MODEL AND APPROACHES FOR TENSION OF PARALLEL

STRUCTURES WITH ELASTIC JOINTS FOR

MICRO AND NANO MANIPULATORS

D. Chakarov, K. Kostadinov and T. Tiankov

Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., bl. 4, 1113, Sofia, Bulgaria

Keywords: Piezo-Actuators, Serial-Parallel Micromanipulator, Elastic Joints, Stiffness Model, Preliminary Tension,

Cell Injection.

Abstract: In this paper piezo actuated micromanipulators are considered with serial-parallel structure including elastic

joints. Such structure allows a preliminary tension of the mechanical system in order to eliminate backlashes

and to improve the performance of the piezo-actuators. A kinematics model of a serial-parallel structure for

local micro manipulators is build here. A pseudo rigid body approach is used, where elastic joints are

modelled as revolute joints. A stiffness model is created to estimate the general stiffness of the manipulator

by means of reduction the stiffness of all elastic joints. Two approaches are presented here for preliminary

tension of parallel manipulator structure: - deflection from the initial manipulator state by introducing of a

driving joints motion during the assembly; - preliminary tensioning of the separate elastic joints. The two

approaches considered are experimented on the manipulator for cell injection. The values of the mechanical

parameters obtained by preliminary tension of the manipulator are pointed out.

1 INTRODUCTION

Micro and nano manipulators are mostly used in

biological and microelectronics research, cellular

technology, chemistry and investigation of thin

films, in atomic force microscopes and scanning

tunnelling microscopes.

There are known micromanipulators with piezo

actuators (Fatikow, 1996; Kasper, 1998). Piezo

actuated micromanipulators with parallel structure

are also developed (Lee, 1999). Robots with parallel

structure possess many advantages. Their small

workspace in the case of cell manipulations is not a

disadvantage, since it is enough for the application

considered. Mechanisms with closed kinematic

chains (Ionescu, 2002; Guergov, 2005, Prusak,

2009) are suitable for high-precision tasks in 3D

space. The high accuracy of such mechanical

systems is due to very high structural stiffness.

From the other hand in order piezo-ceramic

structures to be with high stiffness they can be

realized by parallel or a closed structure which has

to be tensed. It is possible to use deformation in

elastic joints or antagonistic redundant actuators to

achieve tension in closed piezo-ceramic structures

with desired degree of freedom (DoF).

Тo predict the displacements of compliant

mechanisms with elastic joints the pseudo-rigid-

body-model is commonly used (Zhang, 2002). As a

rule it models an elastic joint as a revolute joint with

a torsion spring attached. The pseudo-rigid-body

method is effective and it simplifies the model of

compliant mechanisms. To estimate the mechanism

stiffness with elastic joints an analytical model is

created out taking into account compliances of

elastic joints in all axes. The analytical model is

describing the relationship between input and output

displacements of the mechanism, (Pham, 2005) or

computing the stiffness matrix and estimating the

stiffness performances of the robot (Carbone, 2006).

To increase accuracy of the stiffness matrix

identification alternative methodology is developed

using advantages of analytical and numerical

techniques (Pashkevich, 2009). Such analytical

stiffness models of serial-paralel manipulators with

elastic joints are analyzed in order to synthesize

desired stiffness of the robot end-effector (Chakarov,

2004).

The objective of this paper is to create a stiffness

model and to develop approaches for tension of

serial-parallel structures with elastic joints for micro

135

Chakarov D., Kostadinov K. and Tiankov T.

MODEL AND APPROACHES FOR TENSION OF PARALLEL STRUCTURES WITH ELASTIC JOINTS FOR MICRO AND NANO MANIPULATORS.

DOI: 10.5220/0002171101350140

In Proceedings of the 6th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2009), page

ISBN: 978-989-674-000-9

and nano manipulators with application in cellular

technology, microelectronics, chemistry etc.

2 KINEMATIC MODEL OF

PARALLEL STRUCTURES

FOR LOCAL MICRO AND

NANO MANIPULATORS

Investigated structures are serial-parallel structures

including basic link 0 and some other links 1, … , n

connected in between in a serial chain. The end-

effector M is situated in the end link n of this chain,

which moves in a ν operation space. The driving

chains A

,…, A

, with number m, are attached to

the basic link 0 and to the end link n forming parallel

chains [Chakarov, 2007] as it is shown in Fig.1. The

type of the kinematics joints is not shown in Fig.1,

as they can be elastically or kinematically ones.

