We present the continuous model of a mobile slender mechanism that is intended to be the structure of an autonomous hyper-redundant slender robotic system. Rigid body degrees-of-freedom (DOF) and deformability are coupled through a Lagrangian weak formulation that includes control inputs to achieve forward locomotion and shape tracking. The forward locomotion and the shape tracking are associated to the coupling with a substrate that models a generic environment with which the mechanism could interact. The assumption of small deformations around rigid body placements allows to adopt the floating reference kinematic description. By posing the distributed parameter control problem in weak form, we naturally introduce an approximate solution technique based on Galerkin projection on the linear mode shapes of the Timoshenko beam model that is adopted to describe the body of the system. Simulation results illustrate coupling among forward motion and shape tracking as described by the equations governing the system.

References

References
1.
Shabana
,
A. A.
,
2010
,
Dynamics of Multibody Systems
,
3rd ed.
,
Cambridge University Press
, Cambridge.
2.
Zehetner
,
C.
, and
Irschik
,
H.
,
2005
, “
Displacement Compensation of Beam Vibrations Caused by Rigid-Body Motions
,”
Smart Mater. Struct.
,
14
(
4
), pp.
862
–868.
3.
Altafini
,
C.
,
2002
, “
Following a Path of Varying Curvature as an Output Regulation Problem
,”
IEEE Trans. Autom. Control
,
47
(
9
), pp.
1551
1556
.
4.
Spong
,
M. W.
,
Hutchinson
,
S.
, and
Vidyasagar
,
M.
,
2006
,
Robot Modeling and Control
,
Wiley
, Hoboken, NJ.
5.
Gaul
,
L.
, and
Becker
,
J.
,
2009
, “
Model-Based Piezoelectric Hysteresis and Creep Compensation for Highly-Dynamic Feedforward Rest-to-Rest Motion Control of Piezoelectrically Actuated Flexible Structures
,”
Int. J. Eng. Sci.
,
47
(
11–12
), pp.
1193
1207
.
6.
Boyer
,
F.
,
Ali
,
S.
, and
Porez
,
M.
,
2012
, “
Macrocontinuous Dynamics for Hyperredundant Robots: Application to Kinematic Locomotion Bioinspired by Elongated Body Animals
,”
IEEE Trans. Rob.
,
28
(
2
), pp.
303
317
.
7.
Boyer
,
F.
, and
Primault
,
D.
,
2005
, “
The Poincare-Chetayev Equations and Flexible Multibody Systems
,”
PMM J. Appl. Math. Mech.
,
69
(
6
), pp.
925
942
.
8.
Boyer
,
F.
,
Porez
,
M.
,
Leroyer
,
A.
, and
Visonneau
,
M.
,
2008
, “
Fast Dynamics of an Eel-Like Robot-Comparisons With Navier–Stokes Simulations
,”
IEEE Trans. Rob.
,
24
(
6
), pp.
1274
1288
.
9.
Kim
,
S.-M.
,
Kim
,
H.
,
Boo
,
K.
, and
Brennan
,
M. J.
,
2013
, “
Demonstration of Non-Collocated Vibration Control of a Flexible Manipulator Using Electrical Dynamic Absorbers
,”
Smart Mater. Struct.
,
22
(
12
), p.
127001
.
10.
Bellezza
,
F.
,
Lanari
,
L.
, and
Ulivi
,
G.
,
1999
, “
Exact Modeling of the Slewing Flexible Link
,”
IEEE
International Conference on Robotics and Automation
, Cincinnati, OH, May 13–18, pp.
734
739
.
11.
White
,
M. W. D.
, and
Heppler
,
G. R.
,
1995
, “
Timoshenko Model of a Flexible Slewing Link
,”
American Control Conference
, Seattle, WA, June 21–23, pp.
2815
2819
.
12.
Daniel-Berhe
,
S.
, and
Unbehauen
,
H.
