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An Instantaneous Screw Kinematic Analysis of Parallel Robots
Copn'i!-(hl
©
I F.-\C Thl'ol'\ 01 Rohol s.
\'i e llll
AN INSTANTANEOUS SCREW KINEMATIC ANALYSIS OF PARALLEL ROBOTS Van Jun* and Huang Zhen** * D r/Jarllllflll **DI'/Jarlllll'lIl
Abstract.
.\l n halli(a { EIIWIII'fri llg. SII I/Ih Chilla 11I.\lill/l,. of TI'I//lwillgr. Guallg:hul/. PRC .\lnhalli((li ElIgillrfrillg . .\"urlhl'!l.\1 IlISlill/lf H,'m'Y .\lwhilll'l1'. H ri illll!!,jiallg. PRC
t1 t1
"r
This paper, applying screw theory, presented an effective method to
ana-
lyse multi-loop parallel robot's motion, and established the kinematic relation formulas bstween inputs and outputs. These formulas are explicit, simple, and convenient for use. We have extended the method to general spatial mechanisms. This approach is suitable for 1-6 loop parallel robots and 1-6 degree-of-freedom spatial mechanisms. Robots; vectors; angular velocity; parallel robot; screw.
Keywords. INTRODUCTION
Let L, M, N, be the direction ratios of the vector and P, Q, R be the X, Y, Z direction component
~,
UP to now many published papers have described the
of the moment vector
theory and practice about series robots, but rarely
es can be written ( L, M, N ; P,
about parallel robots.
Combin~ne
K. H. Hunt conceived a six
degree-of-freedom parallel robot model in
1978
~e
a six loop parallel robot
W~.s
R ). into a dual ve-
dual vector
i
is known as a screw. In general,
the vectors ~ and ~o do not satisfY the orthogonality condition. The instantaneous motion of a rigid body in space can be regarded as a rotation
with ball screws on exhibition at the 14th International Robot Conference in Swiden in 1984.
J. Duffy presented using screw theory to describe the
Q.,
the two vectors !!,o£l1,d S !=!!+f;~.
a Florida shoulder which is a three loop spherical There
• Their Plucker co-ordinat-
ctor,
(Fig. 1.) • In 1981 D. Tesar and D. J. Cox designed mechanism.
~
about a fixed axis and a translation along the same axis so that we can use an instantaneous sc-
robots in 12th International Symposium on
Industrial Robots in Paris in 1982, and in the ASME
rew to describe the instantaneous motion of a
18th Mechanism Conference in America in 1984.
ri-
gid body (Fig. 3.) • Let the rotation velocity of the body be
Parallel robots have more stiffness, accuracy of
Yi ,
and
~ibe
~i
and
position,and compactness, which cause more atten-
translation velocity be
tion. They are worth to be further explored. Now a
ctor of the axis, the instantaneou2 motion of the
the unit ve-
effective method is presented to analyse parallel
body can be described in the form
robot's motion in this paper. ~i
INSTANTANF.XlUS SCREWS AND THE MOTIONS OF A
~~.;
where ~d ~I X!i • The Plucker co-ordinates of the motion are
RIGID BODY
(~;;
y.,)
A unit vector ~ bound to a line in space (Fig. 2.).
where V ~ -, V· The instantaneous motion can also be expressed in
The distance from a point in the line to the co-or
the instantaneous screw
dinate system origin is
~
• The moment of the vec-
(5)
tor about the origin is
If the body rotates only, i. e.
(1 ) The vector
~
of the screw is zero so that
and moment vector ~o satisfy the
o
Yi = 0, = Erx ! i
i£.ci = 1..1; !.i X §.i = w'i§.oi If the body only translates in the
orthogonality condition, i.e.
.!t. !. The vectors ( ~ ; ~.
~.;
(2)
i. e.
~;
=0
the pitch and
direction~ ;
•
, the motion of the body can be exp-
ressed as scalar multiple of a free vector\l(O;~i)
are the PlUcker co-ordinates
77
78
Yall .lUll alld HlIall~ Zhl'1l
/
/ v---
o
Fig. 1.
Six degree-or-freedom parallel
Fig. 4.
--
The resultant screw.
robot model •
Fig. 2.
A unit vector in space.
Fig.
5.
A six-loop parallel robot.
o
Fig. 3.