Figure 1: Generalized kinematic scheme of a serial-

parallel manipulator.

All joints are modelled as kinematic joints with

different number of restrictions, which give 6 DoF

for each drive chain. In this way the number DoF of

the structure is defined by the number DoF of the

serial chain h. Let generalized parameters are

accepted to be the parameters of the relative motions

in all joints - elastic and non-elastic of the structure,

presented by (k x 1) vector

]q;q[=θ

(1)

Where

q = [q

, …., q

]

(2)

is an (h x 1) vector of the generalized coordinates in

the joints of the main serial chain with h DoF and

]l;w[q =

(3)

is a (6m x 1) vector of coordinates in the joints of

the actuator chains with number m.

Above

w = [w

,..., w

]

, (4)

is an (5m x1) vector of coordinates in the passive

joints of the actuator chains, and

l = [l

,..., l

]

, (5)

is an (m x 1) vector of coordinates in the motor

linear joints of the actuator chains.

Let the Cartesian coordinates of the end effector

M are denoted as

[

]

6,X,...,XX

ν1

≤ν=

(6)

The relation between the parameters of the basic

serial chain (2) and the parameters of the end

effector (6) is known as a direct problem of the

kinematics of the serial chain. This problem on the

level of displacements and velocities is presented by

the equations

Ψ(q)X

and

qJX

(7)

where

[

]

qXJ

∂

is the (ν x h) matrix of Jacoby.

In the parallel structure each closed loop implies

the appearance of a connection between the

generalized parameters (1). These connections are

expressed by 6m scalar functions for the structure

including m parallel loops:

1,...,6mi ==θΨ ,0)(

The differentiation of above equations gives the

relation

lwq

∂

(8)

The matrix of partial derivations H

, H

and H

with size (6m x h), (6m x 5m) and (6m x m) allows

to produce the summarized matrix of the partial

derivatives

L;W;E

;

D =

∂

(9)

where E is unitary (h x h) matrix, W is a (5m x h)

matrix and L is a (m x h) matrix, or

H;ED =

According to (8) we can reduce the (6m x h)

matrix

HH;H

L;WH

−

−=

∂

== (10)

where

H;H

is a (6m x 6m) invertible matrix.

Using matrix (10) we have the relations between

generalized velocities:

ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics

136

qHq

(11)

qWw

(12)

qLl

(13)

The above equations allow determining the

velocities

with dimension 6m as a function of the

generalized velocities

with dimension equal to the

DoF h of the structure.

When the number of parameters (5) is equal to

the DoF m = h, these parameters can be selected as

independent parameters. In relations (13) L is a (h x

h) matrix and inverse relation is possible:

lLq

−

(14)

Equations (7) and (14) allow determining the

velocities of end-effector, while equations (11) and

(14) - the velocities of passive joints, as function of

velocities of linear actuator joins

lJLX

−

(15)

and

lHLq

−

(16)

By micromanipulations the above equations give

the relations between small motions of

microactuators Δl, small motions of the end-effector

ΔX and small motions in passive joints

Δq

lJLX

Δ=Δ

−

and

lHLq

Δ=Δ

−

3 STIFFNESS MODEL OF

SERIAL – PARALLEL

STRUCTURES FOR MICRO

AND NANO MANIPULATORS

Denote by P = [P

, ... ,Pν ]

the (ν x 1) vector

of the external forces and torques applied to the end-

effector, corresponding to Cartesian coordinates (6).

Denote by Q = [Q

,..., Q

]

the (h x 1) vector

of the generalized forces and torques in the joints of

the main chain corresponding to the general

coordinates (2). According to the

principle of virtual

work and equation (7), the

connection between

forces P and Q is as follows:

Q = J

P (17)

Denote by

qh1qq

F;...;FF =

and

m5w1ww

F;...;FF =

(h x 1) and (5m x 1) vectors

of the forces and torques in the elastic joints,

corresponding to coordinates (2) and (4). Denote by

lll

F;...;FF = the (m x 1) vector of the driving

forces in the linear joints correspond to the

coordinates (5). Above vectors can be summarized

in the (h + 6m) x 1 vector of forces and torques,

corresponding to the coordinates (1) F= [F

; F

]

According to the principle of virtual work and

the equation (12), (13) the relation between forces F

and generalized forces Q, using summarized matrix

(9), is as follows:

Q = D

F (18)

Q = F

+ W

+ L

(19)

Equations (17) and (19) produce

P = F

+ W

+ L

(20)

Differentiation of above equation with respect to

parameters (2) and neglect the second partial

derivatives, gives

∂

(21)

Considering micromanipulator structure as a

system with concentrated compliance in the joints

[Chakarov, 2004] gives

J]LKLWKWK[JK

−−

++=

(22)

where

XPK

∂

is (ν x ν) matrix of the Cartesian

stiffness of the end effector;

qFK

∂∂=

diagonal (h x h) matrix of the shaft stiffness in the

joints of the main serial chain;

wFK

∂∂=

diagonal (5m x 5m) matrix of the shaft stiffness in

the passive joints of the driving chains;

lFK

∂∂=

is diagonal (m x m) matrix of the

shaft stiffness in the driving joints.