,
1997
, “
Physical Parameter Estimation of the Nonlinear Dynamics of a Single Link Robotic Manipulator With Flexible Joint Using the HMF-Method
,”
American Control Conference
, Albuquerque, NM, June 6, Vol.
1–6
, pp.
1504
1508
.
13.
Tomei
,
P.
, and
Tornambe
,
A.
,
1988
, “
Approximate Modeling of Robots Having Elastic Links
,”
IEEE Trans. Syst. Man Cybern.
,
18
(
5
), pp.
831
840
.
14.
Tarumi
,
R.
, and
Oshita
,
Y.
,
2011
, “
Free-Vibration Acoustic Resonance of a Nonlinear Elastic Bar
,”
Philos. Mag.
,
91
(
5
), pp.
772
786
.
15.
Arshad
,
S. H.
,
Naeem
,
M. N.
,
Sultana
,
N.
,
Shah
,
A. G.
, and
Iqbal
,
Z.
,
2011
, “
Vibration Analysis of Bi-Layered FGM Cylindrical Shells
,”
Arch. Appl. Mech.
,
81
(
3
), pp.
319
343
.
16.
Ider
,
S. K.
,
1990
, “
Stability Analysis of Constraints in Flexible Multibody Systems Dynamics
,”
Int. J. Eng. Sci.
,
28
(
12
), pp.
1277
1290
.
17.
Pascal
,
M.
,
2001
, “
Some Open Problems in Dynamic Analysis of Flexible Multi-Body Systems
,”
J. Multibody Syst. Dyn.
,
5
(
4
), pp.
315
334
.
18.
Bopearatchy
,
D. L. P.
, and
Hatanwala
,
G. C.
,
1990
, “
State Space Control of a Multi Link Robot Manipulator by a Translational Modelling Technique
,” 5th
IEEE
International Symposium on Intelligent Control, Philadelphia, PA, Sep. 5–7, Vol.
1
, pp.
285
290
.
19.
Nanayakkara
,
T.
,
Watanabe
,
K.
,
Kiguchi
,
K.
, and
Izumi
,
K.
,
2000
, “
Controlling Multi-Link Manipulators by Fuzzy Selection of Dynamic Models
,” 26th Annual Conference of the
IEEE
Industrial Electronics Society, Nagoya, Japan, Vol.
1
, pp.
638
643
.
20.
Moallem
,
M.
,
Khorasani
,
K.
, and
Patel
,
R.
,
1997
, “
An Inverse Dynamics Sliding Control Technique for Flexible Multi-Link Manipulators
,”
American Control Conference
, Albuquerque, NM, June 4–6, Vol.
3
, pp.
1407
1411
.
21.
Xu
,
J.-X.
,
Pan
,
Y.-J.
, and
Lee
,
T.-H.
,
2001
, “
A Gain Shaped Sliding Mode Control Scheme Using Filtering Techniques With Applications to Multi-Link Robotic Manipulators
,”
American Control Conference
, Vol.
6
, pp.
4363
4368
.
22.
Ower
,
J. C.
, and
Van de Vegte
,
J.
,
1987
, “
Classical Control Design for a Flexible Manipulator: Modeling and Control System Design
,”
J. Rob. Autom.
,
3
(
5
), pp.
485
489
.
23.
Chaolan
,
Y.
,
Jiazhen
,
H.
, and
Guoping
,
C.
,
2006
, “
Modeling Study of a Flexible Hub-Beam System With Large Motion and With Considering the Effect of Shear Deformation
,”
J. Sound Vib.
,
295
(
1–2
), pp.
282
293
.
24.
Chen
,
W.
,
2001
, “
Dynamic Modeling of Multi-Link Flexible Robotic Manipulators
,”
J. Comput. Struct.
,
79
(
2
), pp.
183
195
.
25.
Lee
,
B.-J.
,
2013
, “
Geometrical Derivation of Differential Kinematics to Calibrate Model Parameters of Flexible Manipulator
,”
Int. J. Adv. Rob. Syst.
