In this paper, we present an efficient method based on geometric algebra for computing the solutions to the inverse kinematics problem (IKP) of the 6R robot manipulators with offset wrist. Due to the fact that there exist some difficulties to solve the inverse kinematics problem when the kinematics equations are complex, highly nonlinear, coupled and multiple solutions in terms of these robot manipulators stated mathematically, we apply the theory of Geometric Algebra to the kinematic modeling of 6R robot manipulators simply and generate closed-form kinematics equations, reformulate the problem as a generalized eigenvalue problem with symbolic elimination technique, and then yield 16 solutions. Finally, a spray painting robot, which conforms to the type of robot manipulators, is used as an example of implementation for the effectiveness and real-time of this method. The experimental results show that this method has a large advantage over the classical methods on geometric intuition, computation and real-time, and can be directly extended to all serial robot manipulators and completely automatized, which provides a new tool on the analysis and application of general robot manipulators.

References

References
1.
Siciliano
,
B.
, and
Khatib
,
O.
,
2008
,
Springer Handbook of Robotics
,
Springer-Verlag
,
New York Inc
.
2.
Bingul
,
Z.
,
Ertunc
,
H. M.
, and
Oysu
,
C.
,
2005
, “
Comparison of Inverse Kinematics Solutions Using Neural Network for 6R Robot Manipulator With Offset
,”
Proceedings of the ICSC Congress on Computational Intelligence Methods and Applications
, pp.
1
5
.
3.
Raghavan
,
M.
, and
Roth
,
B.
,
1993
, “
Inverse Kinematics of the General 6R Manipulator and Related Linkages
,”
ASME J. Mech. Des.
,
115
(
3
), pp.
502
508
.10.1115/1.2919218
4.
Manocha
,
D.
, and
Canny
,
J. F.
,
1994
, “
Efficient Inverse Kinematics for General 6R Manipulators
,”
IEEE Trans. Rob. Autom.
,
10
(
5
), pp.
648
657
.10.1109/70.326569
5.
Aspragathos
,
N. A.
, and
Dimitros
,
J. K.
,
1998
, “
A Comparative Study of Three Methods for Robot Kinematics
,”
IEEE Trans. Syst., Man, Cybern., Part B: Cybern.
,
28
(
2
), pp.
135
145
.10.1109/3477.662755
6.
Husty
,
M. L.
,
Pfurner
,
M.
, and
Shrocker
,
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
.10.1016/j.mechmachtheory.2006.02.001
7.
Qiao
,
S.
,
Liao
,
Q.
,
Wei
,
S.
, and
Su
,
H. J.
,
2010
, “
Inverse Kinematic Analysis of the General 6R Serial Manipulators Based on Double Quaternions
,”
Mech. Mach. Theory
,
45
(
2
), pp.
193
199
.10.1016/j.mechmachtheory.2009.05.013
8.
Cheng
,
H.
, and
Gupta
,
K. C.
,
1991
, “
A Study of Robot Inverse Kinematics BasedUupon the Solution of Differential Equations
,”
J. Rob. Syst.
,
8
(
2
), pp.
159
175
.10.1002/rob.4620080203
9.
Olsen
,
A. L.
, and
Petersen
,
H. G.
,
2011
, “
Inverse Kinematics by Numerical and Analytical Cyclic Coordinate Descent
,”
Robotica
,
29
(
3
), pp.
619
626
.10.1017/S026357471000038X
10.
Zhang
,
X.
, and
Nelson
,
C. A.
,
2011
, “
Multiple-Criteria Kinematic Optimization for the Design of Spherical Serial Mechanisms Using Genetic Algorithms
,”
ASME J. Mech. Des.
,
133
, pp.
1
11
.
11.
Olaru
,
A.
,
Olaru
,
S.
, and
Paune
,
D.
,
2011
, “
Assisted Research and Optimization of the Proper Neural Network Solving the Inverse Kinematics Problem
,”
Proceedings of 2011 International Conference on Optimization of the Robots and Manipulators
,
Romania
, pp.
