AAR Rendezvous Algorithm Progress Meeting 10 May 2005

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AAR Rendezvous AlgorithmProgress Meeting10 May
2005

• REID A. LARSON, 2d Lt, USAF
• Control Systems Engineer
• MARK J. MEARS, Ph.D.
• Control Systems Engineer
• AFRL/VACA

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Discussion Outline

• 2-D Rendezvous Formulation
• Algorithm Details
• Example 1
• Example 2
• Lessons Learned
• Future Directions

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2-D Rendezvous Formulation
Control Variables
State Variables
UAV Eqns of Motion
Flight Limits
Terminal Constraints
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Algorithm Details

• Dynamic Optimization Strategy
• Provide initial state, final state, final time,
  and guess for control sequence (interpolation)
• Numerical SolutionMethod of Steepest Descent
• Adapted/modified based on proven strategy
  (Dynamic Optimization, Bryson)
• Solves for local minimum, which is acceptable in
  this case
• Simulation in MATLAB examples to follow

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Algorithm Details

• Provide initial guess for control sequence, u(i)
• Solve state equations s(i) based on guess from
  (1)
• Evaluate terminal constraints, ?
• Back-solve for co-states given final co-states,
  ?(i)
• Back-solve for value of optimality condition at
  each time step (want optimality Hu0 at each
  step)
• ?(i), Hu(i), ?, into steepest descent formula to
  sequence toward optimal solution (calculate
  ?u(i))
• Store u(i)?u(i)?u(i)
• If ?u(i) is small, soln is converged else back
  to (1)
• Iterate until solution converges or max
  iterations reached

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Example 1
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Example 2
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Lessons Learned

• Convergence and solution time fast in most cases
• Steepest Descent works well, even with poor
  initial guess
• Newton-Raphson technique requires very good guess
• Solutions found with flight constraints imposed
  directly
• Other algorithms had varying success
• Ritz-Method solutions (Mark Mears)
• Adjoin constraints to Hamiltonian analytically
  clean but difficult to automate in MATLAB
• Turn-on-dime maneuvers with large final time can
  be cumbersome for solver
• Requires feasible final state for rendezvous
• Focus on generating trajectory for UAV follow
  rendezvous trajectory based on position-error
  control

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Future Directions

• Near Term Focus (Next 2-3 Weeks)
• Convert equations of motion to three dimensions
• Minimum time rendezvous?
• Assess solution time and convergence to optimal
  solutions with added complexity
• MATLAB simulations to validate performance
• Long Term Focus (Next 2 Months)
• Incorporate refueling CONOPS
• AVDS simulations with J-UCAS vehicle model
• Transition algorithms to VACC/VACD/Boeing