Robotics and Computer-Integrated Manufacturing

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Optimal design of a 3-PUPU parallel robot with compliant hinges for

micromanipulation in a cubic workspace

Yuan Yun, Yangmin Li

Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Av. Padre Toma´s Pereira Taipa, Macao SAR, China

a r t i c l e

i n f o

Article history:

Received 23 August 2010 Received in revised form 24 March 2011 Accepted 10 May 2011 Available online 26 May 2011 Keywords: Parallel mechanism Flexure hinge Parasitic rotation Optimal design

a b s t r a c t

In recent years, nanotechnology has been developing rapidly due to its potential applications in various ﬁelds that new materials and products are produced. In this paper, a novel macro/micro 3-DOF parallel platform is proposed for micro positioning applications. The kinematics model of the dual parallel mechanism system is established by the stiffness model with individual wide-range ﬂexure hinge and the vector-loop equation. The inverse solutions and parasitic rotations of the moving platform are obtained and analyzed, which are based on a parallel mechanism with real parameters. The reachable and usable workspace of the macro motion and micro motion of the mechanism are plotted and analyzed. Finally, based on the analysis of parasitic rotations and usable workspace of micro motion, an optimization for the parallel manipulator is presented. The investigations of this paper will provide suggestions to improve the structure and control algorithm optimization for the dual parallel mechanism in order to achieve the features of both larger workspace and higher motion precision.

1. Introduction

Nanotechnology, which is an integration of robotics, control technique, material science, actuator and sensor technology, has been developing rapidly due to its potential for changing the ways in which materials and products are created. Nanotechnology involves the precise manipulation and control of atoms and molecules to create novel structures with remarkable properties[1]. This influen-tial technology requires a new field termed nanomanipulation to deal with how to handle components and structures in nanometer scale by utilizing devices with high positioning accuracy and dexterous manipulation skill and controlling external forces with sensory feedback, which has been enabled by the invention of bio-cell manipulation, optical fibers alignment, micro device assembly, and scanning probe microscopes[2]. Moreover, many applications in precision engineering, such as wafer stepper lithography machines, atomic force microscopes, space telescopes and interferometers, laser communication systems and some other sensitive optical applica-tions, would be impossible to realize without a careful isolation of the instrument from the vibration environment. These disturbances in the form of multiple degree-of-freedom (DOF) vibrations not only degrade the performance of the sensitive instruments and but also lead to failures in some biological experiments and chemical experi-ments. To meet stringent accuracy and performance requirements, vibrations inherent to a flexible structure or generated by on-board disturbance sources will have to be controlled.

In recent years, most of the investigations on 2 or 3-DOF micro parallel manipulators are focusing on the decoupled flexure parallel manipulators and ultra-precision manipulators[3,4]. In fact, for the parallel manipulators, the characteristics of small workspace restricted the development of parallel manipulator in the past decade. Taking into account the requirement of a multi-ple DOF mechanism for multimulti-ple DOF positioning in a larger workspace and effective vibration attenuation, many researchers utilize parallel manipulators with compliant parts for micro positioning and active vibration isolation. A spatial compliant parallel robot with pseudo-elastic flexure hinges is proposed, which uses a kind of shape memory alloy (SMA) as flexure hinges and its workspace is larger than 20020060 mm3_[5]_{. Ouyang}

[6]proposed a new design of a parallel structure for macro–micro systems. The macro motion (DC motor) and micro motion (PZT actuator) are connected by a parallel structure, the two motions are coupled under one compliant mechanism framework. At the same time, a kind of dual parallel mechanism called a 6-PSS parallel mechanism and a 6-SPS one is developed [7]. The experimental prototype is actuated by piezoelectric motors which can realize a centimeter-scale stroke with positioning precision better than 100 nm, and piezoelectric ceramics actuators with a high resolution about 1 nm and non-backlash design ensure nanometer scale precision over the cubic centimeter workspace

[8,9]. A XYZ ﬂexure parallel mechanism (FPM) is also proposed with large displacement and decoupled kinematics structure[4]. The large-displacement FPM has a large motion range over 1 mm. Moreover, the FPM can achieve decoupled X-, Y- and Z-axis translational motions with small cross-axis errors less than 1.9% and small parasitic rotations less than 1.5 mrad. The genetic

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Robotics and Computer-Integrated Manufacturing

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algorithms and artiﬁcial neural networks as an intelligent opti-mization tool are used for the dimensional synthesis of the spatial six DOF parallel manipulator[10].

