Tegoeh Tjahjowidodo Characterization, Modelling and Control of Mechanical Systems Comprising...
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W
P M A
Tegoeh Tjahjowidodo
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
Doctoral Defence
Tegoeh Tjahjowidodo
P M A
W
Overview
Introduction– Nonlinearity sources– Dynamic signal classification
Geometric Nonlinearity (Backlash)
Material Nonlinearity (Friction)
Control of Nonlinear Systems
Application on a Real SystemMechanical 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 SystemMechanical Systems
with Harmonic Drive elements
Conclusions
Introduction Introduction– Nonlinearity sources– Dynamic signal classification
Geometric Nonlinearity (Backlash)
Material Nonlinearity (Friction)
Control of Nonlinear Systems
Application on a Real SystemMechanical Systems
with Harmonic Drive elements
Conclusions
Geometric Nonlinearity
Introduction– Nonlinearity sources– Dynamic signal classification
Geometric Nonlinearity (Backlash)
Material Nonlinearity (Friction)
Control of Nonlinear Systems
Application on a Real SystemMechanical Systems
with Harmonic Drive elements
Conclusions
Material Nonlinearity
Introduction– Nonlinearity sources– Dynamic signal classification
Geometric Nonlinearity (Backlash)
Material Nonlinearity (Friction)
Control of Nonlinear Systems
Application on a Real SystemMechanical Systems
with Harmonic Drive elements
Conclusions
Control of Nonlinear Systems
Introduction– Nonlinearity sources– Dynamic signal classification
Geometric Nonlinearity (Backlash)
Material Nonlinearity (Friction)
Control of Nonlinear Systems
Application on a Real SystemMechanical Systems
with Harmonic Drive elements
Conclusions
Application on Real System
Introduction– Nonlinearity sources– Dynamic signal classification
Geometric Nonlinearity (Backlash)
Material Nonlinearity (Friction)
Control of Nonlinear Systems
Application on a Real SystemMechanical Systems
with Harmonic Drive elements
Conclusions
Conclusions
Tegoeh Tjahjowidodo
P M A
W
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
ConclusionsThere 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.
Identification purposes:– Dynamic behaviour analysis– Design engineering– Damage detection– Control design– Control design
Tegoeh Tjahjowidodo
P M A
W
Characteristic of systems
prescribed displacement (û)
displacement (u)
strain(e)
stress(σ)
body force(b)
prescribed forces(t)
Force B.C
Displacement B.C.
Material nonlinearity
Geometric 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
Tegoeh Tjahjowidodo
P M A
W
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
Tegoeh Tjahjowidodo
P M A
W
Dynamic Signal Classification
Chaotic:• Lyapunov Exponent• Correlation Dimension• etc
‘Well-behaved’
Deterministic
Dynamic Signal
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
Tegoeh Tjahjowidodo
P M A
W
Geometric Nonlinearity
Case study:mechanical system with backlash element
Backlash:
In a mechanical system, any lost motion between driving and driven elements due to clearance between parts.
ck1
k0
x0 x
m
Fin
k1
k1+ k0
x0
-x0
x
k1x+
F(x
)
Introduction
Geometric Nonlinearity
• Well-behaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
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
Tegoeh Tjahjowidodo
P M A
W
Skeleton identification
Free vibration response can be represented in the combination of envelope and instantaneous phase.
SDOF system: 0y)A(y)A(h2y 200
Introduction
Geometric Nonlinearity
• Well-behaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
y(t) = A(t)·cos [(t)-
Tegoeh Tjahjowidodo
P M A
W
Skeleton identification
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).
