Hội nghị toàn quốc về Điều khiển và Tự động hoá - VCCA-2011

VCCA-2011

Structural Synthesis of Decoupled Parallel Mechanisms

Victor Glazunov*, Nguyen Minh Thanh, Tran Quang Nhat

*Mechanical Engineering Research Institute, Russian Academy of Sciences

Hochiminh City University of Transport, Vietnam

e-Mail: vaglznv@mail.ru, minhthanhnguyen@vnn.vn,

tranquangnhat@hcmutrans.edu.vn

Abstract Structural synthesis of decoupled parallel mechanisms

with three parallel kinematic chains is considered.

The synthesis of these manipulators is carried out by

means of screw groups. This approach allows

avoiding satisfied equations by synthesis and

singularity analysis of mechanisms. Kinematic chains

impose the same constraints or one of them imposes

all the constraints and other chains contain actuators.

Keywords Parallel mechanism, theory of screws, decoupled

manipulator, singularities.

1. Introduction Parallel mechanisms are characterized by high stiffness

and payload capacity [1-5], but control of the motions

of the output link is complicated due to the coupling

between kinematic chains. In this paper we consider

parallel manipulators which perform Schoenflies

motions or SCARA motions. The output link has four

degrees of freedom that are three translational motions

and one rotational motion around parallel axes.

There exist different architectures of parallel

manipulators of this type. One approach to design 4 –

DOF parallel mechanism corresponds to the well

known Delta robot which consists of three R-R-P-R

kinematic chains (the P-pair is designed as a four-bar

planar parallelogram) causing translational motions of

the moving platform and of one R-U-P-U kinematic

chain, causing rotation about the vertical axis [6]. This

robot performs Schoenflies motions besides in this

robot three translational motions and one rotational

motion are decoupled. Similar solution can be obtained

by using the robot Ortoglide [7].

Note that translational kinematic pairs can be

represented as planar four-bar parallelograms. By this

approach numerous families of decoupled parallel

mechanisms are obtained [9, 10]. 4 – DOF parallel

manipulators corresponding to Schoenflies motions can

consists of four kinematic chains [8, 11] and of two

kinematic chains [12].

Our approach is based on closed screw groups [13] that

include all the screw products of the members of these

groups. By this the kinematik chains impose the same

constraints. A similar approach is used by different

authors [14-17]. Also another approach is used by

which one chain imposes the constraints and other

chains contain actuators. With regard to the

determination of the singularity the Jacobian matrices

or screws can be applied [18, 19]. We use the screw

groups to describe singularities [20-22] that make it

possible to avoid of complicated mathematical

equations.

The main contribution of this article is that some

decoupled parallel mechanisms are represented. By

this two mentioned approaches are used though in

previous publications [20-22] only one of them was

applied.

2. Structural synthesis of 3 – DOF

parallel mechanisms Let us consider a spherical parallel mechanism (Fig. 1,

a). Each kinematic chain consists of one actuated

rotation pair (rotating actuator) situated on the base and

two passive rotation kinematic pairs. The unit screws of

the axes of these kinematic pairs have coordinates (note

that the origin of the coordinate system is the point O in

which the axes of all the pairs intersect): E11 (1, 0, 0, 0,

0, 0), E12 (e12x, e12y , e12z , 0, 0, 0), E13 (e13x, e13y, e13z , 0,

0, 0), E21(0, 1, 0, 0, 0, 0), E22(e22x, e22y , e22z , 0, 0, 0),

E23(e23x, e23y , e23z, 0, 0, 0), E31 (0, 0, 1, 0, 0, 0), E32(e32x,

e32y , e32z , 0, 0, 0), E33(e33x, e33y , e33z , 0, 0, 0).

All the screws are of zero pitch. All three kinematic

chains impose the same constraints, so that one can

insert other similar chains between the base and

moving platform and the degree of freedom will remain

equal to three. The wrenches of the constraints imposed

by kinematic chains have coordinates (Fig. 1, b): R1 (1,

0, 0, 0, 0, 0), R2 (0, 1, 0, 0, 0, 0), R3 (0, 0, 1, 0, 0, 0),

these wrenches are of zero pitch. All the twists of

motions of the platform can be represented by the

twists reciprocal to the wrenches of the imposed

constraints (Fig. 1, b): 1 (1, 0, 0, 0, 0, 0), 2 (0, 1, 0,

0, 0, 0), 3 (0, 0, 1, 0, 0, 0). All three twists are of zero

pitch.

