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Modelling and Optimization of a
Magnetorheological Fluid Actuator for Footdrop
Mário César dos Santos Caixeirinho
Thesis to obtain the Master of Science Degree in
Mechanical Engineering
Supervisors: Prof. Jorge Manuel Mateus Martins
Prof. Paulo José da Costa Branco
Examination Committee
Chairperson: Prof. Paulo Jorge Coelho Ramalho Oliveira
Supervisor: Prof. Jorge Manuel Mateus Martins
Member of the Committee: Prof. Carlos Baptista Cardeira
June 2017
Abstract
The main aim of this thesis is structurally optimizing an orthosis applied to footdrop pathology. A
variable impedance actuator wearable leg orthosis is suggested composed of two springs and a
magnetorheological fluid linear actuator to restore patient’s gait. It is designed as the orthosis device
due to its variable damping actuation capacity. Both a hydraulic and an electromagnetic system model
are built to posteriori use in a structural and electromagnetic optimization. This is achieved using a multi-
objective genetic algorithm, minimizing the total volume and actuation power of the actuator.
A final actuator is obtained that is theoretically applicable to the problem. Future work and
improvements are suggested.
Keywords: Footdrop, Genetic Algorithm, Linear Actuator, Magnetorheological Fluid, Structural
Optimization.
Resumo
O objetivo principal desta tese é otimizar a estrutura duma prótese com aplicação na patologia do
Pé Pendente. Uma prótese colocada no exterior da perna composta por um atuador de impedância
variável é sugerida de forma a permitir restaurar a locomoção do doente. Este atuador é composto por
duas molas e um amortecedor linear com um fluido magnetoreológico, sendo este usado devido à sua
capacidade de amortecimento variável. Ambos os modelos dos sistemas hidráulico e eletromagnético
são construídos para utilização posterior na otimização estrutural e eletromagnética. Isto é alcançado
usando como optimizador um algoritmo genético multi-objetivo, minimizando o volume total do atuador
e a potência do atuador.
Um atuador final teoricamente aplicável ao problema é obtido. Trabalho futuro e possíveis melhorias
são apresentados.
Palavras-Chave: Algoritmo Genético, Atuador Linear, Fluido Magnetoreológico, Otimização
Estrutural, Pé Pendente.
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Acknowledgments
For all the extensive meetings and patience throughout every step of this work, I would like to thank
firstly my supervisors Professor Jorge Martins and Professor Paulo Branco. I know this wasn’t an easy
road but I’m thankful for all your support and motivation. For inserting me in your work groups to whom
I’m also grateful. Meeting and working with every one of you, made me learn a lot.
Secondly, to my course colleagues and PSEM team members, thank you. I don’t know how to repay
everything you all have done for me, helping me not losing sight of what’s important and comforting me
in the hard times.
Thanks to all my friends who provided me with a normal social life and thus some sanity.
To Nuno Moreira colleague and friend. You were always by my side. I’m sorry if I didn’t help you the
same way. Forever grateful for all you have taught me.
Finally, to my family who had no other choice but to endure. Your love was unconditional and I’m
sorry for all the times I was unfair and short tempered. This thesis is dedicated to you.
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“All wisdom is from the Lord,
and with him it remains forever”
Sir 1, 1
x
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Table of contents
1. Introduction ............................................................................................................................. 1
2. State of the art ......................................................................................................................... 3
2.1. Dropfoot pathology .......................................................................................................... 3
2.2. Magnetorheological fluid actuator.................................................................................... 5
2.3. NSGA-II Program in MATLAB ........................................................................................ 7
3. Design ...................................................................................................................................... 9
3.1. Geometry ......................................................................................................................... 9
3.2. Magnetic circuit .............................................................................................................. 10
3.3. Air-gap hydrodynamics .................................................................................................. 14
4. Validation ............................................................................................................................... 19
4.1. Tibialis anterior .............................................................................................................. 19
4.2. MRF actuator ................................................................................................................. 20
5. Optimization ........................................................................................................................... 25
5.1. Problem definition .......................................................................................................... 25
5.2. Parameter analysis ........................................................................................................ 27
6. Results ................................................................................................................................... 31
7. Discussion ............................................................................................................................. 39
7.1. Other remarks ................................................................................................................ 39
7.2. Battery usage ................................................................................................................. 40
7.3. Improvements ................................................................................................................ 40
8. Conclusions ........................................................................................................................... 41
9. References ............................................................................................................................ 43
Annex A ........................................................................................................................................... 45
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List of figures
Figure 2.1 Normal gait cycle and evolution states with sequential high-speed captures.................. 3
Figure 2.2 Normal ankle movement of angular position, angular velocity and joint torque .............. 4
Figure 2.3 Dropfoot gait deviations (A) foot-slap after CP and (B) toe drag during mid-swing ......... 4
Figure 2.4 Tibialis anterior replaced by a variable impedance actuator [3] ...................................... 5
Figure 2.5 Linear MRF actuator: (A) Structure [9] and (B) Force vs velocity [5] ............................... 6
Figure 2.6 MF fluid actuation: (A) Valve mode, (B) Direct-shear mode and (C) Squeeze mode ...... 7
Figure 3.1 Linear MRF Actuator ........................................................................................................ 9
Figure 3.2 Actuator geometric variables ........................................................................................... 9
Figure 3.3 Piston with coil and electric analogous of the actuator symmetric magnetic circuit ...... 10
Figure 3.4 B-H curve of MRF-132DG [18] ....................................................................................... 12
Figure 3.5 B-H curve of AISI 416 Annealed Stainless Steel [19] .................................................... 12
Figure 3.6 Static shear stress forces representation between fluid-fluid and fluid-structure .......... 16
Figure 3.7 MRF-132DG electromechanical characteristic curve [18] ............................................. 17
Figure 4.1 Various ankle angle samples from SimMechanics simulation ....................................... 19
Figure 4.2 Sample of the ankle measurements from SimMechanics simulation ............................ 19
Figure 4.3 Linear actuator measurements converted from ankle measurements .......................... 20
Figure 4.4 Geometry dimensions of the MRF linear actuator in [23] .............................................. 21
Figure 4.5 Static model validation ................................................................................................... 21
Figure 4.6 Dynamic model validation .............................................................................................. 22
Figure 4.7 B-H curve of MRF-122EG from [23] ............................................................................... 22
