The kinematic differential equation for a spatial point trajectory accepts the time-varying instantaneous screw of a rigid body as input, the time-zero coordinates of a point on that rigid body as the initial condition and generates the space curve traced by that point over time as the solution. Applying this equation to multiple points on a rigid body derives the kinematic differential equations for a displacement matrix and for a joint screw. The solution of these differential equations in turn expresses the trajectory over the course of a finite displacement taken by a coordinate frame in the case of the displacement matrix, by a joint axis line in the case of a screw. All of the kinematic differential equations are amenable to solution by power series owing to the expression for the product of two power series. The kinematic solution for finite displacement of a single-loop spatial linkage may, hence, be expressed either in terms of displacement matrices or in terms of screws. Each method determines coefficients for joint rates by a recursive procedure that solves a sequence of linear systems of equations, but that procedure requires only a single factorization of a 6 by 6 matrix for a given initial posture of the linkage. The inverse kinematics of an 8R nonseparable redundant-joint robot, represented by one of the multiple degrees of freedom of a 9R loop, provides a numerical example of the new analytical technique.

References

References
1.
Lenarcic
,
J.
, and
Stanisic
,
M.
, 2003, “
A Humanoid Shoulder Complex and the Humeral Pointing Kinematics
,”
IEEE Trans. Rob. Autom.
,
19
(
3
), pp.
499
506
.
2.
Newkirk
,
J. T.
,
Watson
,
L. T.
, and
Stanisic
,
M. M.
, 2010, “
Determining the Number of Inverse Kinematic Solutions of a Constrained Parallel Mechanism Using a Homotopy Algorithm
,”
ASME J. Mech. Rob.
,
2
(
2
), p.
024502
.
3.
Kocsis
,
L.
,
Kiss
,
R. M.
, and
Jurák
,
M.
, 2000, “
Determination and Representation of the Helical Axis to Investigate Arbitrary Arm Movements
,”
Facta Universitatis Ser. Phys. Educ.
,
1
(
7
), pp.
31
37
. Available at: http://facta.junis.ni.ac.rs/pe/pe2000/pe2000-04.pdf.
4.
Lee
,
U. K.
, and
Han
,
C. S.
, 2008, “
A Method for Predicting Dynamic Behaviour Characteristics of a Vehicle Using the Screw Theory—Part 1
,”
Proc. Inst. Mech. Eng., Part D (J. Automob. Eng.)
,
222
(
1
), pp.
65
77
.
5.
Simionescu
,
P. A.
,
Talpasanu
,
I.
, and
Di Gregorio
,
R.
, 2010, “
Instant-Center Based Force Transmissivity and Singularity Analysis of Planar Linkages
,”
ASME J. Mech. Rob.
,
2
(
2
), p.
021011
.
6.
Coutsias
,
E. A.
,
Seok
,
C.
, and
Jacobson
,
M. P.
, 2004, “
A Kinematic View of Loop Closure
,”
J. Comput. Chem.
,
25
(
4
), pp.
510
528
.
7.
Lee
,
K.
,
Wang
,
Y.
, and
Chirikjian
,
G. S.
, 2007, “
O (n) Mass Matrix Inversion for Serial Manipulators and Polypeptide Chains Using Lie Derivatives
,”
Robotica
,
25
(
06
), pp.
739
750
.
8.
Shahbazi
,
A.
,
Ilies
,
H. T.
, and
Kazerounian
,
K.
, 2010, “
Hydrogen Bonds and Kinematic Mobility of Protein Molecules
,”
ASME J. Mech. Rob.
,
2
(
2
), p.
021009
.
9.
Dimentberg
,
F. M.
, 1959,
A General Method for the Investigation of Finite Displacements of Spatial Mechanisms and Certain Cases of Passive Joints
(Purdue Translation No. 436),
Purdue University
,
Lafayette, IN
.
10.
Duffy
,
J.
, and
Crane
,
C.
