In this paper, a novel parallel leaf-spring carrying mechanism (PLCM)
is investigated using a compliance-matrix-based approach.
For the analytical modeling and attitude calculation, the geometric errors of the
flexible arm, including the height and the top plane's direction, are considered, and the
displacement method is used to calculate the equilibrium attitude. The
influence of the equilibrium attitude at different heights and the initial tilts of the
top planes are analyzed separately. The validity and effectiveness of the
attitude calculation are illustrated by experimental verification. The laser
triangulation coordinate method is used for attitude measurement. The
deviations of the normal vector between the calculation results and measurement
results are smaller than 2×10-4, which is small enough to satisfy
practical requirements. This can be used to guarantee stable and accurate
wafer transfer in a lithography machine. Therefore, it can be concluded
that the methods employed for analytical model establishment and attitude
calculation can be used as a reference for the analysis and design of a
complex parallel compliant mechanism.
Introduction
A compliant mechanism (CM) is a new type of mechanism that transfers or
transforms force, motion, or energy via the deformation of flexible members. CMs
can reduce the number of components, the assembly time, and maintenance requirements;
simplify the manufacturing process; and improve both precision and reliability.
Industrial examples of precision manipulation are wafer positioning and
transfer in a lithography machine as well as posture adjustment in remote center
compliance (RCC) and microsurgery. The principle of exact constraint design
and kinematic design is often applied to obtain a deterministic behavior
(Yuanqiang and Wangyu, 2014; Smith, 2017). Leaf springs are used in distributed CMs, and
their deflection is not concentrated in a small local region. Thus, higher
stress and a wider range of motion are allowed. However, parasitic error
deterioration in stiffness performance is observed as the range of motion
increases (Smith, 2000).
Parallel leaf springs (PLSs) provide an approximate straight motion over
relatively short strokes. As shown in Fig. 1, PLSs have 1 compliant
degree of freedom (DOF) in the drive direction y. In the other two translational
directions, x and z, and in the rotational directions, Rx, Ry, and Rz, the support
stiffness are several orders of magnitude higher than that in the drive
direction. The material is assumed to be linear elastic in this paper. Since the pioneering work of
Bernoulli and Euler, many related studies have been published for a single
beam (e.g., Timoshenko, 1922; Awtar and Sen, 2010;
Meijaard, 1996). Much work has also been done by Awtar et al. (2007) and Howell (2001) on parallel leaf springs.
Parallel leaf-spring structure.
A parallel leaf-spring carrying mechanism (PLCM) is considered to be a parallel
mechanism. Investigation of the compliance and stiffness of parallel
mechanisms can be dated back to the study of elastically suspended robotic
systems (Patterson and Lipkin, 1993). More recently, a vibratory bowl feeder was modeled as a
parallel mechanism with leaf-spring compliance legs (Dai and Ding, 2006; Ding
and Dai, 2008), and a compliance device was built using parallel slender beams in
remote-center compliance (Ciblak and Lipkin, 2003). Other methods of modeling
compliant mechanisms include finite-element-based approaches
(Pashkevich et al., 2009; Klimchik et al., 2013), and these approaches have been used to model the
complex shapes of elastic limbs, which is usually straightforward and
computationally affordable. In terms of the simple shape of limbs, such
as slender beams and blades, analytical models like the Euler–Bernoulli model or the
Timoshenko model are computationally efficient and can be used to reveal the
intrinsic characteristics of a compliant mechanism.
The position and orientation accuracy of a compliant carrying mechanism is
an important performance index. The geometric error of a rigid connector
influences the terminal trajectory and equilibrium pose. Much work has been done
on the influence of uncontrollable factors, such as the geometric error,
the clearance, and the assembly error. For example, Meijaard et al. (2010) analyzed the consequences of static and dynamic misalignments in
a parallel leaf-spring mechanism; Luo et al. (2015) analyzed the
influence of parasitic displacement and connector deflection on lumped and
distributed compliant parallel-guiding mechanisms; Ding and Dai (2008) considered the mass and the hysteresis damping of flexible members and
proposed a complete model for a vibratory bowl feeder; and Ropponen and Arai (1995) considered the hinge position error, the driving error,
and the clearance of a kinematic joint and proposed an attitude calculation
model for a Stewart platform.