A instantaneous screw in space.
Fig. 6.
Branch i or six-loop parallel robot.
79
An Insta nt a neo lls Sc re w Kin e m a tic ;\nal\' sis
trix.
or Vi (0,0,0, L,R.H). Suppose two instantaneous screws '01,
1, =:!.
are
+ t. y.,
[JiJ
(6)
is the ith branch Jacobin matrix which is
the first order partial derivative matrix (6 x 6). The three components of Wp which is the rotation The sum of two screws is still a screw, and can be expressed as follows
translation velocity of the platform, and
The resultant screw w;i , and the perpendicular distance ~ •• between
w, t , and Wziz are orthogonal.
and
ith branch Jacobin matrix is
r~ ~ !!!!z "W,y l aWl [J'~J==
The position of the resultant screw can be calculated as follows
~~(h;- hl)+ ~hi-h2)COSo(/% + ails inol,.]
'\';
,
0(,&
are shown
=
Ye;
=
L
K~I
~
l!&lI
lW,
JW.
lillI
aw,
~
~~
iIi7" i"IIi
iliz
aVli
!lW,
in Fig. 4. ,and h. is the pitch Substituting equation (14) in motion of the platform is
~;
( 10)
W,!,=
[jl']
et}
lw, , Wz
where (~) =
(13 ) and (12). the (15) (t6 )
• • ••
The motion of the platform can be found i f [ J~)J
of n screws which can be added one by one is
!!;
~
aw,
JWa
~
( 9)
When there are several screws. the resultant screw Il
~PL
T.
~~
of the resultant screw. hi
is the
V~, ~
• V~ are its three components. The superscript(i) means that all elements belong to branch i. The
(8)
r;sinljl;=
w,..,w".... !t'
velocity of the platform are
and {~} are given.
~I(
(11 )
The inverse of equation (15)
is
n.
y.t =l:' .'!':ok
( i=1. 2•••• 6) (17)
,..,1
The ith branch is in special configuration if the KINEMATICS OF MULTI-WOP PARALLEL ROBOTS
Jacobin matrix
[J;1
ia singular.
Equation
(17)
consistes of six element equations. The motions of A six degree-of-freedom and six-loop parallel bot is s hown
ro-
in Fig. 5. , which consists of 6
branches. Each branch connects the platform to the base. The platform parallel to the base is the end effector of the robot.
six revolution pairs can be solved if the motion of the platform is given.
dom parallel robot has six independent inputs.
take out an ith branch from the robot (Fig.
6.)
can also be found. According to equation -I
.,) ["'] J I '; {Wp ! PI1
form and six revolution pairs. It simulates a seri-
~, =
es robot.
~:" = [ ~:J~~{W;.!~
The revolution velocities of the pairs
I.W , W~·
••••••• J~ respectively.
1 (18)
And the d~ections of each pair's axis are ~; •• ~~ , ••.••• '§..~ • The motion of the platform can be expressed as
J
[la ; is the first row of matrix [J~1 . 'tI,JO, ••• Wf! are the rotation velocity of
where
[1,. h, ...
(12)
"">'
i.]
are unit screw respectively, and
their axes coincide with the related pair's axes.
avoid mistakeness. here .,f.J, nated as Q., • Q." be expressed as
1i = !1
as those
...
ti..
;1,". • •• '.W
are desig-
And equation (18) can
+ € '§.oi
And [il' ~ , ••• ~]
is also known as Jacobin ma-
six
first joints adjoin the standing base. In order to
The unit scrsvs depend only on the directions and positions of their related axes of the pairs. i.e.
And
the first joint in each branch respectively. These rotation velocities are absolute ones
••••
(17) .we
write down the first element equation.
which is composed of five links besides the plat-
;t, • iz
If
we exert six independent motions known on the first pairs of each branch, the motion of the platform
In order to analyse the multi-loop parallel robot.
are designated as
The six degree of free-
where
{~}is
the absolute velocity of the first
80
Yan .Iun and Huang Zhen
joint, or input
of the robot, which may be rewri-
tten as
(20) and
[
[
J~I)]~; 'f
(21
j
I
Ij
is not singular, the inverse of equation (19) is
[JlIf{1}
(22)
Equation (19) and ( 22) describe the relations
be-
tween the robot inputs and the motion of the end
(31)
Equation (31) has set up relationship btween six
the six inputs can be found, too. Things worthy of note are the elements of the relative velocity
{.~}
in equation (31) which are in different branches. The instantaneous screw in equation (22) describes the absolute motion of the platform. And the
{i'l in
equations (25) ,( 21) are the relative velocities.
effector or platform • These equations are all linear and solved easily.
