Chapter 2 Preliminary
2.5 Measurement 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 require larger strokes without the decreasing in speed and acceleration.
In the target application, the traveling range in using piezoelectric actuators is just 100 μm [13] due to the constraint of piezoelectric material’s characteristics. In our system, because of fewer physical constraints, we choose electromagnetic actuator to enlarge the traveling range up to several mm.
3.1.3 Fast positioning
For the same reason mentioned in previous section, to increase the throughput of production, motion of any positioning system should be fast as possible so that it can become commercially attractive. For the transient response specification, the rise time and the settling time must be short. To fulfill such objective, we must have both a strong and high speed actuator and a well-designed mechanism which have a higher bandwidth.
3.1.4 Compact system
The positioning system is usually a part of a larger system, which leads to restrictions on the space available for the motor and bearings. These restrictions are even more severe under the circumstance of an expensive ultra clean or vacuum environment. Generally, the outer diameter of a positioning system will be less when there are more moving parts. The implication of the above is that the total positioning should be as compact as possible.
3.2 Electromagnetic Actuation and Damper
This section discusses the design of electromagnetic actuator and damper which will be utilized in this proposed positioning system. Generally, using a rotary motor to generate the linear reciprocating motion must incorporate the use of the transmission mechanism, such as ball-screws, gears, racks, etc., to change the rotating motion into the linear motion. However, the transmission mechanisms usually have problems of backlash and friction which severely influence the precision of positioning. Therefore, it is better to use electromagnetic actuator, so-called voice coil motor (VCM), due to its frictionless contact and non-backlash nature.
VCM is one kind of linear direct-current motor and is composed of permanent
magnets (PMs) and coils which are assembled appropriately. It can produce the linear thrust proportional to the current which flows through the magnetic field.
Fig. 3-1 The VCM actuator of our previous research [8]
Figure 3-1 shows the VCM actuator which is used in our previous research [8].
Two magnets are placed side by side with opposite polarity. The current-carrying coil would generate the force as shown in the above figure. But the assumption on the magnetic field produced by the magnets is not realistic. The real distribution of the magnetic field is highly non-uniform, so that the actuator suffers from the nonlinearity and the current/force relationship of the actuator will vary with the relative position between the magnets and the coils. This nonlinearity also limits the stroke of the actuator. In this section, we will discuss the design of a stronger linear electromagnetic actuator and the built-in electromagnetic damper.
Coils
3.2.1 Near-Uniform Magnetic Field
Since the nonlinearity of the electromagnetic actuator comes from the non-uniform magnetic field, intuitively a construction of uniform magnetic field will solve the issue.
Figures 3-2 and 3-3 show the concept of constructing a near-uniform field. Two plate permanent magnets are installed inside a steel structure. A near-uniform magnetic field will be generated between the two plate permanent magnets and the outer magnetic flux will be confined within the ferromagnetic steel structure to form a closed loop flux path, as shown in Fig. 3-3. The dimension of the plate magnet is 60mm x 60mm x 2mm and its material is neodymium-iron-boron (NdFeB), the powerful rare earth permanent magnet material.
Fig. 3-2 The steel structure and permanent magnets
Fig. 3-3 The near-uniform magnetic field
To verify the near-uniform property of the constructed magnetic field, a finite element analysis is conducted as shown in Fig. 3-4. The outcome of such analysis does validate our viewpoint and the design.
Fig. 3-4 The numerical simulation of magnetic field
3.2.2 The proposed electromagnetic actuator
Figures 3-5 and 3-6 show the configuration and assembly of the proposed electromagnetic actuators. A copper mounting is manufactured with the shape shown in Fig. 3-5 and is placed in the middle between the upper part and the lower part of the steel structure. Note that the two parts of the steel structure are attracted to each other by magnetic force and are connected through four rectangular holes opened on the copper mounting. The clearance between the steel structure and copper mounting is designed for the motion of the moving stage and is sufficient for the needed traveling range. Four square coils are placed on the copper mounting. When current is fed into the coils, the
interaction between the current and the intersected near-uniform magnetic field would generate the electromagnetic force according to Lorentz force principle.
Fig. 3-5 The assembly of the VCM actuator
Fig. 3-6 The forces that are generated by Lorentz force principle
It is noteworthy that only half of the coil intersects the magnetic field, as shown in Fig. 3-6. The part of the coil intersecting the magnetic field can actually be divided into three sub-parts, which result in three corresponding electromagnetic forces, F1, F2
and F3. Due to symmetry, the magnitudes of F2 and F3 are equal but the directions are
opposite. Since the two forces cancel each other, the effective force generated by the coil is only F1. It is important to see that the magnitude of the effective force of a coil is invariant to the translation of the steel structure because the intersection part of the coil corresponding to F1 is always constant and the varying forces F2 and F3 are always equal but opposite. When the stage rotates, the magnitudes of F2 and F3 are different.