On the basis of predecessors' work, a new problem has arisen: the geometric error
of a flexible arm, including the height and the top plane's direction in the parallel
compliant mechanism, causes the attitude of the carrier (wafer) to change after
handover. This issue requires the innovative use of the compliance matrix method combined with
spatial balance and geometric constraints to establish an attitude
calculation model in order to achieve an accurate solution for the spatial flexible
support wafer attitude change in the parallel compliance mechanism.
Therefore, the analytical model of a novel parallel leaf-spring carrying
mechanism is investigated in this paper. Moreover, the geometric errors of
a flexible arm are considered, and the equilibrium attitude is calculated. The
influence of the equilibrium attitude at different heights and the initial tilts of
top planes are also analyzed separately. In addition, attitude measurement
experiments of wafer exchange are used to verify the analytical model and
attitude calculation.
Description of structure
As shown in Fig. 2, a PLCM consists of a fixed base, a voice coil motor, three
flexible arms, an aerostatic slideway, and a carried object (wafer). Each
flexible arm is formed by two PLSs and rigid connections (straws). PLCMs
can realize flexible loading and unloading as well as stable and gentle carrying. A PLCM
is applied to situations that required fast, stable, and accurate
transport of the carried object, such as lithography.
A computer-aided design (CAD) model of a PLCM.
The slideway uses an air-bearing structure, and the moving distance of slideway
is small, which can effectively reduce the inclination and the offset of the
overall movement. Through high-precision machining and assembly, for
example, there are very high requirements for the verticality and
straightness of the slideway, and its influence on the attitude after
exchange is very small and can be ignored. In addition, the initial position
of the carried object is detected using an approach such as edge detection, and adjusting the
corresponding position can ensure accuracy below the micrometer level, so that the object is
in a relatively perfect position after exchange.
Each flexible arm has the drive direction z where the compliance is high,
whereas the other translation directions, x and y, where the compliance is low, and the
rotation directions, Ry and Rx, have suitable compliance to complete
the flexible loading and unloading. The angle between two flexible arms is 120∘, as shown in Fig. 3. As shown in Fig. 4, the global coordinate frame
O(x,y,z) is established at the center of the carried object, and the ith arm coordinate
frame Oi(xi,yi,zi) can be established at the top of the ith
flexible arm. The parallel leaf-spring set coordinate frame is established
at the end of parallel leaf spring set, as shown in Fig. 5.
Top view of a PLCM.
Front view of a flexible arm.
Side view of a leaf-spring set.
Generation of compliance matrix
A PLCM is equivalent to a compliant mechanism. The major components are
three flexible arms. Each arm consists of two parallel leaf springs. A
6-DOF compliance matrix is introduced for the leaf spring. Each leaf spring
can be described as a beam. This combines bending, extension, and torsion. A
small deflection of the leaf spring can be considered as a deflection screw in
the axis coordinates.
ζ(x)={δxδyδzθxθyθz}T,ζ∈se(3),
where δx={δxδyδz}T
represents the three translational deflection elements along the corresponding axes
in the local coordinate frame shown in Fig. 6. θx={θxθyθz}T gives the three
rotational deflection elements about the corresponding axes of the local
coordinate frame. According to the Euler–Bernoulli model, the leaf-spring
compliance matrix can be given in axis coordinates as shown below.
Co=diag[CxoCyoCzoCxCyCz]=diaglEAl312EIzl312EIylGIxlEIylEIz
Here, the beam is assumed to have constant rectangular cross section
A=b∗t and length l. The shear effect is ignored. Moments of inertia Iy and
Iz are given as Iy=112tb3 and Iz=112bt3, the torsion constant Ix is given by Roark et al. (1976), G is the shear
modulus, and E is the elastic modulus.
Structure of a leaf spring.
Coordinate frame {Oe,xe,ye,ze} is located at the end of a
leaf spring. When external wrench is applied at the free end of a
leaf spring, deflection twist T is generated depending on the integrated
compliance Ce. According to the screw theory, the relationship between
Ce and Co can be written as follows:
Ce=AdeTC0Ade=lEA000000l33EIz000l22EIz00l33EIy0-l22EIy0000lGIx0000-l22EIy0lEIy00l22EIz000lEIz,
where Ade is the adjoint transformation matrix between the local coordinate
frame {Oo,xo,yo,zo} and the global
coordinate frame {Oe,xe,ye,ze}.
