Based on the general degree-of-freedom (DOF) formula for spatial mechanisms proposed by the author in 2012, the early single open chain (SOC)-based composition principle for planar mechanisms is extended to general spatial mechanisms in this paper. First, three types of existing mechanism composition principle and their characteristics are briefly discussed. Then, the SOC-based composition principle for general spatial mechanisms is introduced. According to this composition principle, a spatial mechanism is first decomposed into Assur kinematic chains (AKCs) and an AKC is then further decomposed into a group of ordered SOCs. Kinematic (dynamic) analysis of a spatial mechanism can then be reduced to kinematic (dynamic) analysis of AKCs and finally to kinematic (dynamic) analysis of ordered SOCs. The general procedure for decomposing the mechanism into ordered SOCs and the general method for determining AKC(s) contained in the mechanism are also given. Mechanism's kinematic (dynamic) analysis can be reduced to the lowest dimension (number of unknowns) directly at the topological structure level using the SOC-based composition principle. The SOC-based composition principle provides a theoretical basis for the establishment of a unified SOC-based method for structure synthesis and kinematic (dynamic) analysis of general spatial mechanisms.

References

References
1.
Assur
,
L. V.
,
1913
, “
Investigation of Plane Hinged Mechanisms With Lower Pairs From the Point of View of Their Structure and Classification (in Russian)—Part I
,”
Bull. Petrograd Polytech. Inst.
,
20
, pp.
329
386
.
2.
Assur
,
L. V.
,
1914
, “
Investigation of Plane Hinged Mechanisms With Lower Pairs From the Point of View of Their Structure and Classification (in Russian)—Part II
,”
Bull. Petrograd Polytech. Inst
,
21
, pp.
187
283
.
3.
Dobrovolskii
,
V. V.
,
1939
, “
Main Principles of Rational Classification
,” AS USSR.
4.
Verho
,
A.
,
1973
, “
An Extension of the Concept of the Group
,”
Mech. Mach. Theory
,
8
(
2
), pp.
249
256
.
5.
Manolescu
,
N. I.
,
1979
, “
A Unified Method for the Formation of All Planar Jointed Kinematic Chains and Baranov Trusses
,”
Environ. Plann. B
,
6
(
4
), pp.
447
454
.
6.
Galletti
,
C.
,
1986
, “
A Note on Modular Approaches to Planar Linkage Kinematic Analysis
,”
Mech. Mach. Theory
,
21
(
5
), pp.
385
391
.
7.
Fanghella
,
P.
,
1988
, “
Kinematics of Spatial Linkages by Group Algebra: A Structure-Based Approach
,”
Mech. Mach. Theory
,
23
(
3
), pp.
171
183
.
8.
Ceresole
,
E.
,
Fanghella
,
P.
, and
Galletti
,
C.
,
1996
, “
Assur's Groups, AKCs, Basic Trusses, SOCs, etc.: Modular Kinematics of Planar Linkages
,”
ASME
Paper No. 96-DETC/MECH-1027.
9.
Mruthyunjaya
,
T. S.
,
2003
, “
Kinematic Structure of Mechanisms Revisited
,”
Mech. Mach. Theory
,
38
(
4
), pp.
279
320
.
10.
Servatius
,
B.
,
Shai
,
O.
, and
Whiteley
,
W.
,
2010
, “
Combinatorial Characterization of the Assur Graphs From Engineering
,”
Eur. J. Combinatorics
,
31
(
4
), pp.
1091
1104
.
11.
Shai
,
O.
,
Sljoka
,
A.
, and
Whiteley
,
W.
,
2013
, “
Directed Graphs, Decompositions, and Spatial Rigidity
,”
Discrete Appl. Math.
,
161
(
18
), pp.
3028
3047
.
12.
Reuleaux
,
F.
,
1876
,
Theoretische Kinematic
,
Fridrich Vieweg
,
Braunschweig, Germany
(English Translation by A. B. W. Kennedy, The Kinematics of Machinery, Dover, Mineola, NY).