26
28
.
12.
Feng
,
Y.
,
Yao-nan
,
W.
, and
Yi-min
,
Y.
,
2012
, “
Inverse Kinematics Solution for Robot Manipulator Based on Neural Network Under Joint Subspace
,”
Int. of Comput. Commun.
,
7
(
3
), pp.
459
472
.
13.
Hestens
,
D.
, and
Sobczyk
,
G.
,
1987
,
Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics
,
Springer-Verlag
,
Berlin, Heidelberg, New York
.
14.
Hestens
,
D.
,
2001
, “
Old Wine in New Bottles: A New Algebraic Framework for Computational Geometry
,”
Advances in Geometric Algebra With Applications in Science and Engineering
,
E.
Bayro-Corrochano
, and
G.
Sobczyk
, eds., Birkauser, Boston, pp.
1
14
.
15.
Zamora
,
J.
, and
Bayro-Corrochano
,
E.
,
2004
, “
Inverse Kinematics, Fixation and Grasping Using Conformal Geometric Algebra
,”
IROS 2004
,
Sendai, Japan
.
16.
Hildenbrand
,
D.
,
2006
, “
Geometric Computing in Computer Graphics and Robotics using Conformal Geometric Algebra
,” Ph.D. thesis,
Darmstadt University of Technology
, Darmstadt.
17.
Hildenbrand
,
D.
,
Lange
,
H.
, and
Stock
,
F.
,
2008
, “
Efficient Inverse Kinematics Algorithm Based on Conformal Geometric Algebra
,”
Proceedings of the 3rd International Conference on Computer Graphics Theory and Applications
,
Medeira, Portugal
.
18.
Aristidou
,
A.
, and
Lasenby
,
J.
,
2011
, “
Inverse Kinematics Solutions Using Conformal Geometric Algebra
,”
Guide to Geometric Algebra in Practice
, Vol.
1
, Springer Verlag, pp.
47
62
.
19.
Aristidou
,
A.
, and
Lasenby
,
J.
,
2011
, “
FABRIK: A Fast, Iterative Solver for the Inverse Kinematics Problem
,”
Graphical Models
,
73
(
5
), pp.
243
260
.10.1016/j.gmod.2011.05.003
20.
Dorst
,
L.
,
Fontijne
,
D.
, and
Mann
,
S.
,
2007
,
Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry
,
Morgan Kaufmann Publishers/Elsevier
, San Francisco, CA.
21.
Cheng
,
H. H.
,
1994
,
Programming With Dual Numbers and its Applications in Mechanism Design. Engineering With Computers
,
10
(
4
), pp.
212
229
.
22.
Bayro-Corrochano
,
E.
, and
Scheuermann
,
G.
,
2009
,
Geometric Algebra Computing for Engineering and Computer Science
,
Springer–Verlag
,
New York
.
23.
Hestenes
,
D.
,
2010
, “
New Tools for Computational Geometry and Rejuvenation of Screw Theory
,”
Geometric Algebra Computing in Engineering and Computer Science
,
E.
Bayro-Corrochano
, and
G.
Scheuermann
, eds.,
Springer-Verlag
, pp.
3
35
.
24.
Bayro-Corrochano
,
E.
,
2010
,
Geometric Computing for Wavelet Transforms, Robot Vision, Learning, Control and Action
,
Spring-Verlag
,
London
, pp.
169
203
.
25.
Selig
,
J. M.
,
2005
,
Geometric Fundamentals of Robotics, Monographs in Computer Science
,
Springer
,
New York
, pp.
206
278
.
26.
Li
,
H.
,
Hestenes
,
D.
, and
Rockwood
,
A.
,
2001
, “
Generalized Homogeneous Coordinates for Computational Geometry
,”
Geometric Computing With Clifford Algebra
,
G.
Sommer
, ed.,
Springer-Verlag
, pp.
25
58
.
27.
Gohberg
,
I.
,
Lancaster
,
P.
, and
Rodman
,
L.
,
1982
,
Matrix Polynomials
,
Academic Press
,
New York
.
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