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Geometric design considerations for a 3-link robot arm
395
Applications
Geometric Design Considerations for 3-Link Robot Arm A. M a n n a a , F. Dehlawi and M. A k y u r t ('ollege of Engineering, Kmg Abdulaziz Unit~etwitv P.O. Box 9027. Jeddah 21413, Saudi Arabia Geometric design parameters are discussed and the criteria are laid for the proper dimensioning of a three-link robot arm engaged in a given work space and serving workpieces at two different levels. The design procedure is illustrated by an example.
Ko'words: Robotics, Robot manipulator, Robot design, 3-Link arm. Geometric design,
AbduI-Raouf M. Mannaa was born in Jeddah, Saudi Arabia in 1951. He received his B. Sc degree in Mechanical Engineering from the University of Petroleum in Dhahran, Saudi Arabia in 1974. His M.Sc in Mechanical Engineering and his Master of Engineering were received from the University of California at Berkeley in 1976 and 1977 respectively. His Ph.D degree in ,, Mechanical Engineering was received from the University of Washington at Seattle. Dr. Mannaa is presently working as an Assistant Prof. in the Department of Mechanical Design and Production, King Abdulaziz University, Jeddah, Saudi Arabia. Current research areas are robotics, solar cooling, and applied mechanics.
Fouad Dehlawy was born in Makkah, Saudi Arabia in June, 1952. He received his B.Sc degree in Electrical E n g i n e e r i n g from University of Riyadh, Saudi Arabia in 1974, and M.S.E.E. from Stanford University, California in 1977. Another Master in Statistics was received in 1980 and Ph.D in instructional uses of computers in 1983, both from Stanford University. Dr. Dehlawy is currently an Assistant Prof. in the Electrical Engineering Department, King Abdulaziz University, Jeddah, Saudi Arabia. His research interests are Arabization of computers, artificial intelligence and robotics.
a
1. Introduction Robot manipulators comprise essentially an arm and a gripper or hand. The arm moves the gripper within a workspace under computer control. Of the various kinematic systems that produce a workspace, the popular three-link chain equipped with three revolute joints (Fig. 1) is studied below as regards to geometric design constraints and variables. The dynamic analysis of the arm as well as the performance of the prototype arm currently under construction will be presented in subsequent articles.
2. General Design Considerations The design objective for the arm (Figs. 1 and 2) is to pick up (via an unshown gripper located at D) an object positioned at ground level and within a working radius L, and to deliver it on the platform of height H and characteristic width K, a n d / o r vice versa. Lengths of the links need to be chosen in such a way that the entire working length L - K is accessible to the arm at ground level. Similarly, the arm must be able to service the entire platform width K corresponding to an angular location '5 (Fig. 1) at height H.
North-Holland Computers in Industry 7 (1986) 395-400 0166-3615/86/$3.50 ":, 1986 Elsevier Science Publishers B.V. (North-Holland)
Mehmet Akyurt was born in Turkey in 1940. He received his BS and MS degrees in Mechanical Engineering at the Middle East Technical University in Ankara, and his Ph.D from Purdue University, U.S.A. His current interests are mechanisms, heat pipes and solar energy applications.
396
Applications
(omputers in lndust O,
D
-,Vb G3,
G2
Bq~
e 2
11
OB
=
BC CD
= 12 =13
fiG 2
= r2
BG 3
=r3
G1
)¢~\
! \-\
z,/
/
,
i
C
t
.
.
, D
~e
Fig. 1. The three-link arm with triple revolute joints.
Fig. 3 shows versions of a three-link arm with three revolute joints in several different positions and configurations. Two of the arms feature the same L. The arm B C D " is noted to possess two solutions on the ground for given positions of l 2. Fig. 4 shows the top view of the working area of the robot, where the ground level working area is marked as the loading/unloading zone. Note that K in general varies with rotation of the arm about the O B axis and depends on the shape of the platform. A rectangular platform is shown in Fig. 4.
It is observed in Fig. 3 that to be equal to or greater than able to service the immediate form on the ground. For the note that: I~=H+I
it is necessary for 12 K for the arm to be vicinity of the platcase of equality we (1)
1
12+l = [L2+ (u+t,)2]
(2)
For given values of l 2. l 3, t f and L the latter relationship sets a limit to 1~" LH
It >~ L - K
H.
