Abstract

This article unifies the approaches of kinematic and static modeling, and singularity analysis for tendon-driven parallel continuum robots under constant curvature as well as pseudo-rigid-body assumptions with those implemented in conventional rigid parallel robots. Constraint conditions are determined for the legs of this type of parallel continuum robots based on which the velocity equations and Jacobian matrices are derived. These are further exploited for inverse kinematic and singularity analysis. Static models for the robot as well as for each of the continuum links under pseudo-rigid-body assumption are derived. Finally, a simulation example is given to validate the kinematic models. It is shown that singularities can be determined using Grassmann line geometry or by detecting the numerical values of three performance indices.

References

1.
Burgner-Kahrs
,
J.
,
Rucker
,
D. C.
, and
Choset
,
H.
,
2015
, “
Continuum Robots for Medical Applications: A Survey
,”
IEEE Trans. Rob.
,
31
(
6
), pp.
1261
1280
.
2.
Goldman
,
R. E.
,
Bajo
,
A.
,
MacLachlan
,
L. S.
,
Pickens
,
R.
,
Herrell
,
S. D.
, and
Simaan
,
N.
,
2012
, “
Design and Performance Evaluation of a Minimally Invasive Telerobotic Platform for Transurethral Surveillance and Intervention
,”
IEEE Trans. Biomed. Eng.
,
60
(
4
), pp.
918
925
.
3.
Burgner
,
J.
,
Rucker
,
D. C.
,
Gilbert
,
H. B.
,
Swaney
,
P. J.
,
Russell
,
P. T.
,
Weaver
,
K. D.
, and
Webster
,
R. J.
,
2013
, “
A Telerobotic System for Transnasal Surgery
,”
IEEE/ASME Trans. Mechatron.
,
19
(
3
), pp.
996
1006
.
4.
Dong
,
X.
,
Axinte
,
D.
,
Palmer
,
D.
,
Cobos
,
S.
,
Raffles
,
M.
,
Rabani
,
A.
, and
Kell
,
J.
,
2017
, “
Development of a Slender Continuum Robotic System for On-Wing Inspection/Repair of Gas Turbine Engines
,”
Rob. Comput. Integr. Manuf.
,
44
, pp.
218
229
.
5.
Santiago
,
J. L. C.
,
Walker
,
I. D.
, and
Godage
,
I. S.
,
2015
, “
Continuum Robots for Space Applications Based on Layer-Jamming Scales With Stiffening Capability
,”
2015 IEEE Aerospace Conference
,
Big Sky, MT
,
Mar. 7–14
,
IEEE
, pp.
1
13
.
6.
Chirikjian
,
G. S.
, and
Burdick
,
J. W.
,
1994
, “
A Modal Approach to Hyper-Redundant Manipulator Kinematics
,”
IEEE Trans. Rob. Autom.
,
10
(
3
), pp.
343
354
.
7.
Jones
,
B. A.
, and
Walker
,
I. D.
,
2006
, “
Kinematics for Multisection Continuum Robots
,”
IEEE Trans. Rob.
,
22
(
1
), pp.
43
55
.
8.
Webster III
,
R. J.
, and
Jones
,
B. A.
,
2010
, “
Design and Kinematic Modeling of Constant Curvature Continuum Robots: A Review
,”
Int. J. Rob. Res.
,
29
(
13
), pp.
1661
1683
.
9.
Dupont
,
P. E.
,
Lock
,
J.
,
Itkowitz
,
B.
, and
Butler
,
E.
,
2009
, “
Design and Control of Concentric-tube Robots
,”
IEEE Trans. Rob.
,
26
(
2
), pp.
209
225
.
10.
Rucker
,
D. C.
, and
Webster III
,
R. J.
,
2011
, “
Statics and Dynamics of Continuum Robots With General Tendon Routing and External Loading
,”
IEEE Trans. Rob.
,
27
(
6
), pp.
1033
1044
.
11.
Mahvash
,
M.
, and
Dupont
,
P. E.
,
2011
, “
Stiffness Control of Surgical Continuum Manipulators
,”
IEEE Trans. Rob.
,
27
(
2
), pp.
334
345
.
12.
Tang
,
W.
,
Wan
,
T. R.
,
Gould
,
D. A.
,
How
,
T.