,
10
(106), pp.
1
9
.
26.
Chen
,
L.
, and
Deng
,
H.
,
2013
, “
Model Reduction of Rigid-Flexible Manipulators With Experimental Validation
,”
Adv. Materials Res.
,
655–657
, pp.
1101
1107
.
27.
Esfandiar
,
H.
, and
Daneshmand
,
S.
,
2012
, “
Complete Dynamic Modeling and Approximate State Space Equations of the Flexible Link Manipulator
,”
J. Mech. Sci. Technol.
,
26
(
9
), pp.
2845
2856
.
28.
di Castri
,
C.
, and
Messina
,
A.
,
2012
, “
Exact Modeling for Control of Flexible Manipulators
,”
J. Vib. Control
,
18
(
10
), pp.
1526
1551
.
29.
Korayem
,
M. H.
,
Rahimi
,
H. N.
, and
Nikoobin
,
A.
,
2012
, “
Mathematical Modeling and Trajectory Planning of Mobile Manipulators With Flexible Links and Joints
,”
Appl. Math. Modell.
,
36
(
7
), pp.
3229
3244
.
30.
Choi
,
S. B.
,
Kim
,
H. K.
, and
Kim
,
S. C.
,
2000
, “
Position and Force Control of a Two-Link Flexible Manipulator With Piezoelectric Actuators
,”
Appl. Mech. Eng.
,
5
(
1
), pp.
75
80
.
31.
Nguyen
,
P.-B.
, and
Choi
,
S.-B.
,
2010
, “
Open-Loop Position Tracking Control of a Piezoceramic Flexible Beam Using a Dynamic Hysteresis Compensator
,”
Smart Mater. Struct.
,
19
(
12
), p.
125008
.
32.
Lee
,
H. H.
,
1956
, “
New Dynamic Modeling of Flexible-Link Robots
,”
ASME J. Dyn. Syst. Meas. Control
,
127
(
2
), pp.
307
309
.
33.
Zhang
,
X.
,
Xu
,
W.
,
Nair
,
S. S.
, and
Chellaboina
,
V. S.
,
2005
, “
PDE Modeling and Control of a Flexible Two-Link Manipulator
,”
IEEE Trans. Control Syst. Technol.
,
13
(
2
), pp.
301
312
.
34.
Milford
,
R. I.
, and
Asokanthan
,
S. F.
,
1999
, “
Configuration Dependent Eigenfrequencies for a Two-Link Flexible Manipulator: Experimental Verification
,”
J. Sound Vib.
,
222
(
2
), pp.
191
207
.
35.
Hać
,
A.
, and
Liu
,
L.
,
1993
, “
Sensor and Actuator Location in Motion Control of Flexible Structures
,”
J. Sound Vib.
,
167
(
2
), pp.
239
261
.
36.
Book
,
W. J.
,
Maizza-Neto
,
O.
, and
Whitney
,
D.
,
1975
, “
Feedback Control of Two Beam, Two Joint Systems With Distributed Flexibility
,”
ASME J. Dyn. Syst. Meas. Control
,
97
(
4
), pp.
424
431
.
37.
Book
,
W. J.
,
1984
, “
Recursive Lagrangian Dynamics of Flexible Manipulator Arms
,”
Int. J. Rob. Res.
,
3
(
3
), pp.
87
101
.
38.
Uhn Kim
,
J.
, and
Renardy
,
Y.
,
1987
, “
Boundary Control of the Timoshenko Beam
,”
SIAM J. Contr. Optim.
25
(
6
), pp.
1417
1429
.
39.
Meek
,
J.
, and
Liu
,
H.
,
1995
, “
Nonlinear Dynamics Analysis of Flexible Beams Under Large Overall Motions and the Flexible Manipulator Simulation
,”
Comput. Struct.
,
56
(
1
), pp.
1
14
.
40.
Yuan
,
K.
, and
Hu
,
C.