(3)
Computers in Industry
A. Mannaa et al. / 3-Link Robot Arm
397
0 2 "~
03
I-
O
D
-f H
I.
Fig. 2. The arm serving at two different levels.
It m a y be verified that l 2 does not vary strongly with l 1 for any set of H, L and K. Setting l 1 equal to unity, Eq. 3 m a y be re-arranged to yield: L >/K(1 + H),
placement variables 02 and 0 3 of the arm. N o w for a gripper located at D to pick up or deposit an object at ground level, we require that H + l 1 + l 2 sin 02 tan 03 = - / 2 c o s 0 2 + K + ~ "
('7)
(4) \ \
or more generally
\ \
llL >I K(I, + H).
(4a)
The m a x i m u m serviceable platform reach U (Fig. 4) is constrained by:
u
+ 13) - # .
(5)
A further condition that must be satisfied if the a r m is to service the immediate vicinity of its base (point O in Fig. 4) is: l, + 12>~ 13>~ [122 - 12] I/2 .
I I
12
,
,
(6)
~
/
\,/i
/I
,.,.m/--<
//
'
Given H and K, relationships (2), (4), (5) and (6) can be utilized to determine guideline magnitudes for l 2, l 3, L, and U.
H
~
3. Position Control
It is essential for an a r m operating under computer control that precise relationships be established between workpiece locations and the dis-
--
K --
~'1
D"'
D"l
Fig. 3. Versions of 3-1ink arms serving the same K and L.
39g
Applications
(
,mputers m Industry
where 0 ~< 8 ~ L - - K, and 82 + 2 8 ( K
12 COS 02) -Jr- C :-- 0,
(8}
where /
"\
*,
\.\ "
/~ ..... "[ ".
"\
~ "\
",
/
I
f"
% ,/ V
"\
~
/" ,
/
-
/
.
•
/ / / P a t forrn "
.
/7
/
7"
/
/-
" /"
.7
,/
,....
./
\
\,
C=I~ -t~+212[(H+1
"-
\ \'\
,~',
"\
\
\
....
, sin0: .... K c o s 0 ~ ]
+ K : + ( H-+ ll)-'.
,
,~
Similarly for the platform:
L ..
"-. /
.."
0.,4.,/.,
,- J. \ ,.. tan 0 3 -
\
\.
-,
"
",,.,
\\
-
,/
/
\ ~ ,
"\
/
/
~/
",
"".
%._-, \
/
/
/
\
/
/" /
"
,-" /
"\
\
,
\\\k
/
,,, -
/
/N.-,~.~
"\
\
%.
,%
\
/ / Loading
/_
OD
=
L
OG
=
/
Unloading Zone
K
OJ
=
, \
U
Fig. 4. Top view of work area.
Cl
Fig. 5. Four extreme positions of the sample arm.
\
/
,,\
".'v"
D
..... . -
, ,
\
l~ + t z sin 0 z , 8 - l~ cos 02 "
(9)
where 0 ~< 8 ~ K, and 8 2 - - 2 6 l ~ cos 0 2 + C L = 0 ,
(10)
where (71 -- 12 - l~, + 1~ + 21112 sin 02 .
It will be observed that Eqs. 9 and 10 may be obtained from Eqs. 7 and 8 for vanishing H and K. The Eqs. 7--10 establish the required relationships between the displacements 02, 03 and 8. Derivatives of the same would yield corresponding velocities and accelerations.
Computers in lndustrv
A. Mannaa et al./ 3-Li~tk Robot Arm
399
-10
e 3 5.0
- 20
4.5
-30
4.0
-40
co
0') 0,1 "o
3.5
i
-50
(,~
~0
¢I) 3.0
_60
2.5
-70
2
_80 -45
l _30
I -15
I 0
0 2
-
I
1 15
30
deg
Fig. 6. Variation of 6 and 03 with 02 at ground level.
4. Sample Design G i v e n that l I = 2, H = 1 and K = 2 units, let it be required to d e t e r m i n e 12, l 3, L and U. Referring to Eq. 4a, L m a y be assumed as 5. T h e n from Eq. 2: 12 = 3 ~
- 13 .
