Chapter 1 Introduction
1.4 Thesis Organization
There are totally seven chapters in this thesis. It starts with an introductory chapter which motivates this research and introduces the state-of-the-art research results in precision positioning devices. In Chapter 2 , we review some basic theories of electromagnetism, properties of the permanent magnet, the analysis of the energy method, and the introduction on flexure mechanism. The following chapter, Chapter 3 , describes the design concept for fulfilling the desired motion behavior through description of the detailed specifications of various components in the novel 3-DOF positioning stage. Next, the force allocation, the sensing methodology, the mathematic model, the dynamic behavior and the system identification of the 3-DOF positioning stage will be analyzed and derived in Chapter 4 . Then, proper controller design, which based on adaptive sliding-mode control technique, and the numerical simulations are
conducted in 0. Subsequently in order to validate the effectiveness and appealing performance of the design, extensive experimental results are provided in Chapter 6 . Finally, we make some conclusions to sum up the results in this thesis in Chapter 7 .
Chapter 2 Preliminary
Equation Chapter 2 Section 1
2.1 Basic Theories of Electromagnetic
In this section, we will briefly review the basic theories of electromagnetic that will be used in our system design and analysis, specifically concerning Lorentz force principle and eddy current phenomenon.
2.1.1 Lorentz force principle
The Lorentz force equation is the basis for governing all magnetic forces. Magnetic fields are a description of the relativistic effects that occur among moving charges, which are a direct result of the Lorentz transformation of the Coulomb force.
The force on a current element immersed in a magnetic field B is given as:
dF Idl
= ×
dF I Bdl (2.1)
Note that cannot exist by itself as it must be part of a complete loop or circuit. On such a loop, the total summed force is
Idl
B
=
=
∫v
×F Idl (2.2)
As shown in Fig. 2-1, in order to simplify (2.2), a segment with length L of a long straight wire is assumed to be exposed to a uniform magnetic field B that is perpendicular to the wire, and the return path of the wire is to be outside the field. Then, this integral can be expressed as a scalar solution
F ILB (2.3)
where I is the current carried on the conduction wires, L is the length of the conduction wires through the magnetic field, and B is the external magnetic flux density. If there are N-turn wires through the magnetic field, then
=
F NILB (2.4)
It is important to note that the force on the conductor is given only by (2.4) if the field due to the current I can be neglected.
Fig. 2-1 Lorentz force principle
The electromagnetic actuator designed based on Lorentz force principle are often applied when high bandwidth dynamic are to be achieved. Examples are voice coil actuators, loudspeakers, synchronous brushless DC motors, and so on.
2.1.2 Eddy current phenomenon
When a non-ferromagnetic conductor moves in a magnetic field, or a moving and varying magnetic field intersects a non-ferromagnetic conductor, the relative motion causes a circulating current within the conductor. Figure 2-2 shows that when a conductor is moving in the magnetic field B with the velocity v, the eddy current i is generated within the conductor. The interaction between the current and the magnetic field will generate a force to resist the relative motion between the magnetic field and the conductor. Because of the resistance of the conductor, the eddy current will dissipate and energy of the system will be transformed into heat.
N B S
L
F
wire
I
Fig. 2-2 The eddy current phenomenon
For the researches on, say, electric motors, the eddy current phenomenon will reduce the efficiency and should be avoided. However, it can be used to remove the excessive energy from the system to physical contact. There are lots of practical applications of eddy current devices [18], including electromagnetic braking, magnetic damping, and passive vibration control.
Bae et al. [19] proposed the design and modeling of a passive eddy current damper shown in Fig. 2-3. The eddy current damper (ECD) consists of a copper plate and a pair of magnets. The experimental results show that the vibration of the cantilever beam has successfully been suppressed by the additional ECD. The magnitude of the damping force is also described in [19] as:
2
Fd = −σδvB Sα (2.5)
where σ is the thickness of the conductor in the magnetic field, δ is the conductivity, v
is the velocity, B is the magnetic flux density, S is the effective area, and α is the dimension parameter.
Fig. 2-3 Eddy current damper to suppress vibration [19]
2.2 Properties of Permanent Magnet
Since we use the electromagnetic actuators, the magnet characteristics cannot be neglected. In this section, we will review briefly some basic properties of permanent magnet (PM), and then present the detailed data of the used magnets, such as maximum energy product, coercive force, and temperature coefficient, etc.
According to Gauss Law, the magnetic flux continuity law can be described as follows:
0
is applied to a ferromagnetic substance, the material will be magnetized with the internal flux density BG
given by
is the induced magnetization density, defined as the magnetic dipole moment per unit volume, and N is the number of dipoles per unit volume. By (2.7) and (2.8), we can obtain the B-H curve by varying the field HG
and measuring the flux densityBG
. Figure 2-4 is a typical B-H curve of a ferromagnetic material.
