Optimization of Critical Trajectories for Rotorcraft Vehicles

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OPTIMIZATION OF CRITICAL TRAJECTORIES FOR
ROTORCRAFT VEHICLESCarlo L. BottassoGeorgia
Institute of TechnologyAlessandro Croce,
Domenico Leonello, Luca RivielloPolitecnico di
Milano60th Annual Forum of the American
Helicopter SocietyBaltimore, June 710, 2004
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Outline

• 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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Introduction and Motivation

• Goal modeling of critical maneuvers of
  helicopters and tilt-rotors.
• Examples Cat-A certification (Continued TO,
  Rejected TO), balked landing, mountain rescue
  operations, etc.
• But also non-emergency terminal trajectories
  (noise, capacity).
• Applicability
• Vehicle design
• Procedure design.
• Related work Carlson and Zhao 2001, Betts 2001.

TDP
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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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Introduction and Motivation

• Mathematical models of vehicle
• This paper
• Classical flight mechanics model (valid for both
  helicopters and tilt-rotors)
• Paper 8 Dynamics I Wed. 9, 530600
• Comprehensive aeroelastic (multibody based)
  models.

Category A continued take-off with detailed
multibody model.
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Rotorcraft Flight Mechanics Model
Classical 2D longitudinal model for helicopters
and tilt-rotors (MR Main Rotor TR
Tail Rotor) Power balance equation
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Rotorcraft Flight Mechanics Model

• For helicopters, enforce yaw, roll and lateral
  equilibrium
• Rotor aerodynamic forces based on classical blade
  element theory (Bramwell 1976, Prouty 1990).
• In compact form
• where (states),
• (controls)
• helicopter
  
• tilt-rotor add also , , but no ,
  so that

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Trajectory Optimization

• Maneuver optimal control problem
• Cost function
• Boundary conditions
  (initial)
• 
  (final)
• Constraints
• point
  integral
• Bounds
  (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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Numerical Solution Strategies for Optimal Control
Problems
Optimal Control Problem
Optimal Control Governing Eqs.
Indirect
Discretize
Discretize
Direct
Numerical solution
NLP Problem

• 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 all above drawbacks are avoided.

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Trajectory Optimization

• 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.

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Implementation Issues

• 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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Optimization Criteria for Flyable Trajectories
Actuator models not included in flight mechanics
equations (time scale separation argument)
algebraic control
variables Results typically show bang-bang
behavior, with unrealistic control
speeds. Possible excitation of short-period type
oscillations. Simple solution recover control
rates through Galerkin projection Control
rates can now be used in the cost function, or
bounded.
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Optimization Criteria for Flyable Trajectories

• Optimization cost functions
• Index of vehicle performance
• Performance index Minimum control effort from
  a reference trim condition
• Performance index Minimum control velocity
• Control rate bounds

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Minimum Time Obstacle Avoidance

• Optimal Control Problem (with unknown internal
  event at T1)
• Cost function
• Constraints and bounds
• - Initial trimmed conditions at 30 m/s
• - Power limitations

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Minimum Time Obstacle Avoidance
Effect of control rates negligible performance
loss (0.13 sec for a maneuver duration of 13 sec).
Fuselage pitch
Longitudinal cyclic
(Legend w0, w100, w1000)
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Category-A Helicopter Take-Off Procedure
Jar-29
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Optimal Helicopter Multi-Phase CTO

• CTO 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
  constraints.

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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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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
, engine time constants. For
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Optimal Helicopter Multi-Phase CTO
Longitudinal cyclic rate bounds
Free fall (pilot reaction)
Free fall (pilot reaction)


Longitudinal cyclic rate
Longitudinal cyclic
(Legend w0, w100, w1000)
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Optimal Helicopter Multi-Phase CTO
Fuselage pitch
Fuselage pitch rate
(Legend w0, w100, w1000)
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Optimal Helicopter Multi-Phase CTO
Effect of control rates negligible performance
loss.
Trajectory
(Legend w0, w100, w1000)
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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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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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Helicopter HV Diagram

• Fly away (CTO) same as before, with initial
  forward speed as a parameter.
• Rejected TO
• Cost function (max safe altitude)
• Touch-down conditions
• plus usual state and control constraints.

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Helicopter HV Diagram
Deadmans curve
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Helicopter HV Diagram
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 CTO
Formulation 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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Conclusions

• Developed a suite of tools for rotorcraft
  trajectory optimization
• - 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 control
  of large comprehensive maneuvering rotorcraft
  models (Paper 8 Dynamics I Wed. 9, 530600).
• Work in progress
• - Noise as an optimization constraint, through
  Quasi-Static Acoustic Mapping (Q-SAM) method
  (Schmitz 2000).

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Optimal Helicopter Single-Phase CTO Effect of
Control Rates

• Pilot delay (forward integration, 0 ? T01sec)
• Optimal Control Problem (T0 ? T (free))
• Cost function
• Constraints and bounds
• Initial and exit conditions
• Power limitations

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Optimal Helicopter Single-Phase CTO Effect of
Control Rates
Free fall (pilot reaction)

Longitudinal cyclic speed
Longitudinal cyclic
(Legend w0, w100, w1000)
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Optimal Helicopter Single-Phase CTO Effect of
Control Rates
Fuselage pitch
Fuselage pitch rate
(Legend w0, w100, w1000)
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Optimal Helicopter Single-Phase CTO Effect of
Control Rates
Longitudinal cyclic speed bounds
Longitudinal cyclic
Longitudinal cyclic speed
(Legend w0, w100, w1000)
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Optimal Helicopter Single-Phase CTO Effect of
Control Rates
Longitudinal cyclic speed bounds
Fuselage pitch rate
Fuselage pitch
(Legend w0, w100, w1000)
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Optimal Helicopter Single-Phase CTO Effect of
Control Rates
Longitudinal cyclic speed bounds
Trajectory
(Legend w0, w100, w1000)