In this mechanism singularities expressed by loss of

one degree of freedom exist if any three screws Ei1 , Ei2

, Ei3 (i = 1, 2, 3) are linearly dependent which is

possible if they are coplanar (they are situated in the

same plane). In particular if the unit screws E11 (1, 0, 0,

0, 0, 0), E12 (e12x, e12y , e12z , 0, 0, 0), E13 (e13x, e13y, e13z ,

0, 0, 0) are coplanar (Fig. 1, c) then there exist four

wrenches of constraints imposed by kinematic chains:

R1 (1, 0, 0, 0, 0, 0), R2 (0, 1, 0, 0, 0, 0), R3 (0, 0, 1, 0, 0,

0) and R4 (0, 0, 0, 0, r4y, r4z) and only two twists of

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motion of the platform reciprocal to these wrenches 1

(1, 0, 0, 0, 0, 0) and 2 (2x, 2y, 2z , 0, 0, 0), these

twists are of zero pitch. The wrench R4 is of infinite

pitch, it is perpendicular to the axes E11, E12, E13.

If the actuators are fixed then there exist six wrenches

imposed by kinematic chains: R1 (1, 0, 0, 0, 0, 0), R2

(0, 1, 0, 0, 0, 0), R3 (0, 0, 1, 0, 0, 0), R4 (0, 0, 0, r4x , r4y

, r4z), R5 (0, 0, 0, r5x , r5y , r5z) and R6 (0, 0, 0, r6x , r6y ,

r6z). The wrenches R4 , R5, R6 are of infinite pitch.

Singularities corresponding to non-controlled

infinitesimal motion of the moving platform (end -

effector) exist if the wrenches R1, R2, R3, R4 , R5, R6

are linearly dependent which is possible if the

wrenches R4, R5, R6 are coplanar. In this case the

twist of zero pitch (x, y, z , 0, 0, 0) exists which

is perpendicular to the axes of the wrenches R4 , R5,

R6 and therefore reciprocal to all the wrenches R1, R2,

R3, R4, R5, R6. Moreover singularities exist

corresponding both to loss of one degree of freedom

and to non-controlled motion of the moving platform.

By this any three screws Ei1 , Ei2 , Ei3 (i = 1, 2, 3) and

the wrenches R1, R2, R3, R4 , R5, R6 are linearly

dependent.

a)

b) c)

Fig. 1 Spherical parallel mechanism.

Now let us consider a planar parallel mechanism (Fig.

2, a). Each kinematic chain can consist of one rotation

kinematic pair and two prismatic kinematic pairs (the

axis of the rotation pair is perpendicular to the axes of

the prismatic pairs), or of two rotation kinematic pair

and one prismatic kinematic pair (the axes of the

rotation pairs are parallel to each other and are

perpendicular to the axis of the prismatic pair), or of

three rotation kinematic pairs with parallel axes. In our

mechanism two kinematic chains consist of three

rotation kinematic pairs (one of them is actuated and

situated on the base) and one kinematic chains consists

of one actuated rotation kinematic pair situated on the

base (rotating actuator) and two prismatic kinematic

pairs represented as four-bar parallelograms. The unit

screws of the axes of these kinematic pairs have

coordinates: E11 (0, 0, 1, 0, 0, 0), E12 (0, 0, 1, e12x, e12y,

0), E13 (0, 0, 1, e13x, e13y, 0), E21 (0, 0, 1, 0, 0, 0), E22 (0,

0, 1, e22x, e22y, 0), E23 (0, 0, 1, e23x, e23y, 0), E31 (0, 0, 1,

0, 0, 0), E32 (0, 0, 0, e32x, e32y, 0), E33(0, 0, 0, e33x, e33y,

0).

a)

b) c)

Fig. 2 Planar parallel mechanism.

The screws E32 and E33 are of infinite pitch. All other

screws are of zero pitch. All three kinematic chains

impose the same constraints, so that one can insert

other similar chains between the base and moving

platform and the degree of freedom will remain equal

to three. The wrenches of the constraints imposed by

kinematic chains have coordinates (Fig. 2, b): R1 (0, 0,

0, 1, 0, 0), R2(0, 0, 0, 0, 1, 0), R3 (0, 0, 1, 0, 0, 0). All

the twists of motions of the platform can be represented

by the twists reciprocal to the wrenches of the imposed

constraints (Fig. 2, b): 1 (0, 0, 0, 1, 0, 0), 2 (0, 0, 0,

0, 1, 0), 3 (0, 0, 1, 0, 0, 0). The twists 1 and 2 are of

infinite pitch, the twist 3 is of zero pitch.

