
and
nd
is
the disturbance torque (the net unmodeled external
torque).
All
of
the above are expressed in S/C-fied
coordinates except the quaternion.
It
is
expressed in Earth-
fixed
Coordinates. Equation
1
is
Euler's
equation for rigid
body rotational dynamics, and
eq.
2
is
the kinematic
equation for a quaternion
[6].
Equation
3
is special
to
the
filter.
It
represents the modeled disturbance torques.
The net external torque acting on the S/C,
n,
has been
divided into three components. gravity gradient torque,
ngg,
passive magnetically-anchored damper torque,
nhP,
and all
other unmodeled disturbance torques,
n,:
n
=
ngg+ ndamp+nd
(4)
The first two of these torque components, when present,
have
been
explicitly modeled for purposes of filter state and
covariance propagation.
The gravity gradient torque depends on the attitude
quaternion, the ephemeris, and the moments and products of
inertia:
where t
is
the time. The gravity gradient model used in
this
study neglects
J
effects
[6].
The magnetically-anchored damper torque depends on
the SIC-fixed magnetic field unit
vector
and
its
time rate of
change, which in
tum.
depend on the attitude quaternion, the
Earth-relative S/C angular velocity, and the ephemeris
[6]:
where
c-
is
the damping factor,
6
is
the magnetic field
unit vector in S/C-fied coordinates, and the derivative with
respect to time
is
the
total
derivative
(q
is
time varying).
The modeled disturbance torque,
n,,
may include the
effects of atmospheric drag, solar radiation pressure,
residual magnetic dipole moment, S/C dynamics modeling
errors,
or any other unmodeled extemal torques.
No
explicit
physical model of any of these torques
is
included. Rather,
this
term
is
retained in an effort
to
estimate these torques in
the filter by modeling them
as
a random walk process.
The coordinate systems used in this study are a S/C-
fmed
coordinate system, an Earth-fixed coordinate system,
346
and an orbit-following coordinate system. The S/C-fixed
coordinate system
is
a Roll-Pitch-Yaw coordinate system;
the x axis
is
nominally+ parallel
to
the velocity vector, the y
axis
is nominally anti-parallel
to
the orbit normal, and the
z
axis
is
nominally along nadir. This reference frame
is
used
to
define the equations of motion and related equations.
eq.
1-
6,
the inertia matrix,
I,,,
and the pitch wheel angular
momentum,
h,.
The orbit-following coordinate system defines the
nominal orientation of the gravity-gradient-stabilized
S/C.
Its
z
axis
is
exactly along nadir, its
y
axis
is
exactly anti-parallel
to
orbit normal, and its x
axis
is
approximately parallel
to
velocity (exactly parallel in the case
of
circular, nondecaying
orbits),
Its
only purpose in
this
study
is
as
a point
of
reference for measuring roll, pitch, and yaw angles in
reporting attitude results.
The Earth-fixed coordinate system has
its
origin
at
the
Earth's center.
Its
x
axis
passes
through
the equator at the
Greenwich meridian, its y
axis
passes
through
the Equator at
90°
East Longitude, and its
z
axis passes through the North
Pole. It
is
used
to
calculate the
S/C
ephemeris and the
Earth's magnetic field, which are used in torque modeling
and fiter update calculations. Because
this
reference frame
rotates with the Earth, there is a difference between the
S/C's
inertial angular velocity,
yx
and
its
angular velocity with
respect
to
this reference frame,
%sc/E:
(7)
where
A
is
the coordinate transformation matrix from Earth-
fixed to SIC-fixed coordinates defined by
q,
and
me
=
7.29~10-~ rad/sec
is
the
Earth's rotational angular velocity.
The angular velocity of the Earth
as
its revolves about the
Sun
has
been neglected
in
this
transformation.
Table
1
lists
the nominal values of the attitude
dynamics parameters for
two
S/C
examples. Spacecraft
1
is
stabilized by a long gravity gradient boom with a tip
mass,
a
constant
momentum pitch wheel, and a magnetically-
anchored damper. Spacecraft 2
has
a gravity-gradient boom,
but it is left neutrally stable in yaw. The tabulated parameter
values (sometimes with deliberately introduced
perturbations) apply
to
the analyses and simulations
described below.
