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Doctoral Defence. Characterization, Modelling and Control of Mechanical Systems Comprising Material and Geometrical Nonlinearities. Tegoeh Tjahjowidodo Katholieke Universiteit Leuven Departement Werktuigkunde, Div. PMA Thursday 16 November 2006. Overview. Introduction

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Characterization modelling and control of mechanical systems comprising material and geometrical nonlinearities

Doctoral Defence

Characterization, Modelling and Control of Mechanical Systems Comprising Material and Geometrical Nonlinearities

Tegoeh Tjahjowidodo

Katholieke Universiteit Leuven

Departement Werktuigkunde, Div. PMA

Thursday 16 November 2006


Overview

Overview

  • Introduction

    • Nonlinearity sources

    • Dynamic signal classification

  • Geometric Nonlinearity (Backlash)

  • Material Nonlinearity (Friction)

  • Control of Nonlinear Systems

  • Application on a Real System

    Mechanical Systems

    with Harmonic Drive elements

  • Conclusions

  • Introduction

    • Nonlinearity sources

    • Dynamic signal classification

  • Geometric Nonlinearity (Backlash)

  • Material Nonlinearity (Friction)

  • Control of Nonlinear Systems

  • Application on a Real System

    Mechanical Systems

    with Harmonic Drive elements

  • Conclusions

  • Introduction

    • Nonlinearity sources

    • Dynamic signal classification

  • Geometric Nonlinearity (Backlash)

  • Material Nonlinearity (Friction)

  • Control of Nonlinear Systems

  • Application on a Real System

    Mechanical Systems

    with Harmonic Drive elements

  • Conclusions

  • Introduction

    • Nonlinearity sources

    • Dynamic signal classification

  • Geometric Nonlinearity (Backlash)

  • Material Nonlinearity (Friction)

  • Control of Nonlinear Systems

  • Application on a Real System

    Mechanical Systems

    with Harmonic Drive elements

  • Conclusions

  • Introduction

    • Nonlinearity sources

    • Dynamic signal classification

  • Geometric Nonlinearity (Backlash)

  • Material Nonlinearity (Friction)

  • Control of Nonlinear Systems

  • Application on a Real System

    Mechanical Systems

    with Harmonic Drive elements

  • Conclusions

  • Introduction

    • Nonlinearity sources

    • Dynamic signal classification

  • Geometric Nonlinearity (Backlash)

  • Material Nonlinearity (Friction)

  • Control of Nonlinear Systems

  • Application on a Real System

    Mechanical Systems

    with Harmonic Drive elements

  • Conclusions

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

Introduction

  • Introduction

    • Nonlinearity sources

    • Dynamic signal classification

  • Geometric Nonlinearity (Backlash)

  • Material Nonlinearity (Friction)

  • Control of Nonlinear Systems

  • Application on a Real System

    Mechanical Systems

    with Harmonic Drive elements

  • Conclusions


Introduction

Introduction

  • Motivation:

    Having better understanding of a system via appropriate techniques

     System Identification !

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

  • Identification purposes:

    • Dynamic behaviour analysis

    • Design engineering

    • Damage detection

    • Control design

  • There is no general identification method applicable to all systems.

    this depends on the characteristic of the system and the type of the signal involved in the identification.

  • Control design


Characteristic of systems

Characteristic of systems

Force B.C

prescribed displacement (û)

body force

(b)

displacement (u)

Displacement B.C.

Geometric nonlinearity

prescribed forces

(t)

strain

(e)

stress

(σ)

Material nonlinearity

Nonlinearity in a mechanical system can be attributed to different sources:

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions


Geometrical and material nonlinearities

Geometrical and Material Nonlinearities

  • Geometric Nonlinearity

    • the change in geometry as the structure deforms causes a nonlinear change of the parameters in the system

  • hardening spring, softening spring, saturation, …

  • Material Nonlinearity

    • the behaviour of material depends on the current deformation

  • frictional losses, ferromagnetism

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions


Dynamic signal classification

Dynamic Signal Classification

Dynamic Signal

Deterministic

‘Well-behaved’

  • Chaotic:

  • Lyapunov Exponent

  • Correlation Dimension

  • etc

Random

Stationary:

  • Frequency Response Function

  • Volterra

  • etc

Non-stationary:

  • Short Time Fourier Transform

  • Wigner-Ville

  • Wavelet

  • Hilbert

In general dynamic signals can be classified:

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions


Characterization modelling and control of mechanical systems comprising material and geometrical nonlinearities

Geometric Nonlinearity


Geometric nonlinearity

Geometric Nonlinearity

x0

Fin

x

k1x+F(x)

k0

m

k1+ k0

c

k1

k1

-x0

x0

x

Case study:

mechanical system with backlash element

Introduction

Geometric Nonlinearity

•Well-behaved

•Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

Backlash:

In a mechanical system, any lost motion between driving and driven elements due to clearance between parts.


