Navigation of Strategic Submarines

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Effect of Gravity Gradients on Spacecraft in
Near-Earth Orbits January 25, 2007
Alan Zorn, Ph.D. Candidate Dept. of Aeronautics
Astronautics Stanford University
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Gravity Gradients Friend and Foe

• Gravity gradients affect spacecraft attitude and
  orbit when
• Spacecraft is asymmetric (Imax ? Imin)
• Perigee is sufficiently small (
• Beneficial uses
• Attitude control
• Orbit control
• Gravity gradient effect is considered a nuisance
  for many applications, including solar sailing
• Particularly in attitude control, where gravity
  gradient is a major source of torque disturbance
  for large structures operating near the Earth

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Sources of Disturbance Torques

• External
• Solar radiation
• Gravity gradient
• Atmospheric drag
• Magnetic fields
• Internal
• Motions of flexible appendages
• Thruster misalignments and control uncertainties
• Fuel slosh
• Rotating machinery such as solar arrays and
  antenna drives
• Differential thermal loading of a spacecraft

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Gravity Gradient Attitude Control

• Torque induced by gravity gradient on spacecraft
  due to point mass Earth
• Spacecraft will tend to align its axis of minimum
  moment of inertia vertically
• Torque is zero for completely symmetric
  spacecraft (Imax Imin)
• Torque decreases rapidly (1/r3) as spacecraft
  recedes from Earth
• Can be used to steer certain components of
  attitude Tong, 1998
• Momentum dumping of spacecraft in highly
  eccentric orbits
• Can counteract unwanted torque induced by solar
  radiation on asymmetric solar panels

• Case history Geosat
• Nadir-pointing mission (?1?)
• 760 ? 817 km orbit inclined 108.1?
• 635 kg with 45 kg end mass at end of 6 m boom
• Gravity gradient stabilized aided by momentum
  wheels and thrusters

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Gravity Gradient Attitude Dynamics

• Spacecraft attitude dynamics found by solving
  Euler equations
• 1D case 48.7 min period at zero altitude
• Phase plane

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Orbit Control Watanabe Nakamura, 1998

• Gravitational force at spacecraft center of mass
  due to point mass Earth is
• Second term is gravity gradient force (sometimes
  called tidal force)
• Zero for completely symmetric spacecraft (Imax
  Imin)
• Decreases very rapidly (1/r4) as spacecraft
  recedes from Earth
• Can steer certain orbital parameters (p, e, ?) by
  applying one or both types of control inputs
• Vary attitude of a mass-invariant spacecraft
  (constant I)
• Vary mass configuration of spacecraft (variable
  I)
• Orbital parameters related to angular momentum
  vector (i, ?) cannot easily be controlled
• Perhaps because fG is mostly a central force

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Gravity Gradient Torque Disturbance

• Need to distinguish between two cases
• Some approaches to modeling gravity gradient
  disturbance
• One paper addresses elastic flexure with
  parametric models, with gravity gradient as a
  torque disturbance Di Gennaro, 1998
• Another paper addresses thermally-induced
  deformation which alters gravity gradients
  torques acting on the spacecraft Johnston
  Thornton, 1996

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Numerical Examples and Punch Line for Solar
Sailing

• Geosat
• Total mass is 635 kg with 45 kg end mass at end
  of 6 m boom
• Izz 1500 kg-m2 ? Tmax 0.0035 N-m at 45?
  between boom and nadir
• Solar sail
• Sail/mast mass is 14 kg with 40 m ? 40 m square
  sail
• Izz 1867 kg-m2 ? Tmax 0.0044 N-m at 45?
  between sail plane and nadir
• Early validation of solar sail in high LEO (1000
  km) Murphy Wie, 1998
• Gravity gradient is largest torque disturbance
  for this mission
• Gravity gradient torque would saturate any
  reasonably-sized attitude control system unless
  nominal orbit and attitude is carefully selected
• Can be used beneficially for stabilization if
  sails were oriented with nadir
• Sail validation planners are posed with a
  dilemma Can an affordable launch and
  appropriate orbit be found that allows this
  promising inter-planetary propulsion technology
  to set sail beside the dangerous reefs of
  Earths near-space environment?

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References

• Tong, D., Spacecraft Momentum Dumping Using
  Gravity Gradient, Journal of Spacecraft and
  Rockets, vol. 35, no. 5, 1998.
• Watanabe, Y. and Nakamura, Y., Orbit Control for
  a Spacecraft via the Gravity Gradient Force,
  49th International Astronautical Congress, 1998.
• Di Gennaro, S., Adaptive Robust Tracking for
  Flexible Spacecraft in Presence of Disturbances,
  Journal of Optimization Theory and Applications,
  vol. 98, no. 3, 1998.
• Johnston, J. D., and Thornton, E. A., An
  evaluation of thermally-induced structural
  disturbances of spacecraft solar arrays, IEEE
  Energy Conversion Engineering Conference, 1996.
• Murphy, D. and Wie, B., Robust Thrust Control
  Authority for a Scalable Sailcraft, 14th
  AAS/AIAA Space Flight Mechanics Conference, Maui,
  Hawaii February 8-12, 2004.