13.
Franke
,
R.
,
1951
,
Vom Aufbau der Getriebe
,
2nd ed.
, Vol.
1
,
Beuthvertrieb
,
Berlin
.
14.
Beyer
,
R.
,
1963
,
The Kinematic Synthesis of Mechanisms
,
McGraw-Hill
, London.
15.
Hain
,
K.
,
1967
,
Applied Kinematics
,
2nd ed.
,
McGraw-Hill
,
New York
.
16.
Woo
,
L. S.
,
1967
, “
Type Synthesis of Plane Linkages
,”
ASME J. Eng. Ind.
,
89
(
1
), pp.
159
172
.
17.
Tischler
,
C.
,
Samuel
,
A.
, and
Hunt
,
K. H.
,
1995
, “
Kinematic Chains for Robot Hands—I: Orderly Number-Synthesis
,”
Mech. Mach. Theory
,
30
(
8
), pp.
1193
1215
.
18.
Wittenburg
,
J.
,
1977
,
Dynamics of Systems of Rigid Bodies
,
Teubner
,
Stuttgart, Germany
.
19.
Dobrjanskyj
,
L.
, and
Freudenstein
,
F.
,
1967
, “
Some Applications of Graph Theory to the Structural Analysis of Mechanisms
,”
J. Eng. Ind.
,
89
(
1
), pp.
153
158
.
20.
Paul
,
B.
,
1979
,
Kinematics and Dynamics of Planar Machinery
,
Prentice Hall
, Englewood Cliffs,
NJ
.
21.
Kinzel
,
G. L.
, and
Chang
,
C. H.
,
1984
, “
The Analysis of Planar Linkages Using a Modular Approach
,”
Mech. Mach. Theory
,
19
(
1
), pp.
165
172
.
22.
Sohn
,
W. J.
, and
Freudenstein
,
F.
,
1989
, “
An Application of Dual Graphs to the Automatic Generation of the Kinematic Structure of Mechanisms
,”
ASME J. Mech. Trans. Autom. Des.
,
111
(
4
), pp.
494
497
.
23.
Tuttle
,
E. R.
,
Peterson
,
S. W.
, and
Titus
,
J. E.
,
1989
, “
Enumeration of Basic Kinematic Chains Using the Theory of Finite Groups
,”
ASME J. Mech. Trans. Autom. Des.
,
111
(
4
), pp.
498
503
.
24.
Waldron
,
K. J.
, and
Sreenivasan
,
S. V.
,
1996
, “
A Study of the Solvability of the Position Problem for Multi-Circuit Mechanisms by Way of Example of the Double Butterfly Linkage
,”
ASME J. Mech. Des.
,
118
(
3
), pp.
390
395
.
25.
Tsai
,
L.-W.
,
1999
,
Robot Analysis: The Mechanics of Serial and Parallel Manipulators
,
Wiley
,
New York
.
26.
McCarthy
,
J. M.
, and
Soh
,
G. S.
,
2011
,
Geometric Design of Linkages
,
Springer
,
New York
.
27.
Yang
,
T.-L.
, and
Yao
,
F.-H.
,
1988
, “
The Topological Characteristics and Automatic Generation of Structural Analysis and Synthesis of Plane Mechanisms—Part 1: Theory, Part 2: Application
,” ASME Conference, Orlando, FL, pp. 179–190.
28.
Yang
,
T.-L.
, and
Yao
,
F.-H.
,
1992
, “
The Topological Characteristics and Automatic Generation of Structural Analysis and Synthesis of Spatial Mechanisms—Part 1: Topological Characteristics of Mechanical Network; Part 2: Automatic Generation of Structure Types of Kinematic Chains
,” ASME Conference, Phoenix, AZ, pp. 179–190.
29.
Jin
,
Q.
, and
Yang
,
T.-L.
,
2004
, “
Theory for Topology Synthesis of Parallel Manipulators and Its Application to Three Dimension Translation Parallel Manipulators
,”
ASME J. Mech. Des.