,
1996
, “
Nonlinear Modeling and Partial Linearizing Control of a Slewing Timoshenko-Beam
,”
ASME J. Dyn. Syst. Meas. Control
,
118
(
1
), pp.
75
83
.
41.
Zuyev
,
A.
, and
Sawodny
,
O.
,
2006
, “
Observer Design for a Flexible Manipulator Model With a Payload
,” 45th
IEEE
Conference on Decision and Control
, San Diego, CA, Dec. 13–15, Vol.
1–14
, pp.
4490
4495
.
42.
Vukobratovic
,
M.
, and
Tuneski
,
A.
,
1996
, “
Adaptive Control of Single Rigid Robotic Manipulators Interacting With Dynamic Environment—An Overview
,”
J. Intell. Rob. Syst.
,
17
(
1
), pp.
1
30
.
43.
Jia
,
Z. J.
,
Song
,
Y. D.
, and
Cai
,
W. C.
,
2013
, “
Bio-Inspired Approach for Smooth Motion Control of Wheeled Mobile Robots
,”
Cognit. Comput.
,
5
(
2
), pp.
252
263
.
44.
Mahjoubi
,
H.
, and
Byl
,
K.
,
2013
, “
Modeling Synchronous Muscle Function in Insect Flight: A Bio-Inspired Approach to Force Control in Flapping-Wing MAVs
,”
J. Intell. Rob. Syst.
,
70
(
1–4
), pp.
181
202
.
45.
Sun
,
B.
,
Zhu
,
D.
,
Ding
,
F.
, and
Yang
,
S. X.
,
2013
, “
A Novel Tracking Control Approach for Unmanned Underwater Vehicles Based on Bio-Inspired Neurodynamics
,”
J. Mar. Sci. Technol.
,
18
(
1
), pp.
63
74
.
46.
Park
,
Y.
,
Young
,
D.
,
Chen
,
B.
,
Wood
,
R. J.
,
Nagpal
,
R.
, and
Goldfield
,
E. C.
,
2013
, “
Networked Bio-Inspired Modules for Sensorimotor Control of Wearable Cyber-Physical Devices
,”
International Conference on Computing, Networking and Communications (ICNC)
, San Diego, CA.
47.
Zhang
,
J.
,
Qiao
,
G.
,
Song
,
G.
, and
Wang
,
A.
,
2012
, “
Design and Implementation of a Remote Control System for a Bio-Inspired Jumping Robot
,”
Int. J. Adv. Rob. Syst.
,
9
(
117
), pp.
1
9
.
48.
González-Mora
,
J. L.
,
Rodríguez-Hernández
,
A.
,
Rodríguez-Ramos
,
L. F.
,
Díaz-Saco
,
L.
, and
Sosa
,
N.
,
1999
, “
Development of a New Space Perception System for Blind People, Based on the Creation of a Virtual Acoustic Space
,”
Engineering Applications of Bio-Inspired Artificial Neural Networks
(Lecture Notes in Computer Science), Vol.
1607
, J. Mira and J. Sánchez-Andrés, eds., Springer, Berlin, Heidelberg, pp.
321
330
.
49.
Tolu
,
S.
,
Vanegas
,
M.
,
Luque
,
N. R.
,
Garrido
,
J. A.
, and
Ros
,
E.
,
2012
, “
Bio-Inspired Adaptive Feedback Error Learning Architecture for Motor Control
,”
Biol. Cybern.
,
106
(
8–9
), pp.
507
522
.
50.
Gray
,
J.
, and
Lissmann
,
H. W.
,
1938
, “
Studies in Animal Locomotion VII. Locomotory Reflexes in the Earthworm
,”
J. Exp. Biol.
,
15
(
4
), pp.
506
517
.
51.
Dorgan
,
K. M.
,
Law
,
C. J.
, and
Rouse
,
G.
,
2013
, “
Meandering Worms: Mechanics of Undulatory Burrowing in Muds
,”
Proc. R. Soc. B
,
280
(
1757
), pp.