Our motivation in this paper is to develop a novel 3-DOF spatial manipulator with a cubic centimeter-scale workspace and the micron or nanometer scale resolution for 3-D manipulation and vibration isolation during nanometer-scale experiments. A dual parallel manipulator using ﬂexure hinges will be developed, which synthesizes both micro positioning manipulation in a large workspace and active vibration isolation for future possible use in crystal growth experiment in micro gravity environment. Based on our previous research results[11–14], an optimization on the parallel micromanipulator will be carried out through considering the performance of the usable workspace, parasitic rotations and dexterity measures using optimization toolbox in MATLAB for multivariable nonlinear function, PSO method and GA method. Comparing with the existed large-displacement FPMs for 3-DOF micro manipulation, the designed manipulator has the advan-tages of simple structure, easy assembly, and large workspace. The investigations of this paper will provide suggestions to improve the structure and control algorithm optimization for the dual parallel mechanism in order to achieve the features of both large workspace and high motion precision.

In the remainder of the paper, a novel dual translational compliant parallel micromanipulator is designed based on a 3-UPU prototype in Section 2 and the performance analysis is given in Section 3. In Section 4, the structure is optimized to meet the requirement. At last, conclusions will be given in Section 5.

2. Conceptual design of a manipulator with large workspace A compliant parallel manipulator has been designed and fabri-cated as shown inFig. 1in Mechatronics Laboratory of University of Macau. The parameters of the prototype are listed inTable 1. The workspace of end-effector is only about several tens of microns along each axis. In order to enlarge the workspace and guarantee the accuracy, a redundant parallel mechanism combining a 3-PUU parallel mechanism with another spatial 3-UPU one is designed as shown inFig. 2. The parallel platform using flexure hinges at all joints consists of a mobile platform, three guided rails fixed to the base platform, and three limbs with identical kinematic structure. Moreover, since the flexure universal hinge has a complicated structure which is hard for manufacturing and quality control, a

kind of assembled ﬂexure hinge is developed to overcome the difﬁculties in putting together all parts.

2.1. Design of dual motions

Since the 3-PUU and 3-UPU kinematical structures with con-ventional mechanical joints can be arranged to achieve only translational motions with some certain geometric conditions satisﬁed. In this paper, the manipulator is formed by combining together the two structures. Actually, the manipulator is a 3-PUPU parallel manipulator. For convenience, the motion of the end-effector can be divided into two parts, the macro motion and the micro motion. The sliding-blocks are driven by three prismatic actuators, respectively, which offer smooth precision motion along linear guided rails to provide three translational macro motions in a cubic centimeter workspace. At the same time, the micro motion is provided by three precision actuators which can increase the accuracy of the whole system.

2.2. Motor selections

As shown inFig. 2, each limb connects the mobile platform to the fixed base by one prismatic actuator, one flexure universal (U) hinge and a piezoelectric (PZT) actuator followed by another U joint in sequence, where the first U joint is fixed at the sliding block and the second one connects to the moving platform. The end-effector is actuated by three PZT actuators to offer the micro motions. PZT actuators have such merits as sub-nanometer resolution, large force generation, sub-millisecond response, no

Moving Platform Triaxial Accelerometer PZT Actuator Piezo Input Feedback Fixed Base Flexure Hinge

Fig. 1.A prototype of a 3-UPU parallel manipulator.

Table 1

Geometric and material parameters of the 3-UPU prototype.

Item Value

Radius of upper hinges distributionr 20 mm Radius of lower hinges distributionR 90 mm Radius of ﬂexure hingerf 0.8 mm

Length of ﬂexure hingelf 10 mm

Radius of strut with PZTrs 6.35 mm

Length of strut with PZTls 146 mm

Modulus of elasticity of ﬂexure hingeEf 126 GPa

Modulus of elasticity of strutEs 205 GPa

Stroke of PZT actuatorQc 100mm

Moving Platform

PZT Actuator

Linear Guide Rail Sliding Block Flexure Hinge Piezo Input Feedback Macro Motion Micro motion

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magnetic ﬁelds, extremely low steady state power consumption, no wear and tear, vacuum and clean room compatibility.