P
Fig. 2-4 B-H curve of a typical ferromagnetic material
From Fig. 2-4, the curve OP is the initial magnetizing curve, and point P is the saturation point that means the material reaches its maximum magnetization. Once we vary HG
from positive value to negative value and then back to positive value again,
the B-H curve forms a loop called hysteresis. The intersection of the loop and the BG axis is known as the remanence, residual magnetization, or residual flux density, denoted as BGr
, which is the magnetic flux density inside the magnet when the external field HG
is reduced to zero. Moreover, HGc
is known as the coercivity or coercive force which is the external field needed to completely demagnetize the substance.
By (2.6), when external HG
is removed, the residual flux density inside the ferromagnetic material is
BGr MG
(2.9)
=
MG which indicates that the material has become a PM with residual flux density . Then, the magnetization of this PM can be expressed as:
MG BGr
(2.10)
=
G =
Therefore, the dipole moment resulting from the definition of magnetization is then given as:
rV m BG
(2.11)
where V is the volume of this PM.
r
In the following, comparisons among different magnets with their specific properties shown in Table 2-1 will be given. Among the listed magnet materials, Ferrite, also known as ceramic magnet, provides the lowest maximum energy product
and the lowest residual induction
BHmax
BG
. Ceramic materials are hard and brittle and are extensively used in consumer products, and on the back of popular refrigerator magnets.
Rare-earth elements are the most popular materials used to produce the strong magnets. One of the strong magnets is Samarium Cobalt, which has highBGr
, highHGc
, relatively high maximum energy product ( ), and also higher cost than NdFeB.
Commonly, its energy product ranges from 18 BHmax
MGOe (Mega Gauss Oersteds) to about 32 MGOe . The most familiar one of the strong magnets is NdFeB or, for more accurately, sintered NdFeB magnet, whose property is similar to that of SmCo but which belongs to the most powerful class and is commercially available today. Its energy product ranges from 2.8 MGOe to about 48 MGOe . Therefore, NdFeB magnet is the most reliable choice to provide high magnetic force in our system.
Table 2-1 Characteristics of Magnet
Property Unit
AlNiCo Ferrite Rare Earth
AlNiCo
(Data from Spin Technology Corp. in Taiwan.)
We can conclude that NdFeB material is the best choice at present, since AlNiCo has low coercivity, Ferrite has low remanence, and Samarium Cobalt magnets are still expensive. Table 2-2 indicates several characteristics of the NdFeB magnets.
Table 2-2 Specifications of NdFeB
Specifications NdFeB
Remanence (T) 1.29
Coercivity (kA/m) 990
Maximum energy product (kJ/m3) 320
Density (g/cm3) 7.49
Curie temperature (0C) 310
Resistivity (μΩm) 6
2.3 Basic Theories of Energy Methods
In this section, we will introduce how to apply energy method to solve problems involving deflection. Then, Castigliano’s theorem is an important theory to be referred to, which is used here to determine the stiffness of the flexure mechanism. For more details, readers are suggested to refer to the work [17].
2.3.1 External work and strain energy
First, we will define the work caused by an external force and a couple moment.
Work of a force
In mechanics, a force does work when it undergoes a displacement dx that is in the direction the same as that of the force. The work done is a scalar, defined as . If the total displacement is x, the work becomes
dUe =Fdx
x
0 x
Ue =
∫
Fd (2.12)Work of a couple moment
A couple moment M does work when it undergoes a rotational displacement dθ along its direction of action. The work done is defined as dUe=Mdθ . If the total angle of rotational displacement is θ rad., the work becomes
0
Ue M d
θ θ
=
∫
(2.13)When loads are applied to a body, they will deform the material. Provided no energy is lost in the form of heat, the external work done by the loads will be converted into internal work called strain energy. This energy, which is always positive, is stored in the body and is caused by the action of either normal or shear stress.
Normal Stress
In general, if the body is subjected only to a uni-axial normal stress σ, acting in a
specified direction, the strain energy in the body is
i 2
V
U = σε dV
∫
(2.14)Also, if the material behaves in a linear-elastic manner, Hooke’s law suggests σ =Eε , whereby we can express the strain energy in terms of the normal stress as
where E is the Young’s module.