In this mechanism singularities corresponding to loss of

one degree of freedom exist if three screws Ei1 , Ei2 and

Ei3 (i = 1, 2, 3) are linearly dependent which is possible

if three screws Ei1 , Ei2 and Ei3 (i = 1, 2) are situated in

the same plane or if two screws E32 , E33 are parallel. In

particular if E32 = E33 (Figure 4, c) then there exist four

wrenches of constraints imposed by the kinematic

chains: R1 (0, 0, 0, 1, 0, 0), R2(0, 0, 0, 0, 1, 0), R3 (0, 0,

1, 0, 0, 0), R4 (r4x, r4y,, 0, 0, 0, 0) and only two twists of

motion of the platform reciprocal to these wrenches 1

(0, 0, 0, v1x, v1y,, 0) and 2 (0, 0, 1, 0, 0, 0). Note that R4

is perpendicular to E32 and E33, and 1 is parallel to

them.

If the actuators are fixed then there exist six

wrenches imposed by the kinematic chains: R1 (0, 0, 0,

1, 0, 0), R2 (0, 0, 0, 0, 1, 0), R3 (0, 0, 1, 0, 0, 0), R4 (r4x ,

r4y , 0, 0, 0, 1), R5 (r5x , r5y , 0, 0, 0, 1) and R6 (0, 0, 0, 0,

0 , 1). The wrenches R4 and R5, are of zero pitch, they

are situated along the axes of the links connecting

E31 E13 E23

R4

E21

E32

O

2

3

1

R1

R3

R2

z

y x O

R3

R1 R2

1 z

y x

E22 E12

E11

E33

2

E13 E23

O

O

2

3

1

R1

R3

R2

z

y x

E13

E12

E11

E33

E32

E31

E23

E22

E21

O

R4 R3

1

R1 R2

2 z

y x

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passive rotation pairs of the first and the second

kinematic chains. R6 is of infinite pitch. Singularities

corresponding to non-controlled infinitesimal motions

of the moving platform exist if the wrenches R1, R2, R3,

R4 , R5, R6 are linearly dependent which is possible if

the wrenches R4 , and R5 coincide. In this case the twist

of infinite pitch (0, 0, 0, vx , vy , 0) exists which is

perpendicular to the axes of the wrenches R4 and R5

and therefore reciprocal to all the wrenches R1, R2, R3,

R4 , R5, R6 .

Note that singularities exist corresponding both to loss

of one degree of freedom and to non-controlled

infinitesimal motion of the moving platform. By this

any three screws Ei1 , Ei2 , Ei3 (i = 1, 2, 3) and the

wrenches R1, R2, R3, R4 , R5, R6 are linearly dependent.

This mechanism is particularly decoupled. The matter

is that in the third kinematc chain the input link of the

first parallelogram and the output link of the second

parallelogram are connected correspondingly to the

rotating actuator and to the end-effector in the

middles of these links, and the output link of the first

parallelogram and the input link of the second

parallelogram coincide. It causes that the first and the

second actuators drive the position of the end-

effector. The third actuator drives the orientation of

the end-effector.

Note that all these mechanisms correspond to closed

screw groups and all the kinematic chains impose the

same constraints.

3. Design of 4 - DOF decoupled parallel

mechanisms Now let us consider parallel mechanisms of

Schoenflies motions. The first of them (Fig. 3, a)

consists of three kinematic chains. The first and the

second kinematic chains consist of one actuated

prismatic pair (linear actuator) situated on the base,

three rotation kinematic pairs with axes parallel to the

axis of the corresponding actuator and one rotation pair

with axis perpendicular to the axis of the actuator (the

axes of the last rotation pairs of these two chains

coincide). The third kinematic chain consists of one

actuated rotational pair (rotating actuator) situated on

the base, one actuated prismatic pair (linear actuator,

the axes of rotating and linear actuators coincide) and

two prismatic kinematic pairs represented as four-bar

parallelograms. The unit screws of the axes of these

kinematic pairs have coordinates: E11 (0, 0, 0, 1, 0, 0),

E12 (1, 0, 0, 0, eо

12y, eо12z), E13 (1, 0, 0, 0, e

о13y, e

о13z),

E14 (0, 0, 1, eо14x, e

о14y, 0), E21(0, 0, 0, 0, 1, 0), E22 (0, 0,

0, eо22x, 0, e

о22z), E23 (0, 0, 0, e

о23x, 0, e

о23z), E24 (0, 0, 1,

eо24x, e

о24y, 0), E31 (0, 0, 1, 0, 0, 0), E32 (0, 0, 0, 0, 0, 1),

E33(0, 0, 0, eо33x, e

о33y, 0), E34(0, 0, 0, e

о34x, e

о34y, 0).