An ultrasonic motor (USM) is characterized by the absence of noises during operation, with high torque-weight ratio, high accurate speed, and position control, which can provide unlimited stroke with a sub-micron level accuracy. The applications can be found in microscopy, precision motion, robotics and so on. This kind of motor is suitable for the macro motion of rough position-ing. Hence, three USMs are adopted as the prismatic actuators to provide with the macro motion for the mechanism and three groups of guide-way and way-block are adopted as the compo-nents. But the dynamic stall force of the USM is only about 30 N, the designed manipulator adopts the horizontal drive arrange-ment to realize the macro motion in this paper.

2.3. Flexure hinges

In the past several years, more and more designers adopt the flexure hinges instead of conventional mechanism joints since the backlash and frictions in the conventional joints affect the performances of high precision parallel mechanisms remarkably. Although the adoption of flexure hinges in the parallel mechan-ism systems increases the high precision, short stroke actuators with nanometer scale level precision result in a very small workspace. In this paper, the flexure universal hinge is a slender shaft configuration with very high torsional stiffness, which is adopted as passive joint to ensure the large workspace of the whole system and high precision motion. As the wide-range flexure hinge is difficult to manufacture for its slender configura-tion, a kind of assembled flexure hinge is developed, in spite of destroying the monolithic feature[8]. Moreover, considering that it is necessary to choose materials which allow great elastic deformations without undergoing damaging plastic deformations, the materials behaving the greatest ratios between the yield strength and the Young’s modulus (

s

=E) should be selected. The mechanical properties of several common materials are listed in

Table 2. Since the material selection of the ﬂexure hinges will also

affect the natural frequency of the whole system, the beryllium copper alloy CuBe2with high Young’s modulus and

s

=Eis adopted as the ﬂexure hinge material possessing well elasticity and high stiffness.

3. Kinematics model

As shown inFig. 3, a reference frameo-xyzis attached to the ﬁxed platform at the centero. A local coordinate systemo0_x0_y0_z0

is attached to the moving platform at the centero0_{. The coordinate}

system oixiyizi is established on the ith limb of the parallel mechanism. Let l¼[l1 l2 l3]T _{be the vector of the three PZT} actuated length variables and the vector roo0¼ ½x y zT of the reference point o0 _{be the position of the moving platform. In}

addition, letbibe the vectoroo!i andmibe the vectoro0Mi

!

.

3.1. Kinematics model of macro motion

Since the PZT actuators are locked when the 3-PUPU for macro motion is activated, the three PZT actuators can be treated as three struts in this situation. It is obvious that the deformation of the wide-range ﬂexure hinges cannot be ignored for a micro positioning application. The kinematics model is established based on stiffness equation. The wide-range ﬂexure hinges and struts are analyzed using beam elements based on FEM theory. The single limb is divided into four parts as shown inFig. 3. LetTri be the transformation matrix of the ith limb from oixiyizi coordinate system to o-xyz. K0

oi, K0mi and K0ui are the stiffness

matrices of the flexure hinge connecting to the base platform, the flexure hinge connecting to the moving platform and the strut in oixiyizi coordinate system, respectively.poi,p1i,p2iandpmiare nodal load vectors of these nodes shown in Fig. 3 in o-xyz coordinate system. doi, d1i,d2i and dmiare nodal displacement vector ino-xyzcoordinate system. The wide-range flexure hinges and struts can be treated as structural beams. Letp,A, andLbe the axial force, cross sectional area and length of a beam, respectively, E, G, I, and J be the elastic modulus, shear modulus, principal moments of inertia and torsion moment of inertia, respectively. The element tangent stiffness matrices in the local coordinate systems are composed of three components: linear stiffness KL, the geometric stiffness matrix KG, and the large-displacements stiffness matrixKDas follows[15]:

Ke¼KLþKGþKD ð1Þ whereKLis given by EA L 0 0 0 0 0 EAL 0 0 0 0 0 0 12EI L3 0 0 0 6_LEI2 0 12_L3EI 0 0 0 6_LEI2 0 0 12EI L3 0 6_LEI2 0 0 0 12_L3EI 0 6_LEI2 0 0 0 0 GJ_L 0 0 0 0 0 GJ_L 0 0 0 0 6_LEI2 0 4_LEI 0 0 0 6_LEI2 0 2_LEI 0 0 6EI L2 0 0 0 4EI L 0 6EI L2 0 0 0 2EI L EA L 0 0 0 0 0 EAL 0 0 0 0 0 0 12EI L3 0 0 0 6LEI2 0 12L3EI 0 0 0 6LEI2 0 0 12_L3EI 0 6_LEI2 0 0 0 12_L3EI 0 6_LEI2 0 0 0 0 GJ_L 0 0 0 0 0 GJ L 0 0 0 0 6_LEI2 0 2LEI 0 0 0 6LEI2 0 4LEI 0 0 6EI L2 0 0 0 2EI L 0 6EI L2 0 0 0 4EI L 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 Table 2

Mechanical properties of several common materials at room temperature for ﬂexure hinges.