Introduction
Geometric Nonlinearity
• Well-behaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
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
Tegoeh Tjahjowidodo
P M A
W
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
Tegoeh Tjahjowidodo
P M A
W
Parameters identification
Feldman (1994a) formulated parameters of the system based on the instantaneous envelope and phase:
0y)A(y)A(h2y 2
00
22 20 2
0
(t) (t)A A 2A A(t)
m A m A A A
(t) Ah (t)
2 m A 2
20 0y 2h (A)y (A)y f / m
Extending the method to forced vibration problem, Feldman (1994b) proposed the following relations:
F(t)(t) j (t)
X(t)
A
A
A
A2
A
A)A(
2
222
0
2A
A)A(h 0
Introduction
Geometric Nonlinearity
• Well-behaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
Instantaneous information extraction (1)
Hilbert transform:a mathematical transform that shift each frequency component of the
instantaneous spectrum by /2 without affecting the magnitude.
t
1)t(y
dt
)t(y1)}t(y{)t(y~
H
y(t) = A(t)·cos [(t)-
Analytic Signal of y(t)
z(t) y(t) jy(t) A(t) exp[ j (t)]
(+) simple, fast, practical
|z(t)| [z(t)]
y(t) = A(t)·sin [(t)-HT ~
(–) inaccurate! (only suitable for asympotic signal)
Introduction
Geometric Nonlinearity
• Well-behaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
Instantaneous information extraction (2)
Wavelet transform:
dts
t)t(x
s),s(W
1
a time-frequency representation (TFR) technique
(Complex Morlet Wavelet)
b2
c f/ttjb eef)t(
(+) accurate!(–) time consuming, error at the edges, cumbersome
instantaneous frequency(in dilation form):
)τ(
)0(s c
envelope in modulus of wavelet:
))τ(s()(As
),s(W * 1
Introduction
Geometric Nonlinearity
• Well-behaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
Illustration of extraction
Damped-chirp signal
Envelope Estimation
0 2 4 6 8 10-1
-0.5
0
0.5
1
1.5
time (s)
Am
plit
ud
e
HT techniqueWavelet technique
Instantaneous Frequency Estimation
0 2 4 6 8 10-10
-5
0
5
10
15
20
time (s)F
req
ue
ncy
HT techniqueWavelet technique
Introduction
Geometric Nonlinearity
• Well-behaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
Experimental Setup
Encoder #2
2nd link
1st linkEncoder #1
Schematic drawing of a two-link mechanism
• 1st link fixed
• backlash was introduced in the joint
• considered as a base motion system
Introduction
Geometric Nonlinearity
• Well-behaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
Base Motion System
Force balance diagram of link #2
z where;Cz)A(z)A(h2z 2
00
Introduction
Geometric Nonlinearity
• Well-behaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
xz)A(z)A(h2z 200
where z is a relative motion between x and y
k
c
m
x y
: displacement input: displacement output
Tegoeh Tjahjowidodo
P M A
W
Displacement Input
Displacement input measured by encoder #1
0 5 10 15 20 25 30-1.5
-1
-0.5
0
0.5
1
1.5
time (s)
Inp
ut
(de
gre
e)
Input side displacement(encoder input)
Inp
ut
(de
gre
e)
time (s)
Introduction
Geometric Nonlinearity
• Well-behaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
Displacement Output
Displacement output measured by encoder #2
0 5 10 15 20 25 30-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
time (s)
Ou
tpu
t (d
eg
ree
)
Output side displacement(encoder output)
Ou
tpu
t (d
eg
ree
)
time (s)
Introduction
Geometric Nonlinearity
• Well-behaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
Relative Motion
Relative motion between output and input ()
0 5 10 15 20 25 30-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
time (s)
Re
lati
ve
mo
tio
n (
de
gre
e)
Relative motion between two linksR
ela
tive
mo
tion
(d
eg
ree
)
time (s)
Backlash size
Introduction
Geometric Nonlinearity
• Well-behaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
Envelope and instantaneous frequency estimation of input signal using Hilbert and Wavelet Transform
time (s) time (s)F
requ
ency
(H
z)
Envelope extraction Instantaneous frequency extraction
Envelope and Instantaneous Frequency (1)
Introduction
Geometric Nonlinearity
• Well-behaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
Envelope and Instantaneous Frequency (2)
Envelope and instantaneous frequency estimation of output signal using Hilbert and Wavelet Transform
time (s) time (s)F
requ
ency
(H
z)
Envelope extraction Instantaneous frequency extraction
Introduction
Geometric Nonlinearity
• Well-behaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
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
Tegoeh Tjahjowidodo
P M A
W
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
Tegoeh Tjahjowidodo
P M A
W
Identification on Well-behaved Case
Wavelet transform offers better results than the Hilbert transform in skeleton method.
