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.
Issue Section:
Research Papers
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
.
This content is only available via PDF.
Copyright © 2015 by ASME
You do not currently have access to this content.