)
branches. If the six relative motions are given ,
is the second Jacobin matrix. If it
W,ip =
[Jw] L~
inputs and six joint's relative motions in different
[J"r
to
-I
l §J =
[ J~lj-t
J~JJ
The matrix [J~
[JRJ is the forth Jacobin matrix. If it is not singular , the inverse of equation (28) is
Using equation (22), th e
instantaneous screw of the platform can be found if the six inputs are given. The requisite inputs can
In order to calculate the absolute motion screw of a link besides the platform, we have to use equation (12).
For example, we need to find out the
also be found by equation (19) if the motion of the
L-th (L=2-5) link's absolute motion screw in branch
platform is known.
i (i =1-6).
Substituting equation (22) into
vector
equation (11), we get
, W~, ••• W(,{. Then, use
equation
(12) for the absolute motion screw of link L
{ i 'l =[J~r' [J:'J H }
(23)
Defining the product of the inverse matrix
[JlL]
First, we use equation (25) to get the
t::O ",{W,
,12 , ...
[-hJ and
as
,;,1'0', ('
i~
' Wo .
.: I I ,¥il, .:
[J~J = [11j' [j~:r
Using the screw theory,
CJJI] is known as the third Jacobin matrix.
= 1,2,
have established a seri-
al formula which can be used to find out the velo-
So equation (23) can be rewritten as i
~e
city of robot end effector or other link. If the
···6
(25)
The inverse of equation (25) is
B} = [JW) {~;'}
end effector's motion is given, we can find out the requisite inputs. These formulas suitable for 1-6 degree-of-freedom multi-loop parallel robots.
(26)
When the six rotation velocities are known . and MULTI-LOOP SPATIAL MECHANISMS
are set in same branch, the six independent motions or six inputs can be found by equation (26). If the six inputs are set in six different branches, we can
In general, spatial mechanisms may include revolu-
separate equation (25) into six element equations
tion joint (R), prismatic pair (p), cylindric pair
corresponding with each branch
(C), Hook-joint (T), and spherical pair (S) etc.
~;I = ~,
(J;Il;{ U
Revolution and prismatic pair are enssential types
rj"" {o}
which have only single degree-of-freedom.
'f!~ = l 1I~; 9-
.
.
.
(21)
o
sition of revolution and prismatic pairs i.e. C=RP, T=RR, S=RRR. After this replacing, the degree-of-
where
a, b, .•• f are the numbers of the branches,
and
t9, •••
0\ ,
1
As for
the other pairs they can be equivalent to a compo-
are the numbers of the joints in
freedom and the motion of the mechanism would be invariable.
its branch, and [J&)ar iS the a-th row of matrix [Jm]
For example, a two-loop spatial mechanism RSSC-SC
in branch a.
is shown in Fig. 1.
We combine the six element equations
where
;
Its mobility is
W = 6( n-1)-
~ ( 6-~ ) =2. And its equivalent mechanism is
of equation (21 ) and write in the compact form
shown in Fig. 8.
The mobility of its equivalent
{:t} = [J,,1j i}
(28)
{)!}={w':' ,o/i,J
(29)
nism has three
(30)
pairs, we add some pairs (motionless) and links (
mechanism is
W = 2, too. This equivalent
mecha-
branches each of which has only 4-
5 pairs. In order that each branch might have
six
81
.-\11 IIISt;lIlt
pseudo-mechanism instead of the original mechanism, we can also use the previous formulas. For example , a single-loop mechanism is only one branch after replacing pairs.
We can select a
Link in the middle of the branch as a centre link (or platform), then add some links and pairs at two sides of the branch to get a two-loop pseudomechanism, and we can still use the previous formulas.
Fig. 7.