,
126
(
4
), pp.
625
639
.
30.
Jin
,
Q.
, and
Yang
,
T.-L.
,
2004
, “
Synthesis and Analysis of a Group of 3-Degree-of-Freedom Partially Decoupled Parallel Manipulators
,”
ASME J. Mech. Des.
,
126
(
2
), pp.
301
306
.
31.
Yang
,
T.-L.
,
2004
,
Theory and Application of Robot Mechanism Topology
,
China Machine Press
,
Beijing, China
.
32.
Yang
,
T.-L.
, and
Sun
,
D.-J.
,
2012
, “
A General DOF Formula for Parallel Mechanisms and Multi-Loop Spatial Mechanisms
,”
ASME J. Mech. Rob.
,
4
(
1
), p.
011001
.
33.
Yang
,
T.-L.
,
Liu
,
A.-X.
,
Luo
,
Y.-F.
,
Shen
,
H.-P.
,
Hang
,
L.-B.
, and
Jin
,
Q.
,
2009
, “
Position and Orientation Characteristic Equation for Topological Design of Robot Mechanisms
,”
ASME J. Mech. Des.
,
131
(
2
), p.
021001
.
34.
Yang
,
T.-L.
,
Liu
,
A.-X.
,
Shen
,
H.-P.
,
Luo
,
Y.-F.
,
Hang
,
L.-B.
, and
Shi
,
Z.-X.
,
2013
, “
On the Correctness and Strictness of the POC Equation for Topological Structure Design of Robot Mechanisms
,”
ASME J. Mech. Rob.
,
5
(
2
), p.
021009
.
35.
Yang
,
T.-L.
,
Liu
,
A.-X.
,
Shen
,
H.-P.
, and
Hang
,
L.-B.
,
2016
, “
Topological Structure Synthesis of 3T1R Parallel Mechanism Based on POC Equations
,”
Ninth International Conference on Intelligent Robotics and Applications (ICIRA)
, Tokyo, Japan, Aug. 22–24, pp. 147–161.
36.
Huang
,
Z.
, and
Li
,
Q. C.
,
2002
, “
General Methodology for Type Synthesis of Symmetrical Lower-Mobility Parallel Manipulators and Several Novel Manipulators
,”
Int. J. Rob. Res.
,
21
(
2
), pp.
131
145
.
37.
Kong
,
X.
, and
Gosselin
,
C. M.
,
2007
,
Type Synthesis of Parallel Mechanisms
(Springer Tracts in Advanced Robotics, Vol.
33
), Springer, New York.
38.
Hervé
,
J. M.
,
1995
, “
Design of Parallel Manipulators Via the Displacement Group
,”
Ninth World Congress on the Theory of Machines and Mechanisms
, Milan, Italy, Aug. 29–Sept. 2, pp.
2079
2082
.
39.
Gogu
,
G.
,
2008
,
Structural Synthesis of Parallel Robots—Part 1: Methodology
,
Springer
,
Dordrecht, The Netherlands
.
40.
Meng
,
X.
,
Gao
,
F.
,
Wu
,
S.
, and
Ge
,
J.
,
2014
, “
Type Synthesis of Parallel Robotic Mechanisms: Framework and Brief Review
,”
Mech. Mach. Theory
,
78
, pp.
177
186
.
41.
Yang
,
T.-L.
,
1996
,
Basic Theory of Planar Mechanical System: Structure, Kinematic and Dynamic
,
China Machine Press
,
Beijing, China
.
42.
Yang
,
T.-L.
,
Yao
,
F.-H.
,
Zhang
,
M.
, and
Xu
,
Z.
,
1998
, “
A Comparative Study on Some Modular Approaches for Analysis and Synthesis of Planar Linkages—Part I: Modular Structural Analysis and Modular Kinematic Analysis
,”
ASME
Paper No. DETC98/MECH-5920.
43.
Yang
,
T.-L.
,
Yao
,
F.-H.
