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OPTIMIZATION OF CRITICAL TRAJECTORIES FOR ROTORCRAFT VEHICLES Carlo L. Bottasso Georgia Institute of Technology Alessandro Croce, Domenico Leonello, Luca Riviello Politecnico di Milano 60 th Annual Forum of the American Helicopter Society Baltimore, June 7 – 10, 2004. Outline.

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  1. OPTIMIZATION OF CRITICAL TRAJECTORIES FOR ROTORCRAFT VEHICLESCarlo L. BottassoGeorgia Institute of TechnologyAlessandro Croce, Domenico Leonello, Luca RivielloPolitecnico di Milano60th Annual Forum of the American Helicopter SocietyBaltimore, June 7–10, 2004

  2. Outline • Introduction and motivation; • Rotorcraft flight mechanics model; • Solution of trajectory optimization problems; • Optimization criteria for flyable trajectories; • Numerical examples: CTO, RTO, max CTO weight, min CTO distance, tilt-rotor CTO; • Conclusions and future work.

  3. Introduction and Motivation • Goal: modeling of critical maneuvers of helicopters and tilt-rotors. • Examples: Cat-A certification (Continued TO, Rejected TO), balked landing, mountain rescue operations, etc. • But also: non-emergency terminal trajectories (noise, capacity). • Applicability: • Vehicle design; • Procedure design. • Related work: Carlson and Zhao 2001, Betts 2001. TDP

  4. Introduction and Motivation • Tools: • Mathematical models of maneuvers; • Mathematical models of vehicle; • Numerical solution strategy. • Maneuvers are here defined as optimal control problems, whose ingredients are: • A cost function (index of performance); • Constraints: • Vehicle equations of motion; • Physical limitations (limited control authority, flight envelope boundaries, etc.); • Procedural limitations. • Solution yields: trajectory and controls that fly the vehicle along it.

  5. Introduction and Motivation • Mathematical models of vehicle: • This paper: • Classical flight mechanics model (valid for both helicopters and tilt-rotors); • Paper #8 – Dynamics I Wed. 9, 5:30–6:00: • Comprehensive aeroelastic (multibody based) models. Category A continued take-off with detailed multibody model.

  6. Rotorcraft Flight Mechanics Model Classical 2D longitudinal model for helicopters and tilt-rotors: (MR = Main Rotor; TR = Tail Rotor) Power balance equation:

  7. Rotorcraft Flight Mechanics Model • For helicopters, enforce yaw, roll and lateral equilibrium: • Rotor aerodynamic forces based on classical blade element theory (Bramwell 1976, Prouty 1990). • In compact form: • where: (states), • (controls) • helicopter: • tilt-rotor: add also , , but no , so that

  8. Trajectory Optimization • Maneuver optimal control problem: • Cost function • Boundary conditions (initial) • (final) • Constraints • point: integral: • Bounds (state bounds) • (control bounds) • Remark: cost function, constraints and bounds collectively define in a compact and mathematically clear way a maneuver.

  9. Numerical Solution Strategies for Optimal Control Problems Optimal Control Problem Optimal Control Governing Eqs. Indirect Discretize Discretize Direct Numerical solution NLP Problem • Indirect approach: • Need to derive optimal control governing equations; • Need to provide initial guesses for co-states; • For state inequality constraints, need to define a priori constrained and unconstrained sub-arcs. • Direct approach: all above drawbacks are avoided.

  10. Trajectory Optimization • Transcribe equations of dynamic equilibrium using suitable time marching scheme: • Time finite element method (Bottasso 1997): • Discretize cost function and constraints. • Solve resulting NLP problem using a SQP or IP method: • Problem is large but highly sparse.

  11. Implementation Issues • Use scaling of unknowns: • where the scaled quantities are , , , • with , , • so that all quantities are approximately of . • Use boot-strapping, starting from crude meshes to enhance convergence.

  12. Optimization Criteria for Flyable Trajectories Actuator models not included in flight mechanics equations (time scale separation argument) algebraic control variables Results typically show bang-bang behavior, with unrealistic control speeds. Possible excitation of short-period type oscillations. Simple solution: recover control rates through Galerkin projection: Control rates can now be used in the cost function, or bounded.

  13. Optimization Criteria for Flyable Trajectories • Optimization cost functions • Index of vehicle performance: • Performance index + Minimum control effort from a reference trim condition: • Performance index + Minimum control velocity: • Control rate bounds:

  14. Minimum Time Obstacle Avoidance • Optimal Control Problem (with unknown internal event at T1) • Cost function: • Constraints and bounds: • - Initial trimmed conditions at 30 m/s • - Power limitations

  15. Minimum Time Obstacle Avoidance Effect of control rates: negligible performance loss (0.13 sec for a maneuver duration of 13 sec). Longitudinal cyclic Fuselage pitch (Legend:w=0, w=100, w=1000)

  16. Category-A Helicopter Take-Off Procedure Jar-29:

  17. Optimal Helicopter Multi-Phase CTO • CTO formulation: • Achieve positive rate of climb; • Achieve VTOSS; • Clear obstacle of given height; • Bring rotor speed back to nominal at end of maneuver. • All requirements can be expressed as optimization constraints.