, 1980, “
A Displacement Analysis of the General Spatial 7-Link, 7R Mechanism
,”
Mech. Mach. Theory
,
15
(
3
), pp.
153
169
.
11.
Lee
,
H. Y.
, and
Liang
,
C. G.
, 1988, “
Displacement Analysis of the General Spatial 7-Link 7R Mechanism
,”
Mech. Mach. Theory
,
23
(
3
), pp.
219
226
.
12.
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
(
B
), pp.
71
79
.
13.
Qiao
,
S.
,
Liao
,
Q.
, and
Wei
,
S.
, 2010, “
Inverse Kinematic Analysis of the General 6R Serial Manipulators Based on Double Quaternions
,”
Mech. Mach. Theory
,
45
(
2
), pp.
193
199
.
14.
Zoppi
,
M.
, 2002, “
Effective Backward Kinematics for an Industrial 6R Robot
,”
ASME 2002 Design Engineering Technical Conferences and Computers and Information in Engineering Conference
, Montreal.
15.
Manocha
,
D.
, and
Canny
,
J. F.
, 1994, “
Efficient Inverse Kinematics for General 6R Manipulators
,”
IEEE Trans. Rob. Autom.
,
10
(
5
), pp.
648
657
.
16.
McCarthy
,
J. M.
, 2011, “
Kinematics, Polynomials, and Computers—A Brief History
,”
ASME J. Mech. Rob.
,
3
(
1
), p.
010201
.
17.
Wampler
,
C. W.
, and
Sommese
,
A. J.
, 2011, “
Numerical Algebraic Geometry and Algebraic Kinematics
,”
Acta Numerica
,
20
(
1
), pp.
469
567
.
18.
Smith
,
D. R.
, and
Lipkin
,
H.
, 1990, “
Analysis of Fourth Order Manipulator Kinematics Using Conic Sections
,”
Proceeding of the 1990 IEEE International Conference on Robotics and Automation
, pp.
274
278
.
19.
Jin
,
Q.
, and
Yang
,
T.
, 2002, “
Overconstraint Analysis on Spatial 6-Link Loops
,”
Mech. Mach. Theory
,
37
(
3
), pp.
267
278
.
20.
Milenkovic
,
V.
, 1979, “
Computer Synthesis of Continuous Path Robot Motion
,”
Proceedings 5th World Congress Theory of Machines and Mechanisms
,
ASME
, pp.
1332
1335
.
21.
Loo
,
M.
, and
Milenkovic
,
V.
, 1987, “
Multicircular Curvilinear Robot Path Generation
,”
Robots 11 Conference Proceedings and 17th International Symposium Industrial Robots
,
SME
,
Dearborn, MI
, Vol.
18
, pp.
19
27
.
22.
Loo
,
M.
,
Hamidieh
,
Y. A.
, and
Milenkovic
,
V.
, 1990, “
Generic Path Control for Robot Applications
,”
Robots 14 Conference Proceedings
,
SME
,
Dearborn, MI
, Vol.
10
, pp.
49
64
.
23.
Milenkovic
,
P.
, 2011, “
Solution of the Forward Dynamics of a Single-Loop Linkage Using Power Series
,”
ASME J. Dyn. Syst., Meas., Control
,
133
(
6
), p.
061002
.
24.
Rico
,
J. M.
,
Gallardo
,
J.
, and
Duffy
,
J.
, 1999, “
Screw Theory and Higher Order Kinematic Analysis of Open Serial and Closed Chains
,”
Mech. Mach. Theory
,
34
(
4
), pp.
559
586
.
25.
Karger
,
A.
, 1996, “
Singularity Analysis of Serial Robot-Manipulators
,”
ASME J. Mech. Des.
,
118
(
4
), pp.
520
525
.
26.
Müller
,
A.