Backlash

Backlash

Two different cases might appear:

Introduction

Geometric Nonlinearity

•Well-behaved

•Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

  • ‘well-behaved’ response for periodic input

    • Skeleton identification

      • Hilbert transform

      • Wavelet analysis

  • chaotic response for periodic input

    • Chaos quantification

      • Lyapunov exponent

    • Surrogate Data Test


Skeleton identification

Skeleton identification

SDOF system:

y(t) = A(t)·cos [y(t)-f]

  • Free vibration response can be represented in the combination of envelope and instantaneous phase.

Introduction

Geometric Nonlinearity

• Well-behaved

•Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions


Skeleton identification1

Skeleton identification

Introduction

Geometric Nonlinearity

• Well-behaved

•Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

Envelope and instantaneous phase of the free vibration response can be used as a mechanical signature of the dynamic parameters of the system (Feldman 1994a).


Skeleton signature examples 1

Skeleton signature examples (1)

Signatures of restoring forces

Introduction

Geometric Nonlinearity

• Well-behaved

•Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions


Skeleton signature examples 2

Skeleton signature examples (2)

Signatures of damping forces

Introduction

Geometric Nonlinearity

• Well-behaved

•Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions


Parameters identification

Parameters identification

Feldman (1994a) formulated parameters of the system based on the instantaneous envelope and phase:

Introduction

Geometric Nonlinearity

• Well-behaved

•Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

Extending the method to forced vibration problem, Feldman (1994b) proposed the following relations:


Instantaneous information extraction 1

Instantaneous information extraction (1)

~

y(t) = A(t)·sin [y(t)-f]

HT

Analytic Signal of y(t)

[z(t)]

|z(t)|

  • Hilbert transform:

a mathematical transform that shift each frequency component of the instantaneous spectrum by p/2 without affecting the magnitude.

Introduction

Geometric Nonlinearity

• Well-behaved

•Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

y(t) = A(t)·cos [y(t)-f]

(+)simple, fast, practical

(–)inaccurate! (only suitable for asympotic signal)


Instantaneous information extraction 2

Wavelet transform:

Instantaneous information extraction (2)

instantaneous frequency

(in dilation form):

envelope in modulus of wavelet:

a time-frequency representation (TFR) technique

Introduction

Geometric Nonlinearity

• Well-behaved

•Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

(Complex Morlet Wavelet)

(+) accurate!

(–) time consuming, error at the edges, cumbersome


Illustration of extraction

Illustration of extraction

Envelope Estimation

Instantaneous Frequency Estimation

1.5

20

HT technique

HT technique

Wavelet technique

Wavelet technique

15

1

10

0.5

Amplitude

Frequency

5

0

0

-0.5

-5

-1

-10

0

2

4

6

8

10

0

2

4

6

8

10

time (s)

time (s)

Damped-chirp signal

Introduction

Geometric Nonlinearity

• Well-behaved

•Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions


Experimental setup

Experimental Setup

2nd link

Encoder #2

Encoder #1

1st link

Schematic drawing of a two-link mechanism

Introduction

Geometric Nonlinearity

• Well-behaved

•Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

  • 1st link fixed

  • backlash was introduced in the joint

  • considered as a base motion system


Base motion system

Base Motion System

k

m

c

y

x

Introduction

Geometric Nonlinearity

• Well-behaved

•Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

where z is a relative motion between x and y

Force balance diagram of link #2

q: displacement input

f: displacement output


Displacement input

Displacement Input

Displacement input measured by encoder #1

Introduction

Geometric Nonlinearity

• Well-behaved

•Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

Input (degree)

time (s)


Displacement output

Displacement Output

Output (degree)

time (s)

Displacement output measured by encoder #2

Introduction

Geometric Nonlinearity

• Well-behaved

•Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions


Relative motion

Relative Motion

Relative motion (degree)

Backlash size

time (s)

Relative motion between output and input(f-q)

