31 multi-degree of freedom vibration model for a 3-dof hybrid robot

Multi-degree of Freedom Vibration Model for a 3-DOF Hybrid Robot

Yuan Yun and Yangmin Li

Abstract— In recent years, many applications in precisionengineering require a careful isolation of the instrument fromthe vibration sources by adopting active vibration isolationsystem to achieve a very low remaining vibration level especiallyfor the very low frequency under 10Hz vibration signals. Inthis paper, based on our previous research experiences in thesystematical modeling and study of parallel robots, a hybridrobot is described and the vibration model is given by usingLagrange’s Equations. Then a dynamic analysis by using Kane’smethod is presented to verify the result. Finally, numerical sim-ulations on the dynamic characteristics and natural frequencyare presented.

I. INTRODUCTION

Nowadays, a careful isolation of the instrument from the

vibration sources is required in many precision engineering

applications, which is realized by adopting active vibration

isolation system to achieve a very low remaining vibration

level especially for the very low frequency under 10Hz

vibrations.

Parallel manipulators can offer the advantages of high

stiffness, low inertia, and high speed capability which have

been intensively investigated and evaluated by industry and

institutions in recent years. And some designers adopt the

flexure hinges instead of conventional mechanism joints

since the backlash and frictions in the conventional joints in-

fluence the performances of parallel mechanisms remarkably.

Micro/nano positioning manipulators and active vibration

control devices are increasingly being made of parallel

manipulators due to their characteristics of high precision.

Although the adoption of flexure hinges in the parallel

mechanism systems increases the high precision, short stroke

actuators with nanometer scale level precision result in a very

small workspace.

Therefore, a lot of designers focus their attention on multi-

degree of freedom hybrid manipulators or developing wide

range flexure hinges. A spatial compliant 3-DOF parallel

robot with SMA pseudo-elastic flexure hinges was presented

in [1], which has a workspace larger than 200 × 200 × 60mm3 and resolution is better than 1µm. A parallel structure

for macro-micro systems was proposed in [2]. In this new

design, the macro-motion (DC motor) and micro motion

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 was developed

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

Yuan Yun and Yangmin Li are with the Department of ElectromechanicalEngineering, Faculty of Science and Technology, University of Macau,Av. Padre Toms Pereira, Taipa, Macau, China, [email protected],[email protected]

[3], called a 6-PSS parallel mechanism and a 6-SPS one,

which is integrated with wide-range flexure hinges as passive

joints to ensure the large workspace of the whole system and

high precision motion. A XYZ-flexure parallel mechanism is

proposed with large displacement and decoupled kinematics

structure [4], which has a large motion range beyond of

1mm.

Our research is concerned with the development of a

system that can achieve three high accurate translational

positioning or rough positioning with a 3-DOF active vi-

bration isolation, which attenuates the vibration transmission

above some corner frequencies, to protect the payload from

jitters induced by the various disturbance sources. In this

paper, based on our previous research experiences in the

systematical modeling and study of parallel robots, a novel

dual 3-DOF parallel robot with flexure hinges will be de-

scribed. A vibration model of the parallel manipulator will be

presented which will be verified by Kane’s dynamics. Finally,

numerical simulations about the dynamic characteristics and

natural frequency will be presented.

II. SYSTEM DESCRIPTION

The designed manipulator is a 3-DOF dual parallel mecha-

nism combining a 3-PUU parallel mechanism with a spatial

3-UPU one. In the 3-PUU parallel structure, the prismatic

actuators provide three translational macro motions with

micron level accuracy and cubic centimeter workspace. At

the same time, the micro motion is provided by a spatial 3-

UPU structure which can increase the accuracy of the whole

system to the nanometer level or provide a strict acceleration

level for payload placed on the moving platform. 3-PUU and

3-UPU kinematical structures with conventional mechanical

joints can be arranged to achieve only translational motions

with some certain geometric conditions satisfied.

