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Vibration Isolation
Project
Objective
The objective to be achieved is the isolation from
environmental vibration of the LEGO® platform from
the myphotonics project.
This must be done with low cost
instrumentation in order to fit the
purpose of OpenAdaptronik project.
Michelson-Interferometer
Optische Pinzette
Zweistrahl-Interferometer
LEGO® platform
Simple Vibration Problem – Starting point
Equation of motion:
𝑚
𝑑2
𝑥
𝑑𝑡2
+ 𝑟
𝑑𝑥
𝑑𝑡
+ 𝑘𝑥 = 𝑟
𝑑𝑥 𝑝
𝑑𝑡
+ 𝑘𝑥 𝑝
𝑚 𝑥 + 𝑟 𝑥 + 𝑘𝑥 = 𝑟 𝑥 𝑝 + 𝑘𝑥 𝑝
Where: 𝑚𝑒 = 𝑚𝑎𝑠𝑠 𝑡𝑜 𝑏𝑒 𝑖𝑠𝑜𝑙𝑎𝑡𝑒𝑑 ; 𝑟 = 𝑑𝑎𝑚𝑝𝑖𝑛𝑔 ; 𝑘 = 𝑠𝑡𝑖𝑓𝑓𝑛𝑒𝑠𝑠
It is possible to have different behavior
of the system depending on the
damping with fixed mass and stiffness
Effects of damping increasing:
1. Amplitude attenuation at the
resonance frequency (𝜔0 =
𝑘 𝑚 𝑒 )
2. Amplitude increase at
frequencies higher then the
resonance one
• Passing in the Laplace domain is possible to
find the transfer function of the system
𝐺 =
𝑟𝑠 + 𝑘
𝑚 𝑒 𝑠2 + 𝑟𝑠 + 𝑘
=
𝑋
𝑋 𝑝
𝑟1 < 𝑟2 < 𝑟3
Before analyzing the objective problem is mandatory to understand the vibration
problem and the solutions that can be adopted.
Simple Vibration Problem – Skyhook Solution
A solution to this problem is the adoption of an
active damping system.
Skyhook control is a feedback velocity control
widely use in vibration isolation for a vehicle.
It simply introduce into the system a force
proportional to the speed of the mass to be
isolated changing the damping of the system
only form the point of view of the mass and not
from the one of the incoming disturbance.
Equation of motion:
𝑚 𝑥 + 𝑟 𝑥 + 𝑘𝑥 = 𝒇 + 𝑟 𝑥 𝑝 + 𝑘𝑥 𝑝 𝑚 𝑥 + 𝑟 𝑥 + 𝑘𝑥 = −𝑲 𝑥 + 𝑟 𝑥 𝑝 + 𝑘𝑥 𝑝
Choosing: 𝐾 = 2𝑚 𝑒 𝜔0 − 𝑟
the best attenuation at the
resonance frequency is obtained
𝐺 𝑠𝑦𝑠𝐶 =
𝑟𝑠 + 𝑘
𝑚 𝑒 𝑠2 + 𝑟 + 𝑲 𝑠 + 𝑘
1. Amplitude attenuation at
the resonance frequency
2. Amplitude attenuation at
frequencies higher then
the resonance one
+
Real Implementation – Inertial Mass Actuator (IMA)
The actuator that want to be adopted for the purpose is an inertial mass actuator that consists in a coil case that
drive a moving magnetic part through the current flowing in it.
The scheme of the actuator is reported in the figure above showing the intrinsic loop. The equation of motion
governing the actuator are reported below.
𝐿
𝑑𝑖
𝑑𝑡
+ 𝑅𝑖 + 𝛹 𝑥 − 𝑥 𝑝 = 𝑉
𝑚 𝑥 + 𝑟 𝑥 + 𝑘𝑥 = 𝑓 + 𝑟 𝑥 𝑝 + 𝑘𝑥 𝑝
𝑓 = 𝛹𝑖
Equations
governing
the IMA:
Where: 𝐿 = 𝑖𝑛𝑑𝑢𝑐𝑡𝑎𝑛𝑐𝑒
𝑅 = 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝛹 = 𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐𝑎𝑙 𝑚𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝑐𝑜𝑢𝑝𝑙𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡
The electrical and mechanical behavior of the actuator used mast be taken into account during the design of the
control
𝑋 𝑝
EL MECH+
-
𝑉
𝑋
• The actuator chosen for the
purpose is a RockWood
Bass-Shaker (100 W) and
can be directly bought in
internet at a price that goes
from 15 to 30 euro.
Nominal
Value
m 0,2727 𝐾𝑔
r 2 𝑁 𝑚 𝑠
k 22387 𝑁 𝑚
𝛹 1,6444 𝑁 𝐴 | 𝑉 𝑚
R 3,8994 𝑂ℎ𝑚
L 0,5 𝑚𝐻
• Here are reported the
nominal values of the
actuator’s parameters
found experimentally by
the other members of
OpenAdaptronik project.
Real Implementation – Inertial Mass Actuator (IMA)
The eigenfrequency of each actuator are in a range of less than 1 Hz with
respect to the nominal value. They are considered equal from the design
point of view.
It is possible to represent the 4 actuators’ electrical
part equations in matrix form as:
𝐿
𝑑𝑖
𝑑𝑡
+ 𝑅 𝑖 + 𝛹 𝑥 − 𝑥 𝑝 = 𝑉
Where: 𝐿 = 𝑑𝑖𝑎𝑔 𝐿
𝑅 = 𝑑𝑖𝑎𝑔 𝑅
𝛹 = 𝑑𝑖𝑎𝑔 𝛹
This will be useful during control design.
Preliminary Measurements
Before the real problem is taking into account a frequency analysis must be carried out in order to understand if the
system (the platform) could be considered rigid or not.
Laser Vibrometer
Measurement set-up
The measurements were
done with a Laser
Vibrometer on 66 points on
the upper surface of the
platform.
In particular, the corner
identified by the point 6 is
also the excitement point in
order to have a co-located
measurement point.
The measurement source is a
white noise random signal
produced by a shaker.
Scheme measurement set-up points
The signal for each point is measured for a period of 3.2 seconds. A Hanning window is applied to each signal and it
is averaged over 15 records .
A good result on the measure is achieved at least until 400
Hz as shown by the coherence functions of the points 6
and 66, two of the four corners points.
Due to the mounting choice also rigid modes are
measured (red circle) but it is clear that the LEGO® plate
can not be considered a rigid body
Preliminary Measurements
The Prony method is adopted for the identification of the eigenfrequencies and adimensional damping ratios.
1 85,75 0,0103
2 100,42 0,0147
3 150,40 0,0132
4 155,06 0,0170
5 197,39 0,0127
6 214,99 0,0168
7 267,31 0,0157
8 286,21 0,0018
9 327,79 0,0189
10 337,82 0,0158
11 352,91 0,0111
12 370,52 0,0179
13 399,12 0,0161
𝑓0 [𝐻𝑧] 𝜉 [−]
• The stability diagram shows that below 300 Hz is possible to identify in a clear way 8 eigenfrequencies.
• In the region from 300 to 420 Hz ( identified by the cyan bounded region ) the stability diagram is a bit confused.
In fact also the coherence function, for channels different from the co-located one, could reach vary low values
in the mentioned frequency region.
