In this paper, the geometric design problem of serial-link robot manipulators with three revolute (R) joints is solved for the first time using an interval analysis method. In this problem, five spatial positions and orientations are defined and the dimensions of the geometric parameters of the 3-R manipulator are computed so that the manipulator will be able to place its end-effector at these pre-specified locations. Denavit and Hartenberg parameters and 4×4 homogeneous matrices are used to formulate the problem and obtain the design equations and an interval method is used to search for design solutions within a predetermined domain.

1.
Suh, C. H., and Radcliffe, C. W., 1978, Kinematics and Mechanism Design, Wiley and Sons, New York, New York.
2.
Tsai, L. W., 1972, “Design of Open Loop Chains for Rigid Body Guidance,” Ph.D. Thesis, Department of Mechanical Engineering, Stanford University.
3.
Roth, B., 1986, “Analytic Design of Open Chains,” Proceedings of the Third International Symposium of Robotic Research, O. Faugeras and G. Giralt, eds., MIT Press.
4.
Bodduluri, M., Ge, J., McCarthy, M. J., and Roth, B., 1993, “The Synthesis of Spatial Linkages,” in Modern Kinematics: Developments in the Last Forty Years, A. Erdman, ed., John Willey and Sons.
5.
Raghavan
,
M.
, and
Roth
,
B.
,
1995
, “
Solving Polynomial Systems for the Kinematic Analysis and Synthesis of Mechanisms and Robot Manipulators
,”
ASME J. Mech. Des.
,
117
, pp.
71
78
.
6.
Erdman, A. G., and Sandor, G. N., 1997, Mechanism Design Analysis and Synthesis, Vol. 1, Third Edition, Prentice-Hall, Englewood Cliffs, N.J.
7.
Sandor, G. N., and Erdman, A. G., 1984, Advanced Mechanism Design: Analysis and Synthesis, Vol. 2, Prentice-Hall, Englewood Cliffs, N.J.
8.
Tsai
,
L.
, and
Roth
,
B.
,
1973
, “
A Note on the Design of Revolute-Revolute Cranks
,”
Mech. Mach. Theory
,
8
, pp.
23
31
.
9.
Roth, B., 1986, “Analytical Design of Two-Revolute Open Chains,” Proceedings of the Sixth CISM-IFToMM Symposium on the Theory and Practice of Robots and Manipulators, The MIT Press, Cambridge, MA, Malczyk G. and Morecki A., eds., pp. 207–214.
10.
Perez
,
A.
, and
McCarthy
,
J. M.
,
2003
, “
Dimensional Synthesis of Bennett Linkages
,”
ASME J. Mech. Des.
,
125
(
1
), pp.
98
104
.
11.
Mavroidis
,
C.
,
Lee
,
E.
, and
Alam
,
M.
,
2001
, “
A New Polynomial Solution to the Geometric Design Problem of Spatial R-R Robot Manipulators Using the Denavit and Hartenberg Parameters
,”
ASME J. Mech. Des.
,
123
(
1
), pp.
58
67
.
12.
Huang, C., and Chang, Y.-J., 2000, “Polynomial Solution to the Five-Position Synthesis of Spatial CC Dyads via Dialytic Elimination,” Proceedings of the ASME Design Technical Conferences, September 10–13, 2000, Baltimore MD, Paper Number DETC2000/MECH-14102.
13.
Murray
,
A. P.
, and
McCarthy
,
J. M.
,
1999
, “
Burmester Lines of a Spatial Five Position Synthesis from the Analysis of a 3-CPC Platform
,”
ASME J. Mech. Des.
,
121
, pp.
45
49
.
14.
Neilsen, J., and Roth, B., 1995, “Elimination Methods for Spatial Synthesis,” 1995 Computational Kinematics, Merlet J. P. and Ravani, B., eds., Vol. 40 of Solid Mechanics and It’s Applications, pp. 51–62, Kluwer Academic Publishers.
15.
