Abstract

The forward kinematics (FK) of a 6-6 universal-prismatic-spherical (UPS) structure of a parallel robot is highly nonlinear, coupled, and has a one-to-many nature of mapping. There exists no close form solution to a forward kinematic problem (FKP), and real-time kinematic control is extremely difficult. This paper presents the implementation of time efficient and robust solution for FKP using a trajectory modifier algorithm along with a Newton Raphson (NR) method. One micrometer in translation and 0.001 deg in orientation accuracy with an average pose computation time of 2.3 ms are achieved. The novel algorithm is elaborated and the detailed performance parameters are tabulated. The paper presents trajectory following experiments to show robust, real-time FK solution and efficient kinematic control on both standalone and master–slave modes to be used for robot-assisted neurosurgery. The neuro-registration using the FK solutions in real time in a tele-manipulation mode is demonstrated.

References

1.
Patel
,
Y. D.
, and
George
,
P. M.
,
2012
, “
Parallel Manipulators Applications—A Survey
,”
Mod. Mech. Eng.
,
2
(
3
), pp.
57
64
.
2.
Lazard
,
D.
,
1992
, “
Stewart Platform and Grobner Basis
,”
Advances in Robotics Kinematics
,
Ferrara, Italy
,
Sept. 7–9
, pp.
136
142
.
3.
Merlet
,
J.-P.
,
2006
,
Parallel Robots
, 2nd ed.,
Springer Dordrecht
,
Dordrecht, The Netherlands.
.
4.
Zhu
,
G.
,
Wei
,
S.
,
Zhang
,
Y.
, and
Liao
,
Q.
,
2021
, “
Direct Kinematic Analysis of the Spatial Parallel Mechanism With 3-R(P)S Structure Based on the Point Pair Relationship
,”
ASME J. Mech. Rob.
,
13
(
6
), p.
061011
.
5.
Didrit
,
O.
,
Petitot
,
M.
, and
Walter
,
E.
,
1998
, “
Guaranteed Solution of Direct Kinematic Problems for General Configurations of Parallel Manipulators
,”
IEEE Trans. Rob. Autom.
,
14
(
2
), pp.
259
266
.
6.
Shim
,
S.
,
Lee
,
S.
,
Joo
,
S.
, and
Seo
,
J.
,
2022
, “
Denavit-Hartenberg Notation-Based Kinematic Constraint Equations for Forward Kinematics of the 3–6 Stewart Platform
,”
ASME J. Mech. Rob.
,
14
(
5
), p.
054505
.
7.
Griffis
,
M.
, and
Duffy
,
J.
,
1989
, “
A Forward Displacement Analysis of a Class of Stewart Platforms
,”
J. Robot. Syst.
,
6
(
6
), pp.
703
720
.
8.
Nanua
,
P.
,
Waldron
,
K. J.
, and
Murthy
,
V.
,
1990
, “
Direct Kinematic Solution of a Stewart Platform
,”
IEEE Trans. Rob. Autom.
,
6
(
4
), pp.
438
444
.
9.
Wang
,
Y.
,
2006
, “
An Incremental Method for Forward Kinematics of Parallel Manipulators
,”
IEEE Conference on Robotics, Automation and Mechatronics
,
Bangkok, Thailand
,
June 1–3
, pp.
1
5
.
10.
Guglielmetti
,
P.
,
1994
, “
Model-Based Control of Fast Parallel Robots: A Global Approach in Operational Space
,”
Ph.D. dissertation
,
EPFL
,
Lausanne
.
11.
Wen
,
K.
, and
Gosselin
,
C. M.
,
2020
, “
Forward Kinematic Analysis of Kinematically Redundant Hybrid Parallel Robots
,”
ASME J. Mech. Rob.
,
12
(
6
), p.
061008
.
12.
Liu
,
G.
,
Wang
,
Y.
,
Zhang
,
Y.
, and
Xie
,
Z.
,
2015
, “
Real-Time Solution of the Forward Kinematics for a Parallel Haptic Device Using a Numerical Approach Based on Neural Networks
,”
J. Mech. Sci. Technol.
,
29
(
6
), pp.
2487
2499
.
13.
Darvishi
,
M. T.
, and
Barati
,
A.
,
2007
, “
A Third-Order Newton-Type Method to Solve Systems of Nonlinear Equations
,”
Appl. Math. Comput.
,
187
(
2
), pp.
630
635
.
14.
Gonçalves
,
M. L. N.
, and
Oliveira
,
R.
,
2015
, “
Convergence of the Gauss Newton Method for a Special Class of Systems of Equations Under a Majorant Condition
,”
Optimization
,
64
(
3
), pp.
577
594
.
15.
Liu
,
K.
,
Lewis
,
F. L.
, and
Fitzgerald
,
M.
,
1994
, “
Solution of Nonlinear Kinematics of a Parallel-Link Constrained Stewart Platform Manipulator
,”
Circuits Syst. Signal Process.
,
13
(
2–3
), pp.
167
183
.
16.
Vertechy
,
R.
, and
Parenti-Castelli
,
V.
,
2009
, “
A Fast and Robust Hybrid Method for the Solution of the 6-3 Stewart–Gough Platform Direct Position Analysis
,”
ASME J. Mech. Rob.
,
1
(
1
), p.
011014
.
17.
Song
,
S.-K.
, and
Kwon
,