Ade=Re0PeReRe,Pe=00000-l/20l/20Re is the coordinate rotation matrix, which is a 3×3 unit
matrix, and Pe is the antisymmetric matrix of the coordinate
translation vector.
As shown in Fig. 5, two parallel leaf springs constitute a parallel
leaf-spring set. According to the screw theory, the compliance matrix of the
ith leaf spring in the global coordinate frame Og and the corresponding
adjoint mapping can be given by
Cgi=AdgiTCeAdgi,Adgi=Rgi0PgiRgiRgi.Rgi is a 3×3 unit matrix, and Pgi is the antisymmetric
matrix of the ith leaf spring. The latter represents a coordinate translation vector
and can be given by
Pg1=0p+b2-t2-p+b200t200,Pg2=0-p+b2-t2p+b200t200,Pg3=0p+b2q+t2-p+b200-q-t200,Pg4=0-p+b2q+t2p+b200-q-t200.
The PLS set consists of four leaf springs in
parallel. Thus, the relationship between Cg and Cgi can be written as follows:
Cg-1=∑iCgi-1=Cg1-1+Cg2-1+Cg3-1+Cg4-1.
Attitude calculation
When the flexible arms carry an object like a wafer, forces and torques may
occur to make the end of the arm deviate from the ideal position and
orientation; these deviations are related to static compliance
(stiffness). Therefore, the static compliance is used to calculate the
equilibrium attitude of the carried object.
The attitude changes in the PLCM are small under normal operating conditions, comprising only small translation and rotation. Considering the geometric
errors of a flexible arm, including length variation and direction variation
of the top plane, the equilibrium attitude calculation of the carried object is
modeled using the displacement method in order to analyze the influence of
geometric errors. As shown in Fig. 7, global coordinate frame O(x,y,z) is
established at the center of the carried object, and local coordinate frame
Oi(xi,yi,zi), where i=1, 2, 3, is established at the top center of
ith flexible arm. The unknown variables are the direction vector
nf(a,b,c) and the coordinate (xi,yi,zi) of the top center of the ith flexible
arm at equilibrium.
The local coordinate frame Oi(xi,yi,zi) and the global
coordinate frame O(x,y,z).
Displacement and rotation of the flexible arms' top planes
For stiffness matrix K1 of flexible arm 1 in coordinate frame
O1(x1,y1,z1), the relationship between the compliance of a
parallel leaf-spring set and the stiffness matrix K1 can be given by
K1=C1-1=Ad1-1Cg-1(Ad1-1)T,Ad1=R10P1R1R1.
Here, R1=rot(-π/2,z). Translation P1 is completed by shifting the
coordinate along axis x by d and then along axis y by h.
R1=1000010-10,P1=00h00-d-hd0,Ad1=1000000010000-100000-h01000d0001-h0d0-10
The center points of the top planes O1,O2, and O3 are located in plane
xoy. They are uniformly distributed in a circle with radius r0. Arm 2 is in
the same arrangement but rotated about axis z by α. In coordinate
frame O2(x2,y2,z2), stiffness matrix K2 can be given as
follows:
K2=C2-1=TTC1-1T.
Transformation matrix T can be given as follows:
T=-12320000-32-120000001000000-12320000-32-120000001.
In the same way, stiffness matrix K2 in coordinate frame
O2(x2,y2,z2) can be given by
K3=C3-1=TC1-1TT.
The following is an analysis of the displacement and rotation of the flexible arms'
top planes. Local coordinate frame Oi(xi,yi,zi) can follow the
translation and rotation of the ith top plane. Translation δi can be
given by
δi=δxiδyiδzi=xi-xi0yi-yi0zi-zi0,
where (xi0,yi0,zi0) is the initial coordinate of
point Oi; this value can actually be measured.
The rotation of the local coordinate frame
Oi(xi,yi,zi) expresses the rotation of the ith top plane. The initial
normal vector of the ith top plane is
ni0=(ai0,bi0,ci0) and can actually be
measured. After rotation, the normal vector of the carried object is
nf=(a,b,c). According to Rodrigues formula, the rotation matrix can be
given by
rot(ωiθi)=ωxi2(1-cosθi)+cosθiωxiωyi(1-cosθi)-ωzisinθiωxiωzi(1-cosθi)+ωyisinθiωxiωyi(1-cosθi)+ωzisinθiωyi2(1-cosθi)+cosθiωyiωzi(1-cosθi)-ωxisinθiωxiωzi(1-cosθi)-ωyisinθiωyiωzi(1-cosθi)+ωxisinθiωzi2(1-cosθi)+cosθi.
