Kostas Alexis & Christoph Huerzeler 3 July 2012 Body Motion Rotorblade Motion Rotorblade...

Zürich Autonomous Systems Lab

Kostas Alexis & Christoph Huerzeler

3 July 2012

Zürich

Content ◦ Introduction

◦ Modeling Approaches & Challenges

◦ Rotorcraft Modeling

◦ Identification Techniques

Goals ◦ Provide Basic Introduction for Self-Study

◦ Emphasize Core Challenges

◦ Define Research Trends

Introduction Method Results Conclusion

AIROBOTS: Aerial Service Robotics Summer School, 2-6 July, 2012, ETH Zurich 2

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Various Types of UAV Configuration Exist

Underyling Modeling Approaches Similar

Learn From Full-Scale Rotorcraft Theory

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Blackbox

◦ Identify input-output behavior

◦ No understanding of underlying system

Greybox

◦ Based on laws of physics

◦ Identify model parameters

Whitebox

◦ Purely based on laws of physics

◦ All parameters predicted

◦ Numerical or analytical

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Axial Flight

Vortex Ring State (Slow Decent) Normal Working State (Hover)

Turbulent Wake State (Faster Descent) Windmill Break State (Fast Descent)

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Axial Flight

Forward Flight

Retreating Blade

Advancing Blade Reverse

Flow Region

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Source: open.edu/youtube

Main Body Motion

(«slow» dynamics)

Rotorblade Motion

(«fast» dynamics)

Rotorblade Aerodynamics

AIRobots Coaxial

(helicopter type UAV)

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Momentum Conservation

Angular Momentum Conservation

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Blade Feathering («Pitch»)

Blade Flapping

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Blade Feathering

Blade Flapping

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Blade Element Theory

Rotor Torque

Rotor Thrust

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Aerodynamic Coefficients

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Averaged Thrust & Torque

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Pitch Actuation

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Blade Feathering

Blade Flapping

AIROBOTS: Aerial Service Robotics Summer School, 2-6 July, 2012, ETH Zurich

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Blade Flapping

Virtual Hinge with Spring

Flap Moment Couples Body and Rotor Dynamics

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Fully Articulated

Teetering

Hingeless

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Main Body Motion Rotorblade Motion Rotorblade Aerodynamics

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18 States

Only 12 measurable

60 unknown parameters to

identify only based on flight data

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Motivation Parametric ID AIROBOTS Coax Uses of ID Nonparametric ID Trends Conclusions

27 AIROBOTS: Aerial Service Robotics Summer School, 2-6 July, 2012, ETH Zurich

“A helicopter is a collection of

oscillations held together by

differential equations”

White-Box models are often

inaccurate or very difficult

and time consuming to obtain

Blade flapping, coning

and lagging increase

the order of the

system with states that

are not directly

measurable

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Parametric

System Identification

Grey-Box

Black-Box

Frequency Domain Linear System Identification

Time Domain Linear System Identification

Frequency Domain Nonlinear System Identification

Time Domain Nonlinear System Identification

Hybrid Systems Identification Gradient Methods

Evolutionary Methods

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Record Proper

Flight Data

Preparation

System

Identification

Validation

Online ID has

also significant

applications

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Record Proper Flight Data Data Preparation – Use of empirical Metrics!

Excite expected

rotorcraft frequencies

Flight long enough to

capture low frequencies

Start and end at trim

Detrend - Unbias

Estimate Input Delays

Coherence (≥ 0.6) &

Random Error Check

Persistence of Excitation

I/O Spectrogram & Power

Spectral Density

Windowing

Based on trim

Actuation delays

Check Linearity of

the system

Conclusions about

the order

Visual

understanding

Filter

Increase accuracy

over specific

frequencies

Use Chrip Signals

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DC-Brushless Motor AIROBOTS Coaxial Prototype

Use coherence to check Input/Output relations Effect of windowing

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Decide the level of

accuracy and neglect some

phenomena (i.e. blade lag)