D.-S.
,
2001
, “
New Methodology for the Forward Kinematics of 6-DOF Parallel Manipulators Using Tetrahedron Configurations
,”
Proceedings of the IEEE International Conference on Robotics and Automation (ICRA)
,
Seoul, South Korea
,
May 21–26
, Vol. 2, pp.
1307
1312
.
18.
Merlet
,
J. P.
,
1993
, “
Closed-Form Resolution of the Direct Kinematics of Parallel Manipulators Using Extra Sensors Data
,”
IEEE International Conference on Robotics and Automation
,
Atlanta, GA
,
May 2–6
, pp.
200
204
.
19.
Parenti-Castelli
,
V.
, and
Di Gregorio
,
R.
,
1999
, “
Determination of the Actual Configuration of the General Stewart Platform Using Only One Additional Sensor
,”
ASME J. Mech. Des.
,
121
(
1
), pp.
21
25
.
20.
Parikh
,
P. J.
, and
Lam
,
S. S. Y.
,
2005
, “
A Hybrid Strategy to Solve the Forward Kinematics Problem in Parallel Manipulators
,”
IEEE Trans. Rob.
,
21
(
1
), pp.
18
25
.
21.
Parikh
,
P. J.
, and
Lam
,
S. S. Y.
,
2009
, “
Solving the Forward Kinematics Problem in Parallel Manipulators Using an Iterative Artificial Neural Network Strategy
,”
Int. J. Adv. Manuf. Technol.
,
40
(
5–6
), pp.
595
606
.
22.
Zhu
,
Q.
, and
Zhang
,
Z.
,
2019
, “
An Efficient Numerical Method for Forward Kinematics of Parallel Robots
,”
IEEE Access
,
7
, pp.
128758
128766
.
23.
Morell
,
A.
,
Tarokh
,
M.
, and
Acosta
,
L.
,
2013
, “
Solving the Forward Kinematics Problem in Parallel Robots Using Support Vector Regression
,”
Eng. Appl. Artif. Intell.
,
26
(
7
), pp.
1698
1706
.
24.
Petuya
,
V.
,
Gutiérrez
,
J.
,
Alonso
,
A.
,
Altuzarra
,
O.
, and
Hernández
,
A.
,
2008
, “
A Numerical Procedure to Solve Non-Linear Kinematic Problems in Spatial Mechanisms
,”
Int. J. Numer. Methods Eng.
,
73
(
6
), pp.
825
843
.
25.
Bhutani
,
G.
,
2014
, “
Modeling, Design and Development of Frameless Stereotaxy in Robot Assisted Neurosurgery
,”
PhD dissertation
,
HBNI
,
Mumbai
.
26.
Craig
,
J. J.
,
2006
,
Introduction to Robotics: Mechanics and Control
, 3rd ed.,
Pearson/Prentice Hall
,
Hoboken, NJ
.
27.
Merlet
,
J.-P.
,
2004
, “
Solving the Forward Kinematics of Gough-Type Parallel Manipulator With Interval Analysis
,”
Int. J. Rob. Res.
,
23
(
3
), pp.
221
236
.
28.
Dasgupta
,
B.
, and
Mruthyunjaya
,
T. S.
,
1994
, “
A Canonical Formulation of the Direct Position Kinematics Problem for a General 6-6 Stewart Platform
,”
Mech. Mach. Theory
,
29
(
6
), pp.
819
827
.
29.
Murray
,
R.
,
Zexiang
,
L.
, and
Sastry
,
S. S.
,
1994
,
A Mathematical Introduction to Robotic Manipulation
,
CRC Press
,
Boca Raton, FL
.
30.
Siciliano
,
B.
, and
Khatib
,
O.
,
2016
,
Springer Handbook of Robotics
,
Springer
,
New York
.
31.
Yim
,
M.
,
Taylor
,
C. J.
,
Kumar
,
V.
, and
Hsieh
,
A.
,
2017
, “Robotics: Fundamentals, Dynamics and Control,” www.edX.org, Robotics Micromasters Course, https://www.edx.org/micromasters/pennx-robotics, Visited May 30, 2018.
32.
Dwarakanath
,
T. A.
,
Lagoo
,
K. D.
, and
Badodkar
,
D. N.
,
2016
, “
6-PSS Based Parallel Manipulators
,”
The Joint International Conference of the XII International Conference on Mechanisms and Mechanical Transmissions (MTM) and the XXIII International Conference on Robotics (Robotics’16)
,
Aachen, Germany
,
Oct. 26–27
, pp.
331
339
.
33.
Dwarakanath
,
T. A.
, and
Bhutani
,
G.
,
2011
, “
Beam Type Hexapod Structure Based Six Component Force-Torque Sensor
,”
Mechatronics
,
21
(
8
), pp.
1279
1287
.
34.
Yang
,
L.
,
Huan
,
Z.
,
Zhang
,
Y.
, and
Ding
,
H.
,
2018
, “
A Data-Driven Motion Mapping Method for Space Teleoperation of Kinematically Dissimilar Master/Slave Robots
,”
Proceedings of IEEE International Conference on Robotics and Biomimetics
,
Kuala Lumpur, Malaysia
,
Dec. 12–15
, pp.
862
867
.
35.
Advanced Master Slave Manipulators
,” http://barc.gov.in/technologies/msm/msm.html
36.
Kaushik
,
A.
,
Dwarakanath
,
T. A.
, and
Bhutani
,
G.
,
2018
, “
Autonomous Neuro-Registration for Robot-Based Neurosurgery
,”
Int. J. Comput. Assist. Radiol. Surg.
,
13
(
1
), pp.
1807
1817
.
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