Preliminary Measurements - Modal Analysis
Preliminary Measurements - Modal Analysis
It’s now possible to identify the modal residues and, thanks to the co-located measurement, also the mode shapes
of the platform for the 13 modes taken into account and for the 66 points considered can be found.
Only the four corners frequency reconstructions are reported.
The modal residues are found by
means of a least square method
that tries to minimize, for each
point, the error function:
𝐸2
=
ℎ=1
𝑁
𝐻1𝑗𝑘 𝜔ℎ − 𝐻𝑗𝑘 𝜔ℎ
2
Where:
𝐻1𝑗𝑘 𝜔ℎ = measured response
function
𝐻𝑗𝑘 𝜔ℎ = reconstructed response
𝑁 = number of points in the
frequency domain
Hp : Linear system 𝐻𝑗𝑘 𝜔ℎ =
𝑟=1
𝑚
𝐴𝑗𝑘
𝑟
𝜔0,𝑟
2
− 𝜔ℎ
2
+ 𝑖 2𝜉 𝑟 𝜔0,𝑟 𝜔ℎ
=
𝑟=1
𝑚
ϕ 𝑗
𝑟
ϕ 𝑘
𝑟
𝜔0,𝑟
2
− 𝜔ℎ
2
+ 𝑖 2𝜉 𝑟 𝜔0,𝑟 𝜔ℎ
It’s now possible to calculate the mode shapes of the structure ϕ , visualize them and comparing with the results
obtained directly by the software implemented in the Laser Vibrometer used for the measurements.
• Left side: 1st and 2nd eigenmodes reconstructed.
• Right side: 1st and 2nd eigenmodes from L.V.
From left to right and up to down : eigenmodes
reconstructed 3,4,5,6,7,8.
Preliminary Measurements - Modal Analysis
The uncertain eigenfrequencies in the linear reconstruction present a different behavior with respect to the
solutions given by the Laser Vibrometer.
This is particularly evident in the 10th and the 13th mode.
• Left side: 11st and 13th eigenmodes reconstructed
• Right side: 11st and 13th eigenmodes from L.V.
This could be caused by:
• Low coherence for channels
different from the co-located
one in the frequency range of
300 ÷ 420 Hz.
• No more validity of linearity
hypothesis.
• Dynamic interaction between
the LEGO® bricks, still
considering valid the linearity
hypothesis.
• A combination of the last two.
Preliminary Measurements - Modal Analysis
Real System
It is now possible to pass to the analysis of the real configuration of the problem.
The connections between the plate and the actuators are made by other LEGO® bricks. They are glued on the
actuator side and the connection on the plate side is left free in order to have the possibility to easily remove the
LEGO® plate and to not modify the structure.
ϕ
𝜗
𝑥 𝑝1 𝑥 𝑝2
𝑥 𝑝3
𝑓𝑐1
𝑓𝑐2
𝑓𝑐3
k
k
kr
r
r
𝑧𝑐
𝑙 𝑥
𝑙 𝑦
𝑚 𝑒 ; 𝐽φ ; 𝐽ϑ
1
4
3
2
x
yz
 Four actuators are attached on the base of the plate in a
symmetric way and for sensors (accelerometers) are
placed on the top of the plate in a position
corresponding to the center of the actuators.
 In this way is possible to reach a symmetric behavior of
the system.
Actuators Location and Mounting
• In the figure is reported a scheme of the
considered system in which is possible to see
the forces exercised by the four actuators and
the incoming disturbances for each one.
• From now on the moving part of the actuators
are considered integrated with the LEGO® plate
and the mechanical and electrical parameters
are considered equal for each actuator. Scheme of the final configuration
Control Design
What is required for the OpenAdaptronik project is a simple control implementation. Skyhook control offers this
opportunity with an high quality result.
𝐼 𝑞 + 𝑑𝑖𝑎𝑔 2𝜉 𝑟 𝜔 𝑟 𝑞 + 𝑑𝑖𝑎𝑔 𝜔 𝑟
2
𝑞 = 𝜙 𝑇
𝐵 𝑓𝑐 + 𝜙 𝑇
𝐵 (𝑟 𝑥 𝑝 + 𝑘𝑥 𝑝
Where: • 𝑧 = 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 = 𝜙 𝑞
• 𝑦 = 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑜𝑢𝑡𝑝𝑢𝑡𝑠 = 𝐶 𝑧 = 𝐶 𝜙 𝑞
• 𝑓𝑐 = 𝑓𝑜𝑟𝑐𝑒 𝑣𝑒𝑐𝑡𝑜𝑟 𝑝𝑟𝑜𝑣𝑖𝑑𝑒𝑑 𝑏𝑦 𝑡ℎ𝑒 𝑎𝑐𝑡𝑢𝑎𝑡𝑜𝑟𝑠
• 𝑥 𝑝 = 𝑖𝑛𝑝𝑢𝑡 𝑑𝑖𝑠𝑡𝑢𝑟𝑏𝑎𝑛𝑐𝑒 𝑣𝑒𝑐𝑡𝑜𝑟
• 𝐵 = 𝑖𝑛𝑝𝑢𝑡 𝑚𝑎𝑡𝑟𝑖𝑥
• 𝐶 = 𝑜𝑢𝑡𝑝𝑢𝑡 𝑚𝑎𝑡𝑟𝑖𝑥
Considering the system linear, it can be seen in modal form as:
If it were possible to give to the system forces directly in modal form, the best choice would be to give to the system
forces proportional to the modal velocities in order to change, for each mode, the adimensional damping ratio and
to make it critically damped (𝜉 𝑟 = 1) so:
𝐼 𝑞 + 𝑑𝑖𝑎𝑔 2𝜉 𝑟 𝜔 𝑟 𝑞 + 𝑑𝑖𝑎𝑔 𝜔 𝑟
2
𝑞 = 𝑔c + 𝜙 𝑇
𝐵 (𝑟 𝑥 𝑝 + 𝑘𝑥 𝑝
𝑔𝑐 = −𝑑𝑖𝑎𝑔 2𝜔 𝑟 − 2𝜉 𝑟 𝜔 𝑟 𝑞 = − 𝐾 𝑞
𝐼 𝑞 + 𝑑𝑖𝑎𝑔 2𝜔 𝑟 𝑞 + 𝑑𝑖𝑎𝑔 𝜔 𝑟
2
𝑞 = 𝜙 𝑇
𝐵 (𝑟 𝑥 𝑝 + 𝑘𝑥 𝑝
This ideal result is equivalent of doing a Skyhook control on each mode.
A light damped linear system is not completely controllable and not completely observable unless of having an
infinite number of actuators and sensors, due to the orthogonality property of the mode shapes ϕ .
Control Design
Calling: • 𝑁 = 𝑚𝑜𝑠𝑡 𝑐𝑜𝑛𝑡𝑟𝑜𝑙𝑙𝑎𝑏𝑙𝑒 𝑝𝑎𝑟𝑡 𝑠𝑒𝑙𝑒𝑐𝑡𝑜𝑟
• 𝑀 = 𝑚𝑜𝑠𝑡 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑏𝑙𝑒 𝑝𝑎𝑟𝑡 𝑠𝑒𝑙𝑒𝑐𝑡𝑜𝑟
For the control point of view, neglecting the unpredictable disturbance part related to 𝑥 𝑝, it is possible to rewrite the
system in modal form as:
𝐼 𝑞 + 𝑑𝑖𝑎𝑔 2𝜉 𝑟 𝜔 𝑟 𝑞 + 𝑑𝑖𝑎𝑔 𝜔 𝑟
2
𝑞 = 𝜙 𝑇
𝑓𝑐
𝑁 𝐼 𝑀 𝑞 𝑐𝑜 + 𝑁 𝑑𝑖𝑎𝑔 2𝜉 𝑟 𝜔 𝑟 𝑀 𝑞 𝑐𝑜 + 𝑁 𝑑𝑖𝑎𝑔 𝜔 𝑟
2
𝑀 𝑞 𝑐𝑜 = 𝑁 𝜙 𝑇
𝑀 𝑓𝑐
Now the modal coordinates 𝑞 𝑐𝑜 are the controllable and observable ones.
