1. POLI di MI tecnicolano Carlo L. BottassoGiorgio Maisano Francesco Scorcelletti TRAJECTORY OPTIMIZATION PROCEDURES FOR ROTORCRAFT VEHICLES, THEIR SOFTWARE IMPLEMENTATION AND THEIR APPLICABILITY TO MODELS OF VARYING COMPLEXITY Carlo L. Bottasso, Giorgio Maisano Politecnico di Milano Francesco Scorcelletti AgustaWestland & Politecnico di Milano AHS Annual Forum & Technology Display Montréal, Québec, Canada, April 29 – May 1, 2008

2. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA Outline Introduction and motivation The maneuver optimal control problem Solution of maneuver optimization problems: - Direct transcription - Direct multiple shooting Numerical examples and applications Conclusions and future work

3. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA Goal boundaries of the flight envelope Goal: modeling of critical maneuvers of helicopters and tilt-rotors at the boundaries of the flight envelope Examples Examples: Cat-A certification (Continued TO, Rejected TO), ADS-33, autorotation, tail rotor loss, mountain rescue operations, etc. Applicability Applicability: - Vehicle design - Design of procedures, certification Related work Related work: Okuno & Kawachi 1993, Carlson & Zhao 2002, Bottasso et al. 2004 Introduction and Motivation TDP

4. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA 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. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA Vehicle Models Flight mechanics Flight mechanics helicopter and tilt-rotor models (this paper) Comprehensive aeroelastic Comprehensive aeroelastic multibody-based models (not covered here, see Bottasso et al. 2005-2008) ADS-33 sidestep & Category A CTO - multibody model (full-FEM flexible main and tail rotors, main rotor control linkages, Peters-He aerodynamics):

6. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA Direct transcription Actuator disk-type Algebraic inflow Flapping blade Dynamic inflow Full FEM Refined Aerodynamics Model complexity – Computational cost per physical time unit Direct multiple shooting MMSA Preferred Methods for Vehicle Models of Increasing Complexity This paper

7. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA Maneuver Optimal Control Problem (MOCP) Maneuver Optimal Control Problem (MOCP): Cost function Vehicle model 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

8. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA Numerical Solution of Maneuver 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 (impossible for third-party black- box vehicle models) 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

9. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA TOP: Trajectory Optimization Program for Rotorcraft Vehicles Supported vehicle models Supported vehicle models: FLIGHTLAB ©, Europa or other black-box initial value solvers In-house-developed model: Blade element and inflow theory (Prouty, Peters) Quasi-steady flapping dynamics, aerodynamic damping correction Look-up tables for aerodynamic coefficients of lifting surfaces Compressibility effect and downwash at tail due to main rotor Implemented direct solution strategies Implemented direct solution strategies: Direct transcription Direct multiple shooting

10. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA 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 banded Problem is large but highly sparse and banded Direct Transcription

11. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA Remarks Remarks: Rigorous and trivial treatment of state and control constraints Optimality of NLP problem converges to optimality of OC problem as grid id refined Two-point bounday value solver: unstable solution modes do not lead to catastrophic failures as with shooting (ideal for unstable rotorcraft problems) Models with very fast solution components need very small time steps: very large problems, excessive computational cost (size of NLP dictated by time step) best method Typically best method, but applicable only to models of low-moderate complexity with slow solution components Direct Transcription

12. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA 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 Direct Transcription

13. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA Partition time domain into shooting segments: Discretize controls as: Advance solution from to using time steps. Glue segments together with NLP constraints: NLP unknowns NLP unknowns: Direct Multiple Shooting

14. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA Remarks Remarks: Can handle models with fast solution components (size of NLP unrelated to time step) Need special techniques to handle state constraints within shooting segments For state constrained problems, it does not approximate the optimal control problem when the grid is refined (no state constraints within shooting arcs) Need care when dealing with unstable problems: multiple segments necessary for curing (alleviating) instability of single shooting Direct Multiple Shooting

15. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA Grid Refinement Minimum time 90-deg turn, constrained maximum roll rate FLIGHTLAB helicopter model, actuator disk-type rotor Direct transcription Direct transcription method Uniform gridFinal adapted grid

16. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA Grid Refinement Evolution of grid throughout refinement steps

17. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA Grid Refinement multiple shooting Same problem and model, multiple shooting, 16 shooting arcs violations State constraint violations within arcs Zoom ▶ Solution after two refinement steps, 26 total resulting shooting arcs ▼

18. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA Grid Refinement direct transcription multiple shooting Computed control inputs, direct transcription and multiple shooting

19. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA Applications: ADS-33 MTEs Mission Task Elements Mission Task Elements (MTE): assessment of ability to perform critical tasks constrained Optimal Control problems All MTEs can be formulated as constrained Optimal Control problems Example Example: lateral translation MTE Merit function: Good Visual Environment, cargo/utility: Longitudinal and lateral tracking error of ±10 ft, heading error ±10 deg Maneuver duration T≤18 sec

20. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA Applications: ADS-33 MTEs

21. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA Requirements Requirements: Achieve positive rate of climb Achieve V TOSS Clear obstacle of given height Bring rotor speed back to nominal Optimal Helicopter Multi-Phase CTO

22. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA 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

23. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA 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

24. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA Optimal Helicopter Multi-Phase CTO Trajectory (Legend: w=0, w=100, w=1000) negligible performance loss Effect of control rates: negligible performance loss

25. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA 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

26. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA Maneuver flown symmetrically (2D): Side-slipping (3D): Remark Remark: 3D non- symmetric side- slipping Cat-A CTO reduces altitude loss of about 10% Optimal Helicopter Multi-Phase CTO

27. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA 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

28. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA Minimum time 180-deg turn FLIGHTLAB helicopter model Direct transcription Applications: Minimum Time Turn Resulting optimal strategy Resulting optimal strategy: classical bank and turn

29. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA Resulting optimal strategy Resulting optimal strategy: flare, then turn at high side-slip Minimum time 180-deg turn FLIGHTLAB ERICA tilt-rotor model Direct transcription Applications: Minimum Time Turn

30. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA Helicopter Obstacle Avoidance Maneuver

31. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA Conclusions rotorcraft trajectory optimization Developed a suite of tools for rotorcraft trajectory optimization: - Multiple solution strategies: - Direct transcription - Direct multiple shooting - Multiple vehicle models: - In-house-developed model - External black-box models - General, efficient and robust - Adaptive grids for numerical efficiency and accuracy - Applicable to both helicopters and tilt-rotors

32. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA Outlook Transition to industry (AW), and expansion of the applications Pilot models ERICA tilt-rotor: H-V diagrams, Cat A certification Incorporation of Filter Error Parameter Identification Method into TOP (constrained optimization problem solved by multiple shooting) Refinement of MMSA (not covered here) for fine-scale comprehensive models

33. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA Acknowledgements Work supported by: AgustaWestland US Army Research Office EU FP (NICETRIP)

34. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA Reduced model Reduced model: few dofs, captures output response Comprehensive model Comprehensive model: many dofs, captures fine-scale solution details Reduced Models Model Reduction f M M

35. Trajectory Optimization of Rotorcraft Vehicles POLITECNICO di MILANO DIA The Multi-Model Steering Algorithm (MMSA) 1. Maneuver planning problem (reduced model) Reference trajectory 2. Tracking problem (reduced model) consistently approximating A procedure for consistently approximating the fine-scale model MOCP 4. Reduced model update Predictive solutions 3. Steering problem (comprehensive model) Prediction window Steering window Tracking cost Prediction error Prediction window Tracking cost Steering window Prediction error Tracking cost Prediction window Steering window Prediction error 5. Re-plan with updated reduced model Updated reference trajectory Reference trajectory comprehensive model update Fly the comprehensive model along the reference trajectory and, at the same time, update the reduced model (learning).