For a small rotation angle, 1-cosθi≈0, cosθi≈1, sinθi≈θi. Simplifying the above equation yields
rot(ωiθi)=1-ωziθiωyiθiωziθi1-ωxiθi-ωyiθiωxiθi1=1-ξziξyiξzi1-ξxi-ξyiξxi1=rot(xiθi)rot(yiθi)rot(ziθi).
Thus, a small rotation angle θi around axis ωi
equals small rotation transformations ξxi,ξyi, and ξzi. According to Eqs. (13) and (14), rotation ξi can be
given by
ξi=(ξxiξyiξzi)=(ωxiθiωyiθiωziθi),
rotation about axis ωi can be given by
ωi=ωxiωyiωziT=ni0×nf=bi0c-ci0bci0a-ai0cai0b-bi0a,
and rotation angle θi can be given by
θi=sinθi=(1-(ni0⋅nf)2)12=(1-(ai0a+bi0b+ci0c)2)12.
Equilibrium attitude calculation
Equilibrium attitude calculation is used to calculate the carried object's
attitude variation and the top planes' displacements after static equilibrium.
Initial normal vector ni0 and initial coordinate
(xi0,yi0,zi0) of the center point of the ith top plane as well as the
gravity of the carried object are known variables.
Physical parameters of the PLCM in its initial state.
Material properties Parameters of a leaf spring Parameters of a flexible arm Young modulusShear modulusPoisson's ratiobtlpqrhE (GPa)G (GPa)(υ)(mm)(mm)(mm)(mm)(mm)(mm)(mm)117450.350.43020205575
In the global coordinate frame O(x,y,z), according to the stiffness of flexible
arms and the attitude variations of top planes, the forces and couples between the
carried object and each flexible arm can be obtained. The equilibriums of
forces can be expressed as follows:
∑i=13Fxi=0,∑i=13Fyi=0,∑i=13Fzi=-G,
where Fxi, Fyi, and Fzi are the forces between the carried object and
each respective flexible arm.
In the same way, the equilibriums of couples can be expressed as follows:
∑Mx=∑i=13Mxi-Fy1z1+Fz1y1-Fy2z2+Fz2y2-Fy3z3+Fz3y3=-13G∑i=13yi∑My=∑i=13Myi+Fx1z1-Fz1x1+Fx2z2-Fz2x2+Fx3z3-Fz3x3=13G∑i=13xi∑Mz=∑i=13Mzi-Fx1y1+Fy1x1-Fx2y2+Fy2x2-Fx3y3+Fy3x3=0,
where (xi,yi,zi) is the coordinate of the center point of the ith top
plane.
Based on the stiffness definition,
FiMi=Kiδiξi.
From equations above, the equilibrium Eq. (21) can be obtained using Eqs. (18) and (19).
QK1000K2000K3δ1ξ1δ2ξ2δ3ξ3=00-G-13G∑i=13yi13G∑i=13xi0Q=1000001000001000000100000100000100000010000010000010000-z1y11000-z2y21000-z3y3100z1-x10010z2-x20010z3-x30010-y1x10001-y2x20001-y3x30001
Here, Q is the coefficient matrix.
The carried object is a rigid body, and the top planes are in a coplanar
condition after the static equilibrium; the geometric constraint
equations are given as Eqs. (23) and (24). Normal vector nf(a,b,c) is a unit
vector, and this relationship is given by Eq. (22).
22a2+b2+c2=123a(x2-x1)+b(y2-y1)+c(z2-z1)=0a(x3-x1)+b(y3-y1)+c(z3-z1)=024x2-x12+y2-y12+z2-z1212=((x3-x1)2+(y3-y1)2+(z3-z1)2)12=((x3-x2)2+(y3-y2)2+(z3-z2)2)12=s
Normal vector nf(a,b,c) and coordinate (xi,yi,zi) of point
Oi at equilibrium can be obtained by solving the nonlinear equation group.