For the control point of view and for the considered system the input and output matrices are:
𝐵 = 𝐶 = 𝐼 𝑎𝑛𝑑 𝑁 = 𝑀 𝑇
The control logic implementation starts from the considerations done and tries to obtain the wonted behavior
shown in the previous slide taking into account the electrical and magnetic dynamic behavior of the Inertial Mass
Actuators (IMA) used.
Control Design
𝑁 𝐼 𝑀 𝑞 𝑐𝑜 + 𝑁 𝑑𝑖𝑎𝑔 2𝜉 𝑟 𝜔𝑟 𝑀 𝑞 𝑐𝑜 + 𝑁 𝑑𝑖𝑎𝑔 𝜔𝑟
2
𝑀 𝑞 𝑐𝑜 = 𝑁 𝜙 𝑇
𝑀 𝑓𝑐Starting again from:
Defining:
𝑁 𝜙 𝑇
𝑀 = 𝜙 𝐶 → 𝜙 𝐶
𝐿
= 𝜙 𝐶
𝑇
𝜙 𝐶
−1
𝜙 𝐶
𝑇
→ 𝜙 𝐶 𝑓𝑐 = − 𝐾 𝑞 𝑐𝑜 → 𝑓𝑐 = − 𝜙 𝐶
𝐿
𝐾 𝑞 𝑐𝑜
Choosing the control
logic to be:
𝑉 = − 𝐻 𝑦
Being: 𝐻 = 𝑔𝑒𝑛𝑒𝑟𝑎𝑙 𝑐𝑜𝑛𝑡𝑟𝑜𝑙 𝑚𝑎𝑡𝑟𝑖𝑥
Than:
1
𝛹
𝐿
𝑑𝑓𝑐
𝑑𝑡
+ 𝑅 𝑓𝑐 + 𝛹 + 𝐻 𝑦 = 𝛹 𝑥 𝑝
𝑧 = 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 = 𝜙 𝑞
𝑁 𝜙 𝑇
𝑀 𝑓𝑐 = 𝑔 𝑐 = − 𝑁 𝑑𝑖𝑎𝑔 2𝜔𝑟 − 2𝜉 𝑟 𝜔𝑟 𝑀 𝑞 𝑐𝑜 = − 𝐾 𝑞 𝑐𝑜
𝑦 = 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑜𝑢𝑡𝑝𝑢𝑡𝑠 = 𝐶 𝑧 = 𝑀 𝜙 𝑁 𝑞 𝑐𝑜
𝐿
𝑑𝑖
𝑑𝑡
+ 𝑅 𝑖 + 𝛹 𝑦 − 𝑥 𝑝 = 𝑉 ; 𝑖 =
1
𝛹
𝑓𝑐
Passing in the
Laplace domain:
1
𝛹
𝐿 s + 𝑅 𝑓𝑐 + 𝐻 𝛹 𝑦 = 𝛹 𝑥 𝑝 ; 𝐻 𝛹 = 𝛹 + 𝐻
−
1
𝛹
𝐿 s + 𝑅 𝜙 𝐶
𝐿
𝐾 𝑞 𝑐𝑜 + 𝐻 𝛹 𝑀 𝜙 𝑁 𝑞 𝑐𝑜 = 0
𝐻 =
1
𝛹
𝐿 s + 𝑅 𝜙 𝐶
𝐿
𝐾 ϕ 𝑂
𝑅
− 𝛹
Defining: ϕ 𝑂 = 𝑀 𝜙 𝑁 → ϕ 𝑂
𝑅
= ϕ 𝑂
𝑇
ϕ 𝑂 ϕ 𝑂
𝑇 −1
Finally:
Neglecting the contribution
of 𝑥 𝑝 and substituting the
expression of 𝑓𝑐 and 𝑦:
→ 𝐻 𝛹 ϕ 𝑂 =
1
𝛹
𝐿 s + 𝑅 𝜙 𝐶
𝐿
𝐾
Control Design
The control has a non
causal formulation in fact:
𝑉 = − 𝐻 𝑦 = − 𝐷 𝑠 + 𝑃 𝑦
Although, due to the choice
of the sensors, it can be seen
in a causal form as:
𝑉 = − 𝐷 𝑠 + 𝑃 𝑦 = − 𝐷 + 𝑃
1
𝑠
𝑦
Where: 𝑦 = 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑣𝑒𝑐𝑐𝑡𝑜𝑟
Real System – Measurements and Analysis
To implement the control a knowledge of the mode
shapes, eigenfrequencies and adimensional damping
ratio of the structure is needed.
For this reason the reference structure is mounted
on a service structure rigidly connected to the
ground by means of symmetric constraints, as shown
by the red arrows in the figure.
Constraints are needed for two reasons:
1. Find the modal parameters of the structure
considering null the disturbances incoming from
the ground.
2. Neglecting the frequency contribution of the
service structure, at least in the frequency range
of interest.
The measurements are done, in this case, using a dynamometric hammer.
Being not sure about the behavior of the system, due to its particularity, the plate is hammered in each corner
point (near the corresponding accelerometer), identified by the cyan arrows, measuring the Frequency
Response Functions of all the accelerometers.
It made possible to see if significant differences occurs inside the plate that can cause non symmetric
behavior.
Measurement set-up
4
3
2
1
Real System – Measurements and Analysis
The results for the four hammered points are shown and they are superimposed to show that not big
differences occurs at least below 150 Hz.
Real System – Measurements and Analysis
1 19,686 19,701 19,672 19,740
2 21,926 21,889 21,938 21,909
3 27,060 27,122 27,001 27,031
4 75,392 75,452 75,215 75,468
5 78,607 78,603 78,652 78,758
6 109,919 110,004 109,684 109,899
7 124,035 124,396 124,386 124,160
8 133,255 134,186 133,650 133,608
𝑓01 [𝐻𝑧] 𝑓02 [𝐻𝑧] 𝑓03 [𝐻𝑧] 𝑓04 [𝐻𝑧]
1 0,02 0,0143 0,0184 0,0167
2 0,0152 0,0147 0,0147 0,0136
3 0,0152 0,0134 0,0159 0,0131
4 0,0147 0,0101 0,0185 0,0148
5 0,0108 0,0137 0,0108 0,0122
6 0,0129 0,0143 0,0112 0,0145
7 0,0137 0,0139 0,0164 0,0141
8 0,0119 0,0107 0,0116 0,0102
𝜉 1 [−] 𝜉 2 [−] 𝜉 3 [−] 𝜉 4 [−]
For each hammered point the Prony method is applied
to identify the eigenfrequencies and the adimensional
damping ratios.
A good result is achieved, but for lack of information
(only 4 measurement points) an high number of poles
must be used to get a sufficient number of repeated
stable poles.
