1. POLI di MI tecnicolano PROCEDURES FOR ENABLING THE SIMULATION OF MANEUVERS WITH COMPREHENSIVE CODES Carlo L. Bottasso, Alessandro Croce, Domenico Leonello Politecnico di Milano Italy 31st European Rotorcraft Forum Firenze, Italy, 13-15 September 2005

2. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Limiting factors maneuvering flight Limiting factors (maximum loads, vibrations, noise, etc.) are experienced during maneuvering flight and at flight envelope boundaries. impossible to guess the controlscomplex long durationwithin the flight envelope boundaries It is virtually impossible to guess the controls that will fly a complex maneuver of long duration, guaranteeing to stay within the flight envelope boundaries. Maneuvering Multibody Dynamics TDP Example Example: Cat-A continued take-off. Cat-A certification requirements: 1) achieve positive rate of climb; 2) achieve V TOSS ; 3) clear obstacle of given height; 4) bring rotor speed back to nominal at end of maneuver, etc. Many related problems Many related problems: Fixed wing (Frezza), motorcycles (Da Lio, Ambrósio), cars (Minen), sail boats, etc.

3. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Outline MMSA MMBD Overview: the Multi-Model Steering Algorithm (MMSA) for Maneuvering Multibody Dynamics (MMBD); Detailed description of the methodology: adaptive reduced model – Neural-augmented adaptive reduced model; Path planning – Path planning (trajectory optimization); Path tracking – Path tracking (receding horizon model predictive control); Steering – Steering of comprehensive vehicle models; Numerical examples; Conclusions.

4. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Maneuver Definition ManeuversOptimal Control (OC) problems Maneuvers can be formulated as Optimal Control (OC) 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. the solution of optimal control problems with large comprehensive models is not feasible/attractive However, the solution of optimal control problems with large comprehensive models is not feasible/attractive (e.g. flexible vehicle+CFD: cost and problem size, controllability issues). Proposed solution: Proposed solution: design a virtual pilot capable of piloting the virtual vehicle model according to the OC maneuver definition.

5. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Reduced model Reduced model: few dofs, captures flight mechanics solution. Comprehensive model Comprehensive model: many dofs, captures fine scale solution details. The Multi-Model Steering Algorithm (MMSA) Sys Id Motivation Motivation: Solve expensive optimal control problems with reduced model; Use comprehensive model only for initial value problems (known control inputs).

6. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO The Multi-Model Steering Algorithm (MMSA) Reduced model Reduced model: Path planning level 1. Path planning level: solve maneuver optimal control problem to yield reference trajectory (Boundary Value Problem, BVP); Path tracking level 2. Path tracking level: use model predictive control (MPC) to track reference trajectory. - Tracking problem formulated as optimal control problem on a shifting prediction window (BVP); - Apply computed control input to comprehensive model until next prediction (Initial Value Problem, IVP). This decomposition mimics the architecture of advanced control systems for autonomous vehicles (Frazzoli et al. 2000): comprehensive vehicle model = plant automatically adapts minimize tracking errors Iterative system identification automatically adapts reduced model to the comprehensive one, in order to minimize tracking errors.

7. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO The Multi-Model Steering Algorithm (MMSA) 1. Maneuver planning problem (reduced model) Reference trajectory 2. Tracking problem (reduced model) Trajectory flown by comprehensive model 4. Update of reduced model by system identification Predictive solutions 3. Steering problem (comprehensive model) Prediction window Steering window Tracking cost Prediction error

8. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO The Multi-Model Steering Algorithm (MMSA) 1. Maneuver planning problem (reduced model) Reference trajectory 2. Tracking problem (reduced model) Trajectory flown by comprehensive model 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).

9. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO The Multi-Model Steering Algorithm (MMSA) 1. Maneuver planning problem (reduced model) Reference trajectory 2. Tracking problem (reduced model) Trajectory flown by comprehensive model 4. Reduced model update Predictive solutions 3. Steering problem (comprehensive model) Tracking cost Prediction window Steering window Prediction error 5. Re-plan with updated reduced model Reference trajectory

10. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO The Multi-Model Steering Algorithm (MMSA) Highlights Highlights: Computationally feasible Computationally feasible: reduced model for expensive BVP, comprehensive model for IVP; any comprehensive code Applicable to any comprehensive code without the need for modifications; inputoutput constraints MPC can deal with input (limited authority) and output (flight envelope boundary, procedures, etc.) constraints; non-linear MPC is based on non-linear flight mechanics (reduced) models; provably stable MPC is provably stable under reasonable conditions; unstable vehicles Applicable to unstable vehicles; convergence Adaptivity of reduced model ensures convergence, i.e. small tracking errors.

11. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Goal Goal: reduced modelpredicting the behavior of the plant Develop reduced model capable of predicting the behavior of the plant with minimum error (same outputs when subjected to same inputs); self-adaptive Reduced model must be self-adaptive (capable of learning) to adjust to varying flying conditions. model-predictive planning & tracking Reduced model will be used for model-predictive planning & tracking. Predictive solutions Prediction (tracking) window Steering window Prediction error to be minimized Reduced Model Identification

12. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Reduced Model Identification Comprehensive (multibody based) governing equations: where are the states, the controls, the Lagrange multipliers. capture the vehicle flight mechanics Define outputs that capture the vehicle flight mechanics: reduced parametric Find reduced parametric flight mechanics model such that when i.e. captures the gross motion the flight mechanics reduced model captures the gross motion of the comprehensive one (plant). e u e x e ¸ e y = e h ( e x ) : e f ( _ e x ; e x ; e ¸ ; e u ) = 0 ; e c ( _ e x ; e x ) = 0 ; f ( _ y ; y ; u ; p ) = 0 ; e y e y ¼ y e u = u,

13. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Reduced Model Reduced flight mechanics model: - Reference model - Reference model: two-dimensional rigid body model with rotor aerodynamics based on blade element theory with uniform inflow. = CG position vector, CG velocity, pitch angle, pitch rate, rotor angular velocity; = main & tail rotor collective, lateral & longitudinal cyclics, available power. - Augmented reduced model - Augmented reduced model: unknowndefect where is the unknown reference model defect that ensures when f re f ( _ y ; y ; u ) = 0 ; d e y ¼ y yu f re f ( _ y ; y ; u ) = d ( y ( n ) ;:::; y ; u ) ; e u = u.

14. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Reduced Model Identification single-hidden-layer neural networks Approximate with single-hidden-layer neural networks, one for each component: where and = reconstruction error (universal approximator, ); = matrices of synaptic weights and biases; = sigmoid activation functions; = network input. reduced model parameters The reduced model parameters are readily identified with the synaptic weights and biases of the networks: d i ( y ( n ) ;:::; y ; u ) = d i NN ( y ( n ) ;:::;; u ) + " i ; d W i ; V i ; a i ; b i ¾ ( Á ) = ( ¾ ( Á 1 ) ;:::; ¾ ( Á N n )) T x = ( y ( n ) T ;:::; y T ; u T ) T p = ( :::; p i T ;::: ) T ; p i = ( :::; W i j k ; V i j k ; a i j k ; b i j k ;::: ) T : p " i j " i j · C ; 8 C > 0 d i NN ( y ( n ) ;:::; y ; u ) = W i T ¾ ( V i T x + a i ) + b i ;

15. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Reduced Model Identification p i k + 1 = p i k + ¢ p i k ; Minimize Minimize functional reconstruction error (sole function of network parameters ): Steepest descent Steepest descent corrections: = learning rate. p ´ ¢ p i k = ¡ ´ @ E i ( T t rac k k ) @ p i k ; E i ( T a d ap t k ) = ( f i re f ( _ e y ¤ h ; e y ¤ h ; u ¤ h ) ¡ d i NN ( e y ¤ ( n ) h ;:::; e y ¤ h ; u ¤ h )) 2 ¯ ¯ T a d ap t k

16. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Remark Remark: tracking and steering grids are different. coarseflight mechanics scales = coarse grid, captures flight mechanics scales; fineaeroelastic scales = fine grid, captures aeroelastic scales. T t rac k h T s t eer h Reduced Model Identification 1.Filter 1.Filter aeroelastic solution ; 2.Project 2.Project filter outputs onto adaption grid: 3.Compute derivatives interpolation 3.Compute derivatives based on interpolation of filtered and projected outputs. F ( e y ¤ h ) e y ¤ h e y ¤ h j T a d ap t h = P ¡ 1 ( F ( e y ¤ h j T s t eer h )) ; e y ¤ ( n ) h

17. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Reduced Model Identification system identificationmodel defect adaption Effect of system identification by model defect adaption: Output of multibody, reference, and neural-augmented reference with same prescribed inputs. u Short transient = fast adaption

18. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Maneuver optimal control problem Maneuver optimal control problem: solution yields reference to-be-tracked trajectory. Optimize performance index Subjected to: Reduced model equations: Boundary conditions: (initial) (final) Constraints: Remark Remark: cost function, constraints and bounds collectively define in a compact and mathematically clear way a maneuver. Trajectory Planning à ( y ( T 0 )) 2 [ à 0 m i n ; à 0 max ] ; à ( y ( T )) 2 [ à T m i n ; à T max ] ; J p l an = Á ( y ; u ) ¯ ¯ T + Z T T 0 L ( y ; u ) d t ; f ( _ y ; y ; u ; p ¤ ) = 0 ; g p l an ( y ; u ; T ) 2 [ g p l an m i n ; g p l an max ] : Trajectory to be followed by tracking problem y ¤

19. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Model-predictive tracking problem Model-predictive tracking problem: solution yields steering controls. Minimize cost Subjected to: Reduced model equations: Initial conditions: Constraints: Remark same software Remark: formally identical to the planning problem, the two can be solved using the same software. Trajectory Tracking f ( _ y ; y ; u ; p ¤ ) = 0 ; y ( T t rac k 0 ) = e y 0 ; g t rac k ( y ; u ; T ) 2 [ g t rac k m i n ; g t rac k max ] : Tracking window Steering window Tracking cost Tracking trajectory from planning problem u ¤ J t rac k h = Z T t rac k T t rac k 0 ( jj y h ¡ y ¤ h jj S t rac k y + jj _ u h jj S t rac k _ u ) d t ;

20. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Numerical Solution of Trajectory Optimization 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.

21. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO planningtracking Both for planning and tracking: Transcribe Transcribe equations of dynamic equilibrium using suitable time marching scheme: Discretize Discretize cost function and constraints. NLP problem Solve resulting NLP problem using a SQP or IP method: recovery of control rates Galerkin projection allows for the recovery of control rates, otherwise absent from the flight mechanics models: Control rates can now be used in the cost function, or bounded. Direct Transcription of Trajectory Optimization Problems f ( _ y ; y ; u ; p ¤ ) = 0 » h ( y h ; u h ; p ¤ ; T ) = 0 ( m i n y d ; u d ; T J h ( y h ; u h ; T ) ; s. t. : ' h ( y h ; u h ; p ¤ ; T ) 2 [ ' m i n ; ' max ] : R T T 0 ( q h ¡ _ u h ) ¢ v h d t = 0 :

22. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO scaling of unknowns Use scaling of unknowns so that all quantities are approximately of. boot-strapping Use boot-strapping, starting from crude meshes to enhance convergence. Direct Transcription of Trajectory Optimization Problems O ( 1 )

23. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO March forward in timegiven control inputs March forward in time multibody solver with given control inputs as computed by the tracking problem: Project controls Project controls from tracking grid to steering grid : initial value problem Solve initial value problem from current state on steering window: Steering Problem Steering window u ¤ e f ( _ e x h ; e x h ; e ¸ h ; u ¤ h ) = 0 ; e c ( _ e x h ; e x h ) = 0 ; e x ( T s t eer 0 ) = e x 0 : e x 0 u ¤ h j T s t eer h = P ( u ¤ h j T t rac k h ) : T t rac k h T s t eer h Current state (tracking initial condition) e x 0

24. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Requirements Requirements: Achieve positive rate of climb; Achieve V TOSS ; Clear obstacle of given height; Bring rotor speed back to nominal. Cat-A Continued TO

25. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Lower pairs: Sensors; Actuators, controls. Other models: Flexible joints; Unilateral contacts; energy- preserving-decaying Non-linearly stable energy- preserving-decaying scheme. Body models: geometrically exact, composite ready beams and shells; rigid bodies. Rotorcraft Aeroelastic Models Finite element based MB code Finite element based MB code (Bauchau & Bottasso 2001).

26. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Rotorcraft Aeroelastic Models

27. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO 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) Cat-A Continued TO J p l an = ¡ Z ( T 0 ) + w T ( T ¡ T 1 ) + w 1 T ¡ T 0 Z T T 0 _ B 2 1 d t ;

28. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO 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 : Cat-A Continued TO

29. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Cat-A Continued TO Multibody model (one single blade shown, for clarity)

30. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Cat-A Continued TO

31. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Cat-A Continued TO (Legend: comprehensive model, flight mechanics model)

32. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Cat-A Continued TO Effect of re-planning iterations: (Solid line: comprehensive model)

33. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Cat-A Continued TO Effect of re-planning iterations: (Solid line: comprehensive model)

34. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Cat-A Continued TO (Legend: comprehensive model, flight mechanics model) MPC every 1 sec. MPC every 0.2 sec. Effect of MPC activation frequency:

35. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Cat-A Continued TO Effect of MPC: (Legend: comprehensive model, dashed: comprehensive model no MPC (open loop steering), flight mechanics model) Trajectory Pitch vs. time

36. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO 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

37. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Minimum Time Obstacle Avoidance

38. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Minimum Time Obstacle Avoidance (Legend: comprehensive model, flight mechanics model) Trajectories at 1 st iteration Trajectories at 4 th iteration Effect of reduced model adaption:

39. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Minimum Time Obstacle Avoidance (Legend: comprehensive model, flight mechanics model) Pitch vs. time at 1 st iteration Pitch vs. time at 4 th iteration Effect of reduced model adaption:

40. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Minimum Time Obstacle Avoidance Predictive solutions of tracking problem:

41. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Trajectory Optimization Remark: The path planner can be used by itself for the optimization of maneuvers/procedures, performance studies, design optimization, etc.

42. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO 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

43. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO 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.

44. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Helicopter HV Diagram Deadman’s curve

45. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO 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)

46. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Conclusions maneuvering comprehensive flying models Computational procedures for maneuvering comprehensive flying models were proposed, that blend aeroelasticity with flight mechanics; reasonable computational costs Multi-model approach allows reasonable computational costs even for very large aeroelastic models; No modifications No modifications to comprehensive codes are necessary in order to analyze maneuvering flight conditions; steering automobilesmotorcycles Basic idea applicable also to the steering of other vehicle models, e.g. automobiles and motorcycles; unstable systems Receding horizon formulation of MMSA allows for the analysis of unstable systems, such as helicopters; inputoutput constraints MMSA can deal with input and output constraints.

47. Maneuvering Rotorcraft Dynamics POLITECNICO di MILANO Acknowledgements US Army Research Office This work is supported in part by the US Army Research Office, through contract no. 99010 with the Georgia Institute of Technology, and a sub-contract with the Politecnico di Milano (Dr. Gary Anderson, technical monitor). Agusta-Westland Domenico Leonello is supported by a fellowship of Agusta-Westland.