POLI di MI tecnicolano Carlo L. BottassoGiorgio Maisano Francesco Scorcelletti TRAJECTORY...

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TRAJECTORY OPTIMIZATIONPROCEDURES FOR ROTORCRAFT

VEHICLES,THEIR SOFTWARE IMPLEMENTATION

AND THEIR APPLICABILITY TO MODELS OF

VARYING COMPLEXITY

Carlo L. BottassoCarlo L. Bottasso, Giorgio MaisanoGiorgio MaisanoPolitecnico di Milano

Francesco ScorcellettiFrancesco Scorcelletti AgustaWestland & Politecnico di Milano

AHS Annual Forum & Technology DisplayMontréal, Québec, Canada, April 29 – May 1, 2008

TRAJECTORY OPTIMIZATIONPROCEDURES FOR ROTORCRAFT

VEHICLES,THEIR SOFTWARE IMPLEMENTATION

AND THEIR APPLICABILITY TO MODELS OF

VARYING COMPLEXITY

Carlo L. BottassoCarlo L. Bottasso, Giorgio MaisanoGiorgio MaisanoPolitecnico di Milano

Francesco ScorcellettiFrancesco Scorcelletti AgustaWestland & Politecnico di Milano

AHS Annual Forum & Technology DisplayMontréal, Québec, Canada, April 29 – May 1, 2008

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POLITECNICO di MILANO DIA

OutlineOutline

• 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

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GoalGoal: modeling of critical maneuvers of helicopters and tilt-rotors at the boundaries of the flight envelopeboundaries of the flight envelope

ExamplesExamples: Cat-A certification (Continued TO, Rejected TO), ADS-33, autorotation, tail rotor loss, mountain rescue operations, etc.

ApplicabilityApplicability: - Vehicle design

- Design of procedures, certification

Related workRelated work: Okuno & Kawachi 1993, Carlson & Zhao 2002, Bottasso et al. 2004

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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Vehicle ModelsVehicle Models• Flight mechanicsFlight mechanics helicopter and tilt-rotor models (this paper)

• Comprehensive aeroelasticComprehensive 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):

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Direct transcriptionDirect transcription

Actuator disk-type Algebraic inflow

Flapping blade Dynamic inflow

Full FEM Refined Aerodynamics

Model complexity – Computational cost per physical time unit

Direct multiple shootingDirect multiple shootingDirect multiple shootingDirect multiple shooting

MMSAMMSA

Preferred Methods for Vehicle Models of Increasing ComplexityPreferred Methods for Vehicle

Models of Increasing ComplexityTh

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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)

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 of Maneuver Optimal Control Problems

Numerical Solution of Maneuver Optimal Control Problems

Optimal Control Problem

Optimal Control Governing Eqs.

Discretize

Discretize

NLP Problem Numerical solution

DirectIndirec

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Indirect approachIndirect 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 approachDirect approach: all above drawbacks are avoided

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TOP: Trajectory Optimization Program for Rotorcraft Vehicles

TOP: Trajectory Optimization Program for Rotorcraft Vehicles

Supported vehicle modelsSupported 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 strategiesImplemented direct solution strategies:

• Direct transcription

• Direct multiple shooting

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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 sparse sparse and bandedbanded

Direct TranscriptionDirect Transcription

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RemarksRemarks:

• 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)

• Typically best methodbest method, but applicable only to models of low-moderate complexity with slow solution components

Direct Transcription Direct Transcription

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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

Direct Transcription Direct Transcription

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• Partition time domain into shooting segments:

• Discretize controls as:

• Advance solution from to using time

steps.

• Glue segments together with NLP constraints:

• NLP unknownsNLP unknowns:

Direct Multiple Shooting Direct Multiple Shooting

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RemarksRemarks:

• 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 ShootingDirect Multiple Shooting

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Grid RefinementGrid Refinement

• Minimum time 90-deg turn, constrained maximum roll rate

• FLIGHTLAB helicopter model, actuator disk-type rotor

• Direct transcription Direct transcription method

Uniform grid Final adapted grid

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Grid RefinementGrid RefinementEvolution of grid throughout refinement steps

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Grid RefinementGrid RefinementSame problem and model, multiple shootingmultiple shooting, 16 shooting arcs

State constraint violationsviolations within

arcs

Zoom ▶Solution after two refinement

steps, 26 total resulting shooting arcs ▼

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Grid RefinementGrid Refinement

Computed control inputs, direct transcription direct transcription and multiple multiple shootingshooting

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Applications: ADS-33 MTEsApplications: ADS-33 MTEs

Mission Task Elements Mission Task Elements (MTE): assessment of ability to perform critical tasks

All MTEs can be formulated as constrained Optimal Control constrained Optimal Control problemsproblems

ExampleExample: 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

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Applications: ADS-33 MTEsApplications: ADS-33 MTEs

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RequirementsRequirements:• Achieve positive rate of climb• Achieve VTOSS

• Clear obstacle of given height• Bring rotor speed back to nominal

Normal take-off

Continued take-off

Rejected take-off

Take-off decisionpoint

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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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

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

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Maneuver flown symmetrically (2D): Side-slipping (3D):

RemarkRemark: 3D non-symmetric side-slipping Cat-A CTO reduces altitude loss of about 10%



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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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Minimum time 180-deg turn

FLIGHTLAB helicopter model

Direct transcription

Applications: Minimum Time TurnApplications: Minimum Time Turn

Resulting optimal Resulting optimal strategystrategy: classical bank and turn

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Resulting optimal Resulting optimal strategystrategy: flare, then turn at high side-slip

Minimum time 180-deg turn

FLIGHTLAB ERICA tilt-rotor model

Direct transcription

Applications: Minimum Time TurnApplications: Minimum Time Turn

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Helicopter Obstacle Avoidance Maneuver

Helicopter Obstacle Avoidance Maneuver

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ConclusionsConclusions

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

- 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

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OutlookOutlook

• 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

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AcknowledgementsAcknowledgements

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Reduced modelReduced model: few dofs, captures output response

Comprehensive modelComprehensive model: many dofs, captures fine-scale solution details

Reduced ModelsReduced Models

Model

Model

Reductio

Reductio

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The Multi-Model Steering Algorithm (MMSA)The Multi-Model Steering Algorithm (MMSA)

1. Maneuver planning problem (reduced model)

Reference trajectory2. Tracking

problem (reduced model)

A procedure for consistently approximatingconsistently 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 costPrediction window

Steering window

Prediction error

5. Re-plan with updated reduced model

Updated reference trajectory

Reference trajectory

Fly the comprehensive modelcomprehensive model along the reference trajectory and, at the same time, updateupdate the reduced model (learning).

POLI di MI tecnicolano Carlo L. BottassoGiorgio Maisano Francesco Scorcelletti TRAJECTORY...

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Transcript of POLI di MI tecnicolano Carlo L. BottassoGiorgio Maisano Francesco Scorcelletti TRAJECTORY...