Georgia ofTech Institutenology OPTIMIZATION OF CRITICAL TRAJECTORIES FOR ROTORCRAFT VEHICLES Carlo...

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OPTIMIZATION OF CRITICAL TRAJECTORIES FOR ROTORCRAFT

VEHICLES

Carlo L. BottassoGeorgia Institute of Technology

Alessandro Croce, Domenico Leonello, Luca RivielloPolitecnico di Milano

60th Annual Forum of the American Helicopter SocietyBaltimore, June 7–10, 2004

OPTIMIZATION OF CRITICAL TRAJECTORIES FOR ROTORCRAFT

VEHICLES

Carlo L. BottassoGeorgia Institute of Technology

Alessandro Croce, Domenico Leonello, Luca RivielloPolitecnico di Milano

60th Annual Forum of the American Helicopter SocietyBaltimore, June 7–10, 2004

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OutlineOutline

• 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.

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on GoalGoal: modeling of critical maneuvers of helicopters and

tilt-rotors.

ExamplesExamples: Cat-A certification (Continued TO, Rejected TO), balked landing, mountain rescue operations, etc.

But alsoBut also: non-emergency terminal trajectories (noise, capacity).

ApplicabilityApplicability:

• Vehicle design;

• Procedure design.

Related workRelated work: Carlson and Zhao 2001, Betts 2001.

Introduction and MotivationIntroduction and Motivation

TDP

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Introduction and MotivationIntroduction and MotivationToolsTools:

• Mathematical models of maneuvers;

• Mathematical models of vehicle;

• Numerical solution strategy.

ManeuversManeuvers are here defined as optimal control problemsoptimal control problems, whose ingredients are:

• A cost functioncost function (index of performance);

• ConstraintsConstraints:

– Vehicle equations of motion;

– Physical limitations (limited control authority, flight envelope boundaries, etc.);

– Procedural limitations.

SolutionSolution yields: trajectorytrajectory and controlscontrols that fly the vehicle along it.

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Introduction and MotivationIntroduction and MotivationMathematical models of vehicle:

• This paper:

Classical flight mechanicsflight mechanics model (valid for both helicopters and tilt-rotors);

• Paper #8 – Dynamics I Wed. 9, 5:30–6:00:

Comprehensive aeroelasticComprehensive aeroelastic (multibody based) models.

Category A continued take-off with detailed multibody model.

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Classical 2D longitudinal model for helicoptershelicopters and tilt-tilt-rotorsrotors:

(MR = Main Rotor; TR = Tail Rotor)

Power balance equation:

Rotorcraft Flight Mechanics ModelRotorcraft Flight Mechanics Model

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For helicopters, enforce yaw, roll and lateral equilibrium:

Rotor aerodynamic forces based on classical blade classical blade element theoryelement theory (Bramwell 1976, Prouty 1990).

In compact form:

where: (statesstates),

(controlscontrols)

• helicopterhelicopter:

• tilt-rotortilt-rotor: add also , , but no , so that

Rotorcraft Flight Mechanics ModelRotorcraft Flight Mechanics Model

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Maneuver optimal control problemManeuver optimal control problem:

• Cost function

• Boundary conditions (initial)

(final)

• Constraints

point: integral:

• Bounds (state bounds)

(control bounds)

RemarkRemark: cost function, constraints and bounds collectively define in a compact and mathematically clear way a maneuver.

Trajectory Optimization Trajectory Optimization

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Numerical Solution Strategies for Optimal Control Problems

Numerical Solution Strategies for Optimal Control Problems

Optimal Control Problem

Optimal Control Governing Eqs.

Discretize

Discretize

NLP Problem Numerical solution

DirectIndirec

t

Indirect approachIndirect 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 approachDirect approach: all above drawbacks are avoided.

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• TranscribeTranscribe equations of dynamic equilibrium using suitable time marching scheme:

Time finite element method (Bottasso 1997):

• Discretize cost functionDiscretize cost function and constraintsconstraints.

• Solve resulting NLP problemNLP problem using a SQP or IP method:

Problem is largelarge but highly sparsesparse.

Trajectory Optimization Trajectory Optimization

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• Use scaling of unknownsscaling of unknowns:

where the scaled quantities are , , ,

with , ,

so that all quantities are approximately of .

• Use boot-strappingboot-strapping, starting from crude meshes to enhance convergence.

Implementation Issues Implementation Issues

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Optimization Criteria for Flyable Trajectories


Actuator modelsActuator models not included in flight mechanics equations (time scale separation argument)

algebraicalgebraic control variables

Results typically show bang-bang behavior, with unrealisticunrealistic control speeds.