(a)
The design reach L is now established as 6.87. in agreement with Eq. 4. Fig. 5 illustrates the critical positions of the arm. Fig. 6 shows the variation of 6 and 0~ with 02 as c o m p u t e d from Eqs. 7 and 8 when the g r i p p e r is at g r o u n d level. The interrelationships of positional variables at the p l a t f o r m are d e p i c t e d in Fig. 7.
Substituting from Eq. 6 into (a) that
1~>~[l~-4] 1/2,
5. Discussion
we o b t a i n l 2 >~ 3.26. Let l 2 = 3.5. Thus 5.5 >~ 13 >/- 2.87. C h o o s i n g l 3 as 4.0, Eq. 5 now yields U: U ~< 7.23.
It is noted in Fig. 4 that when the arm rotates a b o u t the OB axis to serve the u p p e r extremes of the l o a d i n g / u n l o a d i n g zone, a limit is reached when:
K-
IlL ll+H"
400
('omputers in Industry'
Applications
!30
3.0
~-20
2.5
210
2.0
200
1.5
i
190
1.0
180
0.5
170
0
1 -40
- 30
- 20
- 10
0 2 _ deg
Fig. 7. Variation of 8 and 03 with 02 on the platform.
which, for the sample design, is equal to 4.58. For values of K > 4.58 the design L can no longer be maintained. Fig. 6 illustrates the observation noted above in Fig. 3 that two different configurations of the arm will generally yield the same 8 at ground level. There is, however, a one-to-one relationship between 8 and the angles 02 and 03 at platform level.
Acknowledgement This work was partially sponsored by G r a n t No. 5-411 of the College of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia.
References [1] Paul, R.P. (1981) Robot Manipulator~: Mathematics, Programming, and Control. MIT Press, Cambridge, Mass. [2] Sander, G.N. and Erdman, A.G. (1984) Advanced Mechanism Design: Analysis and Synthesis, l/ol. 2. Prentice Hall. [3] Suh, C.H. and Radcliffe, C.W. (1978) Kinematics and Mechanisms Design. John Wiley. [4] Hall, A.S. (1981) Mechanism Analysis. Bait Publishers, Lafayette, Indiana.
Applications
Geometric Design Considerations for 3-Link Robot Arm A. M a n n a a , F. Dehlawi and M. A k y u r t ('ollege of Engineering, Kmg Abdulaziz Unit~etwitv P.O. Box 9027. Jeddah 21413, Saudi Arabia Geometric design parameters are discussed and the criteria are laid for the proper dimensioning of a three-link robot arm engaged in a given work space and serving workpieces at two different levels. The design procedure is illustrated by an example.
Ko'words: Robotics, Robot manipulator, Robot design, 3-Link arm. Geometric design,
AbduI-Raouf M. Mannaa was born in Jeddah, Saudi Arabia in 1951. He received his B. Sc degree in Mechanical Engineering from the University of Petroleum in Dhahran, Saudi Arabia in 1974. His M.Sc in Mechanical Engineering and his Master of Engineering were received from the University of California at Berkeley in 1976 and 1977 respectively. His Ph.D degree in ,, Mechanical Engineering was received from the University of Washington at Seattle. Dr. Mannaa is presently working as an Assistant Prof. in the Department of Mechanical Design and Production, King Abdulaziz University, Jeddah, Saudi Arabia. Current research areas are robotics, solar cooling, and applied mechanics.
Fouad Dehlawy was born in Makkah, Saudi Arabia in June, 1952. He received his B.Sc degree in Electrical E n g i n e e r i n g from University of Riyadh, Saudi Arabia in 1974, and M.S.E.E. from Stanford University, California in 1977. Another Master in Statistics was received in 1980 and Ph.D in instructional uses of computers in 1983, both from Stanford University. Dr. Dehlawy is currently an Assistant Prof. in the Electrical Engineering Department, King Abdulaziz University, Jeddah, Saudi Arabia. His research interests are Arabization of computers, artificial intelligence and robotics.
a
1. Introduction Robot manipulators comprise essentially an arm and a gripper or hand. The arm moves the gripper within a workspace under computer control. Of the various kinematic systems that produce a workspace, the popular three-link chain equipped with three revolute joints (Fig. 1) is studied below as regards to geometric design constraints and variables. The dynamic analysis of the arm as well as the performance of the prototype arm currently under construction will be presented in subsequent articles.
