A Kinematic Design of 3-PRS Precision Compliant Parallel Platform

Zhiping Kong

1, 2, a

, Wei Zhang

1, 2, b

, Haibo Zhou

1, 2, c

School of Mechanical and Electrical Engineering, Central South University, Changsha, China

State Key Laboratory of High Performance Complex Manufacturing, Central South University, Changsha, China

Keywords: Parallel platform, Kinematics analysis, Large-stroke flexure joints.

Abstract: A kinematic design of the 3-PRS compliant parallel platform with wide-range flexure hinges is presented in

this paper. The manipulator is driven by the piezoelectric motors, which can show the superiority of parallel

platforms and large-stroke flexure hinges. First, taken into account the parasitic motions of the platform, the

kinematics solution is proposed. Then, the influence of parameters to the rotation capacity and relative

rotation error that are used to assess the quality of the revolute and spherical flexure joints. Also, the

reachable travel is observed by ANSYS Workbench. Finally, compare with the FEM and the theoretical

results by several different cases to ensure the efficiency of the kinematics model.

1 INTRODUCTION

High-precision positioning devices are often

required in different areas as scientific, medical and

communication fields for micro-system production

(Ruiz, et al., 2016). For these high-tech applications,

it is necessary to overcome the problems they

usually appear, as clearances, friction and wear,

which greatly affect the accuracy of the platform (Li

and Xu, 2007). A well-known solution for these

problems was put forward in (Bhagat U, et al., 2014),

which introduced the compliant machines with the

flexible hinges to improve the motion accuracy.

Parallel machine were first applied in the test of

tire by Gough, and later were introduced by Stewart

as the simulators (Stewart and D, 1965). Then the

Stewart structure began to be widely used in many

fields. Parallel manipulators with 6-DOF have many

advantages. Dong et al established a kinematics

model via analysing the stiffness model of the whole

6-PSS platform based on FEA method (Dong, et al.,

2005). Recently, parallel platforms with less than 6-

DOF have appeared in many different fields.

Rouhani et al developed a method on the basis of

elastokinematic analysis of a microhexapod

manipulator, in which the elastic deformation of the

flexure hinges was considered for the kinematic

analysis (Rouhani, et al., 2016). Meanwhile, It is

possible to develop 2-DOF mechanisms, as for

example the stage presented in (Huang and Dao

2016), where the stage with flexure elements allows

large displacement along x and y direction. Also, the

paper provides some useful steps for designing and

analyse by FEM. Many 3-DOF parallel manipulators

also have been designed and applied for relevant

areas. For example, the 3-UPU parallel robots with

pure translational motions in (Gregorio, et al., 1998),

other sample of this type is shown in (Liu, et al.,

2005), where the parallel manipulator HANA has

three DOF with one rotational and two translational

degree of freedom. Another 3-DOF mechanism is

presented in (Li,et al., 2005; Zhan, et al., 2005),

where the kinematic of the 3-RPS parallel

mechanism is analysed. Moreover, the necessary

steps to achieve a good design and characterization

are presented. Also 3-PUU mechanism with 2R1T is

proposed in (Wang, et al., 2016), where the mobility

analysis of the platform is carried out by the

reciprocal screw theory.

The 3-PRS parallel platforms were widely

designed and analysed in (Ruiz, et al., 2016; Li and

Xu, 2007; Jia, et al., 2016; Li, et al., 2016; Tsai, et

al., 2003). The 3-PRS parallel platform with large-

stroke flexure joints introduced in this paper. At

present, it is still a challenging task to solve the

design and kinematic problem with flexible joints.

For the mechanisms with flexible joints, they can

overcome many problems, but they also faced some

challenge, like the deformation of the joints and the

parasitic motions. Therefore, it is necessary to study

the design, kinematics and validate these compliant

parallel platform with flexible joints.

Kong, Z., Zhang, W. and Zhou, H.

A Kinematic Design of 3-PRS Precision Compliant Parallel Platform.

DOI: 10.5220/0008849403410346

In Proceedings of 5th International Conference on Vehicle, Mechanical and Electrical Engineering (ICVMEE 2019), pages 341-346

ISBN: 978-989-758-412-1

341

The present will develop the conceptual design,

kinematics study, FEM validation of the 3- PRS

parallel mechanism. Firstly, the design of the 3-PRS

parallel platform are presented in Section 2. Then

the kinematics are analysed in Sections 3, where the

rigid body kinematics of the 3-PRS will be

developed. The FEM of the large-stroke joints will

be explained in section 4, where the kinematics

simulation are concluded. Finally, some concluding

remarks are given in Section 5.