4 APPROACHES FOR

PRELIMINARY TENSIONING

OF PARALLEL MICRO

MANIPULATORS WITH

ELASTIC JOINTS

A preliminary tensioning of the mechanical

micromanipulation system is necessary in order to

eliminate the backlash and to improve the

MODEL AND APPROACHES FOR TENSION OF PARALLEL STRUCTURES WITH ELASTIC JOINTS FOR MICRO

AND NANO MANIPULATORS

137

performance of the piezo-actuators. When only

joints of class five are used for the modeling the

mechanical system, the number of all joints is equal

to the number k of the generalized system

coordinates. In a case when number m of the driving

joints is equal to the number of DoF h (m=h), then

number of all the remaining joins is (k - h).

The following two approaches can be used for

tensioning of the manipulator:

- deflection from the initial manipulator state by

m = h driving joints motion introduced in the

assembly;

- preliminary tensioning of the separate elastic

joints with number j, (k-h ≥ j ≥ h).

4.1 Tensioning by Means of Deflection

from the Initial State

This can be achieved by means of an assembly

deflection δl in the driving joints, which leads to

deflection in all the system joints according to (14),

(12) and deflection of the end-effector according to

(

7) defined by the equations:

lLq

δ=δ

−

(23)

lWLqWw

δ=δ=δ

−

(24)

lLJqJX

δ=δ=δ

−

(25)

These deflections lead to elastic joints forces

defined by the equations:

lLkqkF

qqq

δ=δ=

−

(26)

lWLkwkF

www

δ=δ=

−

(27)

where k

and k

are stiffness matrices of the passive

joints of the basic serial chain and of the driving

chains, respectively. The tensioned elastic system

according to (19) is in a static equilibrium:

Q = F

+ W

+ L

= 0 (28)

The diagonal matrices k

and k

contain non-

zero components, responding to elastic joints and

zero components responding to kinematic joints. The

number of non-zero components j must be bigger or

equal to the DoF j ≥ h in order to achieve full degree

of tension of all the actuators and limbs within the

system. Equation (28) allows definition of the forces

of the driving joints F in number h as a function of

the forces F

, F

in number j ≥ h

= -L

-Т

+ W

] . (29)

4.2 Tensioning by Deformations in the

Elastic Joints

In the manipulator structure with m driving joints

there are k - m passive joints, which can be elastic.

Because the driving joints m = h, by means of which

the piezo-actuators are modelled are hundreds of

times more rigid then the elastic manipulator joints,

it is accepted that the system has 0 DoF. The

tensioning of the elastic joints does not lead to a

change in the manipulator position, but only in a

change of the internal forces. For the actuator

tensioning, the number of the elastic joints j must be

bigger than the number of the DoF j ≥ h. The

preliminary joint deformations can be defined by the

vectors:

δq

= [δq

, …., δq

]

(30)

δw

= [δw

,..., δw

]

, (31)

where the components of which corresponding to

non-elastic (kinematic) joints are equal to 0.

The joint stiffnesses are represented by the

diagonal matrices k

and k

which contain non-zero

components corresponding to the elastic joints and

zero components connected to the kinematic joints.

The preliminary deflections lead to appearance

of forces in the elastic joints defined by equalities:

qkF δ=

(32)

wkF δ=

(33)

The driving elastic joints forces are in a static

equilibrium (19):

Q = F

+ W

+ L

= 0 (34)

The upper equation defines the links among all

the joint forces and allows the derivation of the

driving joints forces in number h as a function of the

elastic forces in number j≥h

= -L

-Т

+ W

] . (35)

The forces in the driving joints can lead to small

deflections in those joints due to the piezo-actuators

high stiffness. Those deflections by the two

approaches can be defined by the equality:

l =δ

(36)

where the diagonal matrix k

includes the stiffness of

the piezo- actuators, and δl

are the deflections in the

driving joints after the system tensioning. Resulting

deflection of the end effector can be received

according to equation (15).

ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics

138

5 NUMERIC

EXPERIMENTATION OF

TENSIONING APPROACHES

IN ROBOTS FOR CELL

INJECTION

A robot has been designed to perform automatic cell

injection. The cells in the range of 10-30 [

μm] are

preliminary positioned in a matrix G. Local robot

structure has serial-parallel structure with 3 DoF as

shown in Fig.2. Base 0, elastic joint J

, manipulator

body 1, actuator А

and working tool 2 with end-

effector M form a serial chain. Actuators А

and А

are located perpendicularly to the manipulator body

1, and they are linked with the base 0 by means of

elastic joints J

and J

, thus forming parallel chains.

The actuators are fixed to the body 1 via universal

joints. Parallel structure comprising actuators А

and

perform orientation motions, while the actuator

performs injection through the pipette 2 attached

to it. The main dimensions of the manipulator are

=0.073[m], b

=0.030[m], b

=0.180[m] as

shown in Fig.2. Piezoactuators and elastic joints

used have parameters specified in Table 1.

Table 1: Parameters of the elements used.

Elements

ravel

[μm]

Resolu-

tion

[nm]

Axial

tiffness

[N/μm]

Reduced Axial

stiffness [N/μm]

Angular

stiffness

[N/rad]

30 0.6 27 23.79

60 1.2 15 13.95

200 40

The end effector stiffness and the characteristics

of the preliminary tensing of the parallel structure

can be found. Since the manipulator under

consideration is assembled with a special rectangle

configuration we can easily derive the scalar

equalities for the characteristic components using

the matrix equalities (22), (23), (24), (25), (29).

Software application based on these matrix

equalities is developed using Microsoft Visual

Studio.Net Express Edition and С++.

To find an estimation of a stiffness component

along axis X, the software application substitutes the

respective matrices in equality (22) for

=40[Nm/rad], J=b

, K

=23.788[N/m], W=b

=40[Nm/rad], L=b

Performing the respective

calculations, we find for the three additives in (22)

that К

=1235+209+660778=0.662 10

[N/m].

Similarly, the rest of the end-effector stiffness

components can be found: K

= 0.662 10

[N/m],

=13.95[N/m]. As seen, the influence of piezo

actuators to the joint stiffness is hundred times larger

than the rest of the elements.

Figure 2: Microrobot for cell injection.

To apply preliminary tension by the actuator,

following the approach outlined in Sub-paragraph

3.1., actuators А

and А

in the parallel structure

should deflect by δl

and δl

, so that the elastic joint

deflection J

should not exceed +/-0.5

, which is the

admissible arbitrary rotation angle. Thus,

δq

=δq

=0.008726[rad].

Using scalar equalities corresponding to (23),

(24), (25), (29) as outlined above, we find the

components of the elastic joint deflections and those

of the actuator tension forces – see Table 2.

To attain preliminary tension in the manipulator

as outlined in Sub-paragraph 3.2, the deflections of

the three elastic joints J

, J

and J

should be less than

or equal to the admissible angles of rotation δw

=δw

=0.008726[rad], δq

=δq

=0.008726 [rad].

Considering equality (35) and its scalar forms,

we find the tension forces of actuators A

and A

, Fl

= Fl

= -16.42[N]. These forces are larger than the

tension forces found by applying the first approach,

due to the tension of joints J

, J

that is equal to the

admissible limit.

Both cases of tension yield small

actuator deformations which can be found using

equality (36).

Thus, considering the first case, those

deformations are δl

= δl

=-0.572.10

-6

[m], while

for the second case we have δl

=δl

=-0.690.10

-6

[m]. The deformations yield deflection of the end

effector with respect to its initial position, equal to

δX

=δY

= -3.432.10

-6

[m] in the first case and to

δX

=δY

= -4.142.10

-6

[m] for the second case.

MODEL AND APPROACHES FOR TENSION OF PARALLEL STRUCTURES WITH ELASTIC JOINTS FOR MICRO

AND NANO MANIPULATORS

139

Table 2: Calculated values of the mechanical parameters.

[m] δq

[rad] δq

[rad] δw

[rad] δX [m] δY [m] Fl

[N] Fl

[N]

δl

= 0.263 10

-3

8.72 10

-3

0 0 0 3.589 10

-3

0 1.572.10

-3

0 -13.6 0

δl

= 0.263 10

-3

0 8.72 10

-3

3.589 10

-3

0 0 0 0 1.572.10

-3

0 -13.6

6 CONCLUSIONS

Piezo actuated micromanipulators with serial-

parallel structure including elastic joints are subject

of this paper. A kinematic model of the micro

manipulators is build using a pseudo rigid body

method, where elastic joints are modelled as

revolute joints.