23
Attitude Determination
Hardware
The only attitude determination sensor used by this
filter is a 3-axis magnetometer. It measures the magnetic
field vector in S/C-fixed coordinates:
b=
Ab
where
bw
the magnetic field in the Earth-Fixed coordinate
system, depends only on the
S/C
ephemeris. The A matrix
depends on
q,
so
eq.
8
defines the nonlinear measurement
equation used by the extended
Kalman
filter.
*
In
the absence
of
orbital eccentricity, librational motion,
disturbance torques, or product of inertia terms.
347
2.4
A
Linearized Altitude
Dynamics
Model
Linearized equations
of
motion and sensor equations
are useful for filter analysis and design.
This
involves
linearization of eq.
1,2,4,5,
6,
and
8.
They are linearized
about the nominal S/C attitude time history: z axis along
nadir, y axis along negative orbit normal, and y-axis angular
velocity equal
to
the orbital rate. The orbit
is
assumed
circular and
J,
effects are neglected.
As
a further
simplification, a dipole model of the Earth’s magnetic field
is
used
[6],
and the field at the
S/C
is
assumed periodic with
the orbital period (the rotation of the field with the
Earth
is
ignored).
The attitude quaternion has been linearized in a special
way. Instead
if
expressing
q
in terms of the sum of a
nominal value plus a perturbation, it is expressed
in
terms
of
a perturbation quaternion times the nominal quaternion using
quaternion multiplication:
rAqli
(9)
where, by definition of the nominal attitude time history,
qnm
defies the attitude of the orbit-following coordinate
system. The perturbational quaternion
is
already normalized
to within fist order in the Aqi.
This
perturbational
expression of the attitude has
just
three
unknowns; the fourth
is
not needed because angles are small, the equations are
linear, and no attitude singularity occurs.
The linearized equations are
A&
=
I&[An
-
AB
x
(IiNt&&,
+
h,)
An
=
An,,+ An,,+nd
(12)
where
Ag
is the perturbational S/C angular velocity
expressed in
s/c-futed
coordinates,
arb
is the orbital
angular velocity expressed in orbit-following coordinates (its
only nonzero element
is
its y element),
p
is
the geocentric
gravitational constant, rsc is the S/C geocentric radius,
Iij
is
the ij element of I,,
1
is
the identity matrix, and
b,,
is the
Earth‘s
magnetic field vector at the S/C expressed in orbit-
following coordinates.
These equations can be combined in standard state
vector format
to
yield a 9th-order system of the form
AX
=
F(t)
AX
+
z(t)
(16)
y
=
H(t)
AX
(17)
where the state is defined as
AxT
=
(Ag$,AqT,ndT) and
where the observation is
y
=
b
x
b
orb.
This
defiiition of
y
retains all of the attitude information in the magnetometer
measurements and gives an H(t) matrix consistent with the
innovation defiition given below (eq.
21).
The 9x9 F(t) and
matrix and the 9-element z(t) vector are derived from eq.
3,
10-14
and the defiition of
Ax.
The 3x9 H(t) matrix is
derived from eq.
15
and the definition of
y.
F(t), H(t), and
z(t) are
all
periodic at the orbital period because the magnetic
field has been assumed periodic at the orbital period. The
periodicity
of
this
linear system can
be
used
to
advantage in
filter design and analysis.
Ah
The presence
of
z
indicates that linearization has not
been done about the nominal motion.
As
can be seen from
eq.
13
and
14,
the nonhomogeneous terms result
from
product of inertia terms (a gravity gradient effect) and from
the time variation
of
the
Earth’s
magnetic field
as
experienced in the orbit following reference frame (a
magnetically-anchored damper effect). Nonzero
z
means that
the
S/C
is
not exactly trimmed at its nominal orientation.
This
out-of-trim condition is not vary large
(S-
lo),
and the
linearized model
is
a good approximation for small
perturbations from
trim.
348