STEERING LAWS FOR CONTROL MOMENT GYROSCOPE SYSTEMS USED IN SPACECRAFTS ATTITUDE CONTROL

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STEERING LAWS FOR CONTROL MOMENT GYROSCOPE
SYSTEMS USED IN SPACECRAFTS ATTITUDE CONTROL
Middle East Technical University Aerospace
Engineering Department

• by Emre YAVUZOGLU
• Supervisor Assoc. Prof. Dr. Ozan TEKINALP

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Outline

• Objectives
• Properties of SGCMGs
• Overview of Steering Laws
• Simulation Work I
• CMG based ACS Model
• Simulation Work II
• Conclusion

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Objectives

• Investigation of the kinematic properties of
  SGCMGs ( singularity problem)
• Steering laws
• Existing steering laws
• Development of new steering laws
• Comparison of steering laws through simulations

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SGCMGs

• Momentum exchange device
• A SGCMG consists of
• Flywheel
• (spinning at a constant rate)
• Gimbal motor
• (to change the direction of h)

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The output torque is
(Torque amplification)
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4-SGCMG Cluster in a Typical Pyramid Mounting
Arrangement

• 3 CMGs to provide full 3-axis attitude control
• 1 CMG to provide extra degree of control
• (min. redundancy for singularity)

ß54.73º to the horizontal
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• Total angular momentum for pyramid configuration

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Total output torque (time rate of change of total
h)
where instantaneous system Jacobian matrix
However, in ACS part we need to determine gimbal
rate, that provides the required torque. Thus, we
need an inversion of torque equation
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• Minimum two-norm solution of this problem gives
  Moore Penrose pseudo inverse

• Most of the steering laws is pseudo inverse
  based. However, the main problem of these methods
  are SINGULARITY (Although CMG cluster is
  redundant).

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What is Singularity?

• Mathematically When J loses rank (rank2),
  (JJT)-1 undefined.
• Physically all output torque vectors remain on
  the same plane (rank2). No output torque can be
  produced along direction, s, normal to this
  plane. (s singularity direction). Three axis
  controllability is lost.

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Singularity Measure

• (System is how much close to the singularity)

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• Singular states directions produces singular
  surfaces in momentum envelope created by mapping
  of gimbal angle set to angular momentum space of
  the cluster.
• Singularity types seen in momentum envelope are
  summarized in a detailed fashion according to
  number of criteria in Chp 3 and Appendix A-3. The
  most dangerous ones are internal elliptic
  singularities.

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Overview of Steering Laws

• MP INVERSE
• (high possibility of encountering singular
  states)

• 2. SR Inverse
• 3. GSR Inverse
• 4. IG Method

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2. Singular Robust Inverse

• Transition method adapted from robotic
  manipulators
• (As singularity is approached small torque
  errors are permitted to transit through it.)

• a, the singularity avoidance parameter to be
  properly selected.
• It can be shown that the matrix within brackets
  is never singular.
• DIS Although singularity measure never becomes
  zero, internal elliptic singularities still can
  not be passed with SR!

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3. Generalized Singular Robust Inverse

• Modified version of SR inverse
• As singularity is approached, deliberate
  deterministic dither signals of increasing
  amplitude are used to get out of singularity
  quickly

where

• DIS Not suitable for precision tracking missions

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4. Inverse Gain

• Previous particular solutions can be combined
  with homogenous solution of torque equation to
  avoid singularities (null motion)

• DIS Null rates may become extremely high, even
  though system is away from singularity.

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New Steering LogicUnified Steering Law

• Starting aim in the development was to find
  gimbal rates both satisfying torque commanded
  and, driving the gimbals to desired nonsingular
  configurations, spontaneously.

• Derived solving the following minimization
  problem

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Resulted gimbal rate equation

• Through simulations we have observed that
  selection of desired gimbal rate, and blending
  coefficient, q, are the key points in the
  utilization of the method. According to this
  selection, 2 approaches are proposed

1. Preplanned Steering
2. Online Steering

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1. Preplanned Steering

• h trajectory to be followed is known beforehand
• Gimbal angle solutions with higher m satisfying
  the h at discrete time points (nodes) are
  computed using SA.
• Then, system is driven to desired gimbal
  solutions at these nodes by adjusting the gimbal
  rates as

• DIS Only requirement to steer desired gimbal set
  is that required h trajectory should be known
  priori.

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2. Online Steering Approaches

• For selection of desired gimbal rate in USL
  Eqn.

• Homogenous gimbal rates found by IG
• Arbitrary constant vector
• Intelligently selected constant vector
• Dynamic vector with randomly changing elements
  (White Noise)

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Simulations I

• Constant torque study

Ideal Profiles
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MP SR Fails at internal elliptic singularity!
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GSR works as transition method(1.5 sec delay,
high gimbal rates)
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USL Preplanned

• 8 nodes are used. Successfully, desired torque is
  realized while accurately achieving desired
  gimbal set at nodes.

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USL Online using Null vector
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USL Online Using Constant Vector (with
dynamically adjusted blending coefficient
q0.5exp(-10m))

• Steering w. arbitrary vector 0,1,0,0

• Steering w. intelligently selected vector

• Steering w. white noise

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USL Preplanned

• Corner maneuver

Cyclic Disturbance
Repeatability maneuver
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CMG based ACS Model

• Three main parts to be considered
• Spacecraft Dynamics
• Quaternion Based Feedback Controller
• CMG Steering Law

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Spacecraft Dynamics
Total angular momentum equation
Corresponding rotational EoM of a rigid S/C
equipped with momentum exchange actuators such as
CMGs, in general given by
Text the external torque vector including the
gravity gradient, solar pressure, and aerodynamic
torques all expressed in the same S/C body axes.
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Combining these 2 equations, we simply obtain
u- u
u Internal control torque input generated by CMG
and transferred to S/C
Rewriting equation in two parts
By using last two equations, and combining them
with S/C kinematics equations (such as
quaternions), an ACS can be designed. Assuming
S/C control torque input is known, the desired
CMG momentum rate is selected as
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Simulations II
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USL Preplanned Results
Desired h profile from ideal system
Gimbal profiles obtained with USL
Attitude profile obtained with USL
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USL Online Results

• Although simulation is started this time at
  internal elliptic singularity (i.e. 90, 0, -90,
  0deg), USL online method effectively takes the
  system away from singularity rapidly, and
  maneuver is completed on time!
• Arbitrarily selected vector 0,1,0,0 is used as
  desired gimbal rate with dynamically adjusted
  blending coefficient.

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Attitude Hold Maneuver
A hypothetical cyclic disturbance torque, Text is
given to the system

• Despite of the disturbance acting about one
  orbital period (5400 s), the spacecraft is
  commanded to maintain its initial attitude of
  RPYinitial 0,0,0 all the time.
• USL is used preplanned and online fashion for
  Attitude Hold maneuver. Both are successful and
  repeatable gimbal histories are observed.

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USL Attitude Hold Results
Attitude profile obtained with USL online
Gimbal History (Repeating pattern)
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CONCLUSION

• A new original robust steering law is presented.
  The steering law combines desired gimbal rates
  with torque requirements in a weighted fashion.
• The law can be employed in a both preplanned and
  spontaneous fashion. The repeatability of the
  approach is demonstrated.
• Singularity is not a problem with this method.
  Through simulations, it is demonstrated that it
  can replace all existing steering laws.
• This method may also be adapted to robotic
  manipulators as a future work.

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