Material Yield strength s(MPa)

Young’s modulus E(GPa)

s=E

Ti-6Al-4V 880 113.8 0.0077

Beryllium copper alloy CuBe2 750 126 0.006

Al alloy 7075 T6 503 72 0.007

Steel ASTM A514 690 205 0.0034

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and the geometric stiffness matrix KG¼p 0 0 0 0 0 0 0 0 0 0 0 0 0 6 5L 0 0 0 1 10 0 6 5L 0 0 0 1 10 0 0 6 5L 0 101 0 0 0 5L6 0 101 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ₁₀1 0 2L 15 0 0 0 101 0 30L 0 0 1 10 0 0 0 2L15 0 101 0 0 0 30L 0 0 0 0 0 0 0 0 0 0 0 0 0 _5L6 0 0 0 ₁₀1 0 6 5L 0 0 0 101 0 0 _5L6 0 1 10 0 0 0 6 5L 0 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ₁₀1 0 ₃₀L 0 0 0 1 10 0 2L 15 0 0 1 10 0 0 0 30L 0 101 0 0 0 2L15 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

The large-displacement stiffness matrix could be neglected when the deformation increment of the element is very small. Dividing the stiffness matrices into 22 sub-matrices, the stiff-ness matrix of the whole limb inoixiyizi coordinate system can be achieved by assembling the nodal stiffness matrices:

K0 i¼ K0 oi11 K 0 oi12 0 0 K0 oi21 K 0 oi22þK 0 ui11 K 0 ui12 0 0 K0 ui21 K 0 ui22þK 0 mi11 K 0 mi12 0 0 K0 mi21 K 0 mi22 2 6 6 6 6 4 3 7 7 7 7 5 Let Ki¼TT riK0iTri pi¼ ½poi p1i p2i pmiT di¼ ½doi d1i d2i dmiT

The Lagrange equation of a single limb can be formulated as

pi¼KidiþMidi€ ð2Þ

whereMiis the mass matrix of single limb, which is similar in form to the stiffness matrix achieved by assembling the element mass matricesMe, deﬁned by

Me¼

Z le

0

NHNdx ð3Þ

whereN is the shape function matrix andH is the generalized density matrix. In addition, letpbe the external load vector acting on the moving platform, the force and moment equilibrium equation can be written as

pX

3

i¼1

pmi¼0 ð4Þ

Since the wide-range ﬂexure hinges are adopted in this parallel mechanism, the system cannot be expressed by a linear equation exactly. The stiffness matrix and nodal load can be described by a function of nodal position. Eq. (2) can be written as

piðdiÞKiðdiÞ diMid€i¼

f

ðdiÞ ð5Þ where

f

ðdiÞ is the imbalance load vector. If d

i is an accurate solution of Eq. (5), we can obtain

piðd iÞKiðd iÞ d iMid€ i ¼

f

ðd iÞ ¼0 ð6Þ

Therefore, the Newton–Raphson method is utilized, which uses an iterative process to approach one root of a function. The speciﬁc root that the process locates depends on the initial and

arbitrarily chosendi-value.

@fðdðnÞ i Þ @dðnÞ i ðdðnÞ i d ðnþ1Þ i Þ ¼

f

ðd ðnÞ i Þ ð7Þ

The acceleration term can be ignored in static analysis. Assume that @fðdðinÞÞ @dðnÞ i

Dd

ðnÞ i ¼

f

ðd ðnÞ i Þ KðnÞ ti ¼ @fðdðinÞÞ @dðnÞ i ¼@piðd ðnÞ i Þ @dðnÞ i @Fiðd ðnÞ i Þ @dðnÞ i where FiðdðinÞÞ ¼K ðnÞ i d ðnÞ i , and @piðd ðnÞ i Þ=@d ðnÞ i ¼0 in conservative system. Eq. (5) can be written as

KðnÞ ti

Dd

ðnÞ i ¼

f

ðd ðnÞ i Þ dðnþ1Þ i ¼d ðnÞ i

Dd

ðnÞ i 8 < : ð8Þ where KðnÞ

t is the tangential stiffness matrix of each limb. The detailed ﬂow chart for this problem is described inFig. 4.