2. General Design Considerations The design objective for the arm (Figs. 1 and 2) is to pick up (via an unshown gripper located at D) an object positioned at ground level and within a working radius L, and to deliver it on the platform of height H and characteristic width K, a n d / o r vice versa. Lengths of the links need to be chosen in such a way that the entire working length L - K is accessible to the arm at ground level. Similarly, the arm must be able to service the entire platform width K corresponding to an angular location '5 (Fig. 1) at height H.
North-Holland Computers in Industry 7 (1986) 395-400 0166-3615/86/$3.50 ":, 1986 Elsevier Science Publishers B.V. (North-Holland)
Mehmet Akyurt was born in Turkey in 1940. He received his BS and MS degrees in Mechanical Engineering at the Middle East Technical University in Ankara, and his Ph.D from Purdue University, U.S.A. His current interests are mechanisms, heat pipes and solar energy applications.
396
Applications
(omputers in lndust O,
D
-,Vb G3,
G2
Bq~
e 2
11
OB
=
BC CD
= 12 =13
fiG 2
= r2
BG 3
=r3
G1
)¢~\
! \-\
z,/
/
,
i
C
t
.
.
, D
~e
Fig. 1. The three-link arm with triple revolute joints.
Fig. 3 shows versions of a three-link arm with three revolute joints in several different positions and configurations. Two of the arms feature the same L. The arm B C D " is noted to possess two solutions on the ground for given positions of l 2. Fig. 4 shows the top view of the working area of the robot, where the ground level working area is marked as the loading/unloading zone. Note that K in general varies with rotation of the arm about the O B axis and depends on the shape of the platform. A rectangular platform is shown in Fig. 4.
It is observed in Fig. 3 that to be equal to or greater than able to service the immediate form on the ground. For the note that: I~=H+I
it is necessary for 12 K for the arm to be vicinity of the platcase of equality we (1)
1
12+l = [L2+ (u+t,)2]
(2)
For given values of l 2. l 3, t f and L the latter relationship sets a limit to 1~" LH
It >~ L - K
H.
(3)
Computers in Industry
A. Mannaa et al. / 3-Link Robot Arm
397
0 2 "~
03
I-
O
D
-f H
I.
Fig. 2. The arm serving at two different levels.
It m a y be verified that l 2 does not vary strongly with l 1 for any set of H, L and K. Setting l 1 equal to unity, Eq. 3 m a y be re-arranged to yield: L >/K(1 + H),
placement variables 02 and 0 3 of the arm. N o w for a gripper located at D to pick up or deposit an object at ground level, we require that H + l 1 + l 2 sin 02 tan 03 = - / 2 c o s 0 2 + K + ~ "
('7)
(4) \ \
or more generally
\ \
llL >I K(I, + H).
(4a)
The m a x i m u m serviceable platform reach U (Fig. 4) is constrained by:
u
+ 13) - # .
(5)
A further condition that must be satisfied if the a r m is to service the immediate vicinity of its base (point O in Fig. 4) is: l, + 12>~ 13>~ [122 - 12] I/2 .
I I
12
,
,
(6)
~
/
\,/i
/I
,.,.m/--<
//
'
Given H and K, relationships (2), (4), (5) and (6) can be utilized to determine guideline magnitudes for l 2, l 3, L, and U.
H
~
3. Position Control
It is essential for an a r m operating under computer control that precise relationships be established between workpiece locations and the dis-
--
K --
~'1
D"'
D"l
Fig. 3. Versions of 3-1ink arms serving the same K and L.
39g
Applications
(
,mputers m Industry
where 0 ~< 8 ~ L - - K, and 82 + 2 8 ( K
12 COS 02) -Jr- C :-- 0,
(8}
where /
"\
*,
\.\ "
/~ ..... "[ ".
"\
~ "\
",
/
I
f"
% ,/ V
"\
~
/" ,
/
-
/
.
•
/ / / P a t forrn "
.
/7
/
7"
/
/-
" /"
.7
,/
,....
./
\
\,
C=I~ -t~+212[(H+1
"-
\ \'\
,~',
"\
\
\
....