Using flexure hinges at all joints, the parallel platform

consists of a mobile platform, a fixed base, and three limbs

with identical kinematic structure. Each limb connects the

mobile platform to the fixed base by one prismatic actuator,

one flexure universal (U) hinge and a piezoelectric (PZT)

ceramics actuator followed by another U joint in sequence

as shown in Fig. 1, where the first U joint is fixed at the

prismatic actuator and the second one connects to the moving

platform actuated by a PZT to offer the micro motions

with merits involving smooth motion, high accuracy, and

fast response, etc. Piezoelectric actuators are adopted as the

prismatic actuators to provide the macro motion for the

mechanism and three groups of guiding-rail and gliding-

block are adopted as the components. The flexure universal

hinge is a slender shaft configuration with very high torsional

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stiffness which is adopted as passive joint to ensure the large

workspace of the whole system and high precision motion.

Fig. 1. The 3-DOF 3-PUU/3-UPU dual parallel manipulator.

III. DYNAMIC MODEL

Since the macro motion is adopted for the rough posi-

tioning, the kinematic analysis is necessary for the 3-PUU

structure. The details have been published in [5]. In this

section, the vibration model using Lagrange’s equations will

be described first for the 3-UPU structrue, and then Kane’s

dynamics modeling will be built up to verify the results.

A. Kinematic Analysis

In this case, the prismatic actuators for macro motion are

self-locked when the micro motion is available.

1) Coordinate System: As shown in Fig. 2, let L=[l1 l2l3]

T be the vector of the three PZT actuated length variables

and the vector roo′=[x y z]T of the reference point o′ be

the position of the moving platform. In addition, let bi be

the vector−−→oBi and mi be the vector

−−→o′Mi. The mass of

moving platform is M . The mass of each strut is ml. C∗

i

(i=1, · · · , 3) is the center of mass for the ith limb. Let

a limb fix right-handed coordinate system with origin C∗

i

(i=1, · · · , 3) located at the center of mass for the ith limb,

with axis directions determined by an orthonormal set of

unit vectors cij (j=1,· · · ,3). The overhat indicates unit length,

the index i corresponds to the ith limb, and the index j

distinguishes the three vectors. ci3 is along the ith limb,

toward the ith flexure hinge which connects the moving

platform. ci2 is perpendicular to the vector

−−→oBi when the

moving platform is in the home position, and ci1 is in the

direction ci2× c

i3. Let a reference frame sj attach to the fixed

platform at the center o with the s1 toward the point B1 and

s3 vertical the fixed platform. Fix a coordinate system fj to

the moving platform at the center o’ with f1 toward the point

Mi and f3 vertical the moving platform. ℜ is a three-order

identity transformation matrix from fj coordinate system

to sj . ℜci is the transformation matrix from cj coordinate

system to sj .

In order to determine the angles of flexure hinges directly,

an initial coordinate system is set on origin C∗

i with axis

directions determined by an orthonormal set of unit vectors

cij0 when the moving platform is in the home position. The

axis directions are located coincident with the corresponding

limb fixed coordinate system cij when the moving platform is

Fig. 2. Coordinate system of the 3-UPU parallel platform

in the initial position. Let the orientation of the cij coordinate

system, relative to the cij0, be described by consecutive

positive rotations qi1 about the c

i10 and qi

2 about the moved

two-axis. The rotation matrix is:

ℜci0 =[

ci10 c

i20 c

i30

]−1 [

ci1 c

i2 c

i3

](1)

It is assumed that the small-angle approximations hold for

angles qik, the rotation matrix:

ℜci0 =

1 0 qi

2

0 1 −qi1

−qi2 qi

1 1

(2)

2) Generalized Velocities for the System: Define general-

ized velocities u for the system as the time rate of change

of the generalized coordinates of q=[x y z]T in the inertial

reference frame:

u =[

x y z]T

(3)

An intermediate reference frame is introduced previously

to permit describing the angular velocity of each limb to

the initial limb position. Designate the unit vectors of these

intermediate reference frames by mij . The expressions for

the angular velocities of the reference frame of each upper

arm with respect to initial reference frame of upper arm are:

ωC∗

i= −qi

1ci10 − qi

2mi2 (4)

The position vectors are:

roo′ = lici3 + bi − mi

rmi= lic

i3 + bi, rC∗

i=

li

2ci3 + bi

Differentiating Eq. (5) with respect to time:

vo′ = lici3 + liωC∗

i× c

i3 (5)

vmi= vo′ , vC∗

i=

lici3 + liωC∗

i× c

i3

2=

vo′

2

Dot-multiplying both sides of Eq. (5) by ci3:

(ci3)

T vo′ = li (i = 1, 2, 3) (6)

which can be assembled into a matrix form:[

l1 l2 l3]T

= A · vo′ (7)

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where A =[

(c1

3) (c2

3) (c3

3)]T

.

Cross-multiplying both sides of Eq. (5) by ci3:

ωC∗

i=

ci3 × vo′

li(8)

The linearized accelerations are:

ao′ = (2liωC∗

i+ liεC∗

i) × c

i3 + liωC∗

i× (ωC∗

i× c

i3) + lic

i3

aC∗

i=

ao′

2(9)

Cross-multiplying both sides of Eq. (30) by ci3, the angular

acceleration of each limb can be obtained by:

εC∗

i=

ci3 × ao′ − 2liωC∗

i− li(ωC∗

i· c

i3)(ωC∗

i× c

i3)

li(10)

B. Vibration Model

In this paper, the vibrational model is established by

using Lagrange’s equation which is a well known tool for

establishing equations of motion of discrete systems. The key

point of Lagrange’s equations is kinetic energy, which can

be formulated favorably with respect to a moving coordinate

system as well.

1) Kinetic Energy of the System: For the moving platform,

the kinetic energy can be given by:

EkM =1

2M(x2 + y2 + z2) (11)

For each limb, assume rotations about a fixed axis, the kinetic

energy can be given by:

Ekli =1

2ml‖vC∗

i‖2 +

1

2ICi

ω2C∗

i(12)

where ICi= 1

3mll

2i and according to Eq. (4), the kinetic

energy of the whole system can be derived by:

Ek = EkM+

3∑

i=1

Ekli =1

2M(x2+y2+z2)+

3∑

i=1

(1

2ml‖vC∗

i‖2

+1

2ICi

((qi1)

2 + (qi2)

2)) = (1

2M +

3

8ml)(x

2 + y2 + z2)

+

3∑

i=1

1

6mll

2i ((q

i1)

2 + (qi2)

2)) (13)

2) Potential Energy of the System: The potential energy

of moving platform is given by:

EpM = Mgz (14)

Since the system is investigated in tiny vibration environ-

ment, according to the Eq. (5), we can obtain:

ci3 =

1

li

x − xbi+ xmi

y − ybi+ ymi

z − zbi+ zmi

(15)

Let

[ci10 c

i20 c

i30

]−1

=

ri11 ri

12 ri13

ri21 ri

22 ri23

ri31 ri

32 ri33

(16)

According to Eq. (1) and (2), the consecutive positive rota-

tions qi1 and qi

2 about the moved two-axis are:

1

li

ri11 ri

12 ri13

ri21 ri

22 ri23

ri31 ri

32 ri33

x − xbi

+ xmi

y − ybi+ ymi

z − zbi+ zmi

=

qi2

−qi1

1

qi1 = H1i −

ri21

lix −

ri22

liy −

ri23

liz

qi2 = H2i +

ri11

lix +

ri12

liy +

ri13

liz (17)

where

H1i =ri21

li(xbi

− xmi) +

ri22

li(ybi

− ymi) +

ri23

li(zbi

− zmi)

H2i =ri11

li(xmi

− xbi) +

ri12

li(ymi

− ybi) +

ri13

li(zmi

− zbi)

For each limb, the potential energy can be written as:

Epli = mlgz − zbi

+ zmi

2+ ki

1(qi1)

2 + ki2(q

i2)

2 (18)

Hence, the potential energy of the whole system can be given

by:

Ep = Mgz +

3∑

i=1

(mlgz − zbi

+ zmi

2+ ki

1(qi1)

2 + ki2(q

i2)

2)

(19)

3) Lagrange’s Equations: The Lagrange’s equations are:

d

dt(∂Ek

∂uj

) −∂Ek

∂qj

+∂Ep

∂qj

= FDj +3∑

i=1

F ai

j (20)

where

d

dt(∂Ek

∂x) = (M +

3

4ml)x +

3∑

i=1

1

3ml(((r

i21)

2 + (ri11)

2)x

+(ri22r

i21 + ri

12ri11)y + (ri

23ri21 + ri

13ri11)z)

d

dt(∂Ek

∂y) = (M +

3

4ml)y +

3∑

i=1

1

3ml((r

i21r

i22 + ri

11ri12)x

+((ri22)

2 + (ri12)

2)y + (ri23r

i22 + ri

13ri12)z)

d

dt(∂Ek

∂z) = (M +

3

4ml)z +

3∑

i=1

1

3ml((r

i21r

i23 + ri

11ri13)x

+(ri22r

i23 + ri

12ri13)y + ((ri

23)2 + (ri

13)2)z)

∂Ek

∂x= 0,

∂Ek

∂y= 0,

∂Ek

∂z= 0

∂Ep

∂x=

3∑

i=1

(2ki1(−

r21

li)(H1i −

r21

lix −

r22

liy −

r23

liz)

+2ki2(

r11

li)(H2i +

r11

lix +

r12

liy +

r13

liz))

∂Ep

∂y=

3∑

i=1

(2ki1(−

r22

li)(H1i −

r21

lix −

r22

liy −

r23

liz)

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+2ki2(

r12

li)(H2i +

r11

lix +

r12

liy +

r13

liz))

∂Ep

∂z= Mg +

3mlg

2+

3∑

i=1

(2ki1(−

r23

li)(H1i −

r21

lix−

r22

liy

−r23

liz) + 2ki

2(r13

li)(H2i +

r11

lix +

r12

liy +

r13

liz)) (21)

The final forward dynamic equation of this parallel multi-

body system can be written as:

M · u + K · q = Q (22)

where M, K and Q are the mass matrix, stiffness matrix and

force vector of the system, among which

M = (M +3

4ml)I +

1

3ml

3∑

i=1

Mi

Mi =

(ri21)

2 + (ri11)

2 ri22r

i21 + ri

12ri11 ri

23ri21 + ri

13ri11

ri21r

i22 + ri

11ri12 (ri

22)2 + (ri

12)2 ri

23ri22 + ri

13ri12

ri21r

i23 + ri

11ri13 ri

22ri23 + ri

12ri13 (ri

23)2 + (ri

13)2

and I is the identity matrix. In this case, the pertinent flexure

hinge stiffness has the relationship of

ki1 = k

j2 = k (i = 1, 2, 3, j = 1, 2, 3).

Hence, the stiffness matrix of the whole system is:

K = 2k

3∑i=1

r2

21+r2

11

l2i

3∑i=1

r21r22+r11r12

l2i

3∑i=1

r21r23+r11r13

l2i

3∑i=1

r21r22+r11r12

l2i

3∑i=1

r2

22+r2

12

l2i

3∑i=1

r22r23+r12r13

l2i

3∑i=1

r21r23+r11r13

l2i

3∑i=1

r22r23+r12r13

l2i

3∑i=1

r2

23+r2

13

l2i

Let Fai be the force exerted by the ith PZT actuator on each

limb at the mass center of the strut. The direct disturbance

forces FD is acting on the center of moving platform. The

generalized external force matrix is:

Q = FD+3∑

i=1

Fai+

2k3∑

i=1

( r21

liH1i −

r11

liH2i)

2k3∑

i=1

( r22

liH1i −

r12

liH2i)

2k3∑

i=1

( r23

liH1i −

r13

liH2i) − Mg − 3mlg

2

Let Fa be the matrix of driving forces Fi given by:

Fa =[

F1 F2 F3

]T(23)

where

Fai = Fici3 (24)

The fundamental natural frequency can be derived by the

global stiffness matrix and mass matrix of the system. First,

the eigenvalues λi ≥ 0 (i = 1, 2, 3) can be given by:

|K − Mλ2| = 0 (25)

The fundamental natural frequency fi is:

fi =λ

2π(26)

C. Kane’s method

In Kane’s method, the dynamics equations are concise

since unwanted reaction forces and torques do not appear

in the final form. The key point underlying Kane’s method

is the derivation of partial velocities.

