This paper presents the linear representation of the screw triangle. The resultant twist of two finite twists is shown to be a linear combination of five screws. The linear representations of all degenerate screw triangles are also derived. The limiting cases of these results of finite displacements confirm the theory of screws in infinitesimal kinematics. The finite kinematic analysis of multi-link serial chains is performed by using the linear representation of the screw triangle to demonstrate a unification of finite and infinitesimal kinematics.

1.
Ball, R. S., 1900, A Treatise of the Theory of Screws, Cambridge University Press.
2.
Bottema, O., and Roth, B., 1979, Theoretical Kinematics, North Holland Publishing Company.
3.
Chen
P.
, and
Roth
B.
,
1969
, “
A Unified Theory for the Finitely and Infinitesimally Separated Position Problems of Kinematic Synthesis
,”
ASME Journal of Engineering for Industry
, Series B, Vol.
91
, No.
1
, pp.
203
208
.
4.
Craig, J. J., 1989, Introduction to Robotics: Mechanics and Control, Second Edition, Addison-Wesley.
5.
Denavit
J.
, and
Hartenberg
R. S.
,
1955
, “
A Kinematic Notation for Lower Pair Mechanisms Based on Matrices
,”
ASME Journal of Applied Mechanics
, Vol.
77
, pp.
215
221
.
6.
Dimentberg, F. M., 1965, The Screw Calculus and Its Application in Mechanics (in Russian Izdat. Nauka, Moscow), English Translation, N.A.S.A., 1968.
7.
Huang, C., 1992, “Application of Linear Algebra to Screw Systems and Position-Force Synthesis of Closed-Loop Linkages,” Ph.D. Thesis, Mechanical Engineering Department, Stanford University, California.
1.
Huang
C.
,
1993
, “
On the Finite Screw System of the Third Order Associated with a Revolute-Revolute Chain
,”
Advances in Design Automation
, Proceedings of the 1993 ASME Design Technical Conferences, Albuquerque, New Mexico, Sept. 19–22, 1993, Vol.
2
, pp.
81
89
.
2.
(Also published in
ASME JOURNAL OF MECHANICAL DESIGN
, Vol.
116
, pp.
875
883
.)
1.
Huang
C.
,
1994
, “
The Finite Screw System Associated with a Prismatic-Revolute Dyad and the Screw Displacement of a Point
,”
Mechanism and Machine Theory
, Vol.
29
, No.
8
, pp.
1131
1142
.
2.
Huang
C.
, and
Roth
B.
,
1994
, “
Analytic Expressions for the Finite Screw Systems
,”
Mechanism and Machine Theory
, Vol.
29
, No.
2
, pp.
207
222
.
3.
Hunt, K. H., 1978, Kinematic Geometry of Mechanisms, Clarendon, Oxford.
4.
James, P., and Roth, B., 1992, “A Unified Theory for Kinematic Synthesis,” Proceedings of the 22nd Biennial Mechanisms Conference, Scottsdale, Arizona, September 13–16, pp. 587–597.
5.
Parkin
I. A.
,
1992
, “
A Third Conformation with the Screw Systems: Finite Twist Displacements of a Directed Line and Point
,”
Mechanism and Machine Theory
, Vol.
27
, No.
2
, pp.
177
188
.
6.
Phillips, J. R., and Zhang, W. X., 1987, “The Screw Triangle and the Cylindroid,” Proceedings of the 7th World Congress, IFToMM, Sevilla, Spain, Pergamon Press, Oxford, pp. 179–182.
7.
Paul, R. P., 1981, Robot Manipulators: Mathematics, Programming, and Control, MIT Press.
8.
Rosenauer, N., and Willis, A. H., 1953, Kinematics of Mechanisms, Associated General Publications, Sydney.
9.
Roth
B.
,
1967
, “
On the Screw Axes and Other Special Lines Associated with Spatial Displacements of a Rigid Body
,”
ASME Journal of Engineering for Industry
, Series B, Vol.
89
, No.
1
, pp.
102
110
.
10.
Roth
B.
,
1968
, “
The Design of Binary Cranks with Revolute, Cylindric, and Prismatic Joints
,”
Journal of Mechanisms
, Vol.
3
, pp.
61
72
.
11.
Sticher
F.
,
1989
, “
On the Finite Screw Axis Cylindroid
,”
Mechanism and Machine Theory
, Vol.
24
, No.
3
, pp.
143
155
.
12.
Sue, C. H., and Radcliffe, C. W., 1978, Kinematics and Mechanisms Design, John Wiley and Sons, New York.
13.
Sugimoto
K.
, and
Duffy
J.
,
1982
, “
Application of Linear Algebra to Screw Systems
,”
Mechanism and Machine Theory
, Vol.
17
, No.
1
, pp.
73
83
.
14.
Tsai, L. W., 1972, “Design of Open Loop Chains for Rigid Body Guidance,” Ph.D. Thesis, Mechanical Engineering Department, Stanford University, California.
15.
Tsai
L. W.
, and
Roth
B.
,
1972
, “
Design of Dyads with Helical, Cylindrical, Spherical, Revolute and Prismatic Joints
,”
Mechanism and Machine Theory
, Vol.
7
, pp.
85
102
.
16.
Tsai
L. W.
, and
Roth
B.
,
1973
, “
Incompletely Specified Displacements: Geometry and Spatial Linkage Synthesis
,”
ASME Journal of Engineering for Industry
, Vol.
95
, Series B, Vol. 2, pp.
603
611
.
17.
Woo
L.
, and
Freudenstein
F.
,
1970
, “
Application of Line Geometry to Theoretical Kinematics and the Kinematic Analysis of Mechanical Systems
,”
Journal of Mechanisms
, Vol.
5
, Pergamon Press, Great Britain, pp.
417
460
.
18.
Yang, A. T., 1963, “Application of Quaternion Algebra and Dual Numbers to the Analysis of Spatial Mechanisms,” Doctoral Dissertation, Dept. of Mechanical Engineering, Columbia University, New York.
19.
Yuan
M. S. C.
,
Freudenstein
F.
, and
Woo
L. S.
,
1971
, “
Kinematic Analysis of Spatial Mechanisms by Means of Screw Coordinates. Part 2—Analysis of Spatial Mechanisms
,”
ASME Journal of Engineering for Industry
, Series B, Vol.
93
, No.
1
, pp.
67
73
.
This content is only available via PDF.
You do not currently have access to this content.