structural synthesis of decoupled parallel mechanisms
DESCRIPTION
Hội nghị toàn quốc về Điều khiển và Tự động hoá - VCCA-2011Structural 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: [email protected], [email protected], [email protected] contain actuators. With regard to the determination of the singularity the Jacobian matrices or screws can be applied [18,TRANSCRIPT
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: [email protected], [email protected],
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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