2.3.2 Strain energy for bending moment
Since a bending moment applied to a straight prismatic member develops normal stress in the member, we can use (2.15) to determine the strain energy stored in the member due to bending. Considering a bending applied to the axis-symmetric beam as shown in Fig. 2-5, the internal moment here is M, and hence the normal stress acting on the arbitrary element at a distance y from the neutral axis is σ =My I/ . If the volume of the element is , where dA is the area of its exposed face and dx is its length, the elastic strain energy in the beam is
dV =dAdx
The integral over the volume can be expressed as the product of an integral over the beam’s cross-sectional area A and an integral over its length L. Thus,
2
Realizing that the area integral represents the moment of inertia I of the beam about the neutral axis, the final result can be re-expressed as:
2
Fig. 2-5 Axis-symmetric beam
2.3.3 Castigiano’s theorem
The internal strain energy for a beam is caused by both bending and shear.
However, if the beam is long and slender, the strain energy due to shear can be neglected compared with that of bending. Assuming this to be the case, the internal strain energy for a beam is given by (2.18). Substituting it into δ = ∂U/∂ , we can get: P
2
Where the variables used are defined below:
δ = displacement of the point caused by the real loads acting on the beam,
P = external force of variable magnitude applied to the beam in the direction of δ, M = internal moment in the beam, expressed as a function of x and is caused by both the force P and the loads on the beam,
E = modulus of elasticity of the material,
I = moment of inertia of cross-sectional area computed about the neutral axis.
If the slope at a point on the elastic curve is to be determined the partial derivative of the internal moment M with respect to the external couple moment M’ acting at the point must be found. For this case,
0L ( M dx)
M M EI
θ= ∂
∂ ′
∫
(2.20)2.4 Flexure Mechanism
The most important advantage of flexure mechanism is frictionless and stictionless which relies on the elastic deformation of material. Sliding and rolling effects are completely eliminated in the devices using flexure mechanism. Flexures have been used
(1) as bearings to provide smooth and guided motion, when in precision motion stages;
(2) as springs to provide preload, when in the brushes of a DC motor or a camera lens cap; (3) to avoid over-constraint, as in the case of bellows or helical coupling; (4) as clamping devices, for example, the collets of a lathe; (5) for elastic averaging as in a windshield wiper; and (6) for energy storage such as, in a bow or a catapult. Above all, it encompasses applications with regard to the transmission of force, displacement as well as energy.
In our work, flexure mechanism is used as the suspension of precision positioning stages. The motion is generated due to molecular level deformation, which results in two primary characteristics of flexures – smooth motion and small range of motion, and the phenomena of friction, stiction and backlash are completely eliminated. On the other hand, flexure mechanisms allow for very clean and precise motion.
FGx
FGx
FGx
FGx
FGy
FGx
Fig. 2-6 Common flexure types: (a) simple cantilever, (b) clamped-clamped, (c) crab-leg, (d) folded-flexure, (e) serpentine
There are several kinds of flexure type such as simple cantilever beam, clamped-clamped flexure, crab-leg flexure, folded flexure, and serpentine flexure as shown in Fig. 2-6. These kinds of flexures generate one DOF motion along the direction of force. In the following, we will give some analyses of characteristics of these flexure mechanisms.
Consider the simple cantilever beam in Fig. 2-7 with a rectangular cross section, and let it be subjected to a load P at its end. Now, we want to determine the displacement of the load.
Fig. 2-7 Cantilevered beam and its free body diagram
It is assumed that we have known the material characteristics, and EI is constant.
Here, E means the Young’s modulus, and I means the inertia mass. Then, clearly, a small deflection on the elastic beam will result in
2 2
( ) d y M x
dx = EI (2.21)
According to free body diagram, we can get
0
Using (2.21) ~ (2.23), and applying the boundary conditions, we can obtain the deflection curve as
so that the tip deflection can be easily derived as 0 3
x 3
y P L
EI k
=
− F
= − . As a result, the
spring constant k can be obtained as
3
k 3EI
= L (2.25)
After discussing the simplest cantilevered beam, now we will discuss the quad-symmetric clamped-clamped flexure mechanism. Due to its symmetric structure, this mechanism only generates one DOF motion. As shown in Fig. 2-8, we can model the structure as four guided-end beams.
Fig. 2-8 Quad-symmetric clamped-clamped flexure and its free body diagram
According to the free body diagram, an external force Fx and a bending moment M0 can be applied to a body to find the displacement δx and the corresponding spring constant kx. Therefore, the bending moment of the beam is found to be
0 x
Now, rewrite the Castigliano’s Theorem, mentioned in (2.19) and (2.20), as follows:
2
which together with the constraint θ = and (2.26), will lead to the following 0 relation:
Again, through use of Castigliano’s Theorem, (2.28), and its substitution into (2.30), we finally derive the displacement as:
0
As a consequence, the spring constant of the beam and of the quad-symmetric clamped-clamped flexure can readily be obtained as:
, 3
Besides the above two kinds, we will analyze another type of flexure, called quad-symmetric crab-leg flexure mechanism, as shown in Fig. 2-9. The same as the clamped-clamped flexure, this mechanism also only generates one DOF motion. It can be modeled as four crab-leg flexure, and we divide every crab-leg into two parts, thigh (beam a) and shin (beam b).