Write the Nonlinear

Differential Equations in

Symbolic Form

Trim and Linearize the

system around an operation

point (i.e. hover, forward

flight)

𝑥 = 𝑓(𝑥, 𝑢, 𝜃) 𝑥→rigid body and rotor states

𝑢 →swashplate and motor inputs

𝜃 →vector of parameters to be identified

Define constraints for the vector of

predicted parameters 𝜃

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Prepared Flight Data over

specific frequencies and

for specific degrees of

freedom

𝑥 = 𝑓(𝑥, 𝑢, 𝜃)

Define Frequency

Response Transform

Fast Fourier Transform

Chirp-Z Transform

Define Objective

Function

Define Optimization

Strategy

Solve constrained

problem

Computes the Z-Transform

of a signal along spiral

contours in the z-plane:

𝐶𝑍𝑇 𝑥 𝑛 = 𝑥 𝑛

𝑁−1

𝑛=0

𝑧𝑘−𝑛

𝑧𝑘 = 𝐴𝑊−𝑘, 𝑘 = 0, … ,𝑀 − 1

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Identified Model

Model Validation

Check Fitting

Check Coherence

I/O Spectrogram & Power

Spectral Density

Dataset not used in

Identification

[𝐹𝐿𝑂𝑊, 𝐹𝐻𝐼𝐺𝐻]

Initialize the

model and run

previous steps Acceptable Fit?

Satisfy constraints?

[𝐹𝐿𝑂𝑊, 𝐹𝐻𝐼𝐺𝐻]

Time & Frequency

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18 States

Only 12 measurable

60 unknown parameters to

identify only based on flight data

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Frequency Area

Optimization Algorithm

0.01,2 𝐻𝑧

Mathworks MATLAB® classic gradient

and adaptive gradient method

Minimum Length

[100 − 120]𝑠𝑒𝑐

Frequency Response

Fast Fourier Transform

Chirp Z Transform

Lower “capturable”

frequency ≈ 0.05𝐻𝑧

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Simplified (Quasi-Steady) Models for Control Computation.

• Identify partial response of the system such as off-axis

responses.

• Identify Closed-loop system response as a step for higher-

level control (closed-loop attitude for velocity, velocity for

trajectory control etc).

Identify actuator dynamics as part of the selection process

System Identification is a research

field but also a tool for the system and

control engineer

Accurate grey-box physically close models

How 𝑢𝑟𝑜𝑙𝑙 excites pitch?

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Estimate the input/output relation of a system without a model but rather

using the recorded data to nonparametrically calculate the frequency response

of the system

• Use of harmonic Windowing

• Empirical Transfer Function

Estimate Frequency Response and Spectrum using analysis

with Frequency-dependent Resolution.

• Estimate Frequency Response with Fixed Frequency

Resolution using Spectral Analysis

Example application: Identify the

Resonance frequency of the coupled

rotors/fuselage dynamics

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Nonlinear Frequency Domain System Identification:

Possibly through the utilization of the generalized frequency response

functions to reconstruct the model.

Nonlinear Identification is until now dominated from time-domain

approaches which lack the advantages of frequency-domain

identification.

Hybrid Systems Identification:

Hybrid systems often appear in robotics

either due to physical interaction or due

to modeling approach.

Identify Piecewise Affine systems

Hinging-Hyperplane AutoRegressive

eXogenous models (HHARX)

PieceWise affine AutoRegressive

eXogenous models (PWARX)

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• Grey-box System Identification can lead to accurate models that preserve

the physicality of the system.

• Frequency-domain System Identification poses significant advantages for

rotorcraft identification.

• All four main steps, flight experiments, data preparation, identification and

model validation require special attention.

• The coupled rotors/fuselage model represents a special and challenging

identification problem.

• Identification can be used as a tool in order to aid in various problems.

• Nonparametric identification can be very if properly used.

• Robotics can be benefitted from the aerospace community experience.

Kostas Alexis & Christoph Huerzeler 3 July 2012 Body Motion Rotorblade Motion Rotorblade...

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