Figure 4.8 MRF-122EG electromechanical characteristic curve from [23] ..................................... 23
Figure 4.9 Geometry measurements of the MRF linear actuator in [3] ........................................... 23
Figure 4.10 Static validation of the model with [3] experimental data ............................................. 24
Figure 5.1 Best result of each population size after 150 generations ............................................. 28
Figure 5.2 Computational time versus population size ................................................................... 28
Figure 5.3 Varying mutation scale on a 1000 individuals’ population after 150 generations .......... 29
Figure 5.4 Mutation scale versus computational time ..................................................................... 30
Figure 6.1 Last Pareto Front ........................................................................................................... 31
Figure 6.2 Best optimized actuator for 110 N after 500 generations .............................................. 32
Figure 6.3 Volume convergence ..................................................................................................... 32
Figure 6.4 Power convergence ....................................................................................................... 33
Figure 6.5 Power volume product convergence.............................................................................. 33
Figure 6.6 Static simulation of the optimized actuator .................................................................... 34
Figure 6.7 Best Pareto Front volume solution after 500 generations.............................................. 34
Figure 6.8 Best Pareto Front power solution after 500 generations ............................................... 35
Figure 6.9 Best optimized actuator for 55 N after 500 generations ................................................ 35
Figure 6.10 Best optimized actuator for 40 N after 500 generations .............................................. 36
Figure 6.11 Best optimized actuator for 55 N with AWG 40 after 500 generations ........................ 36
Figure 6.12 Best optimized actuator for 55 N with AWG 18 after 500 generations ........................ 37
xiv
Figure 7.1 Schematic of piston’s flange future improvement. a) Round tip and b) Ramp tip ......... 40
xv
List of tables
Table 1 – Decision variables ........................................................................................................... 25
Table 2 – Computer system components specification .................................................................. 27
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Abbreviations
AAFO Active Ankle-foot Orthosis
AFO Ankle-foot Orthosis
AWG American Wire Gauge
CD Controlled Dorsiflexion
CP Controlled Plantarflexion
EMO Evolutionary Multi-Objective Optimization
ER Electrorheological Fluid
GA Genetic Algorithm
MMF Magnetomotive Force
MR Magnetorheological
MRF Magnetorheological Fluid
NGPM A NSGA-II Program in MATLAB
NSGA Non-Dominated Sorting Genetic Algorithm
OAT One-at-a-time
PP Powered Plantarflexion
VIA Variable Impedance Actuator
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xix
List of Symbols
Top/Bottom piston’s superficial area
Cross-sectional area of the coil copper wire
Cross-sectional area of the ferromagnetic steel where the magnetic field passes
, Cross-sectional area of the ferromagnetic steel in section 2
Magnetic density flux (magnetic induction field)
, Magnetic density flux (magnetic induction field) of the MR fluid
, Magnetic density flux (magnetic induction field) of the ferromagnetic steel
Piston inner radius
Gap width of the MRF actuator
Coil copper wire diameter
Cylinder wall thickness
Dynamic force of the MRF actuator
Force density of the MR fluid in the axis
Maximum force of the MRF actuator
Fluid-fluid force density on the MR fluid
Seal friction force of the MRF actuator
Static force of the MRF actuator
Fluid-structure force density on the MR fluid
ℎ Piston flange thickness
Magnetic field
, Magnetic field of the MR fluid
, Magnetic field of the ferromagnetic steel
Electric current
Maximum electric current of the MRF actuator
Number of the term in the sum expression
Piston height
Length of the coil copper wire
Length of the middle magnetic path
, Length of the middle magnetic path in section 2
Number of coil turns
Number of coil turns per column
Number of coil turns per row
Pressure of the MR fluid
Power of the MRF actuator
Maximum power of the MRF actuator
Flow rate of the MR fluid
Flow rate dislocated by the actuator moving piston
Axis of the cylindrical coordinate system
xx
Piston outer radius
Piston guide radius
Electric resistance of coil copper wire
Magnetic reluctance
, Magnetic reluctance of the MR fluid
, Magnetic reluctance of ferromagnetic steel in section 1
, Magnetic reluctance of ferromagnetic steel in section 2
, Magnetic reluctance of ferromagnetic steel in section 3
Time
Actual functioning temperature of the coil
Reference working temperature of the coil
Maximum applied voltage of the MRF actuator
MR fluid velocity
First term of the MR fluid velocity in the axis
Second term of the MR fluid velocity in the axis
Velocity of the actuator piston
Volume of the MRF actuator
MR fluid velocity in the axis
MR fluid velocity in the axis
MR fluid velocity in the axis
Axis of the cylindrical coordinate system
Temperature coefficient of resistivity
MR fluid-structure static friction coefficient
Reference resistivity of copper
Electric resistivity of copper
Δ Dynamic pressure difference of the MR fluid
Δ Static pressure difference of the MR fluid
MR fluid viscosity
Axis of the cylindrical coordinate system
Magnetic permeability
, Magnetic permeability of the MR fluid
, Magnetic permeability of the ferromagnetic steel
MR fluid mass density
Magnetic normal stress
MR fluid shear stress by fluid-fluid friction
MR fluid shear stress by fluid-structure friction
Φ Magnetic flux
1
1. Introduction
Nowadays research has evolved greatly in understanding the human body and how it works. There’s
no doubt about how perfectly every part of the body interacts. Although it is a complex system, it rarely
ever has problems in comparison.
Footdrop pathology, known for disabling the person’s ability to lift the foot, resulting in an unusual
walking pattern. Or in medically terms it’s a reduced or even inexistence activity in the dorsiflexor
muscles affecting the human walking gait. Usually, people who live with this pathology correct their gait
by increasing the knee higher than normal when walking and slapping the foot afterwards on the ground.
Otherwise dragging the foot on the floor. Either way, this unusual gait, in time, damages the rest of the
body.
There are a broad range of different solutions on the market, orthosis or not. None of those solutions
are perfect. The least intrusive are ankle-foot orthoses (AFO). Unfortunately, most of them have a single
locking position or a constant damping coefficient, i.e. non-adaptive during gait, called passive orthoses.
These avoid some body wear but not all. Better solutions are adaptive orthoses that can replicate the
normal human gait called active ankle-foot orthosis (AAFO). Magnetorheological fluid (MRF) dampers
show great potential here.
These dampers are embedded with MR fluid and an electromagnetic system. MR fluids are liquid
solutions with micro magnetisable particles. When subjected to a magnetic field the magnetisable
particles in the fluid aggregate and align in the field direction, increasing the fluid viscosity where the
magnetic field passes. This property allows blocking movement in the presence of a high field passing
through the MRF actuator gap sustaining or damping the acting force.
So, this type of actuators are low power variable damping systems, controllable by applied current.
When applied to dropfoot pathology the variable damping coefficient permits to mimic some states of
the human gait. An actuator model and respective validation are presented.
A structural optimization based on the model is done, where the goal is obtaining the smallest
actuator possible and also the lowest power consumption to achieve the required force. Genetic
algorithm (GA) is chosen to accomplish this optimization.
GA is a computational optimization that mimics biological evolution, employing principles of natural
selection. An evolutionary multi-objective optimization (EMO) is used here to optimize two objective
functions. The algorithm code used is called NGPM (version 1.4) abbreviation of “A NSGA-II Program
in MATLAB” which allows parallel computation, reducing overall process time.
Achieving a wearable and suitable optimized MRF linear actuator orthosis application for dropfoot
pathology is the main goal of this thesis. Important results, conclusions, corrections and future work are
presented.
2
3
2. State of the art
In this section are presented the key concepts and all knowledge obtained from previous researches,
explaining where it begun, how it grew and what is being done today. Three major topics were
considered.