A stiffness model is created to estimate the

manipulator stiffness by stiffness reduction of all

elastic joints. In order to eliminate the backlash and

to improve the performance of the piezo-actuators

the parallel structure of the manipulator must be

tensed preliminary. Two approaches are proposed

and presented here for preliminary tension of

parallel manipulator structure:

1. Deflection from the initial manipulator state

by driving joints motion implemented in the

assembly;

2. Preliminary tensioning of separate elastic

joints.

The first approach is easier for realization as the

deviations are performed only in the driving joints.

The elastic joints are tensed to a different degree.

Tensioning of all the elastic joints is realized to

the admissible values by the second approach. Thus,

the maximal values of the tensioning forces are

achieved

The first approach is chosen for the robot

developed for cell injection. The actuators A

and A

are mounted with deviations related to the values

shown in the first column of Table 2. The robot with

mechanical construction preliminary tensioned

experimented achieves motion along the axes X, Y и

Z as follows: ΔX=ΔY= 180

[μm], ΔZ=60 [μm].

Minimal displacement obtained experimentally is

30[nm]

Further numeric experiments and tests are under

consideration for more stiff elastically joints and

structures with arbitrary location of the actuators.

ACKNOWLEDGEMENTS

This work was funded by the European Commission

through the FP6 Integrated Project HYDROMEL

with contract No. FP6 NMP2-CT-2006-026622, to

which the authors are expressing their

acknowledgements.

REFERENCES

Fatikow, S., Munassypov, R., 1996, An Intelligent

Micromanipulation Cell for Industrial and Biomedical

Applications Based on a Piezoelectric Microrobot, 5

Int. Conf. on MicroElectro, Opto, Mechanical Systems

and Components, Berlin, 17-19 Sept., pp. 826-828.

Kasper, R.; Heinemann, W.; Wagner, A., 1998, Modelling

and Control for Piezoelectric Actuators for High

Speed Applications. 4

Inl. Conf. on Motion and

Vibration Control, Vol. 1, ETH Zürich, 231-236.

Lee, K., Arjunan, S., 1999, A three-degree-of-freedom

micromotion in-parallel actuated manipulator, IEEE

Trans. on “Robot. and Autom.”, vol. 7, 238-247.

Ionescu Fl., Kostadinov, K., 2002, Piezo actuated micro

robot for micro and nano manipulations, ARA Journal,

Vol. 2000-2002, Nr.25-27, Montréal, 98-103.

Guergov, S., 2005, Multifunctional technological system

with complex functions, reconfigurable structure, high

level of integration and self-organizing elements.

Proc.of V Int. Congress“Computer Science for Design

and Technology”, STANKIN, Moscow, 31-36.

Prusak, D., Tadeusz Uhl, 2009, Novel Type of Hybrid 3-

DOF Micromanipulator with Piezoelectric Actuators -

Mechanical Construction and Simulations, Solid State

Phenomena, Vols. 147-149, pp25 -30.

Zhang, W., Zou, J., Watson, L., Zhao, W., 2002, The

constant- jacobian method for kinematics of a three

DOF planar micro-motion stage, Journ.of Robotic

Systems 19(2). 63–72.

Pham, H., Chen, I., 2005, Stiffness modeling of flexure

parallel mechanism, Precision. Engineering 29, 467–478.

Carbone, G., Lim, Hun-ok, Takanishi, A., Ceccarelli, M.,

2006, Stiffness analysis of biped humanoid robot,

Mechanism and Machine Theory, 41, 17–40.

Pashkevich, А., Chablat, D. ,Wenger, Ph., 2009, Stiffness

analysis of overconstrained parallel manipulators,

Mechanism and Machine Theory, Vol. 44, 966–982.

Chakarov, D., 2004, Study of the Antagonistic Stiffness of

Parallel Manipulators with Actuation Redundancy,

Mechanism and Machine Theory, Vol. 39/6, 583-601.

Chakarov, D., Abed Al-Wahab, M., Kasper, R.,

Kostadinov, K., 2007, Synthesis of tense piezo

structures for lokal mikro & nano manipulations. 8

Magdeburg days of mechanical engineering, Oct.10-

11, Magdeburg, pp.173-180.

ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics

140