  18. Optimal Helicopter Multi-Phase CTO Cost function: where T1 is unknown internal event (minimum altitude) and T unknown maneuver duration. Constraints: - Control bounds - Initial conditions obtained by forward integration for 1 sec from hover to account for pilot reaction (free fall)

  19. Optimal Helicopter Multi-Phase CTO • Constraints (continued): • - Internal conditions • - Final conditions • - Power limitations • For (pilot reaction): • where: maximum one-engine power in emergency; • one-engine power in hover; • , engine time constants. • For :

  20. Optimal Helicopter Multi-Phase CTO Longitudinal cyclic rate bounds: Free fall (pilot reaction) Free fall (pilot reaction) { { Longitudinal cyclic rate Longitudinal cyclic (Legend:w=0, w=100, w=1000)

  21. Optimal Helicopter Multi-Phase CTO Fuselage pitch rate Fuselage pitch (Legend:w=0, w=100, w=1000)

  22. Optimal Helicopter Multi-Phase CTO Effect of control rates: negligible performance loss. Trajectory (Legend:w=0, w=100, w=1000)

  23. Optimal Helicopter Multi-Phase CTO Free fall (pilot reaction) Rotor angular velocity Power • As angular speed decreases, vehicle is accelerated forward with a dive; • As positive RC is obtained, power is used to accelerate rotor back to nominal speed.

  24. Max CTO Weight Goal: compute max TO weight for given altitude loss ( ). Cost function: plus usual state and control constraints and bounds. Since a change in mass will modify the initial trimmed condition, need to use an iterative procedure: 1) guess mass; 2) compute trim; 3) integrate forward during pilot reaction; 4) compute maneuver and new weight; 5) go to 2) until convergence. About 6% payload increase.

  25. Helicopter HV Diagram • Fly away (CTO): same as before, with initial forward speed as a parameter. • Rejected TO: • Cost function (max safe altitude) • Touch-down conditions • plus usual state and control constraints.

  26. Helicopter HV Diagram Deadman’s curve

  27. Helicopter HV Diagram Rotor angular speed Main rotor collective (Legend:Vx(0)=2m/s, Vx(0)=5m/s, Vx(0)=10m/s)

  28. Optimal Tilt-Rotor Multi-Phase CTO Formulation similar to helicopter multi-phase CTO. Cost function: plus usual state and control constraints and bounds. Trajectory Collective, cyclic, nacelle tilt, pitch

  29. Conclusions • Developed a suite of tools for rotorcraft trajectory optimization: • - Direct transcription based on time finite element discretization; • - General, efficient and robust; • Consistent control rate recovery gives more realistic solutions; • Applicable to both helicopters and tilt-rotors. • Successfully used for model-predictive control of large comprehensive maneuvering rotorcraft models (Paper #8 – Dynamics I Wed. 9, 5:30–6:00). • Work in progress: • - Noise as an optimization constraint, through Quasi-Static Acoustic Mapping (Q-SAM) method (Schmitz 2000).

  30. Optimal Helicopter Single-Phase CTO Effect of Control Rates • Pilot delay (forward integration, 0 T0=1sec) • Optimal Control Problem (T0 T (free)) • Cost function: • Constraints and bounds: • Initial and exit conditions • Power limitations

  31. Optimal Helicopter Single-Phase CTO Effect of Control Rates Free fall (pilot reaction) { Longitudinal cyclic speed Longitudinal cyclic (Legend:w=0, w=100, w=1000)

  32. Optimal Helicopter Single-Phase CTO Effect of Control Rates Fuselage pitch Fuselage pitch rate (Legend:w=0, w=100, w=1000)

  33. Optimal Helicopter Single-Phase CTO Effect of Control Rates Longitudinal cyclic speed bounds: Longitudinal cyclic Longitudinal cyclic speed (Legend:w=0, w=100, w=1000)

  34. Optimal Helicopter Single-Phase CTO Effect of Control Rates Longitudinal cyclic speed bounds: Fuselage pitch rate Fuselage pitch (Legend:w=0, w=100, w=1000)

  35. Optimal Helicopter Single-Phase CTO Effect of Control Rates Longitudinal cyclic speed bounds: Trajectory (Legend:w=0, w=100, w=1000)

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