All needed concepts around the dropfoot pathology are presented, including different human gait
phases and where the MRF actuator will act.
Secondly, every aspect of the MRF actuator is presented. An important focus is made on the MR
fluids and how they work, since they are responsible for variable damping behavior in these actuators.
Finally, a brief explanation of the optimization algorithm used and some current structural
optimization applications.
2.1. Dropfoot pathology
The human gait is composed off two main phases shown in Figure 2.1, taken from [1, 2] for a 74 Kg
healthy male.
When the foot is on the ground it’s called the stance phase, or else the swing phase. There is also
an alternative designation by states as represented. State 1, represented from heel strike until complete
grounded foot plant, called Controlled Plantarflexion (CP). Second state, until foot is in neutral position,
with leg perpendicular to the floor and all toes on the ground. State 3, then ends on maximum ankle
angular position already with some heel off. These two states form the Controlled Dorsiflexion (CD).
Finally, Powered Plantarflexion (PP) occurs in state 4 with total foot plant lift, reaching toe-off. Foot
swing period happens in states 5 and 6.
Figure 2.1 Normal gait cycle and evolution states with sequential high-speed captures
Angular position, velocity and torque of the ankle during human gait states are shown in Figure 2.2
(data from [1] for a 74 Kg healthy male). Positive ankle angles are considered counter clockwise to the
images as convention.
4
Figure 2.2 Normal ankle movement of angular position, angular velocity and joint torque
Foot lifts approximately 10° in the end of CD and -20° in the end of PP.
Dropfoot pathology, as already stated, affects human gait because individuals lack or have reduce
activity in the dorsiflexor muscle, better known as tibialis anterior. Two main problems arise: foot slap
and toe drag (Figure 2.3, taken from [3]).
Figure 2.3 Dropfoot gait deviations (A) foot-slap after CP and (B) toe drag during mid-swing
5
Foot slap occurs after heel strike, due to swing phase attained momentum and body weight. Then
in CP the defective tibialis anterior cannot hold or help damping out the foot, resulting in the usual slap
sound heard from these patient’s gait.
Toe drag on the other hand happens during mid-swing phase which unbalances the body. Normally
the dorsiflexor muscle holds the foot up in the air during this phase.
The use of a variable damping actuator with some attached spring system, also known as a Variable
Impedance Actuator (VIA) is shown in Figure 2.4 and can mimic the behavior of the tibialis anterior.
Figure 2.4 Tibialis anterior replaced by a variable impedance actuator [3]
Two springs are required. One high stiffness spring connected in series with the variable damping
actuator and a low stiffness spring in parallel as shown in the previous figure.
The high stiffness spring and damper will only act in the CP state. This helps control foot damping,
avoiding foot-slap. In contrast, the low stiffness spring is always active supporting the foot weight,
avoiding toe drag.
A linear magnetorheological fluid (MRF) actuator will be used to act as the variable damping
actuator.
The objective is to use an AAFO, in this case a VIA, connected to the foot and leg, helping the
defective tibialis anterior.
2.2. Magnetorheological fluid actuator
As stated in Chapter 1, the goal is modelling and optimizing a linear MRF actuator for dropfoot
orthosis application. This actuator works using MR fluids, which have been studied since World War II.
Research started in 1939 with particle suspensions in low viscosity oils forming an “oil-occluding
fibrous mass” when subjected to an electric field [4], i.e. electrorheological fluids (ER fluids), which was
patented. Almost at the same time, Jacob Rabinow in 1940 at the Bureau of Standards discovered
magnetorheological fluids (MR fluids) and developed the first device applications for it.
Both ER and MR fluids are non-colloidal, i.e. they settle down with time if the fluid is not mixed in
time. The polarizable particles suspensions have a size of micrometers [5]. MR and ER fluids are mostly
different in magnetic saturation values than by the resistance force their polarizable particles can
produce [4].
6
MR fluids are composed off three main components, the base fluid, magnetic metal particles and
stabilizing additives. Maximum fluid viscosity is due to magnetic saturation of the fluid metal particles
and minimum with the carrier fluid viscosity [6].
The base or carrier fluid normally can be of three different types, i.e. hydrocarbon oils, mineral oils
or silicon oils. This fluid carries the magnetic metal particles, lubricating and damping their movement.
Carrier fluid viscosity must not vary much with temperature so that the magnetorheological effect is only
due to the particles inside it. Higher carrier fluid viscosities are justified by higher head losses when no
magnetic field is present, which is undesirable for most applications [6].
Static and kinetic friction coefficients are similar for low magnetic fields. High magnetic fields show
in contrast a dissimilar dry friction behavior. Nevertheless, kinetic friction coefficient is always higher
than the static in these fluids [6]. Static shear resistance forces are consistent with values from known
magnetic tractive forces [4].
Heat transfer is promoted in the active region and has the same direction as the magnetic field lines.
Magnetic particles in the fluid agglomerate increasing the heat transfer and then when the structure
breaks down (magnetic field disactivated), there is a high increase in the heat transfered by particles
dispersion in the carrier fluid [7]. Eddy currents [8] and joule energy loss [5] in the fluid are negligible.
The increase of the superficial area of the actuator and number of active regions, decreases copper
losses [6].
Recently, MR fluids have been applied in many devices: linear dampers (Figure 2.5), rotary brakes,
vibration dampers, etc.
Figure 2.5 Linear MRF actuator: (A) Structure [9] and (B) Force vs velocity [5]
7
There are three modes in which they can be applied in devices: valve mode, direct-shear mode and
squeeze mode [5], Figure 2.6 presents these modes (Adapted from [10]).
Figure 2.6 MF fluid actuation: (A) Valve mode, (B) Direct-shear mode and (C) Squeeze mode
High precision MRF actuators have many leakage problems, rubber O-rings are used to seal and
avoid fluid leakage [6]. MR seals are also used but only in low frequency or static system applications
[11].
Actuators with low nickel steel structures are better for minimizing hysteresis effect [6].
2.3. NSGA-II Program in MATLAB
NGPM is built by Lin [12] based on previous NSGA-II algorithm done by Deb [13]. The Non-
Dominated Sorting Genetic Algorithm (NSGA-II) starts by generating an offspring of size equal to the
initial randomly generated parent population through genetic operators (selection, crossover and
mutation). Then parent and offspring population are combined in sets of non-dominated fronts by fast
non-dominated sorting [14]. Individuals are chosen from best fronts to form the new parent population,
having the same initial parent population size. Process repeats until maximum number of generations
defined by the user is reached. Applications using this elitist and crowd comparison operator algorithm
are shown in optimization design of viscoelastic damping structures [15]. It can iterate problems with
many decision variables and is a tool commonly used in structural optimizations. Also, it doesn’t get
stuck on local minimums or maximums.
The solutions are given by a Pareto front (set of optimal solutions), instead of a single point solution,
this gives more options to the decision maker helping him/her decide what is the best solution.
Computational time can be high but it varies a lot from problem to problem and how the user sets the
algorithm parameters.