, and
Rico
,
J. M.
, 2008, “
Mobility and Higher Order Local Analysis of the Configuration Space of Single-Loop Mechanisms
,”
Advances in Robot Kinematics: Analysis and Design
,
J.
Lenarcic
and
P.
Wenger
, eds.,
Springer
,
The Netherlands
, pp.
215
224
.
27.
Sommer
,
H. J.
, III
, 2008, “
Jerk Analysis and Axode Geometry of Spatial Linkages
,”
ASME J. Mech. Des.
,
130
(
4
), p.
042301
.
28.
Cervantes-Sánchez
,
J. J.
,
Rico-Martínez
,
J. M.
, and
González-Montiel
,
G.
, 2009, “
The Differential Calculus of Screws: Theory, Geometrical Interpretation, and Applications
,”
Proc. Inst. Mech. Eng., Part C
,
223
(
6
), pp.
1449
1468
.
29.
Rico
,
J. M.
,
Gallardo
,
J.
, and
Ravani
,
B.
, 2003, “
Lie Algebra and the Mobility of Kinematic Chains
,”
J. Rob. Syst.
,
20
(
8
), pp.
477
499
.
30.
Milenkovic
,
P. H.
, 2010, “
Mobility of Single-Loop Kinematic Mechanisms Under Differential Displacement
,”
ASME J. Mech. Des.
,
132
(
4
), p.
041001
.
31.
Karsai
,
G.
, 2001, “
Method for the Calculation of the Combined Motion Time Derivatives of Optional Order and Solution for the Inverse Kinematic Problems
,”
Mech. Mach. Theory
,
36
(
2
), pp.
261
272
.
32.
Milenkovic
,
P.
, and
Brown
,
M. V.
, 2011, “
Properties of the Bennett Mechanism Derived From the RRRS Closure Ellipse
,”
ASME J. Mech. Rob.
,
3
(
2
), p.
021012
.
33.
Milenkovic
,
P.
, 2011, “
Series Solution for Finite Displacement of Planar Four-Bar Linkages
,”
ASME J. Mech. Rob.
,
3
(
1
), p.
014501
.
34.
Griewank
,
A.
, and
Walther
,
A.
, 2008,
Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation
,
2nd ed.
,
Society for Industrial and Applied Mathematics (SIAM)
,
.
35.
Koetsier
,
T.
, 1986, “
From Kinematically Generated Curves to Instantaneous Invariants: Episodes in the History of Instantaneous Planar Kinematics
,”
Mech. Mach. Theory
,
21
(
6
), pp.
489
498
.
36.
McCarthy
,
J.
, and
Roth
,
B.
, 1981, “
The Curvature Theory of Line Trajectories in Spatial Kinematics
,”
ASME J. Mech. Des.
,
103
(
4
), pp.
718
724
.
37.
Dooner
,
D. B.
, and
Griffis
,
M. W.
, 2007, “
On Spatial Euler-Savary Equations for Envelopes
,”
ASME J. Mech. Des.
,
129
(
8
), pp.
865
876
.
38.
Woo
,
L.
, and
Freudenstein
,
F.
, 1970, “
Application of Line Geometry to Theoretical Kinematics and the Kinematic Analysis of Mechanical Systems
,”
J. Mech.
,
5
, pp.
417
460
.
39.
Cervantes-Sánchez
,
J. J.
,
Moreno-Báez
,
M. A.
, and
Rico-Martínez
,
J. M.
, 2004, “
A Novel Geometrical Derivation of the Lie Product
,”
Mech. Mach. Theory
,
39
(
10
), pp.
1067
1079
.
40.
Milenkovic
,
V.
, and
Huang
,
B.
, 1983, “
Kinematics of Major Robot Linkages
,”
13th International Symposium on Industrial Robots and Robotics/Robots 7
,
SME
,
Chicago
, Vol.
2
, pp.
31
47
.
41.
Lenarcic
,
J.
, 1998, “
Alternative Computational Scheme of Manipulator Inverse Kinematics
,”
Proceedings of the 1998 IEEE International Conference on Robotics and Automation
, Vol.
