Transcript Diapositiva 1 - Politecnico di Milano

POLItecnico
di MIlano
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
Trajectory Optimization of Rotorcraft Vehicles
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
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Trajectory Optimization of Rotorcraft Vehicles
Introduction and Motivation
Goal: modeling of critical maneuvers of helicopters and tilt-rotors at
the boundaries of the flight envelope
Examples:
Cat-A certification (Continued TO, Rejected TO), ADS-33, autorotation,
tail rotor loss, mountain rescue operations, etc.
TDP
Applicability:
- Vehicle design
- Design of procedures, certification
Related work: Okuno & Kawachi 1993, Carlson & Zhao 2002, Bottasso et al. 2004
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Trajectory Optimization of Rotorcraft Vehicles
Introduction and Motivation
Tools:
• Mathematical models of maneuvers
• Mathematical models of vehicle
• Numerical solution strategy
Maneuvers are here defined as optimal control problems, whose
ingredients are:
• A cost function (index of performance)
• Constraints:
– Vehicle equations of motion
– Physical limitations (limited control authority, flight
envelope boundaries, etc.);
– Procedural limitations
Solution yields: trajectory and controls that fly the vehicle along it
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Trajectory Optimization of Rotorcraft Vehicles
Vehicle Models
• Flight mechanics helicopter and tilt-rotor models (this paper)
• 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):
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MMSA
This paper
Trajectory Optimization of Rotorcraft Vehicles
Preferred Methods for Vehicle Models of
Increasing Complexity
Direct multiple shooting
Direct transcription
Actuator disk-type
Algebraic inflow
Flapping blade
Dynamic inflow
Full FEM
Refined Aerodynamics
Model complexity – Computational cost per physical time unit
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Trajectory Optimization of Rotorcraft Vehicles
Trajectory Optimization
Maneuver Optimal Control Problem (MOCP):
• Cost function
• Vehicle model
• Boundary conditions
(initial)
(final)
• Constraints
point:
• Bounds
integral:
(state bounds)
(control bounds)
Remark: cost function, constraints and bounds collectively define
in a compact and mathematically clear way a maneuver
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Trajectory Optimization of Rotorcraft Vehicles
Numerical Solution of Maneuver Optimal
Control Problems
Optimal Control
Problem
Discretize
Optimal Control
Governing Eqs.
Indirect
Direct
NLP Problem
Discretize
Numerical solution
Indirect approach:
• Need to derive optimal control governing equations (impossible for third-party blackbox 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: all above drawbacks are avoided
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Trajectory Optimization of Rotorcraft Vehicles
TOP: Trajectory Optimization Program
for Rotorcraft Vehicles
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:
• Direct transcription
• Direct multiple shooting
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Trajectory Optimization of Rotorcraft Vehicles
Direct Transcription
• Transcribe equations of dynamic equilibrium using suitable
time marching scheme:
Time finite element method (Bottasso 1997):
• Discretize cost function and constraints
• Solve resulting NLP problem using a SQP or IP method:
Problem is large but highly sparse and banded
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Trajectory Optimization of Rotorcraft Vehicles
Direct Transcription
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)
• Typically best method, but applicable only to models of low-moderate
complexity with slow solution components
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Trajectory Optimization of Rotorcraft Vehicles
Direct Transcription
• Use scaling of unknowns:
where the scaled quantities are
with
,
,
,
,
,
so that all quantities are approximately of
• Use boot-strapping, starting from crude meshes to enhance
convergence
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Trajectory Optimization of Rotorcraft Vehicles
Direct Multiple Shooting
• Partition time domain into shooting segments:
• Discretize controls as:
• Advance solution from
to
using
time steps.