8
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3. Design
3.1. Geometry
Most common MRF actuators are linear. Three important solid components define it: a cylinder, a
piston and a coil. In the cylinder, the piston will move vertically on its axis, as shown in Figure 3.1 from
[16]. MR fluid will surround the piston inside the cylinder. Finally, around the piston inner radius there is
a copper wire coil where external applied electric current activates the actuator. Piston outer radius ,
piston inner radius , piston flange thickness ℎ, gap width , cylinder wall thickness , piston height
and piston guide radius , define the actuator geometry (Figure 3.2).
Figure 3.1 Linear MRF Actuator
Figure 3.2 Actuator geometric variables
10
3.2. Magnetic circuit
By passing an electric current throughout the insulated copper wire coil (left side of Figure 3.3) a
magnetic field is created which actuates the MRF actuator. Two holes on top of the piston work as entry
and exit points for the coil.
The applied magnetomotive force ( ) equal to , where is the number of turns, induces a
constant magnetic flux Φ that passes from the piston to the cylinder wall and back, through the MR
fluid, closing the magnetic circuit. All other magnetic flux leakages are neglected.
Considering magnetic reluctances as resistors, the as the electric potential difference and
Φ as the electric current, an electric analogous is shown (right side of Figure 3.3) for the actuator
symmetric magnetic circuit.
Figure 3.3 Piston with coil and electric analogous of the actuator symmetric magnetic circuit
Number of coil turns is estimated knowing the coil housing size and the copper wire diameter :
= = − 2ℎ
− −
(3.1)
In Equation 3.1, is the number of coil turns per column and the number of coil turns per row.
The clearance is a measure given so that the copper wires can exit the coil housing without passing
through the air-gap.
Due to wire heating, copper electric resistance changes as been stated by the Pouillet’s Law:
= ()
(3.2)
,
,
,
,
,
,
11
Here, is the copper electric resistivity function of the coil working temperature , the length
of it and its cross-sectional area. From [17] a copper wire length approximation is given by:
= 2 + ( − 1)
(3.3)
Electric resistivity changes with temperature so a linear approximation is used near a reference
working temperature equal to 20ºC as stated by Equation 3.4. Here, the reference resistivity at
= equal to 1.72 x 10-8 Ω.m and 3.93 x 10-3 ºC-1 the temperature coefficient of resistivity , both for
annealed copper.
()= [1+ (− )] (3.4)
Nevertheless, for simplification purposes no significant change with varying temperature in the coil
is considered in thus Equation 3.2 reduces to:
= 8 + ∑ ( − 1)
(3.5)
Magnetic reluctances vary with the type of ferromagnetic material where the magnetic density
flux passes, due to its magnetic permeability . Magnetic permeability is not constant for all
values and is defined by the derivative at each point of the material B-H curve. The electric current
passing through the copper wire of the coil generates a magnetic field , which in return due to the
material’s magnetic permeability surrounding it, creates a magnetic induction field . This relation
is seen in Figure 3.4, for the magnetorheological fluid MRF-132DG, and in Figure 3.5 for AISI 416
Annealed Stainless Steel, both used in this work.
12
Figure 3.4 B-H curve of MRF-132DG [18]
Figure 3.5 B-H curve of AISI 416 Annealed Stainless Steel [19]
Due to the observed nonlinearities (magnetic saturation) on the B-H curves, only the linear working
region is considered. Hence, magnetic permeability of the two magnetic materials is considered
constant, for , ≤ 10 A.m-1 and , ≤ 10
A.m-1.
= (3.6)
Each magnetic reluctance can be calculated as
=
(3.7)
13
Where is the length of the magnetic path passing through the middle of the material ( was
assumed uniformly distributed) and is the cross-sectional area normal to it. So, for each section of
the magnetic path in Figure 3.3, one has:
, = − ℎ
, (3.8)
, = − ℎ
,[(+ + ) − (+ )]
(3.9)
Also, because the magnetic path that passes through the MR fluid is very small, the area of
associated magnetic reluctance is considered approximately constant and equal to:
, =
,2ℎ (3.10)
For a differential element of magnetic reluctance ,, the cross sectional area , varies with
, [20] as:
, =1
,
,,
(3.11)
Observing that , is the cross-sectional area of a disk and neglecting the initial magnetic path
,, = ,,
= 1
2ℎ,
, =1
2ℎln
(3.12)
Then:
, =1
2ℎ,ln
(3.13)
Finally, Equation 3.14 describes the relation between the produced magnetic flux Φ, the applied
and the magnetic reluctances of the actuator.
Φ =
,+ 2, + , + 2, (3.14)
Magnetic flux is constant with constant current applied on the coil and because , is considered
approximately constant, they are related by the following magnetic relation:
14
Φ = (3.15)
The magnetic field passing on the MR fluid, ,, can be obtained through the magnetic induction
field using Equation 3.6.
, =Φ
,2ℎ (3.16)
If , ≫ ,, , is reduced to:
, ≈
2 (3.17)
3.3. Air-gap hydrodynamics
The gap between the piston and the inner cylinder wall, called here the “active” region (Figure 3.1),
is analysed next to better understand the MRF actuator hydrodynamics.
Considering the MR fluid a homogeneous and incompressible fluid:
∇∙ = 0 (3.18)
Because it is considered to move only in the direction (Figure 3.2), fluid velocities and are
neglected:
= = 0 (3.19)
Hence:
= 0 (3.20)
That means fluid velocity on the “active” region is constant in the direction.
Considering symmetry in relation to the axis, then:
∂∂
= 0 (3.21)
Using Navier-Stokes equations in cylindrical coordinates (see Annex A) and approximating the MR
fluid to a constant mass density and viscosity fluid:
15
= −
+
∂∂
(3.22)
For steady-state regime the following equation for the hydrodynamics in the “active” region appears:
−
+
∂∂
= 0 (3.23)
This means density forces due to MR fluid acceleration
plus pressure
are equal to the
summation of other force densities , which will be explained further, with forces due to head losses
. MRF-132DG used here has a 0.112 (±0.02) Pa.s viscosity [18]. A final equation for the
hydrodynamics of the “active” region is reached:
∂∂
=1
−
(3.24)
The MR fluid velocity profile in the “active” region can then be estimated solving Equation 3.24,
resulting in:
()=1
2
−
+ + (3.25)
Then, using the respective boundary conditions, ( = )=
( = + )= 0, constants and can now
be obtained. Therefore:
()= ()+ () (3.26)
With
()=(+ − ) (3.27)
And
()=1
2
− [
− (2+ ) + (+ )] (3.28)
16
To obtain the pressure gradient /, a fluid flow rate analysis to the “active” region is done:
= ()
(3.29)
Knowing that must be equal to the flow rate dislocated by the moving piston, , with speed
inside the actuator (no fluid accumulation), then:
= = 2 ()
= (3.30)
Where = ( −
).