4
, pp.
3235
3240
.
42.
Cheng
,
H.
, and
Gupta
,
K. C.
, 1990, “
A Study of the Numerical Robot Inverse Kinematics Based Upon the ODE Solution Method
,”
Mechanism Synthesis and Analysis: Presented at the 1990 ASME Design Technical Conferences—21st Biennial Mechanisms Conference
,
ASME
,
Chicago, IL
, Sep. 16−19, pp.
243
247
.
43.
Lenarcic
,
J.
, 1985, “
An Efficient Numerical Approach for Calculating the Inverse Kinematics for Robot Manipulators
,”
Robotica
,
3
, pp.
21
26
.
44.
Zhao
,
Y.
,
Huang
,
T.
, and
Yang
,
Z.
, 2005, “
A New Numerical Algorithm for the Inverse Position Analysis of all Serial Manipulators
,”
Robotica
,
24
(
3
), pp.
373
376
.
45.
Lucas
,
S. R.
,
Tischler
,
C. R.
, and
Samuel
,
A. E.
, 2000, “
Real-Time Solution of the Inverse Kinematic-Rate Problem
,”
Int. J. Rob. Res.
,
19
(
12
), pp.
1236
1244
.
46.
Siciliano
,
B.
, 1990, “
A Closed-Loop Inverse Kinematic Scheme for on-Line Joint-Based Robot Control
,”
Robotica
,
8
(
3
), pp.
231
243
.
47.
Angeles
,
J.
, 1985, “
On the Numerical Solution of the Inverse Kinematic Problem
,”
Int. J. Rob. Res.
,
4
(
2
), pp.
21
37
.
48.
Sultan
,
I. A.
, 2000, “
On the Positioning of Revolute-Joint Robot Manipulators
,”
J. Rob. Syst.
,
17
(
8
), pp.
429
438
.
49.
Husty
,
M. L.
,
Pfurner
,
M.
, and
Schröcker
,
H. P.
, 2007, “
A New and Efficient Algorithm for the Inverse Kinematics of a General Serial 6R Manipulator
,”
Mech. Mach. Theory
,
42
(
1
), pp.
66
81
.
50.
Milenkovic
,
V.
,
Milenkovic
,
V. J.
, and
Milenkovic
,
P. H.
, 1991, “
Inverse Kinematics of Not Fully Serial Robot Linkages With Nonsingular Wrists
,”
Advances in Robot Kinematics: With Emphasis on Symbolic Computation
,
S.
Stifter
and
J.
Lenarcic
, eds.,
Springer
,
Berlin
, pp.
335
342
.
51.
Stanisic
,
M. M.
, and
Duta
,
O.
, 1990, “
Symmetrically Actuated Double Pointing Systems: The Basis of Singularity-Free Robot Wrists
,”
IEEE Trans. Rob. Autom.
,
6
(
5
), pp.
562
569
.
52.
Wiitala
,
J. M.
, and
Stanisic
,
M. M.
, 2000, “
Design of an Overconstrained and Dextrous Spherical Wrist
,”
ASME J. Mech. Des.
,
122
(
3
), pp.
347
353
.
53.
Milenkovic
,
P.
, 2011, “
Nonsingular Spherically Constrained Clemens Linkage Wrist
,”
ASME J. Mech. Rob.
,
3
(
1
), p.
011014
.
54.
Milenkovic
,
V.
, 1990, “
Non-Singular Industrial Robot Wrist
,” U.S. Patent No. 4,907,937.
55.
Milenkovic
,
V.
, 1987, “
New Nonsingular Robot Wrist Design
,”
Robots 11 Conference Proceedings RI/SME
, pp.
13.29
13.42
.
56.
Milenkovic
,
V.
, 1988, “
Hollow Non-Singular Robot Wrist
,” U.S. Patent No. 4,744,264.
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