For simplicity only the stability diagram obtained for the
hammer in point 1 is reported in the figure.
Instead in the tables below all the found parameters are
reported and they show quite similar values.
Real System – Measurements and Analysis
Also in this case a least square
method is used for reconstruct the
signals response and to find the
modal residues and so the mode
shapes of the system.
The different hammered points
measurement gives, as in the case of
the eigenfrequencies and
adimensional damping ratios,
different values of the modes
shapes.
For the control purpose all the
parameters found are averaged.
1 2 3 4 5 6 7 8
19,7 21,915 27,053 75,382 78,655 109,876 124,244 133,675
0,0173 0,0145 0,0144 0,0145 0,0119 0,0132 0,0145 0,0111
0,3445 0,407 0,4622 0,3821 0,4815 0,1012 0,2613 0,1246
0,3445 -0,407 0,4622 0,3997 -0,4815 -0,0849 -0,2613 0,1081
0,3445 -0,407 -0,4622 0,3997 0,4815 0,0849 -0,2613 -0,1271
0,3445 0,407 -0,4622 0,3997 -0,4815 -0,1132 0,2613 -0,12
𝑓0 [𝐻𝑧]
𝜉 [−]
ϕ1
ϕ2
ϕ3
ϕ4
Real System – Simulation Results
• The analysis done shows that the observable
modes are the 1st, 2nd, 3rd and 5th because they are
linearly independent. Moreover the first three
resonances should correspond to rigid modes.
• The simulation results show a good performance
of the control logic implemented with respect to
the ideal case.
• The comparison is done by introducing in the
system a ground velocity disturbance for each
basement of the actuators that transmit to the
system itself a vector of forces equal to:
𝐹𝐷 = 𝑟 𝑥 𝑝 + 𝑘𝑥 𝑝
• It is possible to see that the implemented control
logic is trying to place the poles of the system in
the positions related to the ideal gain control (it is
shown in the figure the effect of the increase of
the gains on the poles location)
Real System – Implementation
Final Configuration
The system is now removed from the service structure and it is placed directly of the working surface.
This is done because the control must work without regards of the surface where it is placed. This is partially true
because if the system, plate plus actuators, is placed on a structure that with its modes can affects and interacts with
the system, it is possible that the control becomes unstable because, due to the nature of the actuators used, a phase
shift occurs between the sensor measurement and the voltage input.
A careful choice of the placement of the sensors with respect to the actuators must be done. In fact initially they were
left on the top of the LEGO® plate but as it is possible to see in the open loop transfer function reported a thickness
mode occur that make the control no more co-located.
Clearly after 150 Hz (red bounded zone) something strange inside the plate thickness happens, but
the first eight studied resonances are not modified.
Up position
Down position
Real System – Implementation
The control is implemented in a real time system called DSpace and as fist attempt the sampling frequency used is
10 kHz. The effect of sampling will be discussed later.
To solve the integration problem in the real
time implementation a transfer function is
used instead of a adopting an anti wind up
configuration
To solve the offset problem of the sensors an
highpass filter if adopted
Both the transfer functions are digitally implemented and for both the cutoff frequency is chosen to be 1 Hz.
Real System – Experimental Results
A good result is experimentally obtained in the frequency region of interest
Real System – Experimental Results
The results obtained were done hammering the upper surface of the structure because for the best case experiment
it will be needed four shakers, one for each actuator, capable of carrying at least one quarter the weight of the
entire structure.
Transmissibility Measurement set-up
The transmissibility between the accelerometer on the base (cyan arrow) and the accelerometer of the center of the
plate (red arrow) is measured hammering the base.
Despite the high noise, it is possible to see in the controlled transfer function a reduction of the incoming
disturbance in the controlled frequency region and also a gradually phase shift.
Real System – Diagonal Control
Until now the results shown are based on a centralized control. This could cause problem of implementation for the
point of view of the OpenAdaptronik project so a decentralized control could be taken into account.
Choosing the derivative and proportional part of the
control to be:
𝐷 𝑑 = 𝑑𝑖𝑎𝑔
𝑗=1
4
𝐷 𝑘𝑗 ; 𝑃𝑑 = 𝑑𝑖𝑎𝑔
𝑗=1
4
𝑃𝑘𝑗
Where j is the index of the column, a good result is
obtained.
The choice of the decentralized gains is done to ensure
that the same input is given as the control would be
centralized, when the four sensors signals are equal.
In fact it is possible to back calculate the modal gains introduced by the control as:
𝐾 = 𝛹 ϕ 𝐶
𝐿𝑇
ϕ 𝐶
𝐿 −1
ϕ 𝐶
𝐿𝑇
𝐿 𝑠 + 𝑅 −1
𝐻 + 𝛹 ϕ 𝑂
𝑅𝑇
ϕ 𝑂
𝑅
ϕ 𝑂
𝑅𝑇 −1
1 2 3 4 1 2 3 4
243,253875 0 0 0 243,253875 0 0 0
0 271,387391 0 0 0 339,552905 0 0
0 0 335,056031 0 0 0 437,846974 0
0 0 0 976,621053 0 0 0 475,160617
Centralized Control Modal Gains Decentralized Control Modal Gains
Real System – Sampling Effect
A sampling frequency of 10 kHz is definitively to high for
implementing a control on a low cost control board so the
effect of sampling is experimentally studied.
In the two figures the centralized and decentralized controls
effects are compared for two different sampling
frequencies, 1 kHz and 0.5 kHz.
Both the control logics works similar but, despite the
decentralized control formulation is not giving the wonted
modal gains, at least in a theoretical way, it is less affected
by sampling. In fact the reduction needed for stability are:
1 𝑘𝐻𝑧 ∶ 𝐶𝐶 𝐺 𝑟𝑒𝑑𝑢𝑐𝑡𝑖𝑜𝑛
= 78,06%
𝐷𝐶 𝐺 𝑟𝑒𝑑𝑢𝑐𝑡𝑖𝑜𝑛
= 72,90%
0,5 𝑘𝐻𝑧 ∶ 𝐶𝐶 𝐺 𝑟𝑒𝑑𝑢𝑐𝑡𝑖𝑜𝑛
= 94,84%
𝐷𝐶 𝐺 𝑟𝑒𝑑𝑢𝑐𝑡𝑖𝑜𝑛
= 93,55%
The less reduction needed for the stability of the
decentralized control should be caused by the
measurements noise because each one affects only its co-
located input and is not combined with the other signals
noise. More research about this topic should be carried out.
𝐶𝐶 𝐺 𝑟𝑒𝑑𝑢𝑐𝑡𝑖𝑜𝑛
= Centralized Control Gain Reduction
𝐷𝐶 𝐺 𝑟𝑒𝑑𝑢𝑐𝑡𝑖𝑜𝑛
= Decentralized Control Gain Reduction
Conclusions and Future Developments
• It is shown that the control is working as designed and isolation can be carried out with low cost actuators and
without a previous knowledge of the model of the system.
• It is also possible to reduce the complexity of the problem by passing from a centralized control to a
decentralized one that will lead to a more simple implementation.
• It is shown that sampling affects in an heavily way the problem.
• To solve the sampling problem an analogically implementation of the control proposed could be done. For this
specific purpose it is easier and of course cheaper to develop the decentralized control instead of the centralized
one.
• A more careful study of the suggested noise problem relation to the gains reduction between centralized and
decentralized control must be carried out to understand if the problem is really related to this phenomenon.