• Glue segments together with NLP constraints:
• NLP unknowns:
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Trajectory Optimization of Rotorcraft Vehicles
Direct Multiple Shooting
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
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Trajectory Optimization of Rotorcraft Vehicles
Grid Refinement
• Minimum time 90-deg turn, constrained maximum roll rate
• FLIGHTLAB helicopter model, actuator disk-type rotor
• Direct transcription method
Uniform grid
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Final adapted grid
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Trajectory Optimization of Rotorcraft Vehicles
Grid Refinement
Evolution of grid throughout refinement steps
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Trajectory Optimization of Rotorcraft Vehicles
Grid Refinement
Same problem and model, multiple shooting, 16 shooting arcs
State constraint
violations within arcs
Zoom ▶
Solution after two refinement steps,
26 total resulting shooting arcs ▼
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Trajectory Optimization of Rotorcraft Vehicles
Grid Refinement
Computed control inputs, direct transcription and multiple shooting
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Trajectory Optimization of Rotorcraft Vehicles
Applications: ADS-33 MTEs
Mission Task Elements (MTE): assessment of ability to perform critical
tasks
All MTEs can be formulated as constrained Optimal Control problems
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
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Trajectory Optimization of Rotorcraft Vehicles
Applications: ADS-33 MTEs
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Trajectory Optimization of Rotorcraft Vehicles
Optimal Helicopter Multi-Phase CTO
Take-off decision
point
Normal take-off
Continued take-off
Rejected take-off
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Requirements:
• Achieve positive rate of climb
• Achieve VTOSS
• Clear obstacle of given height
• Bring rotor speed back to nominal
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Trajectory Optimization of Rotorcraft Vehicles
Optimal Helicopter Multi-Phase CTO
Cost function:
where T1 is unknown internal event (minimum altitude) and T
unknown maneuver duration
Constraints:
- Control bounds
- Initial conditions obtained by forward integration for 1 sec from
hover to account for pilot reaction (free fall)
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Trajectory Optimization of Rotorcraft Vehicles
Optimal Helicopter Multi-Phase CTO
Constraints (continued):
- Internal conditions
- Final conditions
- Power limitations
For
(pilot reaction):
where:
maximum one-engine power in emergency
one-engine power in hover
,
For
engine time constants
:
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Trajectory Optimization of Rotorcraft Vehicles
Optimal Helicopter Multi-Phase CTO
Effect of control rates: negligible performance loss
Trajectory
(Legend: w=0, w=100, w=1000)
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Trajectory Optimization of Rotorcraft Vehicles
Optimal Helicopter Multi-Phase CTO
Free fall (pilot reaction)
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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Trajectory Optimization of Rotorcraft Vehicles
Optimal Helicopter Multi-Phase CTO
Maneuver flown symmetrically (2D):
Side-slipping (3D):
Remark: 3D nonsymmetric sideslipping Cat-A CTO
reduces altitude
loss of about 10%
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Trajectory Optimization of Rotorcraft Vehicles
Max CTO Weight
Goal: compute max 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 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 increase
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Trajectory Optimization of Rotorcraft Vehicles
Applications: Minimum Time Turn
Minimum time 180-deg turn
FLIGHTLAB helicopter model
Direct transcription
Resulting optimal
strategy: classical
bank and turn
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Trajectory Optimization of Rotorcraft Vehicles
Applications: Minimum Time Turn
Minimum time 180-deg turn
FLIGHTLAB ERICA tilt-rotor model
Direct transcription
Resulting optimal
strategy: flare, then
turn at high side-slip
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Trajectory Optimization of Rotorcraft Vehicles
Helicopter Obstacle Avoidance Maneuver
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Trajectory Optimization of Rotorcraft Vehicles
Conclusions
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
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Trajectory Optimization of Rotorcraft Vehicles
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
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Trajectory Optimization of Rotorcraft Vehicles
Acknowledgements
Work supported by:
• AgustaWestland
• US Army Research Office
• EU FP (NICETRIP)
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Trajectory Optimization of Rotorcraft Vehicles
Reduced Models
Comprehensive model: many dofs,
captures fine-scale solution details
M
f
M
Reduced model: few dofs, captures
output response
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Trajectory Optimization of Rotorcraft Vehicles
The Multi-Model Steering Algorithm
(MMSA)
Updated
reference
trajectory
Reference
trajectory
1. Maneuver planning
problem
(reduced model)
2. Tracking problem
A procedure for consistently approximating the fine-scale
(reducedmodel
model)MOCP
Fly the
model along the 3.
reference
5. Re-plan
withcomprehensive
updated
Steeringtrajectory
problem
and,model
at the same time, update the reduced(comprehensive
model (learning).
reduced
model)
Prediction window
4. Reduced model
Prediction
window
Tracking cost Tracking cost
update
Prediction window
Tracking cost
Prediction error
Prediction error
Steering window
Steering window
Prediction error
Steering window
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Reference
trajectory
Predictive solutions
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