Finally, Equation 3.31 is reached
= + 4
3 − 3− − 3
( + 2) (3.31)
The pressure gradient
is then given by the one undefined force density plus the gap head
losses between the piston and the cylinder wall.
Pressure drop inside the cylinder occurs due to two major forces independent of fluid velocity
(neglecting head losses): one by shear stress (fluid-fluid friction) and another by fluid-structure
friction , both on the active region, as indicated in Figure 3.6. MR fluid weight is considered
negligible in the “active region” in comparison with these forces since the volume is very small.
Figure 3.6 Static shear stress forces representation between fluid-fluid and fluid-structure
17
For a static behaviour (actuator blocking the total force) as magnetic field in the air-gap increases,
magnetic particles in the fluid connect more strongly to each other than to the piston or cylinder walls.
Load is supported by since it is lower than . After a certain point of magnetic field increase,
ferromagnetic particles start to grab to the walls more than to each other, > , and
becomes responsible for supporting the load.
Force density done by the MR fluid as seen is then a minimum between fluid and wall shear stress
friction forces:
= (,) (3.32)
Shear stress between the magnetic particles on the fluid when a magnetic field is present is given
by fluid data sheet [18]. The MRF-132DG electromechanical characteristic used in this work is
shown in Figure 3.7.
Figure 3.7 MRF-132DG electromechanical characteristic curve [18]
Comparing with Figure 3.4, in linear regime ( ≤ 10 A/m) it can be stated that:
= 0.3, (3.33)
Using the area parallel to the flow in the middle of actuator active region and the respective volume
of fluid:
=[2ℎ(2+ )]
2ℎ[(+ ) − ]=
(3.34)
18
The electromagnetic normal stress that attracts the magnetic particles of the fluid into the walls is
given by:
≈1
2,,
(3.35)
Because , ≪ ,. Then wall shear stress is obtained by multiplying by the static
friction coefficient , since is normal to . The largest force acts on the minimum area, so the
piston superficial area on the active region is used instead of the cylinder superficial area. Dividing it by
the volume of fluid passing thought the active region:
=2[2ℎ(2+ )]
2ℎ[(+ ) − ]=,,
(3.36)
Equation 3.32 is then equal to:
= 0.3,
,,,
(3.37)
From Equation 3.31, in static behaviour for constant parameters
is constant and equal to the static
pressure difference ∆ of the MR fluid between the top and bottom of the piston inside the actuator
(no fluid velocity). But since only acts in the “active” region:
Δp = 2ℎ× 0.3,
,,,
(3.38)
In dynamic behaviour, since there are no aligned ferromagnetic particles blocking the whole gap
width , wall shear stress is negligible. So, dynamic pressure difference ∆ of the MR fluid
from Equation 3.31 is given by two components plus head losses. The length where the pressure
difference acts is different for the two components, for head losses and 2ℎ for .
Δp = 2ℎ0.3,
+ 4
3 − 3− − 3
( + 2) (3.39)
The damper strength is known by relating both pressure differences with the moving piston area due
to applied exterior strength plus an added seal friction force . Finally:
= ( −
)∆ + (3.40)
= ( −
)∆ + (3.41)
19
4. Validation
4.1. Tibialis anterior
Based on a previous modelling work by Geyer [21], was possible to replicate human walk in [22]
using Matlab, Simulink and SimMechanics environment. Simulation samples of the ankle angle were
taken during 10 seconds to reach permanent regime and disregard initial transient results (Figure 4.1).
A sample, after initial transient regime of the walking pattern, was chosen to compare with the known
real data (Figure 2.2). Figure 4.2 shows the simulation measurements of the ankle angle, angular
velocity and torque.
Figure 4.1 Various ankle angle samples from SimMechanics simulation
Figure 4.2 Sample of the ankle measurements from SimMechanics simulation
20
Only the angular velocity has different values but similar in behavior compared to Figure 4.1. Since
the proposed linear MRF actuator is going to be attached to the leg of the patient, vertically below the
knee, a transformation of this angular measurements is needed to find the linear displacement, velocity
and acting force. After transformation (Figure 4.3), linear displacement and velocity are shown to have
similar behavior to the ankle angular measurements, as expected.
Figure 4.3 Linear actuator measurements converted from ankle measurements
Linear force presented is only related to the tibialis anterior, instead of the whole forces acting on
the human ankle. This is the only force that is important to replicate in the actuator. Tibialis anterior force
exerted on the actuator is approximately 110 N at the end of the CP state, representing the maximum
force the actuator must block. Also, maximum linear velocity achieved by the actuator is around 400
mm/s and the total displacement 30 mm.
4.2. MRF actuator
Previous experimental data from works with MRF linear actuator is used to validate designed model.
First comparison is between the static model and the Costa’s actuator static experimental data [23]
(Figure 4.5). Geometry dimensions of the actuator are shown in Figure 4.4. Actuator contains MRF-
122EG with a 2380 Kg/m3 density and a 0.07(±0.02) Pa.s viscosity . Static tests were made by
finding the current that makes the actuator hold the weight without the piston falling. Weights were slowly
incremented from 1 until 25 Kg.
21
39
20
10
2
30
2
10
194
Figure 4.4 Geometry dimensions of the MRF linear actuator in [23]
Figure 4.5 Static model validation
22
Figure 4.6 Dynamic model validation
Model fits the experimental points with a static friction coefficient of 2.4 on Equation 3.36, between
the fluid and the piston flange wall. This value is high and comparable to silicon rubber which has also
a friction coefficient higher than unit. In the presence of high magnetic fields the MR fluid has a solid-
viscous behavior justifying this value. Without current, experimental data shows a 15 N seal friction force
that was added to the model.
The dynamic model fits poorly to the Costa’s actuator dynamic experimental data [23] (Figure 4.6).
A constant offset is observed which suggests a missing load in the model.
Used magnetic permeability and electromechanical characteristic of the fluid are presented in Figure
4.7 and Figure 4.8, respectively.
Figure 4.7 B-H curve of MRF-122EG from [23]
23
Figure 4.8 MRF-122EG electromechanical characteristic curve from [23]
A smaller actuator was built by Domingues capable of carrying 0.5 Kg with an actuation of 0.8 A [3]
(Figure 4.10). The structure for this prototype was based on the previously presented Costa prototype.
Built model is represented in Figure 4.9. The piston is movable by thin string lines attached to its tips,
so the piston guide radius is considered approximately null. MRF-132DG was the chosen fluid, with a
density of 2950-3150 Kg/m3 [18]. Figures 3.4 and 3.7, in the previous chapter, show fluid B-H curve and
electromechanical characteristic curve, respectively.
4.5
1.5
0.8
0.5
16.6
0.25
0.5
≈ 240
Figure 4.9 Geometry measurements of the MRF linear actuator in [3]
24
Figure 4.10 Static validation of the model with [3] experimental data
Since no seal friction force was measured, none was introduced in this model. Saturation of the
model around 0.35 A was due to steel magnetic saturation. The steel of the actuator was not specified,
so AISI 416, already stated in chapter 3, was used instead to compare.