The purpose of the positioning stage is the precision motion in x- and y-axis, and the rotation angle of the stage is always regulated to zero. The difference of the magnitudes of F2 and F3 can be neglected when the rotation angle is small.
3.2.3 Eddy current damper
One should also beware the magnetic field intersects the copper mounting as well.
When the steel structure and the magnetic field move relative to the copper mounting, the eddy current will be generated inside the copper structure. Hence, the interaction results in a damping force which resists the mentioned relative motion.
Fig. 3-7 The eddy current inside the copper mounting.
3.3 3-DOF Flexure Mechanism
Numerous multi-DOF flexure mechanisms have been presented in the literature.
There are two well-known configurations adopted in the design of multi- DOF flexure stages – serial kinematics [20]-[25] and parallel kinematics [26]-[28]. Each configuration has its advantages and disadvantages.
In serial design, multiple DOFs are achieved by stacking multiple single DOF systems, one on another. The technical literature has presented several such designs.
Serial kinematical mechanisms are relatively simple to design and have significantly higher inertia, but their weak points are the resulting center of gravity is relatively higher, and the off-axis errors are harder to be corrected. Besides those, the cables which are connected to every stage are sources of disturbance, which is detrimental for nanoscale positioning. Moreover, the actuators, especially when large range of motion is desired, are bulky and may reduce the motion bandwidth of the axes of DOF.
However, parallel kinematical designs are free of these problems due to ground mounted actuators, and are also usually more compact. On the other hand, they provide smaller ranges of motion and exhibit significant cross-axis coupling. Furthermore, the stiffness of one axis varies with motion or force along the other axis. This affects the
static as well as dynamic performance of the mechanism.
In our work, we use the parallel kinematical XY flexure mechanisms as shown in Fig. 3-8 and Fig. 3-9. The thin flexure mechanism is fabricated by electrical discharging wire cutting (EDWC), where the resulting height is 5mm and the width is only 0.3mm.
Fig. 3-8 Parallel kinematical XY flexure mechanism
Fig. 3-9 The detailed view of the thin flexure
Note that the stiffness of the structure should be proportional to the area moment of inertia I. Recalling the cantilever as shown in Fig. 2-7, we can calculate the area moment of inertia I as:
3
12
I =bh (3.1)
Since the height is greater than the width, the stiffness of our designed structure in vertical direction is much greater than that in the lateral direction. Therefore, we assume that the structure is always rigid in the vertical direction even when it has deformations in the lateral direction.
Figure 3-8 shows that the flexure mechanism is made of two kinds of material: the brown outer frame means the frame is made of copper and the gray blue inner flexure means the material is steel, which is used to construct the near-uniform magnetic field.
The reason why the outer frame cannot be made of the same material as the inner flexure is that if the material of outer frame is also ferromagnetic, the moving stage will be attracted and stuck on the outer frame when it moves too close to the outer frame, as shown in Fig. 3-10. Thus, the outer frame should use non-ferromagnetic material. In our design, we choose copper as the material of the outer frame, and then the inner flexure is inlaid into the outer frame.
Fig. 3-10 The stiction on outer frame
3.4 Measurement System
In order to establish the control system of the 3-DOF positioning system, a 3-axis measurement scheme is proposed in this section. Three displacement sensors are placed coplanar with the moving stage to measure the position and posture of the stage. The model of sensors used in the proposed measurement system is OMRON Z4W series LED displacement sensor. The measurement principle of this sensor is based on triangulation methods, which is shown in Fig. 3-11. First, the light is emitted from the LED and then scatters from the surface of object. The scattering of the light will be focused on a position sensitive diode (PSD) inside the sensor. When the object displaces by dx, the focus spot on PSD will also displace by dy which is proportional to dx.
Therefore, the displacement of the object can be measured from the output signal variation of PSD.
Fig. 3-11 The triangulation measurement method
The measurement metrology is shown in Fig. 3-12. One sensor is used to measure the displacement on the X-axis. On the other hands, two sensors are used to measure both the Y-axis displacement and the rotation angle θ of the positioning stage. Using this metrology, we can obtain the position and posture information of the positioning stage by the displacement information between the sensor and the scattering surface on the moving stage. The transformation from the measured signals to the real position and posture information of the positioning stage will be discussed in Section 4.2.
Fig. 3-12 Perspective view of the measuring system
3.5 Integrated Positioning Stage
In the previous sections in this chapter, we have discussed the design of electromagnetic actuators, damper, and the flexure suspension mechanism. Figure 3-13 shows the exploded view of the integrated positioning stage. The outer frames of upper and lower layer and the copper mounting are all fixed on the base layer, and then the entire positioning stage is fixed on a vibration-isolation table to reject the disturbance from the environment.