Possible excitationexcitation of short-period type oscillations.

Simple solution: recover control ratesrecover control rates through Galerkin projection:

Control rates can now be used in the cost functioncost function, or boundedbounded.

t

u h k =1k =0

t

duh

dt

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Optimization cost functionsOptimization cost functions

• Index of vehicle performance:

• Performance index + Minimum control effort from a reference trim condition:

• Performance index + Minimum control velocity:

Control rate boundsControl rate bounds:

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Optimal Control ProblemOptimal Control Problem (with unknown internal event at T1)

• Cost function:

• Constraints and bounds:

- Initial trimmed conditions at 30 m/s

- Power limitations

Minimum Time Obstacle AvoidanceMinimum Time Obstacle Avoidance

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Minimum Time Obstacle AvoidanceMinimum Time Obstacle Avoidance

Fuselage pitch Longitudinal cyclic

(Legend: w=0, w=100, w=1000)

Effect of control rates: negligible performance lossnegligible performance loss (0.13 sec for a maneuver duration of 13 sec).

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Category-A Helicopter Take-Off Procedure

Category-A Helicopter Take-Off Procedure

Jar-29:

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CTO formulationCTO 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 optimization constraintsconstraints.

Z(T )1

TDP

V TOSS

Optimal Helicopter Multi-Phase CTO


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Cost functionCost function:

where T1 is unknown internal event (minimum altitude) and T unknown maneuver duration.

ConstraintsConstraints:

- Control bounds

- Initial conditions obtained by forward integration for 1 sec from hover to account for pilot reaction (free fall)



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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 :



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Longitudinal cyclic Longitudinal cyclic rate

(Legend: w=0, w=100, w=1000)



Free fall (pilot reaction)

{

]/16;/16[1 ssB oo Longitudinal cyclic rate boundsrate bounds:


{

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Fuselage pitch Fuselage pitch rate



(Legend: w=0, w=100, w=1000)

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Trajectory

(Legend: w=0, w=100, w=1000)

Effect of control rates: negligible performance lossnegligible performance loss.

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Power Rotor angular velocity


• 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.

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GoalGoal: compute max TO weightmax 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 procedureiterative 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 increase6% payload increase.

Max CTO WeightMax CTO Weight

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Helicopter HV DiagramHelicopter HV Diagram

Fly away (CTO)Fly away (CTO): same as before, with initial forward speed as a parameter.

Rejected TORejected TO:

• Cost function (max safe altitude)

• Touch-down conditions

plus usual state and control constraints.

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Deadman’s curve

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Main rotor collective Rotor angular speed

(Legend: Vx(0)=2m/s, Vx(0)=5m/s, Vx(0)=10m/s)

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Optimal Tilt-Rotor Multi-Phase CTOOptimal Tilt-Rotor Multi-Phase CTOFormulation similar to helicopter multi-phase CTO.

Cost function:

plus usual state and control constraints and bounds.

Trajectory Collective, cyclic, nacelle tilt, pitch

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ConclusionsConclusions

• Developed a suite of tools for rotorcraft trajectory rotorcraft trajectory optimizationoptimization:

- 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 controlmodel-predictive control of large large comprehensive maneuvering rotorcraft modelscomprehensive maneuvering rotorcraft models (Paper #8 – Dynamics I Wed. 9, 5:30–6:00).

• Work in progress:

- NoiseNoise as an optimization constraint, through Quasi-Static Acoustic Mapping (Q-SAM) method (Schmitz 2000).

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Pilot delay (forward integration, 0 T0=1sec)

Optimal Control ProblemOptimal Control Problem (T0 T (free))

• Cost function:

• Constraints and bounds:

Initial and exit conditions

Power limitations

Optimal Helicopter Single-Phase CTO Effect of Control Rates


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Longitudinal cyclic Longitudinal cyclic speed

(Legend: w=0, w=100, w=1000)




{

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(Legend: w=0, w=100, w=1000)

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Longitudinal cyclic Longitudinal cyclic speed

]/16;/16[1 ssB oo



Longitudinal cyclic speed bounds:

(Legend: w=0, w=100, w=1000)

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]/16;/16[1 ssB oo Longitudinal cyclic speed bounds:

(Legend: w=0, w=100, w=1000)

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Trajectory

]/16;/16[1 ssB oo Longitudinal cyclic speed bounds:

(Legend: w=0, w=100, w=1000)

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