25
5. Optimization
Since the actuator will be used on a patient, the weight is an important aspect that must be
addressed. Also, the additional actuation system must be well thought because it requires a low power
actuation. Low power actuation means smaller and lighter batteries, since the patient must also carry
them. All this conditions justify a multi-objective structural optimization to the actuator explained here.
Nondominated Sorting Genetic Algorithm (NSGA) is a multi-objective genetic algorithm with built in
“binary tournament selection”, “intermediate crossover” and “Gaussian mutation” [12].
“Binary tournament selection” picks pairs of random individuals (solutions), then transforms every
continuous or discrete solution to a corresponding binary value, allowing optimization with both
continuous and discrete decision variables. “Tournaments” between each pair of individuals allow
finding individuals with the best chromosomes for posteriori crossover.
“Intermediate crossover” also known as arithmetic crossover takes pairs of the best individuals from
the parent population and generates one pair of offspring per pair of parents. These offspring are created
using a ratio of random chromosomes from both the parents and building each offspring with these
randomly chosen ratio of chromosomes.
“Gaussian mutation” as the name suggests mutates the offspring population genes by a random unit
Gaussian distribution using a controlled standard deviation specified by the user in the mutation scale
parameter. NSGA also allows for mutation shrinkage (specified by user) controlling the rate at which the
average amount of mutation decreases along the number of generations. This is normally used in
problems that do not get stuck on local solutions, helping to improve the convergence speed.
Software changeable parameters are population size, number of generations, crossover ratio [0,1],
mutation scale (standard deviation of the random mutation number [0,1]) and mutation shrink parameter
(shrinkage rate [0.5,1]).
5.1. Problem definition
Two objective functions, nine constraints and seven decision variables (Table 1) with a top and a
lower boundary for each, formulate this optimization problem.
Table 1 – Decision variables
Piston outer radius
Piston inner radius
Piston flange thickness
Gap width
Piston height
Cylinder wall thickness
Maximum electric current
Minimizing total volume of the actuator (Equation 5.1) and power (Equation 5.2) are the objective
functions. Copper wire electric resistance is obtained from Equation 3.5.
26
To preserve the normal shape of the actuator two geometric constraints are needed (Equation 5.3
and 5.4). Last one imposes a coil housing around the piston inner radius (see Equation 3.1). Also, from
Costa’s actuator a ratio between flange thickness ℎ and gap lower than 5 is used to avoid probable
gap fluid strangulation (Equation 5.5).
Nonlinear magnetic regime (saturation) is avoided in each section of the actuator (Figure 3.3) by
limiting the maximum magnetic field passing through the MR fluid using Equation 3.16 (Equation 5.6)
and the ferromagnetic steel (Equations 5.7, 5.8 and 5.9).
Maximum (power on) and minimum (power off) force constraints of the linear MRF actuator, are
given by Equation 5.10 and Equation 5.11, respectively (see Equation 3.40 and 3.41). Figure 4.3 gives
maximum force and maximum linear velocity for minimum force constraints. The dynamic force
fixed to 100 N was found to be reasonable for plausible gap size results, since the created model did
not explain totally the dynamic behavior. This value was used although it has no practical sense.
Lastly, decision variables lower and upper bounds are shown in Equations 5.12-18.
The optimization problem is formulated as:
Minimize = (+ + ) (5.1)
Minimize = (5.2)
Subject to > 2 (5.3)
≥ 1 (5.4)
≤ 5 (5.5)
, ≤ 100 kA/m (5.6)
, ≤ 1 kA/m (5.7)
,≤ 1 kA/m (5.8)
,[()()]
≤ 1 kA/m (5.9)
≥ 110 N (5.10)
(= 0, = 400 )≤ 100 N (5.11)
Bounded by 1 ≤ ≤ 50 (5.12)
1 ≤ ≤ 50 (5.13)
0.1 ≤ ≤ 30 (5.14)
0.1 ≤ ≤ 10 (5.15)
1 ≤ ≤ 100 (5.16)
0.1 ≤ ≤ 10 (5.17)
0.01 ≤ ≤ 5 (5.18)
27
Seal forces are neglected. Copper wire diameter is set to 0.321 mm (AWG 28), the smallest
diameter of wire available easily hand turned into a coil around the piston. Clearance between the coil
end and the flange end of the piston is set to 1 mm (see Equation 3.1). Although not represented in the
following result schematics, the piston guide radius is also fixed and has a value of 0.5 mm, equal to
a thin string line radius. Piston course inside the actuator is not considered in the optimization because
it is constant. It was verified to be around 30 mm from Figure 4.3 and added to every actuator’s
schematic of the optimization results. Finally, the coefficient of friction is set to 2.4 as concluded in
Chapter 4.
5.2. Parameter analysis
To understand and reach the best results without using too much processing time a parameter
analysis is needed.
All the results were obtained running on a computer with the specifications in Table 2. It was found
that early MATLAB versions already have parallel computing inside its functions, so although specific
functions from older versions exist in the NSGA algorithm to force parallel computing, they are useless.
Table 2 – Computer system components specification
Processor Intel® Core™ i5-4430 CPU @ 3.00GHz
Installed memory (RAM) 8,00 GB
Operating System Windows 10 x64
All optimizations end when they reach a stablished number of generations. Initially the population
size must be fixed. To find out the adequate population size, many optimizations are done varying this
parameter and looking at the best result after 150 generations. This technique is called one-at-a-time
(OAT), varying one parameter and fixing all the others. Figure 5.1 shows this evolution. With increased
population size, so the computational time increases exponentially (Figure 5.2). Smaller populations are
better for this reason. Best results converge to a similar solution around 1000 individuals, so this is the
population size used from here on.
NSGA algorithm is implemented with “intermediate crossover” and the crossover ratio is a parameter
established by the user. The crossover ratio is set to 0.9 to use as much genetic background as possible
from the parents.
28
Pow
er
* V
olu
me [W
.mm
3]
Figure 5.1 Best result of each population size after 150 generations
Tim
e [m
inute
s]
Figure 5.2 Computational time versus population size
29
To improve solutions and avoid local minimums, after crossover there is some offspring genetic
data that suffers mutation, i.e. some genes are generated randomly in the offspring after the crossover.
The percentage of genetic data mutated in the new offspring is controlled by the mutation scale
parameter as stated. After some optimizations varying this parameter, the best percentage that allows
arising to the best solutions without losing much processing time and getting stuck on local solutions is
found. Figure 5.3 shows the best, worst and mean solution of a population with varying mutation scale
parameter. 5% is clearly the best value, although an increase in computational time is seen in Figure
5.4.
Another mutation parameter is the mutation shrinkage, set to zero until now. As the name suggests,
decreases the mutation percentage from generation to generation. This allows for rapid convergence,
but also getting stuck on local minimums if too high, so the value of this parameter must be well thought.
It varies between 0 and 1, but common values suggested by the manual are between 0.5 to 1, so 0.5 is
chosen.