Moreover, because the usage of low pass filters affects in a negative way the phase of the system lowering the
stability margin of the control, in this work they are not used.
• Sensors will be changed from high cost to low costs ones.
• Analogical low cost amplifiers will replace the now used high cost ones.
• An analogical control or a digital one, implemented in a low cost control board, will be developed.
FUTURE DEVELOPMENTS
CONCLUSIONS

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Vibration isolation progect lego(r)

  • 2. Objective The objective to be achieved is the isolation from environmental vibration of the LEGO® platform from the myphotonics project. This must be done with low cost instrumentation in order to fit the purpose of OpenAdaptronik project. Michelson-Interferometer Optische Pinzette Zweistrahl-Interferometer LEGO® platform
  • 3. Simple Vibration Problem – Starting point Equation of motion: 𝑚 𝑑2 𝑥 𝑑𝑡2 + 𝑟 𝑑𝑥 𝑑𝑡 + 𝑘𝑥 = 𝑟 𝑑𝑥 𝑝 𝑑𝑡 + 𝑘𝑥 𝑝 𝑚 𝑥 + 𝑟 𝑥 + 𝑘𝑥 = 𝑟 𝑥 𝑝 + 𝑘𝑥 𝑝 Where: 𝑚𝑒 = 𝑚𝑎𝑠𝑠 𝑡𝑜 𝑏𝑒 𝑖𝑠𝑜𝑙𝑎𝑡𝑒𝑑 ; 𝑟 = 𝑑𝑎𝑚𝑝𝑖𝑛𝑔 ; 𝑘 = 𝑠𝑡𝑖𝑓𝑓𝑛𝑒𝑠𝑠 It is possible to have different behavior of the system depending on the damping with fixed mass and stiffness Effects of damping increasing: 1. Amplitude attenuation at the resonance frequency (𝜔0 = 𝑘 𝑚 𝑒 ) 2. Amplitude increase at frequencies higher then the resonance one • Passing in the Laplace domain is possible to find the transfer function of the system 𝐺 = 𝑟𝑠 + 𝑘 𝑚 𝑒 𝑠2 + 𝑟𝑠 + 𝑘 = 𝑋 𝑋 𝑝 𝑟1 < 𝑟2 < 𝑟3 Before analyzing the objective problem is mandatory to understand the vibration problem and the solutions that can be adopted.
  • 4. Simple Vibration Problem – Skyhook Solution A solution to this problem is the adoption of an active damping system. Skyhook control is a feedback velocity control widely use in vibration isolation for a vehicle. It simply introduce into the system a force proportional to the speed of the mass to be isolated changing the damping of the system only form the point of view of the mass and not from the one of the incoming disturbance. Equation of motion: 𝑚 𝑥 + 𝑟 𝑥 + 𝑘𝑥 = 𝒇 + 𝑟 𝑥 𝑝 + 𝑘𝑥 𝑝 𝑚 𝑥 + 𝑟 𝑥 + 𝑘𝑥 = −𝑲 𝑥 + 𝑟 𝑥 𝑝 + 𝑘𝑥 𝑝 Choosing: 𝐾 = 2𝑚 𝑒 𝜔0 − 𝑟 the best attenuation at the resonance frequency is obtained 𝐺 𝑠𝑦𝑠𝐶 = 𝑟𝑠 + 𝑘 𝑚 𝑒 𝑠2 + 𝑟 + 𝑲 𝑠 + 𝑘 1. Amplitude attenuation at the resonance frequency 2. Amplitude attenuation at frequencies higher then the resonance one
  • 5. + Real Implementation – Inertial Mass Actuator (IMA) The actuator that want to be adopted for the purpose is an inertial mass actuator that consists in a coil case that drive a moving magnetic part through the current flowing in it. The scheme of the actuator is reported in the figure above showing the intrinsic loop. The equation of motion governing the actuator are reported below. 𝐿 𝑑𝑖 𝑑𝑡 + 𝑅𝑖 + 𝛹 𝑥 − 𝑥 𝑝 = 𝑉 𝑚 𝑥 + 𝑟 𝑥 + 𝑘𝑥 = 𝑓 + 𝑟 𝑥 𝑝 + 𝑘𝑥 𝑝 𝑓 = 𝛹𝑖 Equations governing the IMA: Where: 𝐿 = 𝑖𝑛𝑑𝑢𝑐𝑡𝑎𝑛𝑐𝑒 𝑅 = 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝛹 = 𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐𝑎𝑙 𝑚𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝑐𝑜𝑢𝑝𝑙𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 The electrical and mechanical behavior of the actuator used mast be taken into account during the design of the control 𝑋 𝑝 EL MECH+ - 𝑉 𝑋
  • 6. • The actuator chosen for the purpose is a RockWood Bass-Shaker (100 W) and can be directly bought in internet at a price that goes from 15 to 30 euro. Nominal Value m 0,2727 𝐾𝑔 r 2 𝑁 𝑚 𝑠 k 22387 𝑁 𝑚 𝛹 1,6444 𝑁 𝐴 | 𝑉 𝑚 R 3,8994 𝑂ℎ𝑚 L 0,5 𝑚𝐻 • Here are reported the nominal values of the actuator’s parameters found experimentally by the other members of OpenAdaptronik project. Real Implementation – Inertial Mass Actuator (IMA) The eigenfrequency of each actuator are in a range of less than 1 Hz with respect to the nominal value. They are considered equal from the design point of view. It is possible to represent the 4 actuators’ electrical part equations in matrix form as: 𝐿 𝑑𝑖 𝑑𝑡 + 𝑅 𝑖 + 𝛹 𝑥 − 𝑥 𝑝 = 𝑉 Where: 𝐿 = 𝑑𝑖𝑎𝑔 𝐿 𝑅 = 𝑑𝑖𝑎𝑔 𝑅 𝛹 = 𝑑𝑖𝑎𝑔 𝛹 This will be useful during control design.
