1. AAR Rendezvous Algorithm Progress Meeting 10 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 x θ u (i) UAV V u (i) (x u (i),y u (i)) State Variables: UAV Eqn’s of Motion: Control Variables: 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 1)Provide initial guess for control sequence, u(i) 2)Solve state equations s(i) based on guess from (1) 3)Evaluate terminal constraints, ψ 4)Back-solve for co-states given final co-states, λ(i) 5)Back-solve for value of optimality condition at each time step (want optimality H u =0 at each step) 6)λ(i), H u (i), ψ, into steepest descent formula to sequence toward optimal solution (calculate Δu(i)) 7)Store u(i)  u(i)+Δu(i) 8)If Δu(i) is small, sol’n is converged; else back to (1) 9)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