2 SYSTEM DESCRIPTION

The structure of a 3-PRS parallel manipulator is

composed of a moving platform, a fixed base, and

three identical supporting limbs, which is shown in

Figure 1. Each kinematic limb consists of two wide-

range flexure hinges and a rigid rod in the middle.

As shown in Fig.1, one wide-range flexure hinges is

spherical joint (S) that can rotate around three axes,

another is revolute joint (R) that can rotate around

one axe. They can not only have high accuracy, but

also achieve a long strike motion for parallel

platform. The driving components as active joint (P),

and a piezoelectric motor is utilized as actuator,

which has many advantages as high precision, high

driving force and large travel. Here, beryllium

bronze and duralumin are used as the material of the

long-stroke flexible joints and the others.

Figure 1. Basic configuration of the 3-PRS parallel

manipulator.

The main parameters of the platform are

obtained, as shown in Table1.

Table 1. The structure parameters of the 3-PRS parallel

manipulator.

Item

Value

Diameter of moving platform

50mm

Angle between the kinematic chain and

the fixed platform

65°

Length of rigid rod

76mm

Diameter of rigid rod

10mm

chamfering radius of S joint and R joint

1mm

Diameter of S joint

1mm

Length of S joint and R joint

12mm

Thickness of R joint

0.4mm

Width of R joint

8mm

3 KINEMATICS ANALYSIS

The structure of the 3-PRS parallel mechanism is

shown in Figure 2. Three coordinate systems are

considered: the local coordinate systems P(xyz) in

the center of the moving platform; and the global

coordinate systems O(xyz) in the center of the fixed

platform. The position of the center of the moving

platform can be expressed by px , py and pz, and its

direction can be represented by the angles α, β, and γ,

which are the rotation angle around the x, y and z

axes, respectively..

Figure 2. One limb of structure with coordinate systems of

the platform.

Therefore, the displacement vector of the center

of the moving platform in the moving frame P can

be expressed as:

p x y z

d p p p

  







(1)

Also, the position vectors of the S joints Ci (i=1,

2, 3) in P(xyz) frame can be defined by:

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[ 0 0]

[ / 2 3/2 0]

[ / 2 - 3/2 0]

PC r

PC r r





(2)

Then, it is possible to obtain the rotation matrix

T from the moving frame P(xyz) to the reference

frame O(xyz), which can be expressed as:

     

x y z

x x x

y y y

z z z

T R R R

u v w

  













(3)

The position vectors Ci (i=1,2,3) in the reference

frame O can be defined by:

OC OP T PC  

(4)

Substituting Eq.(2) and Eq.(3) into Eq.(4), we

can obtain:

2 1 1 2 2 3 3

3 1 1 2 2 3 3

[ ]

x x y y z z

x y z

OC p u r p u r p u r

OC p u v p u v p u v

   

      

      

(5)

Where



The position vectors of the R joints

( 1,2,3)

Bi

in O frame can be expressed as:

[ 0 ]

[ / 2 3/2 ]

[ / 2 - 3/2 ]

OB R d

OB R R d





(6)

Where

 

3,2,1id

represented the translation

of the passive joint.

Substituting Eq.(5) into Eq.(6) yields, the

parasitic motions can be obtained:

s s

-atan

















(7)

 

c c s s s -c c

      



(8)

 

c s c s s

    

  

(9)

From the Figure 2, we can know the relation:

iiiii

OBOCBCL 

(10)

Where

and

are expressed in Eq.(5) and

Eq.(6).

Since

 

3,2,1id

is the only unknown variable

in Eq.(6), solving Eq.(10) allows us to obtain the

solutions for the inverse kinematic problem.

 

i iz ix ix iy iy

d OC l OB OC OB OC     

(11)

4 SIMULATION RESULTS

4.1 Simulation Design of the Large-

Stroke Flexure Joints

There are many types of flexible hinges. There are

five common types: V-shaped, hyperbolic, parabolic,

elliptical, and rounded straight beam flexible hinges,

which correspond to 1~5 in Figure 3 below. For the

same material, the same boundary and load

conditions of 5N, the deformation and stress of the

flexible hinges by ANSYS workbench were shown

in Figure 4. The maximum displacement is in the

order of: 5 > 4 > 3> 2> 1. The larger the maximum

displacement of the flexible hinge, the more helpful

it is to meet the large stroke requirement of the

motion platform. By comparison, a rounded straight

beam type flexible hinge has better structural

characteristics and mechanical properties.

Figure 3. The section type of the flexible hinge.