A two-loop spatial mechanism
CONCLUTION
RSSC-SC This approach can analyse the motions of multiloop parallel robots, spherical mechanisms, multiloop spatial mechanisms, and single-loop spatial Using screw theory, we have establi-
mechanisms.
shed a serial formulas which are explicit, simple, and convenient to use. The Jacobin matrix [Jx) is the function of the geometry and the physical position of the mechanism only, and has no effect upon the motion of the mechanism.
When the geometry and position of the
mechanism are known, the Jacobin matrix [JJ can be found. And the other Jacobin matrix [JJ[J, [J~ , Fig. 8.
LJ~
Equivalent mechanism
all come from [J~
The presented approach is a general method which enriches the theory of screw. REFERENCES Hunt, K.H.
(1978).
nisms. Cox, D.J.
Kinematic geometry of mecha-
Clarendon press, Oxford. (1981). The dynamic modeling and com-
mand signal formulas for parallel robotic devices. University of Florida, gainesville , FL
U.S.A.
Mohamed, D. , Duffy. J. , Sanger, J. (1983). nematics of fully parallel devices.
Ki-
Procee-
ding of The Sixth World Congress on Theory of Fig. 9.
The pseudo-mechanism
Machines. Mohamed, D. , Duffy, J.
null-length and massless) to make each branch have six pairs. After this changing, the mechanism
is
called pseudo-mechanism shown in Fig. 9. In order to equalise kinematically the axes of added pairs would not have the Jacobin matrix [J~ singularity.
(1984).
A direct deter-
mination of the instantaneous kinematics of fully parallel robot manipulator.
ASME paper
No. 84-DET-114. Sugimoto, K.
Duffy, D.
(1982)
Application of
linear algebra to screw systems. mechanisms theory, VOL. 17
Then we can use the previous formulas to analyse.
Duffy, J.
This approach also suits single-loop spatial me-
Lipkin, H. , Duffy, J.
Machine and
pp.73-83.
(1980). Analysis of mechanisms and ro-
bot manipulators.
Wiley,New York, NY U.S.A. (1982).
Analysis of in-
chanisms or spherical mechanisms, such as RSSR ,
dustrial robots via the theory of screws.
RSCR , RRRRRRR , ···etc. Obviosly, local degree-of
Proceeding of the 12th international symposium
-freedom of the spatial mechanism must be neglec-
on industrial robots.
ted first. Then replacing some pairs and using the
©
I F.-\C Thl'ol'\ 01 Rohol s.
\'i e llll
AN INSTANTANEOUS SCREW KINEMATIC ANALYSIS OF PARALLEL ROBOTS Van Jun* and Huang Zhen** * D r/Jarllllflll **DI'/Jarlllll'lIl
Abstract.
.\l n halli(a { EIIWIII'fri llg. SII I/Ih Chilla 11I.\lill/l,. of TI'I//lwillgr. Guallg:hul/. PRC .\lnhalli((li ElIgillrfrillg . .\"urlhl'!l.\1 IlISlill/lf H,'m'Y .\lwhilll'l1'. H ri illll!!,jiallg. PRC
t1 t1
"r
This paper, applying screw theory, presented an effective method to
ana-
lyse multi-loop parallel robot's motion, and established the kinematic relation formulas bstween inputs and outputs. These formulas are explicit, simple, and convenient for use. We have extended the method to general spatial mechanisms. This approach is suitable for 1-6 loop parallel robots and 1-6 degree-of-freedom spatial mechanisms. Robots; vectors; angular velocity; parallel robot; screw.
Keywords. INTRODUCTION
Let L, M, N, be the direction ratios of the vector and P, Q, R be the X, Y, Z direction component
~,
UP to now many published papers have described the
of the moment vector
theory and practice about series robots, but rarely
es can be written ( L, M, N ; P,
about parallel robots.
Combin~ne
K. H. Hunt conceived a six
degree-of-freedom parallel robot model in
1978
~e
a six loop parallel robot
W~.s
R ). into a dual ve-
dual vector
i
is known as a screw. In general,
the vectors ~ and ~o do not satisfY the orthogonality condition. The instantaneous motion of a rigid body in space can be regarded as a rotation
with ball screws on exhibition at the 14th International Robot Conference in Swiden in 1984.
J. Duffy presented using screw theory to describe the
Q.,
the two vectors !!,o£l1,d S !=!!+f;~.
a Florida shoulder which is a three loop spherical There
• Their Plucker co-ordinat-
ctor,
(Fig. 1.) • In 1981 D. Tesar and D. J. Cox designed mechanism.
