AAR Rendezvous Algorithm Progress Meeting 10 May 2005

1. 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

2. Discussion Outline • 2-D Rendezvous Formulation • Algorithm Details • Example #1 • Example #2 • Lessons Learned • Future Directions

3. 2-D Rendezvous Formulation y Vu(i) θu(i) UAV (xu(i),yu(i)) x Control Variables: State Variables: UAV Eqn’s of Motion: Flight Limits: Terminal Constraints:

4. Algorithm Details • Dynamic Optimization Strategy • Provide initial state, final state, final time, and guess for control sequence (interpolation) • Numerical Solution—Method 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

5. Algorithm Details • Provide initial guess for control sequence, u(i) • Solve state equations s(i) based on guess from (1) • Evaluate terminal constraints, ψ • Back-solve for co-states given final co-states, λ(i) • Back-solve for value of optimality condition at each time step (want optimality Hu=0 at each step) • λ(i), Hu(i),ψ, into steepest descent formula to sequence toward optimal solution (calculate Δu(i)) • Store u(i)u(i)+Δu(i) • If Δu(i) is small, sol’n is converged; else back to (1) • Iterate until solution converges or max iterations reached

6. Example #1

7. Example #2

8. 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

9. 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