Introduction
Geometric Nonlinearity
• Well-behaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
Chaoticity in a system with backlash
m (kg)
k1 (N/m)
k0 (N/m)
c (Ns/m)
x0 (m)
CASE 1 1 0 40000 8 0.005 CASE 2 1 1000 31000 8 0.005
Schematic of mechanical system with backlash component
Example of parameters that lead to chaotic motion:
Introduction
Geometric Nonlinearity
• Well-behaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
Chaotic response (1)
0 50 100 150 200-80
-60
-40
-20
0
20
40
Freq (Hz)
Pow
er S
pect
rum
(dB
)
0 0.5 1 1.5 2 2.5 3-20
-15
-10
-5
0
5
10
15
20
t (sec)
Dis
plac
emen
t (m
m)
CASE 1 was excited with sinusoidal signal with A=100 N and =40 rad/s
time (s) Freq. (Hz)
Dis
pla
cem
ent
(m
m)
Pow
er s
pect
rum
(dB
)
Introduction
Geometric Nonlinearity
• Well-behaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
Chaotic response (2)
Phase plot and Poincaré map of case 1
displacement (mm)
velo
city
(m
m/s
)
Phase plot
displacement (mm)ve
loci
ty (
mm
/s)
Poincaré map
Introduction
Geometric Nonlinearity
• Well-behaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
Chaos Identification
How do we know when the mapping is chaotic?
consider one dimension:• take two initial conditions
differing by a small amount
quantification of chaos Lyapunov exponents
• to identify chaos, observe the evolution in time and compare the differences
• for exponential growth, should see
• is the average rate of the exponential growth
• Lyapunov exponent
et 0t
Introduction
Geometric Nonlinearity
• Well-behaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
EXPONENTIAL GROWTH t
▪ ▪TWO NEARBY
INITIAL TRAJECTORY 0
Tegoeh Tjahjowidodo
P M A
W
Dimensional Analysis
In order to examine the influence of each parameter on the nature of resulting response
02 /cos)p(p'"p F
mkc 02 0xkA 0
t0 0x/xp mk /02
0
primes indicate differentiation with respect to .
where:
1,1
1||,0
1,1
pp
p
pp
)p(F
tcosA)x(Fxcxm
Introduction
Geometric Nonlinearity
• Well-behaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
Effect of Forcing Parameter and Backlash
0 1 2 3 4 5 6 7 8 9 100
0.05
0.1
0.15
0.2
0.25
k0x
0/A
Larg
est
Lyapunov E
xponent
2=0.042=0.02
1/
Larg
est
Lyap
unov
Exp
onen
t
1/=4.6
1/=3.4
Lyapunov exponent vs Forcing Parameter and/or Backlash SizeIntroduction
Geometric Nonlinearity
• Well-behaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
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
Tegoeh Tjahjowidodo
P M A
W
Noise Reduction in Chaotic Signal
Phase plots of output responses for periodic signal before and after noise reduction
• Simple Noise Reduction Methoddeveloped based on near future prediction
Introduction
Geometric Nonlinearity
• Well-behaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
Noise Effect in Estimating Dimension
1 2 3 4 5 6 7 8 9 10
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Dim ens ion (d)
E1
(d)Minim um em bedding dim ens ion us ing Cao's m ethod
cleaned signal
noisy signal
Minimum embedding dimension information for phase-plot reconstructionIntroduction
Geometric Nonlinearity
• Well-behaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
Identification on Chaotic Case
Observing the chaos quantifier, e.g. Lyapunov exponent, could be used, in
principle, to estimate the parameter of a system.