~
about a fixed axis and a translation along the same axis so that we can use an instantaneous sc-
robots in 12th International Symposium on
Industrial Robots in Paris in 1982, and in the ASME
rew to describe the instantaneous motion of a
18th Mechanism Conference in America in 1984.
ri-
gid body (Fig. 3.) • Let the rotation velocity of the body be
Parallel robots have more stiffness, accuracy of
Yi ,
and
~ibe
~i
and
position,and compactness, which cause more atten-
translation velocity be
tion. They are worth to be further explored. Now a
ctor of the axis, the instantaneou2 motion of the
the unit ve-
effective method is presented to analyse parallel
body can be described in the form
robot's motion in this paper. ~i
INSTANTANF.XlUS SCREWS AND THE MOTIONS OF A
~~.;
where ~d ~I X!i • The Plucker co-ordinates of the motion are
RIGID BODY
(~;;
y.,)
A unit vector ~ bound to a line in space (Fig. 2.).
where V ~ -, V· The instantaneous motion can also be expressed in
The distance from a point in the line to the co-or
the instantaneous screw
dinate system origin is
~
• The moment of the vec-
(5)
tor about the origin is
If the body rotates only, i. e.
(1 ) The vector
~
of the screw is zero so that
and moment vector ~o satisfy the
o
Yi = 0, = Erx ! i
i£.ci = 1..1; !.i X §.i = w'i§.oi If the body only translates in the
orthogonality condition, i.e.
.!t. !. The vectors ( ~ ; ~.
~.;
(2)
i. e.
~;
=0
the pitch and
direction~ ;
•
, the motion of the body can be exp-
ressed as scalar multiple of a free vector\l(O;~i)
are the PlUcker co-ordinates
77
78
Yall .lUll alld HlIall~ Zhl'1l
/
/ v---
o
Fig. 1.
Six degree-or-freedom parallel
Fig. 4.
--
The resultant screw.
robot model •
Fig. 2.
A unit vector in space.
Fig.
5.
A six-loop parallel robot.
o
Fig. 3.
A instantaneous screw in space.
Fig. 6.
Branch i or six-loop parallel robot.
79
An Insta nt a neo lls Sc re w Kin e m a tic ;\nal\' sis
trix.
or Vi (0,0,0, L,R.H). Suppose two instantaneous screws '01,
1, =:!.
are
+ t. y.,
[JiJ
(6)
is the ith branch Jacobin matrix which is
the first order partial derivative matrix (6 x 6). The three components of Wp which is the rotation The sum of two screws is still a screw, and can be expressed as follows
translation velocity of the platform, and
The resultant screw w;i , and the perpendicular distance ~ •• between
w, t , and Wziz are orthogonal.
and
ith branch Jacobin matrix is
r~ ~ !!!!z "W,y l aWl [J'~J==
The position of the resultant screw can be calculated as follows
~~(h;- hl)+ ~hi-h2)COSo(/% + ails inol,.]
'\';
,
0(,&
are shown
=
Ye;
=
L
K~I
~
l!&lI
lW,
JW.
lillI
aw,
~
~~
iIi7" i"IIi
iliz
aVli
!lW,
in Fig. 4. ,and h. is the pitch Substituting equation (14) in motion of the platform is
~;
( 10)
W,!,=
[jl']
et}
lw, , Wz
where (~) =
(13 ) and (12). the (15) (t6 )
• • ••
The motion of the platform can be found i f [ J~)J
of n screws which can be added one by one is
!!;
~
aw,
JWa
~
( 9)
When there are several screws. the resultant screw Il
~PL
T.
~~
of the resultant screw. hi
is the
V~, ~
• V~ are its three components. The superscript(i) means that all elements belong to branch i. The
(8)
r;sinljl;=
w,..,w".... !t'
velocity of the platform are
and {~} are given.
~I(
(11 )
The inverse of equation (15)
is
n.
y.t =l:' .'!':ok
( i=1. 2•••• 6) (17)
,..,1
The ith branch is in special configuration if the KINEMATICS OF MULTI-WOP PARALLEL ROBOTS
Jacobin matrix
[J;1
ia singular.