The screws E11, E21, E32, E33, E34 are of infinite pitch.

All other screws are of zero pitch. The first and the

second kinematic chains impose one constraint The

third kinematic chain imposes two constraints. The

wrenches of the constraints imposed by kinematic

chains have coordinates (Fig. 3, b): R1 (0, 0, 0, 1, 0, 0),

R2 (0, 0, 0, 0, 1, 0). All the twists of motions of the

platform can be represented by the twists reciprocal to

the wrenches of the imposed constraints (Fig. 3, b): 1

(0, 0, 0, 1, 0, 0), 2 (0, 0, 0, 0, 1, 0), 3 (0, 0, 0, 0, 0, 1),

4 (0, 0, 1, 0, 0, 0). The twists 1, 2 and 3 are of

infinite pitch, the twist 4 is of zero pitch.

a)

b) c)

Fig. 3 4 - DOF parallel mechanism with planar

parallelograms.

In this mechanism singularities corresponding to loss of

one degree of freedom exist if the orts Ei2 , Ei3 and Ei4

(i = 1, 2) or the orts E33 and E34 are linearly dependent.

This is possible if three orts (unit screws) Ei2 , Ei3 and

Ei4 (i = 1, 2) are situated in the same plane or if the orts

E33 and E34 are parallel. Particularly if three unit

screws Ei2 , Ei3 and Ei4 (i = 1, 2) are situated in vertical

plane then there exist three wrenches of constraints

imposed by kinematic chains: R1 (0, 0, 0, 1, 0, 0), R2 (0,

0, 0, 0, 1, 0) and R3 (0, 0, 1, 0, 0, 0) (Fig. 3, c) and only

three twists of motions of the platform reciprocal to

these wrenches: 1 (0, 0, 0, 1, 0, 0), 2 (0, 0, 0, 0, 1, 0)

and 3 (0, 0, 1, 0, 0, 0). Note that R3 is situated along

the axis z.

If the actuators are fixed then there exist six wrenches

imposed by the kinematic chains: R1 (0, 0, 0, 1, 0, 0),

R2(0, 0, 0, 0, 1, 0), R3 (1, 0, 0, 0, 0, 0), R4 (0, 1, 0, 0, 0,

0), R5 (0, 0, 0, 0, 0, 1) and R6 (0, 0, 1, 0, 0, 0). The

wrenches R3 , R4 , R6 are of zero pitch, the wrench R5

is of infinite pitch.

This mechanism is fully decoupled and isotropic. Each

linear actuator controls the motion of the platform

along one Cartesian coordinate. In the third kinematc

E14

E24

E21 E22

E23

E34

E33

E32

E13

E12

E11

E15 E25

E31

R3

3

z

4

z

O O

1

R1 R2

2 2 3

1

R1 R2

y x

y x

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chain the input link of the first parallelogram and the

output link of the second parallelogram are connected

correspondingly to the rotating actuator ant to the end-

effector in the middles of these links, and the output

link of the first parallelogram and the input link of the

second parallelogram coincide. It causes that the

rotating actuator drives the orientation of the end-

effector. The linear actuators drive the position of the

end-effector.

Now let us consider a parallel mechanism (Fig. 4, a)

corresponding to planar mechanism and containing

parallelograms. The first and the second kinematic

chains consist of one actuated rotation pair (rotating

actuator) situated on the base, one rotation kinematic

pair and one cylindrical kinematic pair with axes

parallel to the axis of the corresponding actuator (the

axes of the cylindrical pairs of these two chains

coincide). The third kinematic chain consists of one

actuated rotation pair situated on the base, one actuated

prismatic pair (the axes of rotating and linear actuators

coincide) and two prismatic kinematic pairs

represented as four-bar parallelograms. The unit screws

of the axes of these kinematic pairs have coordinates:

