1. Georgia ofTech Institutenology 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

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

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

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

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

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

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

9. Critical Trajectory Optimization Numerical Solution Strategies for Optimal Control Problems Optimal Control Problem Optimal Control Governing Eqs. Discretize NLP Problem Numerical solution Direct Indirect Indirect approach 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 Direct approach: all above drawbacks are avoided.

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

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

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

13. Critical Trajectory Optimization 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 Control rate bounds:

14. Critical Trajectory Optimization Optimal Control Problem Optimal Control Problem (with unknown internal event at T 1 ) Cost function: Constraints and bounds: - Initial trimmed conditions at 30 m/s - Power limitations Minimum Time Obstacle Avoidance

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

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

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

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

19. Critical Trajectory Optimization Constraints (continued) 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 : Optimal Helicopter Multi-Phase CTO

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

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

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

23. Critical Trajectory Optimization Optimal Helicopter Multi-Phase CTO Power Rotor angular velocity Free fall (pilot reaction) 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. Critical Trajectory Optimization Goalmax TO weight Goal: compute max TO weight for given altitude loss ( ). Cost function: plus usual state and control constraints and bounds. iterative procedure 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. 6% payload increase About 6% payload increase. Max CTO Weight

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

26. Critical Trajectory Optimization Helicopter HV Diagram Deadman’s curve

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

28. Critical Trajectory Optimization 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. Critical Trajectory Optimization Conclusions rotorcraft trajectory optimization 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. model-predictive controllarge comprehensive maneuvering rotorcraft models 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 - Noise as an optimization constraint, through Quasi-Static Acoustic Mapping (Q-SAM) method (Schmitz 2000).

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

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

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

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

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

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