A noteworthy fact is that the tangential stiffness matrices of each limb have to be updated in order to converge fast to solutions in each sub-step of the iterated algorithm. It is a very huge amount of calculations for a multi-body parallel system. Therefore, a kind of modiﬁed Newton–Raphson method is adopted, which uses the original tangential stiffness matrix instead of renewal one at the beginning of each new iteration. This method generally requires more iterations and sometimes is less stable but it is less cost computationally[16].

3.2. Kinematics model of micro motion

The prismatic actuators for macro motion are self-locked when the micro motion is available. Since the displacement of wide-range ﬂexure hinges can be ignored for several hundreds of micro strokes of actuators, the kinematics equation can be established based on Pseudo-Rigid-Body (PRB) model easily. As shown in

Fig. 5, the ﬂexure hinge can be regarded as a U joint and two

torsional massless springs with stiffnessKbiorKmiwith respect to theo-xyzcoordinate system. The torsion moment of the ﬂexure

N START Input data of structure parameters Y END n=0 n=n+1 Solve T Solve in global coordinate system K Assemble of each limb K

Solve imbalance load vector and p d p Solve T Assemble of each limb K Solve in global coordinate system K

Solve the initial value of d Solve d Solve in local coordinate system K Solve p K d Solve d d d Output d

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hinge caused by the angular displacement

ybi

or

ymi

can be derived by

Mbi¼Kbi

y

bi, Mmi¼Kmi

ymi

ð9Þ

For theith limb, the vector-loop equation can be written as

roo0þmibiliki¼0 ð10Þ

Differentiating Eq. (10) with respect to time _

liki¼r_oo0li

o

_iki ð11Þ

where

o

iis the vector of angular velocities of theith limb. Dot-multiplying both sides of Eq. (11) byki, it can be assembled into a velocity mapping equation

_ l1 _ l2 _ l3 2 6 4 3 7 5¼Jr_oo0 ð12Þ where J¼ kT1 kT2 kT₃ 2 6 6 4 3 7 7 5 ð13Þ

is the Jacobian matrix of top 3-UPU parallel manipulator, which relates the output velocities to the PZT actuated joint rates.

3.3. Objective function for optimization

Considering that the kinematics of the designed parallel manipulator has highly nonlinear characteristics, the genetic algorithm (GA) and particle swarm optimization (PSO) can be used to solve this problem.

The GA can be applied to solve a variety of optimization problems with properties of discontinuous, non-differentiable, sto-chastic, or highly nonlinear, etc. The PSO is a relatively new algorithm proposed by Kennedy and Eberhart, which is a popula-tion-based stochastic optimization technique inspired by the social behavior of bird flocking or fish schooling. PSO method can be employed to solve a variety of optimization problems such as discrete optimization, artificial neural network training, fuzzy sys-tem control and other situations where the evolutionary techniques can be employed[17]. Both the GA and PSO are initialized with random start values within the search space, whereas the PSO has no evolutionary operators of crossover and mutation. As the knowl-edge of the authors, the applications of PSO are first introduced to the issues of manipulator optimal design by Xu and Li[13,14].

The objective function for optimization is deﬁned as a mixed performance index that is a weighted sum of the usable work-space of micro motion (Vw2), the parasitic rotation (

f

max) and the

dexterity measure which can use the condition number as the index. The condition number and conditioning index can be deﬁned as

k

¼_JJ_JJJ1 J

m

¼1=

k

ð14Þ

where_J_Jdenotes the 2-norm of the matrix. A global dexterity index is given by

m

global¼ R V

m

dV V 1 Nw X wAV

m

w ð15Þ

wherewis one ofNwpoints, which is uniformly distributed over the workspace. Therefore, the objective function is deﬁned by following equation f¼ww2 1 Vw2þwf

f

maxþ ð1ww2wfÞ 1

m

global ð16Þ

whereww2A½0,1,wfA½0,1andww2þwfA½0,1. Considering the structure of the dual parallel manipulator, the designable variables are the length of the strut, the radius of the ﬁxed platform and the length of the ﬂexure hinge. Then, the complex method of the nonlinear programming, GA and PSO method in MATLAB toolbox are used to accomplish the parametric optimization by solving the minimum of the objective functionfdescribed in Eq. (16).

