Motivated by the problem of synthesizing a pattern of flexures that provide a desired constrained motion, this paper presents a new screw theory that deals with “line screws” and “line screw systems.” A line screw is a screw with a zero pitch. The set of all line screws within a screw system is called a line variety. A general screw system of rank $m$ is a line screw system if the rank of its line variety equals $m$. This paper answers two questions: (1) how to calculate the rank of a line variety for a given screw system and (2) how to algorithmically find a set of linearly independent lines from a given screw system. It has been previously found that a wire or beam flexure is considered a line screw, or more specifically a pure force wrench. By following the reciprocity and definitions of line screws, we have derived the necessary and sufficient conditions of line screw systems. When applied to flexure synthesis, we show that not all motion patterns can be realized with wire flexures connected in parallel. A computational algorithm based on this line screw theory is developed to find a set of admissible line screws or force wrenches for a given motion space. Two flexure synthesis case studies are provided to demonstrate the theory and the algorithm.

1.
Ball
,
R. S.
, 1998,
The Theory of Screws
,
Cambridge University Press
,
Cambridge, England
(originally published in 1876 and revised by the author in 1900, now reprinted with an introduction by H. Lipkin and J. Duffy).
2.
Hunt
,
K. H.
, 1978,
Kinematic Geometry of Mechanisms
,
Oxford University Press
,
New York, NY
.
3.
Bottema
,
O.
, and
Roth
,
B.
, 1979,
Theoretical Kinematics
,
North-Holland
,
New York
.
4.
Phillips
,
J.
, 1984,
Freedom in Machinery. Volume 1, Introducing Screw Theory
,
Cambridge University Press
,
Cambridge, UK
.
5.
Phillips
,
J.
, 1990,
Freedom in Machinery. Volume 2, Screw Theory Exemplified
,
Cambridge University Press
,
Cambridge, UK
.
6.
Sugimoto
,
K.
, and
Duffy
,
J.
, 1982, “
Application of Linear Algebra to Screw System
,”
Mech. Mach. Theory
0094-114X,
17
(
1
), pp.
73
83
.
7.
Lipkin
,
H.
, and
Duffy
,
J.
, 1985, “
The Elliptic Polarity of Screws
,”
ASME J. Mech., Transm., Autom. Des.
0738-0666,
107
(
3
), pp.
377
386
.
8.
Lipkin
,
H.
, and
Patterson
,
T.
, 1992, Geometric Properties of Modelled Robot Elasticity: Part I—Decomposition, DE-Vol. 45, pp.
187
193
.
9.
Patterson
,
T.
, and
Lipkin
,
H.
, 1992, Geometric Properties of Modelled Robot Elasticity: Part II—Decomposition, DE-Vol. 45, pp.
179
185
.
10.
Patterson
,
T.
, and
Lipkin
,
H.
, 1993, “
Structure of Robot Compliance
,”
ASME J. Mech. Des.
0161-8458,
115
(
3
), pp.
576
580
.
11.
Shuguang
,
H.
, and
Schimmels
,
J. M.
, 1998, “
The Bounds and Realization of Spatial Stiffnesses Achieved With Simple Springs Connected in Parallel
,”
IEEE Trans. Rob. Autom.
1042-296X,
14
(
3
), pp.
466
475
.
12.
Shuguang
,
H.
, and
Schimmels
,
J. M.
, 2000, “
The Bounds and Realization of Spatial Compliances Achieved With Simple Serial Elastic Mechanisms
,”
IEEE Trans. Rob. Autom.
1042-296X,
16
(
1
), pp.
99
103
.
13.
Huang
,
S.
, 1998, “
The Analysis and Synthesis of Spatial Compliance
,” Ph.D. thesis, Marquette University, Milwaukee, WI.
14.
Kim
,
C. J.
, 2008, “
Functional Characterization of Compliant Building Blocks Utilizing Eigentwists and Eigen-Wrenches
,”
Proceedings of ASME IDETC/CIE 2008
, New York, Aug. 3–6.
15.
Dandurand
,
A.
, 1984, “
The Rigidity of Compound Spatial Grids
,”
Structural Topology
,
10
, pp.
41
56
.
16.
Merlet
,
J. -P.
, 1989, “
Singular Configurations of Parallel Manipulators and Grassmann Geometry
,”
Int. J. Robot. Res.
0278-3649,
8
(
5
), pp.
45
56
.
17.
Hao
,
F.
, and
McCarthy
,
J. M.
, 1998, “
Conditions for Line-Based Singularities in Spatial Platform Manipulators
,”
J. Rob. Syst.
0741-2223,
15
(
1
), pp.
43
55
.
18.
McCarthy
,
J. M.
, 2000,
,
Springer-Verlag
,
New York
.
19.
Huang
,
C.
,
Ravani
,
B.
, and
Kuo
,
W.
, 2008, “
A Geometric Interpretation of Finite Screw Systems Using the Bisecting Linear Line Complex
,”
ASME J. Mech. Des.
0161-8458,
130
(
10
), p.
102303
.
20.
Smith
,
S.
, and
Chetwynd
,
D.
, 1992,
Foundations of Ultra-Precision Mechanism Design
,
Taylor & Francis
,
London
.
21.
Smith
,
S. T.
, 2000,
Flexure: Element of Elastic Mechanisms
,
CRC
,
Boca Raton, FL
.
22.
Howell
,
L. L.
, 2001,
Compliant Mechanisms
,
Wiley-Interscience
,
New York, NY
.
23.
Blanding
,
D. L.
, 1999,
Exact Constraint: Machine Design Using Kinematic Processing
,
ASME
,
New York
.
24.
Hale
,
L. C.
, 1999, “
Principles and Techniques for Designing Precision Machines
,” Ph.D. thesis, MIT, Cambridge, MA.
25.
Awtar
,
S.
, and
Slocum
,
A. H.
, 2007, “
Constraint-Based Design of Parallel Kinematic XY Flexure Mechanisms
,”
ASME J. Mech. Des.
0161-8458,
129
(
8
), pp.
816
830
.
26.
Hopkins
,
J. B.
, and
Culpepper
,
M. L.
, 2010, “
Synthesis of Multi-Degree of Freedom, Parallel Flexure System Concepts via Freedom and Constraint Topology (FACT)—Part I: Principles
,”
Precis. Eng.
0141-6359,
34
(
2
), pp.
259
270
.
27.
Hopkins
,
J. B.
, and
Culpepper
,
M. L.
, 2010, “
Synthesis of Multi-Degree of Freedom, Parallel Flexure System Concepts via Freedom and Constraint Topology (FACT). Part II: Practice
,”
Precis. Eng.
0141-6359,
34
(
2
), pp.
271
278
.
28.
Su
,
H. -J.
,
Dorozhkin
,
D. V.
, and
Vance
,
J. M.
, 2009, “
A Screw Theory Approach for the Conceptual Design of Flexible Joints for Compliant Mechanisms
,”
ASME J. Mech. Rob.
1942-4302,
1
(
4
), p.
041009
.
29.
Dai
,
J. S.
, and
Jones
,
J. R.
, 2003, “
A Linear Algebraic Procedure in Obtaining Reciprocal Screw Systems
,”
J. Rob. Syst.
0741-2223,
20
(
7
), pp.
401
412
.
30.
Su
,
H. -J.
, and
Tari
,
H.
, 2010, “
Realizing Orthogonal Motions With Wire Flexures Connected in Parallel
,”
ASME J. Mech. Des.
0161-8458,
132
(
12
), p.
121002
.