Thus, the attitude variation of the carried object caused by geometric errors
can be calculated.
In this paper, the PLCM uses distributed leaf springs as flexible units. The
leaf-spring material is beryllium copper alloy (C17200). It satisfies the
four basic assumptions for deformable solids (continuity, homogeneity,
isotropy, and low deformation). The PLCM generally completes the
bearing and transportation of rigid objects with high precision and low
mass. The attitude changes in the PLCM are small under normal operating
conditions, comprising only small translation and rotation. The connecting
pieces and the carried object are considered to be rigid bodies, and the weight
of the connecting pieces is ignored. The physical parameters of the PLCM in its initial
state are shown in Table 1.
Numerical analysis of the attitude calculation model
Considering the existence of geometric errors, the equilibrium attitude of
the carried object can be calculated using the displacement method. The
geometric parameters of the PLCM are shown in Table 2.
Numerical analysis of the height variation impact.
(x10,y10,z10)(x20,y20,z20)(x30,y30,z30)nfθz(m)(m)(m)(∘)1(0.085,0,0)(-0.0425,0.0736,0)(-0.0425,-0.0736,0)(0,0,1)02(0.085,0,0)(-0.0425,0.0736,0)(-0.0425,-0.0736,0.0005)(0.000011,0.000019,1)0.0012663(0.085,0,0)(-0.0425,0.0736,0)(-0.0425,-0.0736,0.0010)(0.000022,0.000038,1)0.0025324(0.085,0,0)(-0.0425,0.0736,0)(-0.0425,-0.0736,0.0015)(0.000033,0.000057,1)0.0037985(0.085,0,0)(-0.0425,0.0736,0.0005)(-0.0425,-0.0736,0.0005)(0.000022,0,1)0.0012666(0.085,0,0)(-0.0425,0.0736,0.0005)(-0.0425,-0.0736,0.0010)(0.000033,0.000019,1)0.0021937(0.085,0,0)(-0.0425,0.0736,0.0005)(-0.0425,-0.0736,0.0015)(0.000044,0.000038,1)0.0033508(0.085,0,0)(-0.0425,0.0736,0.0010)(-0.0425,-0.0736,0.0010)(0.000044,0,1)0.0025329(0.085,0,0)(-0.0425,0.0736,0.0010)(-0.0425,-0.0736,0.0015)(0.000055,0.000019,1)0.00335010(0.085,0,0)(-0.0425,0.0736,0.0015)(-0.0425,-0.0736,0.0015)(0.000066,0,1)0.003798Height variations in the three flexible arms
The attitude variations cover a small scope, and the mass center of the carried
object is at the origin of the global coordinate O. The three top planes of the flexible arms
are uniformly distributed in a circle with radius r0. The initial normal
vectors are ni0(0,0,1). The height variation range is
0–0.0015 m. Calculating 10 different groups, the results
consist of normal vector nf(a,b,c) as well as angle θz between normal
vector nf and the positive axis z, which are shown in Table 3.
As shown in Table 3, groups 4 and 10 have the same two initial heights of the
same two arms, whereas the other is different; as a result, they have the same angle
θz but different directions of the normal vector nf. Similar
phenomena occur between groups 3 and 8 and between groups 2 and 5. In groups 7 and 9,
the first arm axis z initial values are 0 and 0.0015 m, whereas the second arm values are
0.0005 and 0.0010 m; they have the same angle θz but different
directions. The reason for this is that the three top planes of the arms are uniformly distributed
in a circle, and the plane that goes through the initial center points of the three
top planes has the same angle as the z axis but different directions.
Numerical analysis of the direction variation impact.