4. Simulation results and optimization analysis

In this case, a kind of PAS-series piezoelectric actuator (PAS100) provided a convenient mounting package with a 100

mm stroke is

adopted to provide micron scale accuracy motions. A kind of USM named as HR8-1-S-3 (Nanomotion, Israel) is adopted as macro actuators with the positioning precision better than 100 nm, which can basically satisfy the macro motion demands. In addition, three groups of guide-way and way-block are adopted for the components of the piezoelectric motors. The parameters of the parallel mechan-ism are shown inTable 3. In this section, the inverse solutions and parasitic rotations of moving platform are analyzed and simulated on a parallel mechanism. Whereafter, the reachable and usable workspace of the macro motion and micro motion of the mechan-ism are plotted. Finally, based on the analysis of usable workspace and parasitic rotation, an optimization of the parallel manipulator is presented.

4.1. Parasitic rotations

The inﬂuences of the torsion introduced by the wide-range ﬂexure hinges have to be considered, which may cause the parasitic rotations of end-effector. In Eq. (5), each dbi has one

Fig. 5.PRB model of the parallel mechanism for micro motion.

Table 3

The original geometric and material parameters of the redundant 3-PUPU manipulator.

Item Value

Radius of upper hinges distributionr 20 mm Radius of lower hinges distributionR 90 mm Radius of ﬂexure hingerf 0.8 mm

Length of ﬂexure hingelf 10 mm

Radius of strut with PZTrs 6.35 mm

Length of strut with PZTls 146 mm

Modulus of elasticity of ﬂexure hingeEf 126 GPa

Modulus of elasticity of strutEs 205 GPa

Stroke of PZT actuatorQc 100mm

Stroke of PZT motorQm 10 mm (75 mm)

Mass of the moving platformM 0.140 Kg Mass of the limbmi 0.128 Kg

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input element generated by actuator.d1i and d2iare unknown vectors too,p1iandp2iare zero vectors since no loads act on node 1 and 2, anddmican be solved by the displacement of the moving platform. Utilizing the middle double sub-equations of Eq. (5) for each limb and Eq. (6) together are 42 equations and 42 unknown elements including three input parameters and three parasitic rotations of moving platform. In Table 4, the input motions of actuators and parasitic rotations of moving platform are given by theoretical stiffness model. It indicates that the parasitic rotations are zero when the moving platform only has the displacement of z-axis and the maximal parasitic rotations of the checking points are several micro radians and gradually increase when the moving platform departures from initial position along x-axis and y-axis that cannot be ignored in the design of control algorithms for micro/nanopositioning applications.

4.2. Workspace determination

As the well elastic nature of the selected material of wide-range ﬂexure hinges, the workspace of micro motion only depends on the limits of PZT actuators. Considering the different characteristics between ﬂexure hinge and conventional mechan-ical joint, the parasitic rotations of moving platform have to be calculated and limited in a negligible order of magnitude. In this paper, the limit of the rotation angle is set to 5 mrad. As shown in

Fig. 6, the workspace of macro motion is about several

milli-meters in each axis. As shown inFig. 7, the reachable workspace of micro motion, when the inputs of PZT motors are zero and self-locked at the initial position, is about several tens microns in each axis. It is assumed that the maximal usable inscribed workspace of the macro motion workspace is a column with a radius ofWr1, height of H1 and symmetrical about the initial position. The maximal usable inscribed workspace of the micro motion work-space is a column with a radius of Wr2 and height ofH2. The usable workspace is shown inFigs. 6 and 7denoted byW1and W2, respectively. TheW1is about a radius of 2.46 and 2.14 mm height column and theW2is about a radius of 79.2 and 36:6

mm

height column.

4.3. Position accuracy

Since ﬂexure hinges are adopted by this manipulator at all joints, the position error of macro motion is only related to the attainable resolution of the USMs. The error modeling is estab-lished with the USMs position resolutions using kinematics modeling for the three limbs parallel robot. An error of

d

doi in

actuators movement will produce a positional error of

d

dmi. The

position resolution of the end-effector could be estimated by follows: KðnÞ ti

Dd

ðnÞ i ¼

f

ð

d

d ðnÞ i Þ

d

dðinþ1Þ¼

d

d ðnÞ i

Dd

ðnÞ i 8 < : ð17Þ where

d

di¼ ½

d

doi

d

d1i

d

d2i

d

dmiT

In accordance with the Eq. (17) and the 100 nm attainable resolution of HR8-1-S-3, the position error of macro motion is about 10010070 nm3as shown inFig. 8, which is covered in the micro workspace as presented in the previous subsection.