Innocenti, C., 1994, “Polynomial Solution of the Spatial Burmester Problem,” Mechanism Synthesis and Analysis, ASME DE-Vol. 70, pp. 161–166.
16.
McCarthy, M., 2000, “Chapter 11: Algebraic Synthesis of Spatial Chains,” The Geometric Design of Linkages, New York, McGraw Hill Book Company.
17.
Lee, E., and Mavroidis, C., 2002, “Geometric Design Problem of Spatial PRR Manipulators Using Polynomial Elimination Techniques,” Proceedings of the ASME Design Technical Conferences, September, 2002, Montreal, Canada, Paper Number DETC2002/MECH-34314.
18.
Morgan
,
A. P.
,
Wampler
,
C. W.
, and
Sommese
,
A. J.
,
1990
, “
Numerical Continuation Methods for Solving Polynomial Systems Arising in Kinematics
,”
ASME J. Mech. Des.
,
112
, pp.
59
68
.
19.
Roth
,
B.
, and
Freudenstein
,
F.
,
1963
, “
Synthesis of Path Generating Mechanisms by Numerical Methods
,”
ASME J. Eng. Ind.
,
85B
, pp.
298
306
.
20.
Morgan
,
A. P.
, and
Wampler
,
C. W.
,
1990
, “
Solving a Planar Four-Bar Design Problem Using Continuation
,”
ASME J. Mech. Des.
,
112
, pp.
544
550
.
21.
Wampler
,
C. W.
,
Morgan
,
A. P.
, and
Sommese
,
A. J.
,
1992
, “
Complete Solution of the Nine-Point Path Synthesis Problem for Four-Bar Linkages
,”
ASME J. Mech. Des.
,
114
, pp.
153
159
.
22.
Dhingra
,
A. K.
,
Cheng
,
J. C.
, and
Kohli
,
D.
,
1994
, “
Synthesis of Six-Link, Slider-Crank and Four-Link Mechanisms for Function, Path and Motion Generation Using Homotopy With m-Homegenization
,”
ASME J. Mech. Des.
,
116
, pp.
1122
1130
.
23.
Lee
,
E.
, and
Mavroidis
,
C.
,
2002
, “
Solving the Geometric Design Problem of Spatial 3R Robot Manipulators Using Polynomial Continuation
,”
ASME J. Mech. Des.
,
124
, pp.
652
661
.
24.
Lee, E., and Mavroidis, C., 2003, “Four Precision Points Geometric Design of Spatial 3R Manipulators,” Proceedings of IFTOMM’s 11th World Congress in Mechanism and Machine Science, August 18–21, 2003 Tianjin-China.
25.
Moore, R. E., 1966, Interval Analysis, Prentice Hall, Englewood Cliffs, NJ.
26.
Moore, R. E., 1959, “Automatic Error Analysis in Digital Computation,” Lockheed Missiles and Space Co. Technical Report LMSD-48421, Palo Alto, CA.
27.
Neumaier, A., 1990, Interval Methods for Systems of Equations, PHI Series in Computer Science, Cambridge University Press, Cambridge.
28.
Collavizza
,
M.
,
Deloble
,
F.
, and
Rueher
,
M.
,
1999
, “
Comparing Partial Consistencies
,”
Reliable Computing
,
5
, pp.
1
16
.
29.
Tapia
,
R. A.
,
1971
, “
The Kantorovitch Theorem for Newton’s Method
,”
Am. Math. Monthly
,
78
, pp.
389
392
.
30.
Pieper, D., and Roth, B., 1969, “The Kinematics of Manipulators under Computer Control,” Proceedings of the Second World Congress on the Theory of Machines and Mechanisms, Zakopane, Poland, Vol. 2, pp. 159–169.
31.
Yamamura
,
K.
,
Kawata
,
H.
, and
Tokue
,
A.
,
1998
, “
Interval Solution of Nonlinear Equations Using Linear Programming
,”
BIT
,
38
(
1
), pp.
186
199
.
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