  • 7. Preliminary Measurements Before the real problem is taking into account a frequency analysis must be carried out in order to understand if the system (the platform) could be considered rigid or not. Laser Vibrometer Measurement set-up The measurements were done with a Laser Vibrometer on 66 points on the upper surface of the platform. In particular, the corner identified by the point 6 is also the excitement point in order to have a co-located measurement point. The measurement source is a white noise random signal produced by a shaker. Scheme measurement set-up points
  • 8. The signal for each point is measured for a period of 3.2 seconds. A Hanning window is applied to each signal and it is averaged over 15 records . A good result on the measure is achieved at least until 400 Hz as shown by the coherence functions of the points 6 and 66, two of the four corners points. Due to the mounting choice also rigid modes are measured (red circle) but it is clear that the LEGO® plate can not be considered a rigid body Preliminary Measurements
  • 9. The Prony method is adopted for the identification of the eigenfrequencies and adimensional damping ratios. 1 85,75 0,0103 2 100,42 0,0147 3 150,40 0,0132 4 155,06 0,0170 5 197,39 0,0127 6 214,99 0,0168 7 267,31 0,0157 8 286,21 0,0018 9 327,79 0,0189 10 337,82 0,0158 11 352,91 0,0111 12 370,52 0,0179 13 399,12 0,0161 𝑓0 [𝐻𝑧] 𝜉 [−] • The stability diagram shows that below 300 Hz is possible to identify in a clear way 8 eigenfrequencies. • In the region from 300 to 420 Hz ( identified by the cyan bounded region ) the stability diagram is a bit confused. In fact also the coherence function, for channels different from the co-located one, could reach vary low values in the mentioned frequency region. Preliminary Measurements - Modal Analysis
  • 10. Preliminary Measurements - Modal Analysis It’s now possible to identify the modal residues and, thanks to the co-located measurement, also the mode shapes of the platform for the 13 modes taken into account and for the 66 points considered can be found. Only the four corners frequency reconstructions are reported. The modal residues are found by means of a least square method that tries to minimize, for each point, the error function: 𝐸2 = ℎ=1 𝑁 𝐻1𝑗𝑘 𝜔ℎ − 𝐻𝑗𝑘 𝜔ℎ 2 Where: 𝐻1𝑗𝑘 𝜔ℎ = measured response function 𝐻𝑗𝑘 𝜔ℎ = reconstructed response 𝑁 = number of points in the frequency domain Hp : Linear system 𝐻𝑗𝑘 𝜔ℎ = 𝑟=1 𝑚 𝐴𝑗𝑘 𝑟 𝜔0,𝑟 2 − 𝜔ℎ 2 + 𝑖 2𝜉 𝑟 𝜔0,𝑟 𝜔ℎ = 𝑟=1 𝑚 ϕ 𝑗 𝑟 ϕ 𝑘 𝑟 𝜔0,𝑟 2 − 𝜔ℎ 2 + 𝑖 2𝜉 𝑟 𝜔0,𝑟 𝜔ℎ
  • 11. It’s now possible to calculate the mode shapes of the structure ϕ , visualize them and comparing with the results obtained directly by the software implemented in the Laser Vibrometer used for the measurements. • Left side: 1st and 2nd eigenmodes reconstructed. • Right side: 1st and 2nd eigenmodes from L.V. From left to right and up to down : eigenmodes reconstructed 3,4,5,6,7,8. Preliminary Measurements - Modal Analysis
  • 12. The uncertain eigenfrequencies in the linear reconstruction present a different behavior with respect to the solutions given by the Laser Vibrometer. This is particularly evident in the 10th and the 13th mode. • Left side: 11st and 13th eigenmodes reconstructed • Right side: 11st and 13th eigenmodes from L.V. This could be caused by: • Low coherence for channels different from the co-located one in the frequency range of 300 ÷ 420 Hz. • No more validity of linearity hypothesis. • Dynamic interaction between the LEGO® bricks, still considering valid the linearity hypothesis. • A combination of the last two. Preliminary Measurements - Modal Analysis
  • 13. Real System It is now possible to pass to the analysis of the real configuration of the problem. The connections between the plate and the actuators are made by other LEGO® bricks. They are glued on the actuator side and the connection on the plate side is left free in order to have the possibility to easily remove the LEGO® plate and to not modify the structure. ϕ 𝜗 𝑥 𝑝1 𝑥 𝑝2 𝑥 𝑝3 𝑓𝑐1 𝑓𝑐2 𝑓𝑐3 k k kr r r 𝑧𝑐 𝑙 𝑥 𝑙 𝑦 𝑚 𝑒 ; 𝐽φ ; 𝐽ϑ 1 4 3 2 x yz  Four actuators are attached on the base of the plate in a symmetric way and for sensors (accelerometers) are placed on the top of the plate in a position corresponding to the center of the actuators.  In this way is possible to reach a symmetric behavior of the system. Actuators Location and Mounting • In the figure is reported a scheme of the considered system in which is possible to see the forces exercised by the four actuators and the incoming disturbances for each one. • From now on the moving part of the actuators are considered integrated with the LEGO® plate and the mechanical and electrical parameters are considered equal for each actuator. Scheme of the final configuration
  • 14. Control Design What is required for the OpenAdaptronik project is a simple control implementation. Skyhook control offers this opportunity with an high quality result. 𝐼 𝑞 + 𝑑𝑖𝑎𝑔 2𝜉 𝑟 𝜔 𝑟 𝑞 + 𝑑𝑖𝑎𝑔 𝜔 𝑟 2 𝑞 = 𝜙 𝑇 𝐵 𝑓𝑐 + 𝜙 𝑇 𝐵 (𝑟 𝑥 𝑝 + 𝑘𝑥 𝑝 Where: • 𝑧 = 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 = 𝜙 𝑞 • 𝑦 = 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑜𝑢𝑡𝑝𝑢𝑡𝑠 = 𝐶 𝑧 = 𝐶 𝜙 𝑞 • 𝑓𝑐 = 𝑓𝑜𝑟𝑐𝑒 𝑣𝑒𝑐𝑡𝑜𝑟 𝑝𝑟𝑜𝑣𝑖𝑑𝑒𝑑 𝑏𝑦 𝑡ℎ𝑒 𝑎𝑐𝑡𝑢𝑎𝑡𝑜𝑟𝑠 • 𝑥 𝑝 = 𝑖𝑛𝑝𝑢𝑡 𝑑𝑖𝑠𝑡𝑢𝑟𝑏𝑎𝑛𝑐𝑒 𝑣𝑒𝑐𝑡𝑜𝑟 • 𝐵 = 𝑖𝑛𝑝𝑢𝑡 𝑚𝑎𝑡𝑟𝑖𝑥 • 𝐶 = 𝑜𝑢𝑡𝑝𝑢𝑡 𝑚𝑎𝑡𝑟𝑖𝑥 Considering the system linear, it can be seen in modal form as: If it were possible to give to the system forces directly in modal form, the best choice would be to give to the system forces proportional to the modal velocities in order to change, for each mode, the adimensional damping ratio and to make it critically damped (𝜉 𝑟 = 1) so: 𝐼 𝑞 + 𝑑𝑖𝑎𝑔 2𝜉 𝑟 𝜔 𝑟 𝑞 + 𝑑𝑖𝑎𝑔 𝜔 𝑟 2 𝑞 = 𝑔c + 𝜙 𝑇 𝐵 (𝑟 𝑥 𝑝 + 𝑘𝑥 𝑝 𝑔𝑐 = −𝑑𝑖𝑎𝑔 2𝜔 𝑟 − 2𝜉 𝑟 𝜔 𝑟 𝑞 = − 𝐾 𝑞 𝐼 𝑞 + 𝑑𝑖𝑎𝑔 2𝜔 𝑟 𝑞 + 𝑑𝑖𝑎𝑔 𝜔 𝑟 2 𝑞 = 𝜙 𝑇 𝐵 (𝑟 𝑥 𝑝 + 𝑘𝑥 𝑝 This ideal result is equivalent of doing a Skyhook control on each mode.