Table 4

Input motions of actuators and parasitic rotations of moving platform.

Displacement(mm) Input motions(mm) Parasitic rotation(mrad)

[0 0 2] (4.68174.68174.6817) (0 0 0) [0 02.5] (4.9272 4.9272 4.9272) (0 0 0) [1 21] (4.0473 0.1252 1.9405) (4.0298 1.7811 0.2102) [21 1] (3.2976 0.12333.6609) (1.84223.29390.1423) [1 1 1] (1.26153.66921.8330) (1.8136 1.5896 0.1348) [111] (1.0620 3.4973 1.6163) (2.00421.83790.1070) [2 2 0] (1.9357 0.74922.8619) (3.79943.4233 0.2356) [22 0] (2.04530.9839 2.8512) (3.8694 3.50470.2157)

Fig. 6.Reachable and usable workspace of the macro motion.

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4.4. Conditioning index

From the Eq. (14), the conditioning index of the micro motion with the USMs for macro motions being locked at the initial position can be plotted as shown inFig. 9. The global dexterity index is about 0.3288288299 with the 57 720 sampling points selected in the workspace.

4.5. Optimization results

The three designable variables are the length of the strut, the radius of the initial radius of lower hinges distribution and the length of the ﬂexure hinge with the limit of: 140 mmrlsr200 mm, 50 mmrRr120 mm and 9 mmrlfr20 mm. Considering the empirical numerical values of three designable variables, the vari-ables ww2 and wf are 0.44 and 0.33 in order to attach equal importance to the three designable variables. In this paper, the traditional optimization toolbox in MATLAB for multivariable non-linear function is adopted ﬁrst. The program is developed on a personal computer (Intel Pentium 4 CPU 3.00 GHz, 1 GB RAM) with six independent runs. The initial values are set as shown in the

Table 5with the corresponding results of optimized variables and

elapsed time of the program. It is noticeable that the optimization method is sensitive to the initial values. Since thefminconfunction may only give the local solutions, big discrepancy between number of results can be observed inTable 5.

In addition, the convergent processes of GA and PSO are illustrated. Both the GA and PSO are initialized with random starting guesses vectors within the given parameters ranges. The population size is assigned as 70. For the GA approach, the genetic operators are chosen to be non-uniform mutation with ratio of 0.08 and arithmetic crossover with the ratio of 0.8, respectively. For the PSO, the initial inertia weight and the final inertia weight are selected as 0.9 and 0.4, respectively. The local and global acceleration constants are assigned as 2. The maximum genera-tion number, the minimum global error gradient and the optimi-zation results are listed inTable 6for contrast. The optimization results solved by GA are more acceptable in comparison with the traditional optimization method. With reference to the GA, all of the three independent processes of PSO produce the same results. With reference to the three optimization approaches, the PSO gives the best fitness value of 8.5781 and the GA gives the better fitness value of 8.5879 than the traditional optimization method (8.7858). Moreover, the convergent processes of GA and PSO are illustrated inFigs. 10 and 11, respectively. It is noticeable that the convergence rate of the PSO method is much sharper than that of the GA method. Whereas, in view of the calculational time shown

inTables 5and6, the PSO requires the longest time to solve the

ﬁnal results.

Choosing the result of the best ﬁtness produced by PSO, the designed dual parallel manipulator is optimized to take the advantage of larger micro motion workspace and smaller parasitic rotation with certain sacriﬁce of the global dexterity index. The volume of maximal usable inscribed workspace of the micro motion workspace is 1.2451e3 mm3_{. The radio of the} optimiza-tion for the volume of the usable workspace is 72.63%. The optimization ratios of parasitic rotations of the optimal parallel manipulator are listed in Table 7. It is shown that the

−5 0 5 x 10−5 −5 0 5 x 10−5 −8 −7 −6 −5 −4 −3 −2 −1 x 10−5 x (mm)

Position error of macro motion

y (mm)

z (mm)

Fig. 8.The position error of the macro motion.

−0.1 −0.05 0 0.05 0.1 −0.1 −0.05 0 0.05 0.1 0.32877 0.32878 0.32879 0.3288 0.32881 x (mm) y (mm) Conditioning index

Fig. 9.The conditioning index distribution.