,
Zhang
,
M.
, and
Xu
,
Z.
,
1998
, “
A Comparative Study on Some Modular Approaches for Analysis and Synthesis of Planar Linkages—Part II: Modular Dynamic Analysis and Modular Structural Synthesis
,”
ASME
Paper No. DETC98/MECH-6058.
44.
Shen
,
H.-P.
,
Ting
,
K.-L.
, and
Yang
,
T.-L.
,
2000
, “
Configuration Analysis of Complex Multiloop Linkages and Manipulators
,”
Mech. Mach. Theory
,
35
(
3
), pp.
353
362
.
45.
Yang
,
T.-L.
,
1986
, “
Structural Character of Planar Complex Mechanisms and Methods of Kinematic and Kinetostatic Analysis by Imaginary Unknown Parameters
,”
ASME
Paper No. 86-DET-180.
46.
Hang
,
L. B.
,
Jin
,
Q.
,
Jin
,
J.
,
Wu
., and
Yang
,
T.-L.
,
2000
, “
A General Study of the Number of Assembly Configurations for Multi-Circuit Planar Linkages
,”
J. Southeast Univ.
,
16
(
1
), pp.
46
51
.
47.
Nicolas
,
R.
, and
Federico
,
T.
,
2012
, “
On Closed-Form Solutions to the Position Analysis of Baranov Trusses
,”
Mech. Mach. Theory
,
50
(2), pp.
179
196
.
48.
Hahn
,
E.
, and
Shai
,
O.
,
2016
, “
A Single Universal Construction Rule for the Structural Synthesis of Mechanisms
,”
ASME
Paper No. IDETC/CIE 2016-59133.
49.
Hahn
,
E.
, and
Shai
,
O.
,
2016
, “
Construction of Baranov Trusses Using a Single Universal Construction Rule
,”
ASME
Paper No. IDETC/CIE 2016-59134.
50.
Shi
,
Z.-X.
,
Luo
,
Y.-F.
,
Hang
,
L.-B.
, and
Yang
,
T.-L.
,
2007
, “
A Simple Method for Inverse Kinematic Analysis of the General 6R Serial Robot
,”
ASME J. Mech. Des.
,
129
(
8
), pp.
793
798
.
51.
Jin
,
Q.
, and
Yang
,
T.-L.
,
2002
, “
Over-Constraint Analysis on Spatial 6-Link Loops
,”
Mech. Mach. Theory
,
37
(
3
), pp.
267
278
.
52.
Feng
,
Z.-Y.
,
Zhang
,
C.
, and
Yang
,
T.-L.
,
2006
, “
Direct Displacement Solution of 4-DOF Spatial Parallel Mechanism Based on Ordered Single-Open-Chain
,”
Chin. J. Mech. Eng.
,
42
(
7
), pp.
35
38
.
53.
Shi
,
Z.-X.
,
Luo
,
Y.-F.
, and
Yang
,
T.-L.
,
2006
, “
Modular Method for Kinematic Analysis of Parallel Manipulators Based on Ordered SOCs
,”
ASME
Paper No. DETC2006-99089.
54.
Shen
,
H.
,
Shao
,
G.
,
Deng
,
J.
, and
Yang
,
T.-L.
,
2017
, “
A Novel 3T1R Parallel Robot 2 PaRSS: Design and Kinematics
,”
ASME
Paper No. DETC2017-67265.
55.
Zhang
,
J.-Q.
, and
Yang
,
T.-L.
,
1994
, “
A New Method and Automatic Generation for Dynamic Analysis of Complex Planar Mechanisms Based on the SOC
,” ASME Design Technical Conference, pp. 215–220.
56.
Yang
,
T.-L.
,
Li
,
H.-L.
, and
Luo
,
Y.-F.
,
1991
, “
On the Structure of Dynamic Equation of Any Mechanical System
,”
Chin. J. Mech. Eng.
,
27
(
4
), pp.
1
15
.
You do not currently have access to this content.