n10n20n30nfθz (∘)1(0,0,1)(0,0,1)(0,0,1)(0,0,1)02(0,0,1)(0,0,1)(0.010000,0,0.99995)(0.001539,0.002437,0.999996)0.1651633(0,0,1)(0,0,1)(-0.005,0.00866,0.99995)(0.001341,0.002552,0.999996)0.1651634(0,0,1)(0,0,1)(-0.005,-0.00866,0.99995)(-0.004215,-0.007301,0.999964)0.4830605(0,0,1)(0.01,0,0.99995)(0.010000,0,0.99995)(0.002828,0,0.999996)0.1620086(0,0,1)(0.01,0,0.99995)(-0.005,0.00866,0.99995)(0.002645,0.000148,0.999996)0.1517807(0,0,1)(0.01,0,0.99995)(-0.005,-0.00866,0.99995)(-0.001851,-0.009497,0.999953)0.5543708(0,0,1)(-0.005,0.00866,0.99995)(-0.005,0.00866,0.99995)(-0.002096,0.009459,0.999953)0.5551229(0,0,1)(-0.005,0.00866,0.99995)(-0.005,-0.00866,0.99995)(-0.0113978,0,0.999935)0.65305710(0,0,1)(-0.005,-0.00866,0.99995)(-0.005,-0.00866,0.99995)(-0.002096,-0.009459,0.999953)0.555122Direction variations of the top planes of the three flexible arms
The initial coordinates of the top planes' center points are (0.085, 0, 0),
(-0.0425, 0.0736, 0), and (-0.0425, -0.0736, 0). The initial normal vectors of the
top planes, ni0, are changed in each group, and there are four cases: (0, 0, 1), (0.01, 0, 0.99995), (-0.005, 0.00866, 0.99995), and
(-0.005, -0.00866, 0.99995). Calculating 10 different groups, the results
consist of the normal vector nf(a,b,c) as well as the angle θz between
nf and the z axis, as shown in Table 4.
It can be found from Table 4 that the direction variation impact is larger than the
height variation impact. The first group only shows translation in the z direction. Group 9
has the largest θz in Table 4, as the initial directions
of tilt are separate along the length direction of the leaf spring. It can be
seen that groups 2 and 3 have same two initial angles between ni0
and axis z but different directions; as a result, they have the same
angle θz but different directions of the normal vector nf. Arms 2 and 3 of groups 8 and 10 have the same initial angles between
ni0 and the z axis but different directions, and the three arms are
circularly symmetric; thus, they have the same angle θz but different
directions of the normal vector nf.
The three top planes realize a coplanar condition when equilibrium is established, and each
top plane has a different translation and rotation. It can be seen from
analysis that the normal vector nf of the carried object relates to the
initial tilts of the normal vectors of the top planes. As shown in Fig. 4, the
compliance of a single flexible arm in direction Ry is relatively large, so
the initial tilt in direction Ry has a great influence on the normal vector
nf. Thus, the compliance of a single flexible arm in direction Rx is
relatively small, so the initial tilt in direction Rx has little influence on the
normal vector nf.
Experimental verification of the attitude calculation
As shown in Figs. 8 and 9, an experimental wafer exchange system was
established. The attitude measurement of the wafer exchange is the main purpose of this system,
which is used for the validation of the attitude calculation model. The system
includes the following parts: the PLCM, a sensing module, a vacuum adsorption
module, a moving module, a signal-collecting module, and a computer.
Schematic illustration of the experiment.
Experimental setup for attitude measurement of wafer exchange.
The sensing module is divided into three parts: z-direction displacement
detection, wafer edge detection, and vacuum pressure detection. It mainly
measures the attitudes of the wafer and the top planes of the arms. Laser triangulation
displacement measurement is used, which (combined with the 3D moving stages)
constitutes a 3D coordinate measurement system. The laser displacement sensor
is a LK-G30 sensor (KEYENCE) that has a 0.01 µm repeat accuracy and a
±5 mm measurement range. The 3D moving stages are the xy axis moving stages
and the Rz rotary stage (BOCIC), which have a respective 0.1 µm and 1.4 µrad resolution and a respective 3 µm and 9.7 µrad positioning accuracy. Wafer edge detection
selects a racer linear array charge-coupled device (CCD) with a pixel size of 7 µm, an 8 bit
output, and 1024 or 2048 units corresponding to a frequency of 18.35 or
9.42 kHz, respectively (BASLER). The position and orientation of the top planes were measured and fitted using the planar fit method
without loading the carried object (Li et al., 2016).
The attitude measurement of wafer exchange includes three main processes: the
attitude measurement of the three top planes of the PLCM arms, the wafer exchange, and
the wafer exchange attitude measurement. There are three sets of PLCMs for the wafer exchange
experiment. The results of the attitude measurement of the three
top planes are given in Table 5. The wafer exchange process is that the
moving part of the mechanism (slider mover and three flexible arms) proceeds to the
handover position. The tops of the flexible arms then use vacuum to absorb
the wafer. After reaching the vacuum threshold, the moving part continues to
rise to the fixed position. Finally, the wafer is brought down to the
specified position of the lithography workpiece table, and the vacuum
adsorption is removed to complete the wafer exchange. The control program of
the motion part adopts the deadbeat control with constraint and no ripple
method (Li et al., 2015).