, and
John
,
N. W.
,
2012
, “
A Stable and Real-Time Nonlinear Elastic Approach to Simulating Guidewire and Catheter Insertions Based on Cosserat Rod
,”
IEEE Trans. Biomed. Eng.
,
59
(
8
), pp.
2211
2218
.
13.
Howell
,
L. L.
,
2001
,
Compliant Mechanisms
,
John Wiley & Sons, Inc.
,
New York
.
14.
Su
,
H.-J.
,
2009
, “
A Pseudorigid-Body 3R Model for Determining Large Deflection of Cantilever Beams Subject to Tip Loads
,”
ASME J. Mech. Rob.
,
1
(
2
), p.
021008
.
15.
Khoshnam
,
M.
,
Khalaji
,
I.
, and
Patel
,
R. V.
,
2015
, “
A Robotics-Assisted Catheter Manipulation System for Cardiac Ablation With Real-Time Force Estimation
,”
2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)
,
Hamburg, Germany
,
Sept. 28–Oct. 2
,
IEEE
, pp.
3202
3207
.
16.
Roesthuis
,
R. J.
, and
Misra
,
S.
,
2016
, “
Steering of Multisegment Continuum Manipulators Using Rigid-Link Modeling and FBG-Based Shape Sensing
,”
IEEE Trans. Rob.
,
32
(
2
), pp.
372
382
.
17.
Huang
,
S.
,
Meng
,
D.
,
Wang
,
X.
,
Liang
,
B.
, and
Lu
,
W.
,
2019
, “
A 3D Static Modeling Method and Experimental Verification of Continuum Robots Based on Pseudo-Rigid Body Theory
,”
2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)
,
Macau, China
,
Nov. 3–8
,
IEEE
, pp.
4672
4677
.
18.
Rao
,
P.
,
Peyron
,
Q.
,
Lilge
,
S.
, and
Burgner-Kahrs
,
J.
,
2021
, “
How to Model Tendon-Driven Continuum Robots and Benchmark Modelling Performance
,”
Front. Rob. AI
,
7
, p.
223
.
19.
Till
,
J.
,
Bryson
,
C. E.
,
Chung
,
S.
,
Orekhov
,
A.
, and
Rucker
,
D. C.
,
2015
, “
Efficient Computation of Multiple Coupled Cosserat Rod Models for Real-Time Simulation and Control of Parallel Continuum Manipulators
,”
2015 IEEE International Conference on Robotics and Automation (ICRA)
,
Seattle, WA
,
May 26–30
,
IEEE
, pp.
5067
5074
.
20.
Hopkins
,
J. B.
,
Rivera
,
J.
,
Kim
,
C.
, and
Krishnan
,
G.
,
2015
, “
Synthesis and Analysis of Soft Parallel Robots Comprised of Active Constraints
,”
ASME J. Mech. Rob.
,
7
(
1
), p.
011002
.
21.
Singh
,
I.
,
Singh
,
M.
,
Pathak
,
P. M.
, and
Merzouki
,
R.
,
2017
, “
Optimal Work Space of Parallel Continuum Manipulator Consisting of Compact Bionic Handling Arms
,”
2017 IEEE International Conference on Robotics and Biomimetics (ROBIO)
,
Macau, China
,
Dec. 5–8
,
IEEE
, pp.
258
263
.
22.
Altuzarra
,
O.
,
Caballero
,
D.
,
Zhang
,
Q.
, and
Campa
,
F. J.
,
2018
, “
Kinematic Characteristics of Parallel Continuum Mechanisms
,”
International Symposium on Advances in Robot Kinematics
,
Bologna, Italy
,
July 1–5
,
Springer
, pp.
293
301
.
23.
White
,
E. L.
,
Case
,
J. C.
, and
Kramer-Bottiglio
,
R.
,
2018
, “
A Soft Parallel Kinematic Mechanism
,”
Soft Rob.
,
5
(
1
), pp.
36
53
.
24.
Yang
,
Z.
,
Zhu
,
X.
, and
Xu
,
K.
,
2018
, “
Continuum Delta Robot: A Novel Translational Parallel Robot With Continuum Joints
,”
2018 IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM)
,
Auckland, New Zealand
,
July 9–12
,
IEEE
, pp.
748
755
.
25.
Young
,
E. M.
, and
Kuchenbecker
,
K. J.
,
2019
, “
Implementation of a 6-DOF Parallel Continuum Manipulator for Delivering Fingertip Tactile Cues
,”
IEEE Trans. Haptic
,
12
(
3
), pp.