Equation
(17)
consistes of six element equations. The motions of A six degree-of-freedom and six-loop parallel bot is s hown
ro-
in Fig. 5. , which consists of 6
branches. Each branch connects the platform to the base. The platform parallel to the base is the end effector of the robot.
six revolution pairs can be solved if the motion of the platform is given.
dom parallel robot has six independent inputs.
take out an ith branch from the robot (Fig.
6.)
can also be found. According to equation -I
.,) ["'] J I '; {Wp ! PI1
form and six revolution pairs. It simulates a seri-
~, =
es robot.
~:" = [ ~:J~~{W;.!~
The revolution velocities of the pairs
I.W , W~·
••••••• J~ respectively.
1 (18)
And the d~ections of each pair's axis are ~; •• ~~ , ••.••• '§..~ • The motion of the platform can be expressed as
J
[la ; is the first row of matrix [J~1 . 'tI,JO, ••• Wf! are the rotation velocity of
where
[1,. h, ...
(12)
"">'
i.]
are unit screw respectively, and
their axes coincide with the related pair's axes.
avoid mistakeness. here .,f.J, nated as Q., • Q." be expressed as
1i = !1
as those
...
ti..
;1,". • •• '.W
are desig-
And equation (18) can
+ € '§.oi
And [il' ~ , ••• ~]
is also known as Jacobin ma-
six
first joints adjoin the standing base. In order to
The unit scrsvs depend only on the directions and positions of their related axes of the pairs. i.e.
And
the first joint in each branch respectively. These rotation velocities are absolute ones
••••
(17) .we
write down the first element equation.
which is composed of five links besides the plat-
;t, • iz
If
we exert six independent motions known on the first pairs of each branch, the motion of the platform
In order to analyse the multi-loop parallel robot.
are designated as
The six degree of free-
where
{~}is
the absolute velocity of the first
80
Yan .Iun and Huang Zhen
joint, or input
of the robot, which may be rewri-
tten as
(20) and
[
[
J~I)]~; 'f
(21
j
I
Ij
is not singular, the inverse of equation (19) is
[JlIf{1}
(22)
Equation (19) and ( 22) describe the relations
be-
tween the robot inputs and the motion of the end
(31)
Equation (31) has set up relationship btween six
the six inputs can be found, too. Things worthy of note are the elements of the relative velocity
{.~}
in equation (31) which are in different branches. The instantaneous screw in equation (22) describes the absolute motion of the platform. And the
{i'l in
equations (25) ,( 21) are the relative velocities.
effector or platform • These equations are all linear and solved easily.
)
branches. If the six relative motions are given ,
is the second Jacobin matrix. If it
W,ip =
[Jw] L~
inputs and six joint's relative motions in different
[J"r
to
-I
l §J =
[ J~lj-t
J~JJ
The matrix [J~
[JRJ is the forth Jacobin matrix. If it is not singular , the inverse of equation (28) is
Using equation (22), th e
instantaneous screw of the platform can be found if the six inputs are given. The requisite inputs can
In order to calculate the absolute motion screw of a link besides the platform, we have to use equation (12).
For example, we need to find out the
also be found by equation (19) if the motion of the
L-th (L=2-5) link's absolute motion screw in branch
platform is known.
i (i =1-6).
Substituting equation (22) into
vector
equation (11), we get
, W~, ••• W(,{. Then, use
equation
(12) for the absolute motion screw of link L
{ i 'l =[J~r' [J:'J H }
(23)
Defining the product of the inverse matrix
[JlL]
First, we use equation (25) to get the
t::O ",{W,
,12 , ...
[-hJ and
as
,;,1'0', ('
i~
' Wo .
.: I I ,¥il, .:
[J~J = [11j' [j~:r
Using the screw theory,
CJJI] is known as the third Jacobin matrix.
= 1,2,
have established a seri-
al formula which can be used to find out the velo-
So equation (23) can be rewritten as i
~e
city of robot end effector or other link. If the
···6
(25)
The inverse of equation (25) is
B} = [JW) {~;'}
end effector's motion is given, we can find out the requisite inputs. These formulas suitable for 1-6 degree-of-freedom multi-loop parallel robots.
