Geometric Attitude Control of a Small Satellite for Ground Tracking Maneuvers Barry Goeree and Brian

1
Geometric Attitude Control of a Small Satellite
for Ground Tracking Maneuvers Barry Goeree and
Brian ShuckerUniversity of Arizona
2
Outline of the presentation

• Introduction
• Ground tracking kinematics
• Attitude control law
• Simulation results
• Summary and discussion

3
UASat fact sheet

• The UASat is being designed to be launched from
  the Hitchhiker Ejection System (HES).
• Space shuttle orbit is used to design the
  satellite.
• altitude 407 km
• inclination 51.6o
• The HES restricts the dimensions of the
  satellite.
• Max. height 52 cm (20.5 in)
• Max. diameter 50 cm (20 in)
• Max. weight 68 kg (150 lb)
• There are three missions.
• Detection and imaging of lightning and sprites.
• Photometering of stars.
• Up-link communication by laser signals.

4
Attitude control system overview

• Three-axis stabilization is required.
• Attitude will be determined using Kalman
  filtering and several sensors
• Magnetometer
• GPS (spatial)
• Sun sensors
• Horizon sensor
• Assume that an accurate attitude estimate is
  available.
• The following actuators will be used
• Three torque rods
• Four reaction wheels

5
Ground tracking maneuver
6
Control law overview
7
Elastic and viscous term (1)

• The elastic and viscous term provide feedback.
• The viscous term is a damping term.
• Elastic term implements a spatial spring-like
  behavior.
• The relative rotation between desired and actual
  attitude is
• Define stored energy as function of this relative
  rotation
• where P is a user selectable matrix of
  co-stiffnesses.
• Energy is minimized when frame vectors are
  aligned.

8
Elastic and viscous term (2)

• The torque can be found using a virtual work
  argument.
• Virtual perturbation attitude
  
• Virtual work equals change in potential function
• Much computation yields a simple result
• For small attitude errors
  the control parameters have an intuitive
  interpretation.
• One can show that for small errors
  where the stiffness matrix K is defined as

9
Model based term

• The attitude dynamics are given by
  where the reaction wheel
  configuration matrix is defined as
• Assuming that the external disturbances are
  negligible, one can solve for the torque to match
  the desired acceleration

z
e4,rw
e2,rw
e3,rw
y
x
e1,rw
10
Wheel speed management term

• The wheel speeds should be kept close to a
  specified desired speed to avoid avoid zero
  crossings and to let the bearings operate in the
  hydro-dynamic lubrication regime.
• Let then the term does not contribute
  to the torque on the satellite core
• The vector eN follows from Erw but how to pick
  scalar a ?
• Define the error as
• It is easy to show that
• The second term is negative when

11
Selection of control parameters

• For rotations about one of the principal axes of
  stiffness, the attitude dynamics are approximated
  by a second order system
• How to pick the stiffness and damping
  coefficients?
• Let torque saturate at 2?
• Let the damping be critical
• This corresponds to a natural frequency of about
  0.06Hz

12
Whats included in the simulations?

• Attitude dynamics and orbital kinematics are
  simulated.
• Perfect attitude knowledge is assumed.
• The control laws are sampled at 4Hz.
• The aerodynamic drag torques are modeled. Solar
  pressure, gravity gradient and residual magnetic
  moment are not modeled.
• The reaction wheels models include Coulomb and
  viscous friction, limited torque capability
  (7.410-3 Nm), misalignment (4o), uncertainty in
  gain (10) and uncertainty in inertia (4).
• The satellite core model includes uncertainty in
  the moments of inertia (5) and principal axes of
  inertia (4o).

13
Telescope axis pointing error
Pointing error (degrees)
Time (s)
14
Reaction wheel torques
-3
1
Reaction Wheel Torques (Nm)
Time (s)
15
Reaction wheel speeds
1000
900
800
700
600
Reaction Wheel Speeds (rpm)
500
400
300
200
100
0
0
200
400
600
800
1000
1200
Time (s)
16
Summary and discussion

• A geometric attitude controller was designed for
  the control of ground tracking maneuvers.
• Attitude error is defined intrinsically and
  intuitively as the relative rotation between
  actual and desired attitude.
• The control law is nonlinear.
• Control law consists of elastic and viscous
  terms, a model based compensation term, and a
  wheel speed management term.
• For small errors, the control parameters have an
  intuitive interpretation (stiffness and damping
  around axes) and can be chosen using linear
  controller design techniques.
• Control law is stable for large errors if torques
  do not saturate.
• Control laws were verified in simulation.
• Attitude estimation error will dominate overall
  error.

17
Questions? Comments?
18
The Student Satellite Project at the University
of Arizona is the best evidence I have discovered
anywhere of the creative initiative of Americans
committed to the Space Program, which has been an
important part of my life for forty years. When
I joined JPL as a young engineer in 1958, soon
after the launch of America's first satellite,
the adventure of space exploration had captured
the imagination of young people all over
America, and there was no bureaucracy to slow us
down. The new NASA in 1998 recognizes the
importance of youthful energy and innovative
capacity, and welcomes such initiatives as the
SSP. This is a very exciting development,
heralding as new day for both NASA and our
students. They have done their part, with the
encouragement of the University of Arizona. Now
it is time for the community to step up to the
challenge of demonstrating that all of Arizona
stands behind this incredible initiative of the
young men and women of the SSP who are reaching
beyond the skies. Peter Likins, June 17, 1998.
19
Reaction wheel specifications
20
Power drawn by reaction wheels
21
Atmospheric Drag

• Satellite is modeled as collection of faces
• For each contributing face (Wertz)
• Torques on the order of 1e-5 Nm