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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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Tegoeh Tjahjowidodo
Katholieke Universiteit Leuven
Departement Werktuigkunde, Div. PMA
Thursday 16 November 2006
Mechanical Systems
with Harmonic Drive elements
Mechanical Systems
with Harmonic Drive elements
Mechanical Systems
with Harmonic Drive elements
Mechanical Systems
with Harmonic Drive elements
Mechanical Systems
with Harmonic Drive elements
Mechanical Systems
with Harmonic Drive elements
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Introduction
Mechanical Systems
with Harmonic Drive elements
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
this depends on the characteristic of the system and the type of the signal involved in the identification.
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
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Dynamic Signal
Deterministic
‘Wellbehaved’
Random
Stationary:
Nonstationary:
In general dynamic signals can be classified:
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
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
•Wellbehaved
•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.
Two different cases might appear:
Introduction
Geometric Nonlinearity
•Wellbehaved
•Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
SDOF system:
y(t) = A(t)·cos [y(t)f]
Introduction
Geometric Nonlinearity
• Wellbehaved
•Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Introduction
Geometric Nonlinearity
• Wellbehaved
•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).
Signatures of restoring forces
Introduction
Geometric Nonlinearity
• Wellbehaved
•Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Signatures of damping forces
Introduction
Geometric Nonlinearity
• Wellbehaved
•Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Feldman (1994a) formulated parameters of the system based on the instantaneous envelope and phase:
Introduction
Geometric Nonlinearity
• Wellbehaved
•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:
~
y(t) = A(t)·sin [y(t)f]
HT
Analytic Signal of y(t)
[z(t)]
z(t)
a mathematical transform that shift each frequency component of the instantaneous spectrum by p/2 without affecting the magnitude.
Introduction
Geometric Nonlinearity
• Wellbehaved
•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 frequency
(in dilation form):
envelope in modulus of wavelet:
a timefrequency representation (TFR) technique
Introduction
Geometric Nonlinearity
• Wellbehaved
•Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
(Complex Morlet Wavelet)
(+) accurate!
(–) time consuming, error at the edges, cumbersome
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)
Dampedchirp signal
Introduction
Geometric Nonlinearity
• Wellbehaved
•Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
2nd link
Encoder #2
Encoder #1
1st link
Schematic drawing of a twolink mechanism
Introduction
Geometric Nonlinearity
• Wellbehaved
•Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
k
m
c
y
x
Introduction
Geometric Nonlinearity
• Wellbehaved
•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 measured by encoder #1
Introduction
Geometric Nonlinearity
• Wellbehaved
•Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Input (degree)
time (s)
Output (degree)
time (s)
Displacement output measured by encoder #2
Introduction
Geometric Nonlinearity
• Wellbehaved
•Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Relative motion (degree)
Backlash size
time (s)
Relative motion between output and input(fq)
Introduction
Geometric Nonlinearity
• Wellbehaved
•Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Envelope and instantaneous frequency estimation of input signal using Hilbert and Wavelet Transform
Introduction
Geometric Nonlinearity
• Wellbehaved
•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 estimation of output signal using Hilbert and Wavelet Transform
Introduction
Geometric Nonlinearity
• Wellbehaved
•Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Instantaneous frequency extraction
Envelope extraction
Frequency (Hz)
time (s)
time (s)
Reconstruction of restoring force using Hilbert and Wavelet Transform
Introduction
Geometric Nonlinearity
• Wellbehaved
•Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Reconstruction of damping force using Hilbert and Wavelet Transform
Introduction
Geometric Nonlinearity
• Wellbehaved
•Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Introduction
Geometric Nonlinearity
• Wellbehaved
•Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Wavelet transform offers better results than the Hilbert transform in skeleton method.
Schematic of mechanical system with backlash component
Introduction
Geometric Nonlinearity
• Wellbehaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Example of parameters that lead to chaotic motion:
CASE 1 was excited with sinusoidal signal with A=100 N and w=40 rad/s
Introduction
Geometric Nonlinearity
• Wellbehaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Displacement (mm)
Power spectrum (dB)
Freq. (Hz)
time (s)
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
• Wellbehaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
EXPONENTIAL GROWTH dt
d = d e
lt
0
t
How do we know when the mapping is chaotic?
quantification of chaos Lyapunov exponents
Introduction
Geometric Nonlinearity
• Wellbehaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
▪▪
TWO NEARBY
INITIAL TRAJECTORY d0
In order to examine the influence of each parameter on the nature of resulting response
Introduction
Geometric Nonlinearity
• Wellbehaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
where:
primes indicate differentiation with respect to .
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
• Wellbehaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Phase plots of output responses for periodic signal with different excitation level
Introduction
Geometric Nonlinearity
• Wellbehaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Introduction
Geometric Nonlinearity
• Wellbehaved
• 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
Minimum embedding dimension information for phaseplot reconstruction
Introduction
Geometric Nonlinearity
• Wellbehaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Introduction
Geometric Nonlinearity
• Wellbehaved
• 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.
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.
Coulomb model
Two different friction regimes have been distinguished:
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
appears predominantly as a function of displacement
function of sliding velocity
(ArmstrongHélouvry, 1991,
Canudas de Wit et al., 1995,
Swevers et al., 2000,
AlBender et al., 2004)
Friction force
Fm
1
1
3
y(q)
2
qm
qm
0
(displacement)
x
y(q)
y(q)
2
2
x
Presliding friction
hysteresis with nonlocal memory
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
4
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
F
Wi
ki
x
parallel connection of N elastoslide elements
(MaxwellSlip elements)
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Mathematical representation of MaxwellSlip elements
dSPACE® 1104
acquisition board
servo
DC Motor
amp
Experimental setup of DC motor ABB M19S
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Encoder
Load
Timing Belt
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)
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 physicsbased model:
2.06% (0.3360)
Coulomb model
Exponential Coulomb model
GMS
1.97% (0.3470)
torque (Nm)
MSE (max.err.)
Coulomb
2.06% (0.3770)
ExpCoulomb
LuGre
2.03% (0.3530)
GMS model
LuGre model
torque (Nm)
time (s)
time (s)
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Exponential Coulomb model
Coulomb model
9.59% (1.2604)
torque (Nm)
GMS4
1.39% (0.5711)
MSE (max.err.)
Coulomb
17.92% (1.2993)
ExpCoulomb
GMS model
LuGre model
LuGre
4.30% (0.6466)
GMS10
1.19% (0.5177)
torque (Nm)
time (s)
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
time (s)
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.
Control
input (u)
Qh
Qh
Position error (e)
Modelbased controllers
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 presliding region
developed based on the equivalent dynamic parameters of the system.
treats two different regimes of friction in separated modes.
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
The corresponding gains are designed and optimized at some points (of amplitude of motion) regarding a certain performance criteria.
1st mode
Gain Scheduling Strategy
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Lookup table of the gains
2nd mode
Based on the obtained profile of the hysteresis in the presliding regime, the equivalent dynamic parameters can be constructed:
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Performance criteria:
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
In the pointtopoint (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.
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 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
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.
(Harmonic Drive)
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
• Identification
• Control
Conclusions
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 principle of harmonic drive
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
• Identification
• Control
Conclusions
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
• Identification
• Control
Conclusions
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
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
• Identification
• Control
Conclusions
Lock the
wavegenerator
load cell
Bentley probe
×103
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
×103
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 triangularwave excitation
(with varying amplitudes):
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
• Identification
• Control
Conclusions
F
k1
k0
x0
x
x0
F
x
Two different approach of models:
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
• Identification
• Control
Conclusions
Proposed model of torsional stiffness:
Maxwellslip elements
+
hardening spring
Presliding friction
elementary
stickslip 1
x
elementary
stickslip N
T
hardening spring
S
S
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
• Identification
• Control
Conclusions
Torsional Stiffness
×103
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
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%)
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
• Identification
• Control
Conclusions
FHAC mini servo actuator
load inertia
circularspline inertia
armature inertia
completeclose package
torsional stiffness of the HD
friction in the motor
kS
kH
m3
m1
m2
T
TF
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.
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
meq
T
Tf
Assumption:
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.
PD
du
.=0
and
dt
x

x
memorize
x
=x
x

x
h
x1

x2
x2
+
SYSTEM
xd

+
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
• Identification
• Control
Conclusions
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
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
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
• Identification
• Control
Conclusions
A piecewise linear model together with nonlocal memory hysteresis resolve the difficulties in determining the model of torsional stiffness in harmonic drive.
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Thank you for your attention