Spacecraft Translational Motion

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Spacecraft Translational Motion

• The two-body problem involves the motion of a
  satellite caused by the gravitational attraction
  of a central body
• The circular orbit solution is well-known and is
  useful for many satellite applications
• This analysis is about the translational motion
  of a satellite close to a circular orbit

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Pop Quiz
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Pop Quiz (2)
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Pop Quiz (3)
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Generalized Eigenvectors
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Repeated Eigenvalues with Complete Set of
Eigenvectors
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Repeated Eigenvalues with Complete Set of
Eigenvectors (2)
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Repeated Eigenvalues with Incomplete Set of
Eigenvectors
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Repeated Eigenvalues with Incomplete Set of
Eigenvectors (2)
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Translational Motion Equations for a Satellite
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Orbital Frame

• Same as roll-pitch-yaw frame, for spacecraft
• The o3 axis is in the nadir direction
• The o2 axis is in the negative orbit normal
  direction
• The o1 axis completes the triad, and is in the
  velocity vector direction for circular orbits

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Translational Motion Equations for a Satellite
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Translational Motion Equations for a Satellite
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Verify that Circular Orbit is an Equilibrium
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Standard Form for EOM
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Linearize About Circular Orbit
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Linearize About Circular Orbit (2)
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Linearized System Near Circular Orbit

• Completing the partial derivatives and
  simplifying leads to

Clearly this system is controllable and
stabilizable (exercise) Also, the system is in a
decoupled form that is common for mechanical
systems
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Linearized System Is Decoupled

• Two rows and columns are decoupled from the rest

These two states are decoupled from the rest and
so the system can be arranged into a 2?2 system
and a 4?4 system
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Linearized System Is Decoupled (2)