E11 (0, 0, 1, eо11x, e

о11y, 0), E12 (0, 0, 1, e

о12x, e

о12y, 0),

E13 (0, 0, 1, eо13x, e

о13y, 0), E14 (0, 0, 0, 0, 0, 1), E21 (0,

0, 1, eо21x, e

о21y, 0), E22 (0, 0, 1, e

о22x, e

о22y, 0), E23 (0, 0,

1, eо23x, e

о23y, 0)= E13 (0, 0, 1, e

о13x, e

о13y, 0), E24 (0, 0, 0,

0, 0, 1)= E14 (0, 0, 0, 0, 0, 1), E31 (0, 0, 1, 0, 0, 0), E32

(0, 0, 0, 0, 0, 1), E33(0, 0, 0, eо33x, e

о33y, 0), E34(0, 0, 0,

eо34x, e

о34y, 0).

a)

b) c)

Fig. 4 4 - DOF parallel mechanism corresponding to

planar mechanism.

The screws E14, E24, E32, E33 and E34 are of infinite

pitch. All other screws are of zero pitch. All the

kinematic chains impose two constraints. The wrenches

of the constraints imposed by kinematic chains have

coordinates (Fig. 4, b): R1 (0, 0, 0, 1, 0, 0), R2 (0, 0, 0,

0, 1, 0). All the twists of motions of the platform can be

represented by the twists reciprocal to the wrenches of

the imposed constraints (Fig. 4, b): 1 (0, 0, 0, 1, 0, 0),

2 (0, 0, 0, 0, 1, 0), 3 (0, 0, 0, 0, 0, 1), 4 (0, 0, 1, 0, 0,

0). The twists 1, 2 and 3 are of infinite pitch, the

twist 4 is of zero pitch.

In this mechanism singularities corresponding to loss of

one degree of freedom exist if the orts Ei1 , Ei2 and Ei3

(i = 1, 2) or the orts E33 and E34 are linearly dependent.

This is possible if three orts (unit screws) Ei1 , Ei2 and

Ei3 (i = 1, 2) are situated in the same plane or if the orts

E33 and E34 are parallel. Particularly if three unit screws

Ei1 , Ei2 and Ei3 (i = 1, 2) are situated in the plane

parallel to the axis x then there exist three wrenches of

constraints imposed by kinematic chains: R1 (0, 0, 0, 1,

0, 0), R2 (0, 0, 0, 0, 1, 0) and R3 (1, 0, 0, 0, 0, 0) (Fig. 4,

c) and only three twists of motions of the platform

reciprocal to these wrenches: 1 (0, 0, 0, 0, 1, 0), 2 (0,

0, 0, 0, 0, 1) and 3 (0, 0, 1, 0, 0, 0). Note that R3 is

situated along the axis x.

If the actuators are fixed then there exist six wrenches

imposed by the kinematic chains: R1 (0, 0, 0, 1, 0, 0),

R2(0, 0, 0, 0, 1, 0), R3 (1, 0, 0, 0, 0, 0), R4 (0, 1, 0, 0, 0,

0), R5 (0, 0, 0, 0, 0, 1) and R6 (0, 0, 1, 0, 0, 0).

This mechanism is particularly decoupled. The rotating

actuators of the first and the second kinematic chains

control the motion of the platform in the horizontal

plane. In the third kinematc chain the input link of the

first parallelogram and the output link of the second

parallelogram are connected correspondingly to the

rotating actuator and to the end-effector in the middles

of these links, and the output link of the first

parallelogram and the input link of the second

parallelogram coincide. It causes that the rotating

actuator drives the orientation of the end-effector. The

linear actuator drives the position of the end-effector on

the vertical axis. Note that the kinematic chains impose

different constraints.

4. Conclusions In this paper, structural synthesis of decoupled

mechanisms containing three kinematic chains is

presented. The synthesis and the singularity analysis is

carried out by using of Plücker coordinates of twists

and wrenches corresponding to the kinematic chains.

We considered closed screw groups and corresponding

mechanisms and after this we represented 4 – DOF

mechanisms. One of them is fully decoupled and

isotropic, other mechanisms are particularly decoupled.

We used the approach based on closed screw groups

[13]. This approach allows avoiding complicated

equations by synthesis and singularity analysis of

mechanisms.

Kinematic chains impose the same constraints or one of

them imposes all the constraints and other chains

E14 E24

E21

E33

E22

E12

E11

E34

E31

E32

4

z

O

2 3

1

R1 R2

y x

3

E2ER

R

RRO

z

O

R3 R1 R2

1

y x

E13 E23

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contain actuators. The main contribution of this article

is that some decoupled parallel mechanisms are

represented.

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