Pow
er
* V
olu
me [W
.mm
3]
Figure 5.3 Varying mutation scale on a 1000 individuals’ population after 150 generations
30
Tim
e [m
inute
s]
Figure 5.4 Mutation scale versus computational time
31
6. Results
As stated in Chapter 5, the goal is to optimize an actuator that can damp and stop a maximum load
of 110 N. 500 generations are used as the stopping criteria for the NSGA algorithm, since there’s no
other stopping criteria implemented.
Figure 6.1 presents the first optimization with all the parameters set. A Pareto Front formed by all
the individuals (solutions) of the population after the 500 generations is obtained. During this process
the solutions converge to the origin, because the objective functions are being minimized, creating a
curve like the one represented.
The best solution for this Pareto Front is given by the minimum product between objective functions.
Figure 6.2 shows a schematic of the best solution size and its respective measures and characteristics
alongside. Proved convergence of the best Pareto Front solution’s volume, power and product of these
two, in every generation is showed in Figure 6.3, 6.4 and 6.5, respectively.
A static simulation of the optimized actuator is presented in Figure 6.6.
Volume 104 [mm3]
0 1 2 3 4 5 6 7 8 90
0.2
0.4
0.6
0.8
1
1.2
500 Generations1000 Individuals
Least Volume
Least Power
Best Solution
Figure 6.1 Last Pareto Front
32
13.86
6.08
2.66
0.54
8.29
1.19
0.59
0.56
1.06
162
110.32
98.85
5.93 x 103
Figure 6.2 Best optimized actuator for 110 N after 500 generations
Generations0 50 100 150 200 250 300 350 400 450 500
0
1
2
3
4
5
Figure 6.3 Volume convergence
33
Figure 6.4 Power convergence
Generations0 50 100 150 200 250 300 350 400 450 500
0
5
10
15
20
Figure 6.5 Power volume product convergence
Pow
er
[W]
34
Current [A]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0
50
100
150
Figure 6.6 Static simulation of the optimized actuator
Figure 6.7 and 6.8 present the best volume solution and the best power solution from the last
generated Pareto Front, respectively.
12.81
6.06
2.59
0.53
7.21
1.12
1.07
1.00
1.08
96
110.60
79.12
4.73 x 103
Figure 6.7 Best Pareto Front volume solution after 500 generations
35
24.02
8.88
6.91
1.46
29.04
3.20
0.09
0.01
1.93
2021
117.54
99.10
7.51 x 105
Figure 6.8 Best Pareto Front power solution after 500 generations
Optionally to a single actuator, two or three actuators with less damping capacity can be set to work
in parallel, decreasing the volume and actuation power. To test this hypothesis, Figure 6.9 and 6.10
present the best volume solution of an optimized actuator with a maximum strength of 55 N and another
of 40 N, respectively.
12.02
4.78
1.33
0.39
3.41
0.57
1.05
1.70
0.62
36
58.92
79.59
1.81 x 103
Figure 6.9 Best optimized actuator for 55 N after 500 generations
36
8.54
3.88
1.61
0.33
4.33
0.88
0.54
1.60
0.34
30
40.03
59.56
1.30 x 103
Figure 6.10 Best optimized actuator for 40 N after 500 generations
Changing coil wire diameter can also help in reaching smaller volumes and low power actuators.
Previous optimizations have an AWG 28 copper wire diameter (0.321 mm). In Figure 6.11 the best
volume solution for an optimized actuator with maximum strength of 55 N and an AWG 18 copper wire
diameter (1.024 mm) is presented. Also, an AWG 40 copper wire diameter (0.0799 mm) optimized
actuator is presented in Figure 6.12.
9.07
5.24
1.46
0.36
3.85
0.79
2.05
0.20
10.42
374
55.85
48.24
1.26 x 103
Figure 6.11 Best optimized actuator for 55 N with AWG 40 after 500 generations
37
13.36
4.71
1.73
0.53
7.73
0.79
0.20
2.93
0.07
24
55.08
96.53
5.23 x 103
Figure 6.12 Best optimized actuator for 55 N with AWG 18 after 500 generations
38
39
7. Discussion
Results prove the methodology used and how it’s possible to optimize similar linear MRF actuators.
A linear MRF cylindric actuator that can damp a maximum force of 110 N was initially obtained. In Figure
6.2 was presented the new 110 N actuator which can be compared with the initial 250 N actuator (Figure
4.4). Unfortunately, the optimized actuator was still too large in diameter for a leg orthosis application.
So, other solutions were searched and presented.
One first solution was looking in the last Pareto Front population for an individual with the smallest
volume, increasing in return, the actuation power. Pareto Front’s present a set of solutions, instead of
only a single solution, this allows the decision maker to choose what is best for the application in hand.
So, actuator in Figure 6.7 with a total diameter of 28,92 mm and a maximum current of 1 A, is the
smallest actuator of the last population, but it is still too big for the needed application.
Another possible solution was dividing the total strength in parallel in two or three actuators. An
actuator with 55 N of damping force was optimized and the best volume solution presented in Figure
6.9. This actuator has a total diameter of 25.96 mm and a maximum current input of 1.7 A. The same
was made for a 40 N actuator (Figure 6.10), with a total diameter of 19.5 mm reached and a maximum
current of 1.6 A. These solutions approach the plausible maximum diameter size of 25 mm, although
the power increase. Because the difference from two actuators in parallel to three wasn’t substantial,
the two parallel actuators are taken as the better solution.
Finally, the smallest wire possible (AWG 40) for two actuators in parallel was used, expecting a high
decrease in volume. From this optimization (Figure 6.12) a total actuator diameter of 20.44 mm and a
maximum current of 0.2 A was obtained. In this case, although current and diameter values decreased,
the required voltage increased.
Two actuators solution with AWG 28 coil wire is thus pointed out for future improvements,
recognizing that a set of new research lines are specified in the end of this thesis.
7.1. Other remarks
From the optimization point of view and as foreseen, it is important to notice that with decreasing
volume the actuator input power increases significantly. The inverse is also verified and decisions
variables boundaries are reached, like the lowest permittable current for example (Figure 6.8).
Also, important to mention that steel sectional areas stay practically equal to each other’s, which
proves the optimizer is working well avoiding a section magnetic saturation prior to other sections.
Two main constraints control the optimization process. Equation 5.10 forces the optimizer to close
the gap and in contrast Equation 5.11 tries to open it. This is because not only the actuator must work
with acting current, but it must also let the piston move freely without much friction when no current is
applied.
40
7.2. Battery usage
The goal of this application is a mobile orthosis, so the battery usage must be thought of. From a
simple market study, lithium ion batteries for cellphones are the most used for mobile applications. This
type of battery has around 3.6 V and 3 Ah of capacity each. Other type of batteries have less capacity
and occupy much more space in comparison, to give the same energy.