Position and orientation of the top planes.
Top plane's center position Top plane's direction vector (x10,y10,z10) (m)(x20,y20,z20) (m)(x30,y30,z30) (m)n10n20n301(0.085, 0, -0.000009)(-0.041962, 0.07286, 0.000207)(-0.040182, -0.073438, 0.000316)(-0.002239, -0.00683, 0.999974)(0.002609, -0.002913, 0.999992)(0.008343, 0.000717, 0.999965)2(0.085, 0, -0.000212)(-0.042135, 0.072899, -0.000073)(-0.040309, -0.073538, -0.000035)(-0.001624, -0.010369, 0.999945)(0.002711, -0.002215, 0.999994)(0.009183, -0.001019, 0.999957)3(0.085, 0, -0.00018)(-0.042121, 0.072815, -0.000071)(-0.040665, -0.073794, -0.000289)(-0.003653, -0.00963, 0.999947)(0.00126, -0.003304, 0.999994)(0.002445, -0.000358, 0.999997)
Results of the attitude calculation and attitude measurement.
With respect to exchanging the wafer and loading the carried object, the results of the attitude
calculation and attitude measurement are given in Table 6, which includes
the normal vector nf(a,b,c) of the carried object as well as the angle θz between the
normal vector nf and the positive z axis. The deviation δf
of the normal vector nf of the attitude calculation and the normal vector
nf′ of attitude measurement can be given by
δf=‖nf-nf′‖=(a-a′)2+(b-b′)2+(c-c′)2.
It can be seen from Tables 5 and 6 that both the position and orientation of top
planes can cause a different equilibrium attitude of the carried object.
Different combinations of the position and orientation of the three top planes change the
directions of the equilibrium attitude of the carried object. For example,
it can be seen from Table 6 that group 1 is located in the first quadrant, group 2 is located in the fourth quadrant, and group 3 is located in the third quadrant.
The normal vectors of the attitude calculation and attitude measurement are
located in the same quadrant; they are very close, and the angles between the
normal vector and the positive z axis are also very close. The deviations
δf of the normal vector between calculation results nf and
measurement results nf′ are 1.92×10-4 for group 1,
1.60×10-4 for group 2 and 1.67×10-4 for group 3, which are small enough
to satisfy practical requirements. The effectiveness of the attitude calculation
model is illustrated by experimental verification. Using the Euler–Bernoulli
model, which ignores shear deformation and rotary inertia, makes calculation
result relatively small. Measurement error may be another reason for the
deviations.
Conclusion
This paper presents a comprehensive study of a new parallel leaf-spring
carrying mechanism, covering analytical modeling to attitude calculation, and provides a
better understanding of the characteristics of PLCMs. With respect to the analytical modeling, this
paper establishes a compliance matrix of the parallel leaf-spring set. Regarding the
attitude calculation, the respective carried object eccentricity and geometric errors of the
flexible arm are considered, and the displacement method is used to
calculate the equilibrium attitude. Moreover, the laser triangulation coordinate
method is used to carry out the attitude measurement. The effectiveness of the attitude
calculation model is illustrated by numerical analysis and experimental
verification. The proposed methods to address the mobility characteristic
analysis and attitude calculation are not only available for PLCM but also
for complex parallel compliant mechanisms.
Data availability
All of the data used in this paper can be obtained upon request from the
corresponding author.
Author contributions
PL conceived of the presented idea and carried out the experiments. PL and
ZRT undertook the numerical analysis and wrote the paper. WHZ checked the article and
made suggestions.
Competing interests
The contact author has declared that neither they nor their co-authors have any competing interests.
Disclaimer
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Financial support
This research has been supported by the National Natural Science Foundation of
China (grant no. 62003237), the Tianjin Enterprise Technology Commissioner
Project (grant no. 20YDTPJC01700), and the State Key Laboratory of applied
optics (grant no. SKLA02020001A02).
Review statement
This paper was edited by Daniel Condurache and reviewed by Zhixuan Cao and Petr Chalupa.
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