  • 15. A light damped linear system is not completely controllable and not completely observable unless of having an infinite number of actuators and sensors, due to the orthogonality property of the mode shapes ϕ . Control Design Calling: • 𝑁 = 𝑚𝑜𝑠𝑡 𝑐𝑜𝑛𝑡𝑟𝑜𝑙𝑙𝑎𝑏𝑙𝑒 𝑝𝑎𝑟𝑡 𝑠𝑒𝑙𝑒𝑐𝑡𝑜𝑟 • 𝑀 = 𝑚𝑜𝑠𝑡 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑏𝑙𝑒 𝑝𝑎𝑟𝑡 𝑠𝑒𝑙𝑒𝑐𝑡𝑜𝑟 For the control point of view, neglecting the unpredictable disturbance part related to 𝑥 𝑝, it is possible to rewrite the system in modal form as: 𝐼 𝑞 + 𝑑𝑖𝑎𝑔 2𝜉 𝑟 𝜔 𝑟 𝑞 + 𝑑𝑖𝑎𝑔 𝜔 𝑟 2 𝑞 = 𝜙 𝑇 𝑓𝑐 𝑁 𝐼 𝑀 𝑞 𝑐𝑜 + 𝑁 𝑑𝑖𝑎𝑔 2𝜉 𝑟 𝜔 𝑟 𝑀 𝑞 𝑐𝑜 + 𝑁 𝑑𝑖𝑎𝑔 𝜔 𝑟 2 𝑀 𝑞 𝑐𝑜 = 𝑁 𝜙 𝑇 𝑀 𝑓𝑐 Now the modal coordinates 𝑞 𝑐𝑜 are the controllable and observable ones. For the control point of view and for the considered system the input and output matrices are: 𝐵 = 𝐶 = 𝐼 𝑎𝑛𝑑 𝑁 = 𝑀 𝑇 The control logic implementation starts from the considerations done and tries to obtain the wonted behavior shown in the previous slide taking into account the electrical and magnetic dynamic behavior of the Inertial Mass Actuators (IMA) used.
  • 16. Control Design 𝑁 𝐼 𝑀 𝑞 𝑐𝑜 + 𝑁 𝑑𝑖𝑎𝑔 2𝜉 𝑟 𝜔𝑟 𝑀 𝑞 𝑐𝑜 + 𝑁 𝑑𝑖𝑎𝑔 𝜔𝑟 2 𝑀 𝑞 𝑐𝑜 = 𝑁 𝜙 𝑇 𝑀 𝑓𝑐Starting again from: Defining: 𝑁 𝜙 𝑇 𝑀 = 𝜙 𝐶 → 𝜙 𝐶 𝐿 = 𝜙 𝐶 𝑇 𝜙 𝐶 −1 𝜙 𝐶 𝑇 → 𝜙 𝐶 𝑓𝑐 = − 𝐾 𝑞 𝑐𝑜 → 𝑓𝑐 = − 𝜙 𝐶 𝐿 𝐾 𝑞 𝑐𝑜 Choosing the control logic to be: 𝑉 = − 𝐻 𝑦 Being: 𝐻 = 𝑔𝑒𝑛𝑒𝑟𝑎𝑙 𝑐𝑜𝑛𝑡𝑟𝑜𝑙 𝑚𝑎𝑡𝑟𝑖𝑥 Than: 1 𝛹 𝐿 𝑑𝑓𝑐 𝑑𝑡 + 𝑅 𝑓𝑐 + 𝛹 + 𝐻 𝑦 = 𝛹 𝑥 𝑝 𝑧 = 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 = 𝜙 𝑞 𝑁 𝜙 𝑇 𝑀 𝑓𝑐 = 𝑔 𝑐 = − 𝑁 𝑑𝑖𝑎𝑔 2𝜔𝑟 − 2𝜉 𝑟 𝜔𝑟 𝑀 𝑞 𝑐𝑜 = − 𝐾 𝑞 𝑐𝑜 𝑦 = 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑜𝑢𝑡𝑝𝑢𝑡𝑠 = 𝐶 𝑧 = 𝑀 𝜙 𝑁 𝑞 𝑐𝑜 𝐿 𝑑𝑖 𝑑𝑡 + 𝑅 𝑖 + 𝛹 𝑦 − 𝑥 𝑝 = 𝑉 ; 𝑖 = 1 𝛹 𝑓𝑐 Passing in the Laplace domain: 1 𝛹 𝐿 s + 𝑅 𝑓𝑐 + 𝐻 𝛹 𝑦 = 𝛹 𝑥 𝑝 ; 𝐻 𝛹 = 𝛹 + 𝐻
  • 17. − 1 𝛹 𝐿 s + 𝑅 𝜙 𝐶 𝐿 𝐾 𝑞 𝑐𝑜 + 𝐻 𝛹 𝑀 𝜙 𝑁 𝑞 𝑐𝑜 = 0 𝐻 = 1 𝛹 𝐿 s + 𝑅 𝜙 𝐶 𝐿 𝐾 ϕ 𝑂 𝑅 − 𝛹 Defining: ϕ 𝑂 = 𝑀 𝜙 𝑁 → ϕ 𝑂 𝑅 = ϕ 𝑂 𝑇 ϕ 𝑂 ϕ 𝑂 𝑇 −1 Finally: Neglecting the contribution of 𝑥 𝑝 and substituting the expression of 𝑓𝑐 and 𝑦: → 𝐻 𝛹 ϕ 𝑂 = 1 𝛹 𝐿 s + 𝑅 𝜙 𝐶 𝐿 𝐾 Control Design The control has a non causal formulation in fact: 𝑉 = − 𝐻 𝑦 = − 𝐷 𝑠 + 𝑃 𝑦 Although, due to the choice of the sensors, it can be seen in a causal form as: 𝑉 = − 𝐷 𝑠 + 𝑃 𝑦 = − 𝐷 + 𝑃 1 𝑠 𝑦 Where: 𝑦 = 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑣𝑒𝑐𝑐𝑡𝑜𝑟
  • 18. Real System – Measurements and Analysis To implement the control a knowledge of the mode shapes, eigenfrequencies and adimensional damping ratio of the structure is needed. For this reason the reference structure is mounted on a service structure rigidly connected to the ground by means of symmetric constraints, as shown by the red arrows in the figure. Constraints are needed for two reasons: 1. Find the modal parameters of the structure considering null the disturbances incoming from the ground. 2. Neglecting the frequency contribution of the service structure, at least in the frequency range of interest. The measurements are done, in this case, using a dynamometric hammer. Being not sure about the behavior of the system, due to its particularity, the plate is hammered in each corner point (near the corresponding accelerometer), identified by the cyan arrows, measuring the Frequency Response Functions of all the accelerometers. It made possible to see if significant differences occurs inside the plate that can cause non symmetric behavior. Measurement set-up 4 3 2 1
  • 19. Real System – Measurements and Analysis The results for the four hammered points are shown and they are superimposed to show that not big differences occurs at least below 150 Hz.
  • 20. Real System – Measurements and Analysis 1 19,686 19,701 19,672 19,740 2 21,926 21,889 21,938 21,909 3 27,060 27,122 27,001 27,031 4 75,392 75,452 75,215 75,468 5 78,607 78,603 78,652 78,758 6 109,919 110,004 109,684 109,899 7 124,035 124,396 124,386 124,160 8 133,255 134,186 133,650 133,608 𝑓01 [𝐻𝑧] 𝑓02 [𝐻𝑧] 𝑓03 [𝐻𝑧] 𝑓04 [𝐻𝑧] 1 0,02 0,0143 0,0184 0,0167 2 0,0152 0,0147 0,0147 0,0136 3 0,0152 0,0134 0,0159 0,0131 4 0,0147 0,0101 0,0185 0,0148 5 0,0108 0,0137 0,0108 0,0122 6 0,0129 0,0143 0,0112 0,0145 7 0,0137 0,0139 0,0164 0,0141 8 0,0119 0,0107 0,0116 0,0102 𝜉 1 [−] 𝜉 2 [−] 𝜉 3 [−] 𝜉 4 [−] For each hammered point the Prony method is applied to identify the eigenfrequencies and the adimensional damping ratios. A good result is achieved, but for lack of information (only 4 measurement points) an high number of poles must be used to get a sufficient number of repeated stable poles. For simplicity only the stability diagram obtained for the hammer in point 1 is reported in the figure. Instead in the tables below all the found parameters are reported and they show quite similar values.