(26)
When the six rotation velocities are known . and MULTI-LOOP SPATIAL MECHANISMS
are set in same branch, the six independent motions or six inputs can be found by equation (26). If the six inputs are set in six different branches, we can
In general, spatial mechanisms may include revolu-
separate equation (25) into six element equations
tion joint (R), prismatic pair (p), cylindric pair
corresponding with each branch
(C), Hook-joint (T), and spherical pair (S) etc.
~;I = ~,
(J;Il;{ U
Revolution and prismatic pair are enssential types
rj"" {o}
which have only single degree-of-freedom.
'f!~ = l 1I~; 9-
.
.
.
(21)
o
sition of revolution and prismatic pairs i.e. C=RP, T=RR, S=RRR. After this replacing, the degree-of-
where
a, b, .•• f are the numbers of the branches,
and
t9, •••
0\ ,
1
As for
the other pairs they can be equivalent to a compo-
are the numbers of the joints in
freedom and the motion of the mechanism would be invariable.
its branch, and [J&)ar iS the a-th row of matrix [Jm]
For example, a two-loop spatial mechanism RSSC-SC
in branch a.
is shown in Fig. 1.
We combine the six element equations
where
;
Its mobility is
W = 6( n-1)-
~ ( 6-~ ) =2. And its equivalent mechanism is
of equation (21 ) and write in the compact form
shown in Fig. 8.
The mobility of its equivalent
{:t} = [J,,1j i}
(28)
{)!}={w':' ,o/i,J
(29)
nism has three
(30)
pairs, we add some pairs (motionless) and links (
mechanism is
W = 2, too. This equivalent
mecha-
branches each of which has only 4-
5 pairs. In order that each branch might have
six
81
.-\11 IIISt;lIlt
pseudo-mechanism instead of the original mechanism, we can also use the previous formulas. For example , a single-loop mechanism is only one branch after replacing pairs.
We can select a
Link in the middle of the branch as a centre link (or platform), then add some links and pairs at two sides of the branch to get a two-loop pseudomechanism, and we can still use the previous formulas.
Fig. 7.
A two-loop spatial mechanism
CONCLUTION
RSSC-SC This approach can analyse the motions of multiloop parallel robots, spherical mechanisms, multiloop spatial mechanisms, and single-loop spatial Using screw theory, we have establi-
mechanisms.
shed a serial formulas which are explicit, simple, and convenient to use. The Jacobin matrix [Jx) is the function of the geometry and the physical position of the mechanism only, and has no effect upon the motion of the mechanism.
When the geometry and position of the
mechanism are known, the Jacobin matrix [JJ can be found. And the other Jacobin matrix [JJ[J, [J~ , Fig. 8.
LJ~
Equivalent mechanism
all come from [J~
The presented approach is a general method which enriches the theory of screw. REFERENCES Hunt, K.H.
(1978).
nisms. Cox, D.J.
Kinematic geometry of mecha-
Clarendon press, Oxford. (1981). The dynamic modeling and com-
mand signal formulas for parallel robotic devices. University of Florida, gainesville , FL
U.S.A.
Mohamed, D. , Duffy. J. , Sanger, J. (1983). nematics of fully parallel devices.
Ki-
Procee-
ding of The Sixth World Congress on Theory of Fig. 9.
The pseudo-mechanism
Machines. Mohamed, D. , Duffy, J.
null-length and massless) to make each branch have six pairs. After this changing, the mechanism
is
called pseudo-mechanism shown in Fig. 9. In order to equalise kinematically the axes of added pairs would not have the Jacobin matrix [J~ singularity.
(1984).
A direct deter-
mination of the instantaneous kinematics of fully parallel robot manipulator.
ASME paper
No. 84-DET-114. Sugimoto, K.
Duffy, D.
(1982)
Application of
linear algebra to screw systems. mechanisms theory, VOL. 17
Then we can use the previous formulas to analyse.
Duffy, J.
This approach also suits single-loop spatial me-
Lipkin, H. , Duffy, J.
Machine and
pp.73-83.
(1980). Analysis of mechanisms and ro-
bot manipulators.
Wiley,New York, NY U.S.A. (1982).
Analysis of in-
chanisms or spherical mechanisms, such as RSSR ,
dustrial robots via the theory of screws.
RSCR , RRRRRRR , ···etc. Obviosly, local degree-of
Proceeding of the 12th international symposium
-freedom of the spatial mechanism must be neglec-
on industrial robots.
ted first. Then replacing some pairs and using the