Introduction

Geometric Nonlinearity

• Well-behaved

•Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions


Envelope and instantaneous frequency 1

Envelope and Instantaneous Frequency (1)

Envelope and instantaneous frequency estimation of input signal using Hilbert and Wavelet Transform

Introduction

Geometric Nonlinearity

• Well-behaved

•Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

Instantaneous frequency extraction

Envelope extraction

Frequency (Hz)

time (s)

time (s)


Envelope and instantaneous frequency 2

Envelope and Instantaneous Frequency (2)

Envelope and instantaneous frequency estimation of output signal using Hilbert and Wavelet Transform

Introduction

Geometric Nonlinearity

• Well-behaved

•Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

Instantaneous frequency extraction

Envelope extraction

Frequency (Hz)

time (s)

time (s)


Restoring force estimation

Restoring force estimation

Reconstruction of restoring force using Hilbert and Wavelet Transform

Introduction

Geometric Nonlinearity

• Well-behaved

•Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions


Damping force estimation

Damping force estimation

Reconstruction of damping force using Hilbert and Wavelet Transform

Introduction

Geometric Nonlinearity

• Well-behaved

•Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions


Identification on well behaved case

Identification on Well-behaved Case

Introduction

Geometric Nonlinearity

• Well-behaved

•Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

Wavelet transform offers better results than the Hilbert transform in skeleton method.


Chaoticity in a system with backlash

Chaoticity in a system with backlash

Schematic of mechanical system with backlash component

Introduction

Geometric Nonlinearity

• Well-behaved

• Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

Example of parameters that lead to chaotic motion:


Chaotic response 1

Chaotic response (1)

CASE 1 was excited with sinusoidal signal with A=100 N and w=40 rad/s

Introduction

Geometric Nonlinearity

• Well-behaved

• Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

Displacement (mm)

Power spectrum (dB)

Freq. (Hz)

time (s)


Chaotic response 2

Chaotic response (2)

Phase plot

Poincaré map

velocity (mm/s)

velocity (mm/s)

displacement (mm)

displacement (mm)

Phase plot and Poincaré map of case 1

Introduction

Geometric Nonlinearity

• Well-behaved

• Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions


Chaos identification

Chaos Identification

EXPONENTIAL GROWTH dt

  • for exponential growth, should see

  • is the average rate of the exponential growth

  • Lyapunov exponent

d = d e

lt

0

t

How do we know when the mapping is chaotic?

quantification of chaos Lyapunov exponents

Introduction

Geometric Nonlinearity

• Well-behaved

• Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

  • consider one dimension:

  • take two initial conditions differing by a small amount

  • to identify chaos, observe the evolution in time and compare the differences

▪▪

TWO NEARBY

INITIAL TRAJECTORY d0


Dimensional analysis

Dimensional Analysis

In order to examine the influence of each parameter on the nature of resulting response

Introduction

Geometric Nonlinearity

• Well-behaved

• Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

where:

primes indicate differentiation with respect to .


Effect of forcing parameter and backlash

Effect of Forcing Parameter and Backlash

Largest Lyapunov Exponent

1/a=3.4

1/a

1/a=4.6

Lyapunov exponent vs Forcing Parameter and/or Backlash Size

Introduction

Geometric Nonlinearity

• Well-behaved

• Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions


Chaotic response in experimental setup

Chaotic response in experimental setup

Phase plots of output responses for periodic signal with different excitation level

Introduction

Geometric Nonlinearity

• Well-behaved

• Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions


Noise reduction in chaotic signal

Noise Reduction in Chaotic Signal

  • Simple Noise Reduction Method

    • developed based on near future prediction

Introduction

Geometric Nonlinearity

• Well-behaved

• Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

Phase plots of output responses for periodic signal before and after noise reduction


Noise effect in estimating dimension

Noise Effect in Estimating Dimension

Minimum embedding dimension information for phase-plot reconstruction

Introduction

Geometric Nonlinearity

• Well-behaved

• Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions


Identification on chaotic case

Identification on Chaotic Case

Introduction

Geometric Nonlinearity

• Well-behaved

• Chaotic

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

Observing the chaos quantifier, e.g. Lyapunov exponent, could be used, in principle, to estimate the parameter of a system.


Characterization modelling and control of mechanical systems comprising material and geometrical nonlinearities

Material Nonlinearity


Material nonlinearity

Material Nonlinearity

friction

velocity

Case study:

mechanical system with friction element

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

Friction is the result of a complex interaction between two contact surfaces.

