poli di mi tecnicolano carlo l. bottassogiorgio maisano francesco scorcelletti trajectory...
TRANSCRIPT
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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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POLITECNICO di MILANO DIA
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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POLITECNICO di MILANO DIA
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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POLITECNICO di MILANO DIA
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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POLITECNICO di MILANO DIA
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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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)
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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POLITECNICO di MILANO DIA
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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POLITECNICO di MILANO DIA
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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POLITECNICO di MILANO DIA
• 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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POLITECNICO di MILANO DIA
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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POLITECNICO di MILANO DIA
• 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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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 unknownsNLP unknowns:
Direct Multiple Shooting Direct Multiple Shooting
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POLITECNICO di MILANO DIA
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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POLITECNICO di MILANO DIA
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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POLITECNICO di MILANO DIA
Grid RefinementGrid RefinementEvolution of grid throughout refinement steps
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POLITECNICO di MILANO DIA
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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POLITECNICO di MILANO DIA
Grid RefinementGrid Refinement
Computed control inputs, direct transcription direct transcription and multiple multiple shootingshooting
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POLITECNICO di MILANO DIA
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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POLITECNICO di MILANO DIA
Applications: ADS-33 MTEsApplications: ADS-33 MTEs
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POLITECNICO di MILANO DIA
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
Optimal Helicopter Multi-Phase CTO
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POLITECNICO di MILANO DIA
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)
Optimal Helicopter Multi-Phase CTO
Optimal Helicopter Multi-Phase CTO
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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
Optimal Helicopter Multi-Phase CTO
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POLITECNICO di MILANO DIA
Optimal Helicopter Multi-Phase CTO
Optimal Helicopter Multi-Phase CTO
Trajectory
(Legend: w=0, w=100, w=1000)
Effect of control rates: negligible performance lossnegligible performance loss
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POLITECNICO di MILANO DIA
Optimal Helicopter Multi-Phase CTO
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
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POLITECNICO di MILANO DIA
Maneuver flown symmetrically (2D): Side-slipping (3D):
RemarkRemark: 3D non-symmetric side-slipping Cat-A CTO reduces altitude loss of about 10%
Optimal Helicopter Multi-Phase CTO
Optimal Helicopter Multi-Phase CTO
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POLITECNICO di MILANO DIA
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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POLITECNICO di MILANO DIA
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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POLITECNICO di MILANO DIA
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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POLITECNICO di MILANO DIA
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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fM
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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).