Fig. 2-9 Quad-symmetric crab-leg flexure and its free body diagram
For the purpose of analysis, likewise we apply Fx , Fy , and M0 at the end of the thigh. According to the free body diagram, the bending moments of the thigh (Ma) and shin (Mb) can be respectively expressed as:
0
a y
M =M −F
b b
x (2.33)
0 x ( 0 y a) x
M =M −Fξ = M −F L −Fξ (2.34)
We now use Castigliano’s Theorem, (2.28), and apply the boundary conditions (θ0 = ,0
y 0
δ = ) to find M0 and Fy as:
0
Now, we use Castigliano’s Theorem again to derive δ as:
0 0
After substituting (2.36) and (2.38) into (2.39), we readily have
2 2
from which the spring constant of the quad-symmetric crab-leg flexure can be readily derived as:
2.5 Measurement Error
There is no perfect measurement system in the real world. The error comes from imperfect sensor assembly and alignment, measurement methodology, and signal noise.
This section would discuss two common errors, namely, Abbe error and cosine error.
The compensation of the two errors will be described in the following section so that an
accurate and reliable measurement methodology can be properly derived.
2.5.1 Abbe principle and Abbe error
When the axis to be measured and the axis of measurement are not coaxial, a measurement error will occur due to the offset between two axes. In late 1800s, Dr.
Ernest Abbe investigated the issue and proposed this principle: the measuring system should be placed coaxially with the axis which is to be measured. Figure 2-10 shows the Abbe error, which can be estimated as:
Abbe tan
e =D θ (2.42)
where D is the offset between the two axes.
Fig. 2-10 Abbe error
2.5.2 Cosine error
Cosine error occurs when the axis of measurement and axis which is to measured are not completely parallel. In Fig. 2-11, the included angel θ causes that the measured displacement x’ is different from the real displacement x. The cosine error hence results from inadequate alignment between the motion stage and the sensor.
Fig. 2-11 Cosine error
From the above figure, the relationship between the measured displacement x’ and the real displacement x can be expressed as:
' 1
x sin x
= θ (2.43)
Then, the cosine error can be estimated as:
cos
(
' 1
e x x sin x x x θ
= − = θ − = sec −1 (2.44)
)
Chapter 3
Mechatronic Design
Equation Chapter (Next) Section 1
The aiming target of our research is a positioning stage with high positioning accuracy, large moving range with multiple DOFs. To realize these properties, we adopt the planar parallel flexure mechanism as the suspension of the moving stage due to its frictionless effect, electromagnetic actuator for its low cost, and appropriate arrangement of measurement system in order to precisely measure the 3 DOF displacements of the three degrees of freedom of the designed system.
The related researches and needed background knowledge have been reviewed and introduced in the previous chapter. In this chapter, the design concept of the proposed positioning stage will be introduced, including the flexure suspension mechanism, electromagnetic actuator and damper, measuring system, and the integration of all the components.
3.1 Design Strategies
Now, we list all the design objectives that we want to accomplish as follows:
1. high positioning accuracy,
2. long planar stroke, 3. fast positioning, 4. compact system.
The following subsections will translate these goals into actuator level requirements. Since most objectives are strongly coupled, we are not able to consider respective design separately.
3.1.1 High positioning accuracy
To attain high positioning accuracy, either the system needs a high disturbance rejection, or the external noise sources need to be shielded off. Moreover, the bits-resolution of AD/DA cards and the resolution of sensor are also some major factors to be concerned. Therefore, instead of investing on installation of expensive equipments, we set our design goal on how to utilize commonly available sensors and AD/DA cards on how to optimize the integrated performance up to respective performance limits of individual components.
To reject the large disturbance and obtain a high bandwidth, performance of the actively controlled system is usually limited by the controllability, linearity, and response time of the actuator and controller.
3.1.2 Long planar stroke
A larger positioning range within the same outer dimensions is advantageous.
Normally, to reduce costs of production, parallel or batch processing is often applied to increase throughput of samples or products. Consequently, those larger specimens
Normally, to reduce costs of production, parallel or batch processing is often applied to increase throughput of samples or products. Consequently, those larger specimens