1
9
.
52.
Daltorio
,
K. A.
,
Boxerbaum
,
A. S.
,
Horchler
,
A. D.
,
Shaw
,
K. M.
,
Chiel
,
H. J.
, and
Quinn
,
R. D.
,
2013
, “
Efficient Worm-Like Locomotion: Slip and Control of Soft-Bodied Peristaltic Robots
,”
Bioinspiration Biomimetics
,
8
(
3
), 035003.
53.
Yapp
,
W. B.
,
1956
, “
Locomotion of Worms
,”
Nature
,
177
(
4509
), pp.
614
615
.
54.
Martins
,
J. M.
,
Botto
,
M. A.
, and
da Costa
,
J. M. S.
,
2003
, “
A Newton–Euler Model of a Piezo-Actuated Nonlinear Elastic Manipulator Link
,”
Proceedings of the 11th International Conference on Advanced Robotics
, U. Nunes, A. T. deAalmeida, A. K. Bejczy, K. Kosuge, and J. A. T. Macgado, eds., Coimbra, Portugal, Vol.
1–3
, pp.
935
940
.
55.
Jayne
,
B. C.
,
1986
, “
Kinematics of Terrestrial Snake Locomotion
,”
Copeia
,
1986
(
4
), pp.
915
927
.
56.
Umetani
,
Y.
, and
Hirose
,
S.
,
1974
, “
Biomechanical Study of Serpentine Locomotion
,”
1st ROMANSY Symposium
, Udine, Vol.
177
, pp.
171
184
.
57.
Hirose
,
S.
,
1985
, “
Connected Differential Mechanism and Its Applications
,”
2nd International Conference in Advances in Robotics
, pp.
319
326
.
58.
Hirose
,
S.
, and
Umetani
,
Y.
,
1976
, “
Kinematic Control of an Active Cord Mechanism With Tactile Sensors
,”
CISM-ZFToM
Symposium on Theory and Practice of Robots and Manipulators
, pp.
241
252
.
59.
Hirose
,
S.
,
Ikuta
,
K.
,
Tsukamoto
,
M.
, and
Sato
,
K.
,
1987
, “
Considerations in Design of the Actuator Based in the Shape Memory Effect
,”
6th IGToMM Congress
, pp.
1549
1556
.
60.
Hirose
,
S.
, and
Umetani
,
Y.
,
1979
, “
The Kinematics and Control of a Soft Gripper for the Handling of Living and Fragile Objects
,”
IGToMM Congress
, pp.
1549
1556
.
61.
Ijspeert
,
A. J.
,
Hallam
,
J.
, and
Willshaw
,
D.
,
1998
, “
From Lampreys to Salamanders: Evolving Neural Controllers for Swimming and Walking
,”
From Animals to Animat, Proceedings of the Fifth International Conference of the Society for Adaptive Behaviour (SAB98)
, R. Pfeifer, B. Blumberg, J. A. Meyer, and S. W. Wilson, eds., MIT Press, pp.
390
399
.
62.
Yang
,
S.
,
Jin
,
X.
,
Liu
,
K.
, and
Jiang
,
L.
,
2013
, “
Nanoparticles Assembly-Induced Special Wettability for Bio-Inspired Materials
,”
Particuology
,
11
(
4
), pp.
361
370
.
63.
Liu
,
K.
,
Tian
,
Y.
, and
Jiang
,
L.
,
2013
, “
Bio-Inspired Superoleophobic and Smart Materials: Design, Fabrication, and Application
,”
Prog. Mater. Sci.
,
58
(
4
), pp.
503
564
.
64.
Saavedra
,
F.
,
Erick
,
I.
,
Friswell
,
M. I.
, and
Xia
,
Y.
,
2013
, “
Variable Stiffness Biological and Bio-Inspired Materials
,”
J. Intell. Mater. Syst. Struct.
,
24
(
5
), pp.
529
540
.
65.
Sugawara-Narutaki
,
A.