Introduction
Geometric Nonlinearity
• Well-behaved
• Chaotic
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
Material Nonlinearity
Case study:mechanical system with friction element
Conventional friction model:
Coulomb modeldiscontinuity at zero velocity
velocity
frictionFriction is the result of a complex interaction
between two contact surfaces.
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
Friction Characterisation
Two different friction regimes have been distinguished:
• 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)
f ,
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
2
Pre-sliding regime (1)
0 (displacement)
Friction force
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
11
2
y(q)
-y(-q)
y(q)
qm
Fm
-qm
2
3
x
x
4
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
Pre-sliding regime (2)Equivalent dynamic parameters
The Describing Function technique is used to obtain the equivalent stiffness and damping:
y(q) is the virgin curve of the hysteresis
d)cos()cos(1yA
4k
02A
e
A
02e dq
2
)A(y)q(y
A
8c
displacement
fric
tion
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
Pre-sliding friction model
ki
Wi
x
F
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
Tegoeh Tjahjowidodo
P M A
W
Sliding regime
• When the motion is entering the sliding regime, in most cases, the Maxwell-Slip model is no longer suitable.
• The friction usually has a maximum value at the beginning and then continues to decrease with increasing velocity
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
GMS friction model
Generalized Maxwell-Slip (GMS) model developed at PMA/KULeuven
vii k
dt
dF
– If the model is slipping:
)s(
FC)(
dt
dF
vvsgn i
ii
Mathematical representation of Maxwell-Slip elements
– If the model is sticking:
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
DC Motor
Experimental setup of DC motor ABB M19-S
Load
Timing Belt
Encoder
DC Motorservo
dSPACE® 1104acquisition board
amp
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
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
0 2 4 6 8 10-50
-40
-30
-20
-10
0
10
20
30
40
50
time (s)
Ve
loc
ity
(ra
d/s
ec
)
1st velocity signal
0 2 4 6 8 10-30
-20
-10
0
10
20
30
time (s)
2nd velocity signal
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
1st velocity signal 2nd velocity signal
time (s) time (s)
velo
city
(ra
d/s
)
Tegoeh Tjahjowidodo
P M A
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Identification Strategy
The optimization is based upon minimization of cost function:
N
iii
y
)yy(N
)y(MSE1
22
100
Identification technique for the physics-based model:
• Genetic Algorithm• Nelder-Mead Downhill Simplex Method
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
Identification Results #1Identification results of low frequencies experiment:
1 2 3 4 5 6
-10
-5
0
5
10
Coulomb model
time (s)
To
rqu
e (
Nm
)
eps=0.1
1 2 3 4 5 6
-10
-5
0
5
10
Exponential Coulomb model
time (s)
eps=0.1
Coulomb model
torq
ue
(N
m)
1 2 3 4 5 6
-10
-5
0
5
10
LuGre model
time (s)
To
rqu
e (
Nm
)
1 2 3 4 5 6
-10
-5
0
5
10
GMS model
time (s)
GMS model
time (s)
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
ConclusionsLuGre model
time (s)
torq
ue
(N
m)
1 2 3 4 5 6
-10
-5
0
5
10
LuGre model
time (s)
To
rqu
e (
Nm
)
1 2 3 4 5 6
-10
-5
0
5
10
GMS model
time (s)
Exponential Coulomb model
1 2 3 4 5 6
-10
-5
0
5
10
Coulomb model
time (s)
To
rqu
e (
Nm
)
eps=0.1
1 2 3 4 5 6
-10
-5
0
5
10
Exponential Coulomb model
time (s)
eps=0.1Coulomb 2.06% (0.3770)
Exp-Coulomb 2.06% (0.3360)
LuGre 2.03% (0.3530)
GMS 1.97% (0.3470)
MSE (max.err.)