  • Conventional friction model:

    Coulomb model

    • discontinuity at zero velocity


Friction characterisation

Friction Characterisation

Two different friction regimes have been distinguished:

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

  • the pre-sliding regime:

appears predominantly as a function of displacement

  • the sliding regime:

function of sliding velocity

(Armstrong-Hélouvry, 1991,

Canudas de Wit et al., 1995,

Swevers et al., 2000,

Al-Bender et al., 2004)


Pre sliding regime 1

Pre-sliding regime (1)

Friction force

Fm

1

1

3

y(q)

2

-qm

qm

0

(displacement)

x

-y(-q)

y(q)

2

2

x

Pre-sliding friction

  • Friction force in pre-sliding regime not only depends on the output at some time instant in the past and the input, but also on past extremum values of the input or output as well.

     hysteresis with non-local memory

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

4


Pre sliding regime 2

Pre-sliding regime (2)

friction

displacement

Equivalent dynamic parameters

The Describing Function technique is used to obtain the equivalent stiffness and damping:

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

y(q) is the virgin curve of the hysteresis


Pre sliding friction model

Pre-sliding friction model

F

Wi

ki

x

  • Alternative representation of hysteresis function with non-local memory for pre-sliding friction:

     parallel connection of N elasto-slide elements

    (Maxwell-Slip elements)

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions


Sliding regime

Sliding regime

  • When the motion is entering the sliding regime, in most cases, the Maxwell-Slip model is no longer suitable.

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

  • The friction usually has a maximum value at the beginning and then continues to decrease with increasing velocity


Gms friction model

GMS friction model

  • Generalized Maxwell-Slip (GMS) model

    • developed at PMA/KULeuven

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

  • If the model is sticking:

Mathematical representation of Maxwell-Slip elements

  • If the model is slipping:


Dc motor

DC Motor

dSPACE® 1104

acquisition board

servo

DC Motor

amp

Experimental setup of DC motor ABB M19-S

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

Encoder

Load

Timing Belt


Friction identification

Friction Identification

  • Two sets of experiments were carried out for friction identification in the DC motor (ABB M19-S):

    • Low frequencies signal

    • High frequencies signal

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

1st velocity signal

2nd velocity signal

velocity (rad/s)

time (s)

time (s)


Identification strategy

Identification Strategy

The optimization is based upon minimization of cost function:

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

Identification technique for the physics-based model:

  • Genetic Algorithm

  • Nelder-Mead Downhill Simplex Method


Identification results 1

Identification Results #1

2.06% (0.3360)

Coulomb model

Exponential Coulomb model

GMS

1.97% (0.3470)

torque (Nm)

MSE (max.err.)

Coulomb

2.06% (0.3770)

Exp-Coulomb

LuGre

2.03% (0.3530)

GMS model

LuGre model

torque (Nm)

time (s)

time (s)

  • Identification results of low frequencies experiment:

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions


Identification results 2

Identification Results #2

Exponential Coulomb model

Coulomb model

9.59% (1.2604)

torque (Nm)

GMS4

1.39% (0.5711)

MSE (max.err.)

Coulomb

17.92% (1.2993)

Exp-Coulomb

GMS model

LuGre model

LuGre

4.30% (0.6466)

GMS10

1.19% (0.5177)

torque (Nm)

time (s)

  • Identification results of high frequencies experiment:

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

time (s)


Identification on material nonlinearity case

Identification on Material Nonlinearity Case

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

Friction identification is possible to be conducted using a single experiment. However, selection of the excitation signal plays an important role for the identification step.


Characterization modelling and control of mechanical systems comprising material and geometrical nonlinearities

Control of Nonlinear Systems


Overview of controllers

Overview of Controllers

Control

input (u)

-Qh

Qh

Position error (e)

Model-based controllers

  • Linear Controllers

    • PD controller

    • Cascade controller (combine position-speed loops)

  • Nonlinear Controllers

    • Discontinuous Nonlinear Proportional Feedback (DNPF) controller

    • Gain Scheduling controller

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

 adds an extra compensating torque when the position error is within pre-sliding region

 developed based on the equivalent dynamic parameters of the system.


Gain scheduling controller 1

Gain Scheduling Controller (1)

 treats two different regimes of friction in separated modes.