295
306
.
26.
Mauzé
,
B.
,
Dahmouche
,
R.
,
Laurent
,
G. J.
,
André
,
A. N.
,
Rougeot
,
P.
,
Sandoz
,
P.
, and
Clévy
,
C.
,
2020
, “
Nanometer Precision With a Planar Parallel Continuum Robot
,”
IEEE Rob. Autom. Lett.
,
5
(
3
), pp.
3806
3813
.
27.
Nuelle
,
K.
,
Sterneck
,
T.
,
Lilge
,
S.
,
Xiong
,
D.
,
Burgner-Kahrs
,
J.
, and
Ortmaier
,
T.
,
2020
, “
Modeling, Calibration, and Evaluation of a Tendon-Actuated Planar Parallel Continuum Robot
,”
IEEE Rob. Autom. Lett.
,
5
(
4
), pp.
5811
5818
.
28.
Lilge
,
S.
,
Nuelle
,
K.
,
Boettcher
,
G.
,
Spindeldreier
,
S.
, and
Burgner-Kahrs
,
J.
,
2021
, “
Tendon Actuated Continuous Structures in Planar Parallel Robots: A Kinematic Analysis
,”
ASME J. Mech. Rob.
,
13
(
1
), p.
011025
.
29.
Böttcher
,
G.
,
Lilge
,
S.
, and
Burgner-Kahrs
,
J.
,
2021
, “
Design of a Reconfigurable Parallel Continuum Robot With Tendon-actuated Kinematic Chains
,”
IEEE Rob. Autom. Lett.
,
6
(
2
), pp.
1272
1279
.
30.
Gosselin
,
C.
, and
Schreiber
,
L.-T.
,
2016
, “
Kinematically Redundant Spatial Parallel Mechanisms for Singularity Avoidance and Large Orientational Workspace
,”
IEEE Trans. Rob.
,
32
(
2
), pp.
286
300
.
31.
Merlet
,
J.-P.
,
2004
, “
Solving the Forward Kinematics of a Gough-type Parallel Manipulator With Interval Analysis
,”
Int. J. Rob. Res.
,
23
(
3
), pp.
221
235
.
32.
Kong
,
X.
,
2008
, “Forward Kinematics and Singularity Analysis of a 3-RPR Planar Parallel Manipulator,”
Advances in Robot Kinematics: Analysis and Design
,
J.
Lenarčič
and
P.
Wenger
, eds.,
Springer, Cham
, pp.
29
38
.
33.
Wen
,
K.
, and
Gosselin
,
C. M.
,
2020
, “
Forward Kinematic Analysis of Kinematically Redundant Hybrid Parallel Robots
,”
ASME J. Mech. Rob.
,
12
(
6
), p.
061008
.
34.
Gosselin
,
C. M.
,
Sefrioui
,
J.
, and
Richard
,
M. J.
,
1992
, “
Solutions Polynomiales Au Problème De La Cinématique Directe Des Manipulateurs Parallèles Plans à Trois Degrés De Liberté
,”
Mech. Mach. Theory
,
27
(
2
), pp.
107
119
.
35.
Wen
,
K.
, and
Gosselin
,
C.
,
2019
, “
Kinematically Redundant Hybrid Robots With Simple Singularity Conditions and Analytical Inverse Kinematic Solutions
,”
IEEE Rob. Autom. Lett.
,
4
(
4
), pp.
3828
3835
.
36.
Gosselin
,
C.
, and
Angeles
,
J.
,
1990
, “
Singularity Analysis of Closed-Loop Kinematic Chains
,”
IEEE Trans. Rob. Autom.
,
6
(
3
), pp.
281
290
.
37.
Briot
,
S.
, and
Goldsztejn
,
A.
,
2021
, “
Singularity Conditions for Continuum Parallel Robots
,”
IEEE Trans. Rob.
,
38
(
1
), pp.
507
525
.
38.
Cardou
,
P.
,
Bouchard
,
S.
, and
Gosselin
,
C.
,
2010
, “
Kinematic-Sensitivity Indices for Dimensionally Nonhomogeneous Jacobian Matrices
,”
IEEE Trans. Rob.
,
26
(
1
), pp.
166
173
.
You do not currently have access to this content.