On average a person walks 7000 steps a day, equivalent to approximately 6 kilometers [24]. Each
step takes about 0.06 seconds during CP (Figure 2.1). For simplification, it’s considered that the actuator
is on maximum current throughout the CP state, which is around 5.1% of the step duration. Considering
70% of battery efficiency due to external factors that may affect the battery performance, the total
number of steps until battery is totally drained, is given by:
=
×× ×
=3×3600
2×1,7×0,06×0.7 = 51 941 ~ 1
It is proven that two actuators can work a full week before the battery has to be recharged again,
by this simple battery life calculation.
7.3. Improvements
An improvement is the addiction of a non-magnetic material in the tip of the piston’s flange to form
a cork like behavior when the magnetic field acts on the MR fluid. Figure 7.1 presents the schematic for
two options of tips. This improvement is expected to increase greatly the damping force of the respective
actuator, because the MR fluid is forced to go through a gradually smaller gap where the next set of
aligned ferromagnetic particles are even more connected than the ones before. The tip must be of non-
magnetic material because otherwise the magnetic flux Φ is strangulated and the fluid loses
connection strength between particles.
Figure 7.1 Schematic of piston’s flange future improvement. a) Round tip and b) Ramp tip
41
8. Conclusions
This thesis proves that a linear MRF actuator model can be structurally optimized to work as an
orthosis for footdrop pathology patients. The best solution was concluded to be two actuators working
in parallel, having approximately 25 mm of exterior diameter and with a AWG 28 coil wire. One cellphone
lithium ion battery per actuator should be enough to power up the actuators for a full work week.
A validated static model of the actuator has been built and used successfully in the optimization
algorithm. The dynamic model although built and implemented, presents incoherent results when
compared to real experimental data. Future work should focus on correcting the dynamic model by firstly
modelling the actuator without actuation. Both dynamic and static models were created based on a
magnetic and a hydrodynamic air-gap analysis of the actuator.
Constructing and comparing the optimized actuator real data to the model should be the second
priority in future work, for both static and dynamic models. Afterwards, a non-magnetic tip like the ones
presented in the previous chapters should be added to close reinforce the gap strength and test the
desired cork behavior in the MR fluid, repeating the same tests. Other future improvements should be
building a model for low velocities studies, improving actual model robustness.
The best individuals of the Pareto Front after some generations have minimum volume and power,
but as seen the user may want to improve the volume increasing the power of the actuator. In these
cases, because the copper wire diameter is fixed, the temperature increases in the actuator due to Joule
effect and may influence the decision maker to choose bad solutions because the optimizer doesn’t
account for heat generation. Two improvement options arise, or a temperature model is built or the
copper wire diameter is set in the optimization as a discrete decision variable. Both can also work
together instead of implementing just one of the options.
42
43
9. References
1. Winter, D.A., Biomechanics & Motor Control of Human Gait. 2nd ed. 1991. 2. Winter, D.A., Biomechanics and Motor Control of Human Movement. 3rd ed. 2004. 3. Domingues, A.R.C. and J.M.M.s. Martins, On the development of an active soft second-skin
orthotic for dropfoot patients. 2014, UTL. Instituto Superior Técnico: Lisboa. 4. Winslow, W.M., Induced Fibration of Suspensions. Journal of Applied Physics, 1949: p. 1137-
1140. 5. Carlson, J.D., D.M. Catanzarite, and K.A. St. Clair, Commercial Magneto-Rheological Fuid
Devices. International Journal of Modern Physics B, 1996: p. 2857-2865. 6. Rabinow, J., The magnetic fluid clutch. Journal of Electrical Engineering, 1948: p. 1167-1167. 7. Shulman, Z.P., V.I. Kordonsky, and S.A. Demchuk, The mechanism of heat transfer in
magnetorheological systems. International Journal of Heat and Mass Transfer, 1979: p. 389-394.
8. Jacob, R., Magnetic fluid torque and force transmitting device. 1951, Google Patents. 9. Dyke, S.J., et al., Modeling and control of magnetorheological dampers for seismic response
reduction. Smart Materials and Structures, 1996: p. 565. 10. Poynor, J.C., Innovative designs for magneto-rheological dampers. 2001, Virginia Tech. 11. Kordonsky, W.I., Magnetorheological effect as a base of new devices and technologies. Journal
of Magnetism and Magnetic Materials, 1993: p. 395-398. 12. Lin, S. NGPM – a NSGA-II program in Matlab v1.4. 2011; Available from:
www.mathworks.com/matlabcentral/fileexchange/31166-ngpm-a-nsga-ii-programin-matlab-v1-4.
13. Deb, K. and J. Sundar, Reference point based multi-objective optimization using evolutionary algorithms, in Proceedings of the 8th annual conference on Genetic and evolutionary computation. 2006, ACM: Seattle, Washington, USA. p. 635-642.
14. Srinivas, N. and K. Deb, Muiltiobjective optimization using nondominated sorting in genetic algorithms. Evol. Comput., 1994: p. 221-248.
15. Xu, C., S. Lin, and Y. Yang, Optimal design of viscoelastic damping structures using layerwise finite element analysis and multi-objective genetic algorithm. Computers & Structures, 2015: p. 1-8.
16. Costa, E. and P.J.C. Branco, Continuum electromechanics of a magnetorheological damper including the friction force effects between the MR fluid and device walls: Analytical modelling and experimental validation. Sensors and Actuators A: Physical, 2009: p. 82-88.
17. Jardineiro, V.L.D., Projecto de Actuador Electromagnético para o Ajuste de Caudal em Contadores de Água com regime de Pré-Pagamento utilizando a norma STS, in Departamento de Engenharia Electrotécnica e de Computadores. 2015, Instituto Superior Técnico. p. 93.
18. LORD. Lord Corporation. 2017; Available from: www.lord.com/. 19. MagWeb. World's Largest Database Soft Magnetic Materials. [cited 2017; Available from:
magweb.us/free-bh-curves/. 20. George P. Gogue & Joseph J. Stupak, J. Theory & Practice of Electromagnetic Design of DC
Motors & Actuators. Available from: www.consult-g2.com/course/chapter3/chapter.html. 21. Geyer, H. and H. Herr, A Muscle-Reflex Model That Encodes Principles of Legged Mechanics
Produces Human Walking Dynamics and Muscle Activities. IEEE Transactions on Neural Systems and Rehabilitation Engineering, 2010: p. 263-273.
22. Ferreira, F., Development of a human walking model comprising springs and positive force feedback to generate stable gait, in Dep. of Mechanical Engineering. 2013, IST, Lisbon, Portugal.
23. Costa, E. and P. Branco, Construção de um dispositivo amortecedor magnetoreológico para uma suspensão activa. 2008, UTL, Instituto Superior Técnico: Lisboa.
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25. Appendix B, Navier-Stokes Equations, in Chemically Reacting Flow: Theory & Practice. 2003, John Wiley & Sons, Inc.,. p. 763-773.
44
45
Annex A
Navier-Stokes equations in cylindrical coordinates, constant viscosity [25]
z component
+
+
+
= −
+ + ( + )
(.)
r component
+
+
+
−
= −
+ −
−2
+ ( + )
(.)
component
+
+
+
+ = −
1
+ −
−2
+ ( + )
1
(.)