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

  • The first mode

  • when the system is moving in the sliding region

  • linear controller + equivalent Coulomb friction model

  • The second mode

  • when the system is moving in the pre-sliding region

  • adjusts proportional (kp) and derivative (kd) gain based on the equivalent dynamic parameter

The corresponding gains are designed and optimized at some points (of amplitude of motion) regarding a certain performance criteria.


Gain scheduling controller 2

Gain Scheduling Controller (2)

1st mode

Gain Scheduling Strategy

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

Look-up table of the gains

2nd mode


Equivalent dynamic parameters

Equivalent Dynamic Parameters

Based on the obtained profile of the hysteresis in the pre-sliding regime, the equivalent dynamic parameters can be constructed:

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions


Gain scheduling design

Gain Scheduling Design

  • By using the obtained equivalent dynamic parameters, some PD gains at different selected amplitudes are optimized.

  • The optimal gains are interpolated for intermediate points.

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions


Control objectives

Control Objectives

Performance criteria:

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

In the point-to-point (PTP) positioning system, a high accuracy and a short transition time are the most important performance criteria, while the path of the motion is less significant.

High accuracy and fast response speed with no overshoots are desired.


Step input

Step Input

The step responses to a 0.4 rad step input are appraised.

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

-


Step response result

Step Response Result

Step responses of the system using the proposed gain scheduling controller in comparison with the PD, cascade and DNPF controllers.

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions


Control of nonlinear system

Control of Nonlinear System

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

Model based control is able to yield good results, depending on the models used and the control strategy.


Characterization modelling and control of mechanical systems comprising material and geometrical nonlinearities

Application on a Real System

(Harmonic Drive)


The harmonic drive

The Harmonic Drive

  • Invented by C. Walton Musser in 1955.

  • Originally labeled ‘strain-wave gearing’, which employs a continuous deflection wave along a non-rigid gear to allow gradual engagement of gear teeth.

  • Can provide very high reduction ratios in a very small package.

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

• Identification

• Control

Conclusions


The harmonic drive components

The Harmonic Drive components

WAVE DRIVE®

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

• Identification

• Control

Conclusions

(has 2 more teeth than flexspline)


Operating mechanism

Operating mechanism

Operating principle of harmonic drive

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

• Identification

• Control

Conclusions


Scope of harmonic drive research

Scope of Harmonic Drive Research

  • Torsional Stiffness:

    • apparently due to deformation of the wave generator, and an increase in gear-tooth contact area with increasing load (Nye, 1991).

    • displays a ‘soft wind-up’ behavior, which is characterized by very low stiffness at low applied load (Kircanski, 1997).

  • Frictional Losses:

    • occurs primarily at the gear-tooth interface

    • Friction in Harmonic Drive is strongly position dependent due to kinematic error (Kennedy, 2003).

  • Kinematic error:

    • caused by a number of factors, such as tooth-placement errors, out-of-roundness in HD components, and misalignment during assembly.

    • The error signature can display frequency components at two cycles per wave-generator revolution (Tuttle, 1992).

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

• Identification

• Control

Conclusions


Torsional stiffness definition

Torsional Stiffness - definition

Torsional stiffness is measured by locking the wave generator to the circular spline and applying loads to a link subjected to the flexspline.

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

• Identification

• Control

Conclusions


Torsional stiffness experiments 1

Torsional Stiffness – experiments (1)

  • Experiment was carried out on WAVE DRIVE® component:

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

• Identification

• Control

Conclusions

Lock the

wave-generator

load cell

Bentley probe


Torsional stiffness experiments 2

Torsional Stiffness – experiments (2)

×10-3

4

3

2

1

0

Torsion (rad)

-1

-2

-3

-4

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Torque (Nm)

Stiffness curve obtained from sine excitation:

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

• Identification

• Control

Conclusions


Torsional stiffness experiments 3

Torsional Stiffness – experiments (3)

×10-3

3.75

3.75

2.50

2.50

1.25

1.25

Torsion (rad)

0

0

-1.25

-1.25

-2.50

-2.50

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0

2

4

6

8

time (s)

Torque (Nm)

Stiffness curve obtained from triangular-wave excitation

(with varying amplitudes):

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

• Identification

• Control

Conclusions


Torsional stiffness models

Torsional Stiffness Models

F

k1

k0

-x0

x

x0

F

x

  • Piecewise linear model

  • Hysteresis models

Two different approach of models:

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

• Identification

• Control

Conclusions


Schematic of torsional stiffness model

Schematic of torsional stiffness model

Proposed model of torsional stiffness:

Maxwell-slip elements

+

hardening spring

Pre-sliding friction

elementary

stick-slip 1

x

elementary

stick-slip N

T

hardening spring

S

S

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

• Identification

• Control

Conclusions


Torsional stiffness hysteresis model 1

Torsional Stiffness – hysteresis model (1)

Torsional Stiffness

×10-3

3

2

1

Torsion (rad)

0

-1

-2

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Torque (Nm)

Identification results:

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

• Identification

• Control

Conclusions


Torsional stiffness hysteretic model 2

Torsional Stiffness – hysteretic model (2)

0.4

0.2

0

Torque (Nm)

-0.2

-0.4

0

5

10

15

20

25

30

0.4

0.2

Error (Nm)

0

-0.2

-0.4

0

5

10

15

20

25

30

Time (sec)

Identification results:

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

• Identification

• Control

Conclusions

1.14% MSE

(MSE of piecewise linear model: 8.7%)


Control of mechanical system with hd

Control of Mechanical System with HD

  • System apparatus:

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

• Identification

• Control

Conclusions

FHA-C mini servo actuator

  • gear set with AC servo motor in one compact package


Fha c mini servo actuator

FHA-C mini servo actuator

load inertia

circular-spline inertia

armature inertia

complete-close package

torsional stiffness of the HD

friction in the motor

kS

kH

m3

m1

m2

T

TF

  • Schematic drawing of FHA-C

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

• Identification

• Control

Conclusions

Friction in the DC motor and torsional stiffness in the gear set cannot be identified separately.


Control of fha c mini servo actuator

Control of FHA-C mini servo actuator

keq

m3

m1

T

Two different approaches are considered for control purposes:

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

• Identification

• Control

Conclusions

  • First approach: a mass on a frictional surface

meq

T

Tf

  • Second approach: two masses connected by a hysteresis torsional spring


2 nd approach two masses and hysteresis spring 1

2nd Approach - Two masses and hysteresis spring (1)

Assumption:

  • Neglect the external hysteretic nonlinearity source.

  • Nonlinearity mainly comes from the torsional stiffness

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

• Identification

• Control

Conclusions

The torsional stiffness is identified by locking the output shaft and measuring the motor current.


2 nd approach two masses and hysteresis spring 2

2nd Approach - Two masses and hysteresis spring (2)

PD

du

|.|=0

and

dt

|x-

|

x

memorize

x

=x

|x-

|<Q

x

h

x1

-

x2

x2

+

SYSTEM

xd

-

+

  • Resulting the equivalent dynamic parameters:

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

• Identification

• Control

Conclusions

  • A modified gain scheduling strategy:


Control results of 2 nd approach 1

Control Results of 2nd Approach (1)

Step response to a 0.2 rad step input

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

• Identification

• Control

Conclusions


Control results of 2 nd approach 2

Control Results of 2nd Approach (2)

Step response to a 0.2 rad step input + load

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

• Identification

• Control

Conclusions


Application on a system with hd

Application on a System with HD

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

• Identification

• Control

Conclusions

A piecewise linear model together with non-local memory hysteresis resolve the difficulties in determining the model of torsional stiffness in harmonic drive.


Conclusions

Conclusions

  • Two cases for a nonlinear system with ‘well-behaved’ and chaotic response for a periodic input have been addressed and appropriate identification methods are developed for each.

  • Identification of systems with material nonlinearity (friction) utilizing single experiment is feasible to be carried out.

  • Detailed understanding of a physical system is playing an important role in achieving a controller with high performance.

  • Identification of a system, in which the two nonlinearity sources are manifested, has been conducted successfully.

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions


Future work

Future work

  • Combining the advantages of Hilbert and wavelet transform in order to improve the skeleton techniques.

  • Implementation of the skeleton technique for a real higher-degree-of-freedom system.

  • Further study for the applicability of the GMS model to any friction conditions.

  • Extension of the identification and control methods to higher-degree-of-freedom systems with two (or more) hysteresis (material) nonlinearities.

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions


Characterization modelling and control of mechanical systems comprising material and geometrical nonlinearities

Introduction

Geometric Nonlinearity

Material Nonlinearity

Control of Nonlinear Systems

Application on Real System

Conclusions

Thank you for your attention


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