, sin0: .... K c o s 0 ~ ]
+ K : + ( H-+ ll)-'.
,
,~
Similarly for the platform:
L ..
"-. /
.."
0.,4.,/.,
,- J. \ ,.. tan 0 3 -
\
\.
-,
"
",,.,
\\
-
,/
/
\ ~ ,
"\
/
/
~/
",
"".
%._-, \
/
/
/
\
/
/" /
"
,-" /
"\
\
,
\\\k
/
,,, -
/
/N.-,~.~
"\
\
%.
,%
\
/ / Loading
/_
OD
=
L
OG
=
/
Unloading Zone
K
OJ
=
, \
U
Fig. 4. Top view of work area.
Cl
Fig. 5. Four extreme positions of the sample arm.
\
/
,,\
".'v"
D
..... . -
, ,
\
l~ + t z sin 0 z , 8 - l~ cos 02 "
(9)
where 0 ~< 8 ~ K, and 8 2 - - 2 6 l ~ cos 0 2 + C L = 0 ,
(10)
where (71 -- 12 - l~, + 1~ + 21112 sin 02 .
It will be observed that Eqs. 9 and 10 may be obtained from Eqs. 7 and 8 for vanishing H and K. The Eqs. 7--10 establish the required relationships between the displacements 02, 03 and 8. Derivatives of the same would yield corresponding velocities and accelerations.
Computers in lndustrv
A. Mannaa et al./ 3-Li~tk Robot Arm
399
-10
e 3 5.0
- 20
4.5
-30
4.0
-40
co
0') 0,1 "o
3.5
i
-50
(,~
~0
¢I) 3.0
_60
2.5
-70
2
_80 -45
l _30
I -15
I 0
0 2
-
I
1 15
30
deg
Fig. 6. Variation of 6 and 03 with 02 at ground level.
4. Sample Design G i v e n that l I = 2, H = 1 and K = 2 units, let it be required to d e t e r m i n e 12, l 3, L and U. Referring to Eq. 4a, L m a y be assumed as 5. T h e n from Eq. 2: 12 = 3 ~
- 13 .
(a)
The design reach L is now established as 6.87. in agreement with Eq. 4. Fig. 5 illustrates the critical positions of the arm. Fig. 6 shows the variation of 6 and 0~ with 02 as c o m p u t e d from Eqs. 7 and 8 when the g r i p p e r is at g r o u n d level. The interrelationships of positional variables at the p l a t f o r m are d e p i c t e d in Fig. 7.
Substituting from Eq. 6 into (a) that
1~>~[l~-4] 1/2,
5. Discussion
we o b t a i n l 2 >~ 3.26. Let l 2 = 3.5. Thus 5.5 >~ 13 >/- 2.87. C h o o s i n g l 3 as 4.0, Eq. 5 now yields U: U ~< 7.23.
It is noted in Fig. 4 that when the arm rotates a b o u t the OB axis to serve the u p p e r extremes of the l o a d i n g / u n l o a d i n g zone, a limit is reached when:
K-
IlL ll+H"
400
('omputers in Industry'
Applications
!30
3.0
~-20
2.5
210
2.0
200
1.5
i
190
1.0
180
0.5
170
0
1 -40
- 30
- 20
- 10
0 2 _ deg
Fig. 7. Variation of 8 and 03 with 02 on the platform.
which, for the sample design, is equal to 4.58. For values of K > 4.58 the design L can no longer be maintained. Fig. 6 illustrates the observation noted above in Fig. 3 that two different configurations of the arm will generally yield the same 8 at ground level. There is, however, a one-to-one relationship between 8 and the angles 02 and 03 at platform level.
Acknowledgement This work was partially sponsored by G r a n t No. 5-411 of the College of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia.
References [1] Paul, R.P. (1981) Robot Manipulator~: Mathematics, Programming, and Control. MIT Press, Cambridge, Mass. [2] Sander, G.N. and Erdman, A.G. (1984) Advanced Mechanism Design: Analysis and Synthesis, l/ol. 2. Prentice Hall. [3] Suh, C.H. and Radcliffe, C.W. (1978) Kinematics and Mechanisms Design. John Wiley. [4] Hall, A.S. (1981) Mechanism Analysis. Bait Publishers, Lafayette, Indiana.