  • 21. Real System – Measurements and Analysis Also in this case a least square method is used for reconstruct the signals response and to find the modal residues and so the mode shapes of the system. The different hammered points measurement gives, as in the case of the eigenfrequencies and adimensional damping ratios, different values of the modes shapes. For the control purpose all the parameters found are averaged. 1 2 3 4 5 6 7 8 19,7 21,915 27,053 75,382 78,655 109,876 124,244 133,675 0,0173 0,0145 0,0144 0,0145 0,0119 0,0132 0,0145 0,0111 0,3445 0,407 0,4622 0,3821 0,4815 0,1012 0,2613 0,1246 0,3445 -0,407 0,4622 0,3997 -0,4815 -0,0849 -0,2613 0,1081 0,3445 -0,407 -0,4622 0,3997 0,4815 0,0849 -0,2613 -0,1271 0,3445 0,407 -0,4622 0,3997 -0,4815 -0,1132 0,2613 -0,12 𝑓0 [𝐻𝑧] 𝜉 [−] ϕ1 ϕ2 ϕ3 ϕ4
  • 22. Real System – Simulation Results • The analysis done shows that the observable modes are the 1st, 2nd, 3rd and 5th because they are linearly independent. Moreover the first three resonances should correspond to rigid modes. • The simulation results show a good performance of the control logic implemented with respect to the ideal case. • The comparison is done by introducing in the system a ground velocity disturbance for each basement of the actuators that transmit to the system itself a vector of forces equal to: 𝐹𝐷 = 𝑟 𝑥 𝑝 + 𝑘𝑥 𝑝 • It is possible to see that the implemented control logic is trying to place the poles of the system in the positions related to the ideal gain control (it is shown in the figure the effect of the increase of the gains on the poles location)
  • 23. Real System – Implementation Final Configuration The system is now removed from the service structure and it is placed directly of the working surface. This is done because the control must work without regards of the surface where it is placed. This is partially true because if the system, plate plus actuators, is placed on a structure that with its modes can affects and interacts with the system, it is possible that the control becomes unstable because, due to the nature of the actuators used, a phase shift occurs between the sensor measurement and the voltage input. A careful choice of the placement of the sensors with respect to the actuators must be done. In fact initially they were left on the top of the LEGO® plate but as it is possible to see in the open loop transfer function reported a thickness mode occur that make the control no more co-located. Clearly after 150 Hz (red bounded zone) something strange inside the plate thickness happens, but the first eight studied resonances are not modified. Up position Down position
  • 24. Real System – Implementation The control is implemented in a real time system called DSpace and as fist attempt the sampling frequency used is 10 kHz. The effect of sampling will be discussed later. To solve the integration problem in the real time implementation a transfer function is used instead of a adopting an anti wind up configuration To solve the offset problem of the sensors an highpass filter if adopted Both the transfer functions are digitally implemented and for both the cutoff frequency is chosen to be 1 Hz.
  • 25. Real System – Experimental Results A good result is experimentally obtained in the frequency region of interest
  • 26. Real System – Experimental Results The results obtained were done hammering the upper surface of the structure because for the best case experiment it will be needed four shakers, one for each actuator, capable of carrying at least one quarter the weight of the entire structure. Transmissibility Measurement set-up The transmissibility between the accelerometer on the base (cyan arrow) and the accelerometer of the center of the plate (red arrow) is measured hammering the base. Despite the high noise, it is possible to see in the controlled transfer function a reduction of the incoming disturbance in the controlled frequency region and also a gradually phase shift.
  • 27. Real System – Diagonal Control Until now the results shown are based on a centralized control. This could cause problem of implementation for the point of view of the OpenAdaptronik project so a decentralized control could be taken into account. Choosing the derivative and proportional part of the control to be: 𝐷 𝑑 = 𝑑𝑖𝑎𝑔 𝑗=1 4 𝐷 𝑘𝑗 ; 𝑃𝑑 = 𝑑𝑖𝑎𝑔 𝑗=1 4 𝑃𝑘𝑗 Where j is the index of the column, a good result is obtained. The choice of the decentralized gains is done to ensure that the same input is given as the control would be centralized, when the four sensors signals are equal. In fact it is possible to back calculate the modal gains introduced by the control as: 𝐾 = 𝛹 ϕ 𝐶 𝐿𝑇 ϕ 𝐶 𝐿 −1 ϕ 𝐶 𝐿𝑇 𝐿 𝑠 + 𝑅 −1 𝐻 + 𝛹 ϕ 𝑂 𝑅𝑇 ϕ 𝑂 𝑅 ϕ 𝑂 𝑅𝑇 −1 1 2 3 4 1 2 3 4 243,253875 0 0 0 243,253875 0 0 0 0 271,387391 0 0 0 339,552905 0 0 0 0 335,056031 0 0 0 437,846974 0 0 0 0 976,621053 0 0 0 475,160617 Centralized Control Modal Gains Decentralized Control Modal Gains
  • 28. Real System – Sampling Effect A sampling frequency of 10 kHz is definitively to high for implementing a control on a low cost control board so the effect of sampling is experimentally studied. In the two figures the centralized and decentralized controls effects are compared for two different sampling frequencies, 1 kHz and 0.5 kHz. Both the control logics works similar but, despite the decentralized control formulation is not giving the wonted modal gains, at least in a theoretical way, it is less affected by sampling. In fact the reduction needed for stability are: 1 𝑘𝐻𝑧 ∶ 𝐶𝐶 𝐺 𝑟𝑒𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 78,06% 𝐷𝐶 𝐺 𝑟𝑒𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 72,90% 0,5 𝑘𝐻𝑧 ∶ 𝐶𝐶 𝐺 𝑟𝑒𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 94,84% 𝐷𝐶 𝐺 𝑟𝑒𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 93,55% The less reduction needed for the stability of the decentralized control should be caused by the measurements noise because each one affects only its co- located input and is not combined with the other signals noise. More research about this topic should be carried out. 𝐶𝐶 𝐺 𝑟𝑒𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = Centralized Control Gain Reduction 𝐷𝐶 𝐺 𝑟𝑒𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = Decentralized Control Gain Reduction
  • 29. Conclusions and Future Developments • It is shown that the control is working as designed and isolation can be carried out with low cost actuators and without a previous knowledge of the model of the system. • It is also possible to reduce the complexity of the problem by passing from a centralized control to a decentralized one that will lead to a more simple implementation. • It is shown that sampling affects in an heavily way the problem. • To solve the sampling problem an analogically implementation of the control proposed could be done. For this specific purpose it is easier and of course cheaper to develop the decentralized control instead of the centralized one. • A more careful study of the suggested noise problem relation to the gains reduction between centralized and decentralized control must be carried out to understand if the problem is really related to this phenomenon. Moreover, because the usage of low pass filters affects in a negative way the phase of the system lowering the stability margin of the control, in this work they are not used. • Sensors will be changed from high cost to low costs ones. • Analogical low cost amplifiers will replace the now used high cost ones. • An analogical control or a digital one, implemented in a low cost control board, will be developed. FUTURE DEVELOPMENTS CONCLUSIONS