Table 5

Optimization results of independent runs by using multivariable nonlinear method. Initial guess ½lsR lf(mm) ls(mm) R(mm) lf(mm) f(n) t(min) [146 90 10] 145.9941 90.0294 9.0999 9.1334 30.12 [160 80 15] 155.0338 90.0375 9.0000 9.0775 25.43 [140 90 9.5] 140.2766 89.9530 9.0275 9.1887 32.06 [150 60 15] 144.9992 90.0006 9.0500 9.1413 48.58 [160 70 10] 149.9948 94.9574 9.0501 9.1997 42.66 [140 66 9] 140.0000 72.7634 9.0000 8.7858 19.22 Table 6

Optimization results of independent runs by using GA and PSO.

Method Number of iterations Minimum global error gradient ls (mm) R (mm) lf (mm) f(n) t (min) GA 86 1e9 140.2200 66.7000 9.0000 8.5905 455.29 51 1e9 140.6585 66.7763 9.2032 8.6172 292.82 51 1e9 140.3602 66.8965 9.0042 8.6172 260.09 86 1e9 140.2234 66.7013 9.0020 8.5905 471.75 70 1e9 140.2684 66.7278 9.0158 8.5879 363.82 PSO 1330 1e25 140.0000 66.7105 9.0000 8.5781 4282.60 200 1e9 140.0000 66.7105 9.0000 8.5781 613.25 100 1e9 140.0000 66.7105 9.0000 8.5781 336.57

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optimization ratios of the maximum parasitic rotations of test points are about 23.30–70.39%. But the best performances of the workspace and parasitic rotations are achieved with certain sacriﬁce of the global dexterity index decreased about 33.46%.

5. Conclusions

In this paper, a novel compliant dual parallel platform for micro positioning applications is designed. The designed manip-ulator has the advantages of simple structure, easy assembly and large workspace compared with the existed large-displacement FPMs for 3-DOF micro manipulation. The stiffness model of the macro parallel mechanism with wide-range ﬂexure hinges is established. The roots of the equations are obtained by using Newton–Raphson method. Then the kinematics model is estab-lished for the micro motion system. Referring to the parameters of a 3-UPU prototype, the inverse kinematic results and the parasitic rotations are given on some checking points in workspace. It is found that the parasitic rotations are zero when the moving

platform is moving alongz-axis. If the moving platform depar-tures from initial position alongx-axis and y-axis, the maximal parasitic rotations of the checking points will gradually increase, which are about several micro radians. Then the workspaces of macro and micro motions are plotted. Finally, a 3-DOF micro parallel manipulator is optimized by considering the usable workspace of micro motion, the parasitic rotation and the dexterity. According to the objective function for optimization defined by a mixed performance index, an optimization result is derived by the optimization toolbox of MATLAB using multi-variable nonlinear function, GA and PSO method. Generally speaking, using PSO method can achieve the best optimization results among the three methods, but the runtime for the PSO is much longer than the normal optimization method. For the optimized parallel manipulator, the volume of the usable work-space and the maximum parasitic rotations of test points are significantly improved, but the best performances of the work-space and parasitic rotations are achieved after certain sacrifice of the global dexterity index.

The investigations of this paper may provide suggestions for the structure optimization to achieve the features of both larger workspace and higher motion precision. The results will be useful in modifying the structure of the prototype platform with high dynamic properties. A prototype of the upper 3-UPU parallel robot has been developed, and the experimental investigations will be carried out to verify the positioning performance of the prototype.

Acknowledgment

This work is supported by the Macao Science and Technology Development Fund under Grant no. 016/2008/A1 and research committee of University of Macau under Grant no. UL016/08-Y2/ EME/LYM01/FST.

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PSO Model: Common PSO

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Green = Personal Bests Blue = Current Positions

Red = Global Best

Fig. 11.Convergent processes of PSO.

Table 7

Optimization ratios of maximal parasitic rotations.

Displacement(mm) Parasitic rotation(mrad) Ratio(%)

[1 21] (3.0910 1.1896 0.0423) 23.30 [21 1] (0.80821.37020.0408) 58.40 [1 1 1] (0.5371 0.3187 0.0258) 70.39 [111] (1.48651.45580.0234) 25.83 [2 2 0] (2.20351.6399 0.0397) 42.00 [22 0] (2.2729 2.5248 0.0019) 34.75

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