1) Linearized Partial Velocities: Since the velocity of

mass center of moving platform can also be obtained by:

vo′ = x · f1 + y · f2 + z · f3 = x · s1 + y · s2 + z · s3 (27)

the partial velocities and partial angular velocities are formed

by inspection of the relevant velocity vectors:

vo′,u1= s1, vo′,u2

= s2, vo′,u3= s3

vC∗

i,u1

=s1

2, vC∗

i,u2

=s2

2, vC∗

i,u3

=s3

2, vmi,uj

= vo′,uj

ωC∗

i,u1

=ci3 × s1

li, ωC∗

i,u2

=ci3 × s2

li, ωC∗

i,u3

=ci3 × s3

li(28)

2) Generalized Active Forces: Let Fmi and Mmi repre-

sent the force and moment exerted by the ith limb on the

moving platform, at the ith flexure hinge. Since Fmi is a

noncontributing force, it can be ignored in the analysis. The

total moment Mm due to the three flexure hinges connecting

the moving platform is:

Mm =

3∑

i=1

Mmi = k

3∑

i=1

(qi1c

i10 + qi

2mi2) (29)

where ki1 and ki

2 is pertinent flexure hinge stiffness. Let

FD and MD represent the disturbance force and moment

acting on the mass center o′ of moving platform. The moving

platform’s contribution to the set of generalized active forces,

for the jth generalized velocity, is:

Qm,uj= vo′,uj

· (Mg + FD) (30)

The forces and moments acting on the ith limb are due to

the moving platform (through the ith upper flexure hinge),

gravity of each limb, the PZT actuator and the moments

through the lower flexure hinge. The contributing loads of

the limbs are as follows:

QC∗

i,uj

= vC∗

i,uj

·mlg+(li ·ci3)uj

·Fai +ωC∗

i ,uj·(Mli−Mmi)

Mli = −k(qi1c

i10 + qi

2mi2) (31)

where (li · ci3)uj

= (vo′ − liωC∗

i× c

i3)uj

= (vo′ − (ci3 × vo′) × c

i3)uj

= ((vo′ · ci3)c

i3)uj

= (sj · ci3)c

i3

3) Generalized Inertia Forces: The contribution Q∗

m,ujto

the generalized inertia forces for the jth generalized velocity

is:

Q∗

m,uj= vo′,uj

· (−Mao′) (32)

The contribution Q∗

C∗

i,uj

to the generalized inertia forces

due to the ith limb for the jth generalized velocity is:

Q∗

C∗

i,uj

= vC∗

i,uj

· (−muaC∗

i) + ωC∗

i ,uj· (−IC∗

iεC∗

i

−ωC∗

i× IC∗

iωC∗

i) (33)

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where IC∗

iis the inertia matrix of each limb with respect to

its mass center C∗

i in global coordinate system sj . Let I′C∗

i

be the inertia matrix of each limb with respect to its mass

center C∗

i in the coordinate system cij . Hence,

IC∗

i=

[ci1 c

i2 c

i3

]· I′C∗

i·[

ci1 c

i2 c

i3

]T

4) Dynamical Equations: The holonomic generalized ac-

tive force for the jth (j = 1, · · · , 3) generalized velocity

is:

Fj = Qm,uj+

3∑

i=1

QC∗

i,uj

(34)

Likewise, the contribution to the set of holonomic general-

ized inertia force is:

F∗

j = Q∗

m,uj+

3∑

i=1

Q∗

C∗

i,uj

(35)