,
2013
, “
Bio-Inspired Synthesis of Polymer-Inorganic Nanocomposite Materials in Mild Aqueous Systems
,”
Polym. J.
,
45
(
3
), pp.
269
276
.
66.
Stevens
,
M. M.
, and
Mecklenburg
,
G.
,
2012
, “
Bio-Inspired Materials for Biosensing and Tissue Engineering
,”
Polym. Int.
,
61
(
5
), pp.
680
685
.
67.
Menciassi
,
A.
, and
Dario
,
P.
,
2003
, “
Bio-Inspired Solutions for Locomotion in the Gastrointestinal Tract: Background and Perspectives
,”
Philos. Trans. R. Soc., A
,
361
(
1811
), pp.
2287
2298
.
68.
Dario
,
P.
,
Ciarletta
,
P.
,
Menciassi
,
A.
, and
Kim
,
B.
,
2003
, “
Modelling and Experimental Validation of the Locomotion of Endoscopic Robots in the Colon
,”
Experimental Robotics VIII
(Springer Tracts in Advanced Robotics)
, B. Siciliano and P. Dario, eds., Springer, Berlin, Heidelberg, Vol.
5
, pp.
445
453
.
69.
Huston
,
D.
,
Miller
,
J.
, and
Esser
,
B.
,
2004
, “
Adaptive, Robotic and Mobile Sensor Systems for Structural Assessment
,”
Proceedings of the Smart Structures and Materials 2004: Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems
, S. C. Liu, ed., San Diego, CA, Vol.
5391
, pp.
189
196
.
70.
Esser
,
B.
, and
Huston
,
D. R.
,
2005
, “
Versatile Robotic Platform for Structural Health Monitoring and Surveillance
,”
Smart Struct. Syst.
,
1
(
4
), pp.
325
338
.
71.
Cha
,
Y.
,
Kim
,
H.
, and
Porfiri
,
M.
,
2013
, “
Energy Harvesting From Underwater Base Excitation of a Piezoelectric Composite Beam
,”
Smart Mater. Struct.
,
22
(
11
), p.
115026
.
72.
Cha
,
Y.
,
Shen
,
L.
, and
Porfiri
,
M.
,
2013
, “
Energy Harvesting From Underwater Torsional Vibrations of a Patterned Ionic Polymer Metal Composite
,”
Smart Mater. Struct.
,
22
(
5
), p.
055027
.
73.
Kuroda
,
S.
,
Kunita
,
I.
,
Tanaka
,
Y.
,
Ishiguro
,
A.
,
Kobayashi
,
R.
, and
Nakagaki
,
T.
,
2014
, “
Common Mechanics of Mode Switching in Locomotion of Limbless and Legged Animals
,”
J. R. Soc., Interface
,
11
(
95
), p.
20140205
.
74.
Chadwick
,
P.
,
1998
,
Continuum Mechanics: Concise Theory and Problems
,
1 ed.
,
Dover
, Mineola, NY.
75.
Bender
,
C. M.
, and
Orszag
,
S. A.
,
1999
,
Advanced Mathematical Methods for Scientists and Engineers I
,
Springer
, New York.
76.
Timoshenko
,
S.
,
1974
,
Vibration Problems in Engineering
,
D. Van Nostrand Company, Inc.
, New York.
77.
Hetnarski
,
R. B.
, and
Ignaczak
,
J.
,
2010
,
The Mathematical Theory of Elasticity
,
2 ed.
,
CRC Press
, Boca Raton.
78.
van Rensburg
,
N. F. J.
, and
van der Merwe
,
A. J.
,
2006
, “
Natural Frequencies and Modes of a Timoshenko Beam
,”
Wave Motion
,
44
(
1
), pp.
58
69
.
79.
Majkut
,
L.
,
2009
, “
Free and Forced Vibrations of Timoshenko Beam Described by Single Differential Equation
,”
J. Theor. Appl. Mech.
,
47
(
1
), pp.
193
210
.
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