Tegoeh Tjahjowidodo
P M A
W
Identification Results #2Identification results of high frequencies experiment:
time (s)
0 2 4 6 8 10-15
-10
-5
0
5
10
15
time (s)
Err
or
Coulomb Model
0 2 4 6 8 10-15
-10
-5
0
5
10
15
time (s)
Exponential-Coulomb ModelCoulomb model
torq
ue
(N
m)
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
0 2 4 6 8 10-15
-10
-5
0
5
10
15
time (s)
To
rqu
e (
Nm
)
LuGre Model
0 2 4 6 8 10-15
-10
-5
0
5
10
15
time (s)
GMS ModelLuGre model
torq
ue
(N
m)
Exponential Coulomb model
0 2 4 6 8 10-15
-10
-5
0
5
10
15
time (s)
Err
or
Coulomb Model
0 2 4 6 8 10-15
-10
-5
0
5
10
15
time (s)
Exponential-Coulomb Model
GMS model
time (s)
0 2 4 6 8 10-15
-10
-5
0
5
10
15
time (s)
To
rqu
e (
Nm
)
LuGre Model
0 2 4 6 8 10-15
-10
-5
0
5
10
15
time (s)
GMS Model
MSE (max.err.)
Coulomb 17.92% (1.2993)
Exp-Coulomb 9.59% (1.2604)
LuGre 4.30% (0.6466)
GMS4 1.39% (0.5711)
GMS10 1.19% (0.5177)
Tegoeh Tjahjowidodo
P M A
W
Identification on Material Nonlinearity Case
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.
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
Overview of Controllers
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
Qh
-Qh
Position error (e)
Controlinput (u)
adds an extra compensating torque when the position error is within pre-sliding region
developed based on the equivalent dynamic parameters of the system.
Tegoeh Tjahjowidodo
P M A
W
Gain Scheduling Controller (1)
treats two different regimes of friction in separated modes.
The first mode
The corresponding gains are designed and optimized at some points (of amplitude of motion) regarding a certain
performance criteria.
• when the system is moving in the sliding region• linear controller + equivalent Coulomb friction model
• when the system is moving in the pre-sliding region• adjusts proportional (kp) and derivative (kd) gain based on
the equivalent dynamic parameter
The second mode
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
Gain Scheduling Controller (2)
Look-up table of the gains
System SWITCHING FUNCTION
x
du dt |.|=0
memorize =x |x- |<Q
h
and |x- |
-
+
xd
PD
F du dt
- +
+ -
PD
1st m
od
e2n
d m
od
e
Gain Scheduling StrategyIntroduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
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
Tegoeh Tjahjowidodo
P M A
W
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
Tegoeh Tjahjowidodo
P M A
W
Control Objectives
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.
Performance criteria:Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
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
Tegoeh Tjahjowidodo
P M A
W
Step Response ResultStep 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
Tegoeh Tjahjowidodo
P M A
W
Control of Nonlinear System
Model based control is able to yield good results, depending on the models used and the control
strategy.
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
Conclusions
Tegoeh Tjahjowidodo
P M A
W
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
Tegoeh Tjahjowidodo
P M A
W
The Harmonic Drive components
WAVE DRIVE®
(has 2 more teeth than flexspline)
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
• Identification
• Control
Conclusions
Tegoeh Tjahjowidodo
P M A
W
Operating mechanism
Operating principle of harmonic drive
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
• Identification
• Control
Conclusions
Tegoeh Tjahjowidodo
P M A
W
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
Tegoeh Tjahjowidodo
P M A
W
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
Tegoeh Tjahjowidodo
P M A
W
Torsional Stiffness – experiments (1)
load cell
Bentley probe
Lock thewave-generator
Experiment was carried out on WAVE DRIVE® component:
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
• Identification
• Control
Conclusions
Tegoeh Tjahjowidodo
P M A
W
Torsional Stiffness – experiments (2)
Stiffness curve obtained from sine excitation:Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
• Identification
• Control
Conclusions
-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Torque