Kane’s dynamical equations are:

F∗

j + Fj = 0(j = 1, 2, 3) (36)

Eq. (36) can be expanded as:

vo′,uj·(Mg+FD−Mao′)+

3∑

i=1

((li ·ci3)uj

·Fai +vC∗

i,uj

·ml(g

−aC∗

i)+ωC∗

i,uj

·(Mli−Mmi−IC∗

iεC∗

i−ωC∗

i×IC∗

iωC∗

i)) = 0

The equations for the inverse dynamics can be arranged to

the form:

D · Fa = H (37)

where

D =

s1 · c1

3 s1 · c2

3 s1 · c3

3

s2 · c1

3 s2 · c2

3 s2 · c3

3

s3 · c1

3 s3 · c2

3 s3 · c3

3

(38)

H =

−vo′,u1· (Mg + FD − Mao′)

−3∑

i=1

(vC∗

i,u1

· ml(g − aC∗

i) + ωC∗

i ,u1

·(Mli − Mmi − IC∗

iεC∗

i− ωC∗

i× IC∗

iωC∗

i))


−3∑

i=1

(vC∗

i,u2

· ml(g − aC∗

i) + ωC∗

i,u2


iεC∗

i− ωC∗

i× IC∗

iωC∗

i))


−3∑

i=1

(vC∗

i,u3

· ml(g − aC∗

i) + ωC∗

i,u3


iεC∗

i− ωC∗

i× IC∗

iωC∗

i))

(39)

Here we noticed that the forward dynamic result is easy to

be obtained by using Lagrange’s equation. And the Kane’s

method is a convenient and rapid approach to get the inverse

dynamic analytical solution, even though they all lead to the

same model.

Fig. 3. Displacements and accelerations of the moving platform (a).

IV. NUMERICAL SIMULATION

Due to the requirement of the cubic centimeter scale large

workspace, the parameters of the parallel mechanism are

shown in Table I. As the well elastic nature of the selected

material beryllium bronze of wide-range flexure hinges, the

workspace of micro motion only depends on the limits

of PZT ceramics. Considering the different characteristics

between flexure hinge and conventional mechanical joint, the

parasitic rotations of moving platform have to be calculated

and limited in 5mrad in this paper. The maximal usable

inscribed workspace of micro motion, when the inputs of

PZT motors are zero and self-locked at the initial position,

is a column with a radius of 66.6µm and 38.6µm height

by using PRB model considering the strokes of selected

PZT ceramics. For the details of the theoretical derivation

of macro/micro usable workspace, please refer to the [5] of

our previous work.

TABLE I

GEOMETRIC AND MATERIAL PARAMETERS

Item Value

Radius of moving platform r 20mm

Radius of fixed platform R 90mm

Radius of flexure hinge rf 0.9mm

Length of flexure hinge lf 10mm

Radius of strut rs 6.35mm

Length of strut ls 120mm

Modulus of elasticity of flexure hinge Ef 130GPa

Modulus of elasticity of strut Es 70GPa

Stroke of PZT ceramic Qc 100µm

Stroke of PZT motor Qm 10mm

A. Dynamic Simulation

In order to confirm the dynamic model, the moving

platform is moving along the z axis with the displacements

and accelerations given by Fig. 3. In this state, it is obvious

that the actuators should output the same forces as shown in

Fig. 4. The results solved by Lagrange’s equations are almost

the same with those obtained by Kane’s method.

Then another motion is simulated as following curves

shown in Fig. 5 and the three input forces of actuators

with respect to time are shown in Fig. 6. Since the mass

center of moving platform has movement along each axis,

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Fig. 4. Driving forces of the three actuators solved by Lagrange’s equationsand Kane’s method (a).

Fig. 5. Displacements and accelerations of the moving platform (b).

the three driving forces appear different in amplitude, phase

and period.

B. Natural Frequency

The distributions of three fundamental natural frequencies

are simulated when the mass center of moving platform is

located in z = 121.2435mm and moving in a plane with a

radius of 66.6µm as shown in Fig. 7 according to the Eq.