Fig. 3-13 The exploded view of the positioning stage
Since the center steel structure is suspended by the two layer flexure mechanism, so it is movable, and the allowed traveling range is 3mm x 3mm which is determined by the clearance between the steel structure and the copper mounting. Since the coils are fixed on the mounting and the steel structure is movable, the reacting forces of the
effective forces which have been mentioned before will act on the movable structure as the propelling forces. With the four forces acting on the movable positioning stage as shown in Fig. 3-14, the sum of the forces can drive the stages to move along x- and y-axis and rotate along θ-axis.
Fig. 3-14 The arrangement of four propelling forces
Note that all the forces, which include the reacting propelling forces and the damping force, are located at the same positions while the movable structure is moving.
On the other hand, the arm between respective force exerting point to the center of mass changes with the displacement of the movable stage. This phenomenon will result the extra torque acting on the stage, but it can be overcome by careful force distribution.
Chapter 4
Modeling and System Identification
Equation Chapter (Next) Section 1
4.1 Force Allocation
Fy
Fig. 4-1 The allocation of the forces
Referring to the end of last section, the arm of each force exertion to the center of mass changes with the displacement of the movable stage. As a result, an adequate force allocation strategy should be investigated to compensate the unexpected excessive torque. Figure 4-1 shows the force allocation. Let the arms of the four electromagnetic forces be the same when the movable stage is located at the equilibrium. Consider the situation that the movable stage has displacements along both x- and y-axis, denote as x and y. The arms of the four forces are thus changed and the equation of the force
d
summation and the associated torque can be expressed as:
which can then be re-arranged into the following matrix form:
,1
A force allocation relationship should be established to distribute the control efforts to each electromagnetic actuator. From (4.2), the three control efforts are known and the four forces are unknown. This is an underdetermined linear system problem and the solution is not unique. A general solution can be expressed as:
,1
redundant in (4.3). A carefully chosen q can decouple the forces along the x- and y-axis and distribute current of each actuator more uniformly. A more adequate q is:
0 1
Substituting (4.4) into (4.3), we attain the following force allocation equation:
,1
4.2 Sensing Methodology
Recall the measurement system mentioned in Section 3.4, in order to get the position and posture information of the positioning stage, including the position in the x- and y-axis and the rotating angle, three displacement sensors are used to measure three distinct displacements of the positioning stage. Therefore, we appropriately arrange the locations of the sensors. The proper arrangement of the sensors is illustrated in Fig. 4-2.
Fig. 4-2 The arrangement of sensors
The variables and l d mean the half width of the square moving stage and the s
distance between two sensors in y-axis, respectively. The measured distances of the three sensors are denoted as x y and ˆ ˆ, 1 ˆy . When the moving stage has no 2
displacement and rotation, which is considered as equilibrium state, the constant distances are denoted as x y0, 1,0 and . Therefore, the differences of the distances
can be expressed as:
y2,0
0 ˆ, , 1 1,0 ˆ1 2 2,0 ˆ2
x x x y y y y y y
Δ = − Δ = − Δ = − (4.6)
With the simple geometric relationships, the rotation of the moving stage can be expressed as:
1 1
tan
s
y y
θ = − Δ − Δd 2 (4.7)
In Section 2.5, the error of measurement has been discussed. The compensation of the error should be considered in order to get accurate information about the position and posture of the positioning stage. In Fig. 4-2, even when the position of center of mass is kept at the origin, the measurement signals of sensors will still shift due to the rotation of the stage. The error is similar to cosine error. On the other hands, when there is a displacement of the moving stage in the y-axis as shown in Fig. 4-3, the Abbe error occurs because the motion axis and measurement axis are not in line.
Fig. 4-3 Abbe error in the measurement system
Therefore, after compensation of Abbe error and cosine error terms, the position of the center of mass of the moving stage can be expressed as:
( )
where x and y are coupled in the above equation. To get a decoupled expression of position information, we rearrange some terms in (4.8) as follows:
( )
4.3 Dynamic Formulation
Figure 4-4 shows the free body diagram of the positioning system. The rigid body’s dynamics of this positioning system are three degrees of freedom consisting of the translations in the x- and y-axis, respectively, and the rotation around the z-axis, defined as θ. Through the help of Newton’s Law, the equations of motion can be expressed in the following equations:
1
Fig. 4-4 The free body diagram of the system
where m is the mass of the moving stage, is the moment of inertia, J kx y/ /θare the spring constants, bx y/ /θ are the damping constants, F Fx, ,y and τ are the control efforts which are mentioned in Section 4.1. Notice that the damping forces and are invariant to the motion of the mover. However, when the center of mass of
where m is the mass of the moving stage, is the moment of inertia, J kx y/ /θare the spring constants, bx y/ /θ are the damping constants, F Fx, ,y and τ are the control efforts which are mentioned in Section 4.1. Notice that the damping forces and are invariant to the motion of the mover. However, when the center of mass of