A Kinematic Design of 3-PRS Precision Compliant Parallel Platform

343

A theoretical mechanical model is established for

the flexible hinge, and the evaluation indexes

includes rotational capacity θ and relative rotational

error s, which can be seen in (Shi, et al., 2013).

Figure 4. The deformation and stress of the flexible hinges.

It can be seen that the main parameters affecting

the performance of the flexible spherical joint are:

the length of the hinge deformation part L and the

minimum diameter of the middle elongation t. A

group of typical values L=12mm, t=1mm are

selected, then change a variable near the typical

value with other variables fixed , the data graphs of

the influence on rotational capacity and relative

rotational error are shown in Figure 5.

Figure 5. The performance of the flexible spherical joints

in terms of L and t.

In the same way, the main parameters affecting

the performance of the flexible revolute joint are the

length L, the minimum thickness t, and the width b.

A group of typical values L=12mm, t=0.4mm, and

b=8mm were selected, and the data graphs of

influence are shown in Fig 6.

Figure 6. The performance of the flexible rotating joints in

terms of L, t and b.

As shown in Figure 5 and Figure 6, with the

increase of the length of the joints, the rotational

capacity and the relative rotational error increased.

The increase in the diameter of S joints, the

thickness and width or R joints causes the increased

rotational capacity and the educed relative rotational

error. At the same time, the influence of the hinge

size on the stiffness and stroke of the platform is

also analyzed. The influence on the stroke of the

platform and the law of the hinge size and the

rotation ability are the same. By contrast, the

influence on the stiffness of the platform is opposite

to the rotation ability.

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4.2 Inverse Kinematics Simulation

As shown in Table 2, the five pose values of the

given platform are input into the pseudo rigid body

kinematics model and the FEM model respectively

to obtain the displacement of the driver. It can be

seen from the above table that the results of Matlab

and Ansys calculations are very small, the error is

within 2%, which verifies the correctness of the

kinematics model, and also shows that the flexible

parallel motion platform is regarded as a rigid body

under small deformation conditions. The feasibility

of the analysis. And the motion range of the moving

platform are in Table 3.

Table 2. Comparison of FEA and Theoretical analysis.

z,α,β

(mm,°,°)

Theoretical

d1,d2,d3

(mm,mm,mm)

FEM

d1,d2,d3

(mm,mm,mm)

(5,0,0)

(5,5,5)

(0,0,0.5)

(-0.2373,0.1193,

0.1193)

(-0.2385,0.1208,

0.1207)

(0,0.5,0)

(-0.0001,0.2070,

-0.2056)

(0.0005,0.2091,

-0.2066)

(0,1,0.5)

(-0.2383,0.5355,

-0.2931)

(-0.2367,0.5425,

-0.2912)

(0.5,0.5,1)

(0.0264,0.9447,

0.5309)

(0.0262,0.9532,

0.5344)

Table 3. The motion range of the moving platform.

Positive

limit

Negative

limit

Angular displacement

around x axes(°)

9.2

-9.3

Angular displacement

around y axes(°)

-9.5

Linear displacement

along z axes(mm)

-10

4.3 Forward Kinematic Simulation

Given a trajectory of the platform,

)2sin(5.0 t





calculate the movements of the driving units by the

FEA model and the inverse kinematic model. Then

the values were employed to the forward kinematic

model by FEA respectively. The movements in the

actuator are shown in Figure 7, and the trajectory of

the platform by theoretical modal and FEA modal

are shown in Figure 8. The comparison between two

models shows that the maximum positioning error is

0.0075 degrees in the range of 0.5 degrees. The

results shows that the errors between two models are

very small.

5 CONCLUSION

In this paper, a new 3-PRS compliant micro parallel

platform with large-stroke hinges is presented. The

platform is driven by a piezoelectric motor, and the

Figure 7. The movements of the actuators.

Figure 8. The end pose of the platform.

wide-range flexure hinges including spherical hinges

and rotating hinges, which can improve the accuracy

and increase travel. The FEA model of the large-

stroke hinges are developed to analyze the

characteristics and mechanical properties.

Meanwhile, by using of the ANSYS Workbench,

obtain the workspace of the manipulator, which can

reach about 18° around x axes, 18° around y axes,

more than 20mm along z axes depending on the

travel of the drive part. An inverse kinematic

solution is given, several comparison results

between theoretical and simulation show that the

model is effective for modeling compliant parallel

platforms with wide-range flexible joints.

A Kinematic Design of 3-PRS Precision Compliant Parallel Platform

345

ACKNOWLEDGMENTS

This work was supported in part by the National Key

Research and Development Program of China under

Grant 2017YFB1104800, in part by the National

Natural Science Foundation of China under Grant

51575534.

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