To
rsio
nT
orsi
on (
rad)
Torque (Nm)
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
3
0
2
1
-1
-2
-3
-4
4
×10-3
Tegoeh Tjahjowidodo
P M A
W
Torsional Stiffness – experiments (3)
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
-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04
-0.1
-0.05
0
0.05
0.1
0.15
Torque
To
rsio
n
Tor
sion
(ra
d)
Torque (Nm)
3.75
0
2.50
1.25
-1.25
-2.50
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
time (s)
3.75
0
2.50
1.25
-1.25
-2.50
0 2 4 6 8
×10-3
Tegoeh Tjahjowidodo
P M A
W
Piecewise linear model
Hysteresis models
Torsional Stiffness Models
x
F
k1
x0
-x0
x
F
k0
Two different approach of models:Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
• Identification
• Control
Conclusions
Tegoeh Tjahjowidodo
P M A
W
Schematic of torsional stiffness model
Proposed model of torsional stiffness:Maxwell-slip elements
+hardening spring
elementarystick-slip 1
x
T
Pre-sliding friction
hardening spring
elementarystick-slip N
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
• Identification
• Control
Conclusions
Tegoeh Tjahjowidodo
P M A
W
Torsional Stiffness – hysteresis model (1)
Identification results:
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
• Identification
• Control
Conclusions
-0.03 -0.02 -0.01 0 0.01 0.02 0.03
-0.1
-0.05
0
0.05
0.1
0.15
Torque
To
rsio
n
Torsional Stiffness of Flexspline Component
Tor
sion
(ra
d)
Torque (Nm)
Torsional Stiffness
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
3
0
2
1
-1
-2
×10-3
Tegoeh Tjahjowidodo
P M A
W
Torsional Stiffness – hysteretic model (2)
Identification results:
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
0 5 10 15 20 25 30-0.04
-0.02
0
0.02
0.04
To
rqu
e
ActualModel
0 5 10 15 20 25 30-0.04
-0.02
0
0.02
0.04
Err
or
Time (s)
Err
or (
Nm
)T
orqu
e (N
m)
Time (sec)
0.4
0.2
0
-0.2
-0.4
0.4
0.2
0
-0.2
-0.40 5 10 15 20 25 30
0 5 10 15 20 25 30
Tegoeh Tjahjowidodo
P M A
W
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
Tegoeh Tjahjowidodo
P M A
W
FHA-C mini servo actuator
Schematic drawing of FHA-C
TkS kH
TF
m1 m2m3
armature inertia circular-spline inertiaload inertia
friction in the motor torsional stiffness of the HDcomplete-close package
Friction in the DC motor and torsional stiffness in the gear set cannot be
identified separately.
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
• Identification
• Control
Conclusions
Tegoeh Tjahjowidodo
P M A
W
Control of FHA-C mini servo actuator
Two different approaches are considered for control purposes:
First approach: a mass on a frictional surface
meqT
Tf
Second approach: two masses connected by a hysteresis torsional spring
m3keqm1T
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
• Identification
• Control
Conclusions
Tegoeh Tjahjowidodo
P M A
W
2nd Approach - Two masses and hysteresis spring (1)
Assumption:Neglect the external hysteretic nonlinearity source.
Nonlinearity mainly comes from the torsional stiffness
The torsional stiffness is identified by locking the output shaft and measuring the motor current.
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
• Identification
• Control
Conclusions
Tegoeh Tjahjowidodo
P M A
W
2nd Approach - Two masses and hysteresis spring (2)
Resulting the equivalent dynamic parameters:
A modified gain scheduling strategy:
xd
SYSTEM
memorize =x|x-|<Qh
dudt
|.|=0and
PD
|x-|
+ -
x2
x1
- + x2
Introduction
Geometric Nonlinearity
Material Nonlinearity
Control of Nonlinear Systems
Application on Real System
• Identification
• Control
Conclusions
Tegoeh Tjahjowidodo
P M A
W
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
Tegoeh Tjahjowidodo
P M A
W
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
Tegoeh Tjahjowidodo
P M A
W
Application on a System with HD
A piecewise linear model together with non-local 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
• Identification
• Control
Conclusions
Tegoeh Tjahjowidodo
P M A
W
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
Tegoeh Tjahjowidodo
P M A
W
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