(26). The fist-order natural frequency is about 9.5811Hz, and

the second-order and third order natural frequency is about

16.478Hz.

V. CONCLUSIONS AND FUTURE WORKS

In this paper, a novel dual 3-DOF parallel robot with flex-

ure hinges has been presented. This system can achieve three

high accurate translational positioning or rough positioning

with a 3-DOF active vibration isolation and excitation func-

tion to the payload placed on the moving platform. Based

on our previous research experiences in the systematical

modeling and study of parallel robots, a vibrational model

Fig. 6. Driving forces of the three actuators solved by Lagrange’s equationsand Kane’s method (b).

Fig. 7. The three fundamental natural frequency.

of the parallel manipulator has been presented which has

been verified by Kane’s dynamics equation. Finally, some

simulation results by MATLAB are shown to verify the

correctness of the formulae derivation and natural frequency

are presented. The investigations of this paper are expected

to make contributions to the research on active vibration

isolation based on parallel manipulators. After this work,

some control modeling for active vibration control will

be given and optimal control algorithm will be discussed.

Finally, a prototype of the parallel robot will be developed,

and the experimental investigation will be carried out to

verify the vibration control performance of the prototype.

REFERENCES

[1] J. Hesselbach, A. Raatz, J. Wrege, and S.Soetebier, “Design and anal-ysis of a macro parallel robot with flexure hinges for micro assemblytasks,” in Proc. of 35th International Symposium on Robotics, Paris,France, 2004, No.TU14-041fp.

[2] P. R. Ouyang, “Hybrid intelligent machine systems: Design, Modelingand Control,” Ph.D. Thesis, University of Saskatchewan, Canada,2005.

[3] W. Dong, L. N. Sun, and Z. J. Du, “Design of a precision compliantparallel positioner driven by dual piezoelectric actuators,” Sensors and

Actuators A: Physical, Vol. 135, pp. 250-256, 2007.[4] X. Tang and I. M. Chen, “A large-displacement 3-DOF flexure

parallel mechanism with decoupled kinematics structure,” in IEEE/RSJ

International Conference on Intelligent Robots and Systems, Beijing,China, pp. 1668-1673, 2006.

[5] Y. Yun and Y. Li, “Performance Analysis and Optimization of a NovelLarge Displacement 3-DOF Parallel Manipulator”, in Proc. of IEEE

International Conference on Robotics and Biomimetics, Bangkok,Thailand, pp. 246-251, 2008.

[6] R. D. Hampton and G. S. Beech, “A Kane’s Dynamics Model for theActive Rack Isolation System”, NASA/TMł2001C2110063, MarshallSpace Flight Center, AL, 2001.

[7] Y. Yun, Y. Li and Q. Xu, “Active vibration control on A 3-DOF parallelplatform based on Kanes dynamic method”, in Proc. of SICE Annual

Conference, Chofu, Tokyo, Japan, pp. 2783-2788, 2008.[8] L. W. Tsai and S. Joshi, “Kinematics analysis of 3-DOF position

mechanisms for use in hybrid kinematic machines,” ASME Journalof Mechanical Design, Vol. 124, No. 2, pp. 245C253, 2002.

[9] Y. Li and Q. Xu, “Kinematic analysis and dynamic control of a 3-PUUparallel manipulator for cardiopulmonary resuscitation,” Proc. of 12thInt. Conf. on Advanced Robotics, pp. 344C351, 2005.

[10] M. S. Whorton, H. Buschek and A. J. Calise, “Homotopy Algorithmfor Fixed Order Mixed H2/H∞ Design,” Journal of Guidance,

Control, and Dynamics, Vol. 19, No. 6, pp. 1262C1269, 1996.[11] M. Whorton, “Robust Control for Microgravity Vibration Isolation Us-

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and Control Conference and Exhibit, Austin, Texas, Aug. 11-14, 2003.

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31 multi-degree of freedom vibration model for a 3-dof hybrid robot

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