This paper presents a general algorithm for solving the dynamic of tree structure robots with rigid and flexible links, active and passive joints, and with a fixed or floating base. The algorithm encompasses in a unified approach both the inverse and direct dynamics. It addresses also the hybrid case where each active joint is considered with known joint torque as in the direct dynamic case, or with known joint acceleration as in the inverse dynamic case. To achieve this goal, we propose an efficient recursive approach based on the generalized Newton–Euler equations of flexible tree-structure systems. This new general hybrid algorithm is easy to program either numerically or using efficient customized symbolic techniques. It is of great interest for studying floating base systems with soft appendages as those currently investigated in soft bio-inspired robotics or when a robotic system has to modify its structure for some particular tasks, such as transforming an active joint into a compliant flexible one, or modifying a task with a floating base into one with fixed base.
Issue Section:
Research Papers
References
1.
Luh
, J. Y. S.
, Walker
, M. W.
, and Paul
, R. C. P.
, 1980
, “On-Line Computational Scheme for Mechanical Manipulators
,” ASME J. Dyn. Syst., Meas., Control
, 102
(2
), pp. 69
–76
.2.
Featherstone
, R.
, 1983
, “The Calculation of Robot Dynamics Using Articulated-Body Inertias
,” Int. J. Rob. Res.
, 2
(3
), pp. 87
–101
.3.
Featherstone
, R.
, 2008
, Rigid Body Dynamics Algorithms
, Springer
, New York
.4.
Jain
, A.
, and Rodriguez
, G.
, 1992
, “Recursive Flexible Multibody System Dynamics Using Spatial Operators
,” J. Guid. Control Dyn.
, 15
(6
), pp. 1453
–1466
.5.
Boyer
, F.
, and Khalil
, W.
, 1998
, “An Efficient Calculation of Flexible Manipulator Inverse Dynamics
,” Int. J. Rob. Res.
, 17
(3
), pp. 282
–293
.6.
Hughes
, P. C.
, and Sincarsin
, G. B.
, 1989
, “Dynamics of Elastic Multibody Chains: Part B-Global Dynamics
,” Dyn. Stab. Syst.
, 4
(3–4
), pp. 227
–243
.7.
Sharf
, I.
, and D’Eleuterio
, G. M. T.
, 1992
, “Parallel Simulation Dynamics for Elastic Multibody Chains
,” IEEE Trans. Rob. Autom.
, 8
(5
), pp. 597
–606
.8.
Anderson
, K. S.
, 1990
, “Recursive Derivation of Explicit Equations of Motion for Efficient Dynamic/Control Simulation of Large Multibody Systems
,” Ph.D. thesis, Stanford University, Stanford, CA.9.
Khalil
, W.
, and Kleinfinger
, J.-F.
, 1986
, “A New Geometric Notation for Open and Closed-Loop Robots
,” IEEE International Conference on Robotics and Automation
(ICRA
), San Francisco, CA, Apr. 7–10, pp. 1174
–1180
.10.
Verl
, A.
, Albu-Schaeffer
, A.
, and Brock
, O.
, eds., 2015
, Soft Robotics: Transferring Theory to Application
, Springer
, New York
.11.
Roberts
, T. J.
, and Azizi
, E.
, 2011
, “Flexible Mechanisms: The Diverse Roles of Biological Springs in Vertebrate Movement
,” J. Exp. Biol.
, 214
(3
), pp. 353
–361
.12.
Pfeifer
, R.
, Lungarella
, M.
, and Iida
, F.
, 2007
, “Self-Organization, Embodiment, and Biologically Inspired Robotics
,” Science
, 318
(5853
), pp. 1088
–1093
.13.
Khosla
, P. K.
, 1986
, “Real-Time Control and Identification of Direct-Drive Manipulators
,” Ph.D. thesis, Carnegie Mellon, Pittsburgh, PA.14.
Khalil
, W.
, and Kleinfinger
, J.-F.
, 1987
, “Minimum Operations and Minimum Parameters of the Dynamic Model of Tree Structure Robots
,” IEEE J. Rob. Autom.
, 3
(6
), pp. 517
–526
.15.
Khalil
, W.
, and Dombre
, E.
, 2002
, Modeling Identification and Control of Robots
, Hermes, Penton-Sciences
, London
.16.
Canavin
, J.
, and Likins
, P.
, 1977
, “Floating Reference Frames for Flexible Spacecraft
,” J. Spacecr. Rockets
, 14
(12
), pp. 724
–732
.17.
Bae
, D. S.
, et al, 2001
, “A Generalized Recursive Formulation for Constrained Flexible Multibody Dynamics
,” Int. J. Numer. Methods Eng.
, 50
(8
), pp. 1841
–1859
.18.
Meirovitch
, L.
, 1989
, Dynamics and Control of Structures
, Wiley
, New York
.19.
Craig
, J. J.
, 1986
, Introduction to Robotics: Mechanics and Control
, Addison Wesley Publishing Company
, Reading, UK
.20.
Angeles
, J.
, 2002
, Fundamentals of Robotic Mechanical Systems
, 2nd ed., Springer-Verlag
, New York
.21.
Jain
, A.
, 2011
, Robot and Multibody Dynamics: Analysis and Algorithms
, Springer-Verlag
, Berlin
.22.
Walker
, M. W.
, and Orin
, D. E.
, 1982
, “Efficient Dynamic Computer Simulation of Robotics Mechanism
,” ASME J. Dyn. Syst., Meas., Control
, 104
(3
), pp. 205
–211
.23.
Rodriguez
, G.
, Kreutz
, K.
, and Jain
, A.
, 1991
, “A Spatial Operator Algebra for Manipulator Modeling and Control
,” Int. J. Rob. Res.
, 10
(4
), pp. 371
–381
.24.
Featherstone
, R.
, 1999
, “A Divide-and-Conquer Articulated Body Algorithm for Parallel O(log(n)) Calculation of Rigid Body Dynamics—Part 1: Basic Algorithm
,” Int. J. Rob. Res.
, 18
(9
), pp. 867
–875
.25.
Featherstone
, R.
, 1999
, “A Divide-and-Conquer Articulated Body Algorithm for Parallel O(log(n)) Calculation of Rigid Body Dynamics—Part 2: Trees, Loops and Accuracy
,” Int. J. Rob. Res.
, 18
(9
), pp. 876
–892
.26.
Mukherjee
, R.
, and Anderson
, K. S.
, 2007
, “A Logarithm Complexity Divide-and-Conquer Algorithm for Multiflexible Articulated Body Systems
,” Comput. Nonlinear Dyn.
, 2
(1
), pp. 10
–21
.27.
Mukherjee
, R.
, and Anderson
, K. S.
, 2007
, “An Orthogonal Complement Based Divide-and-Conquer Algorithm for Constrained Multibody Systems
,” Comput. Nonlinear Dyn.
, 48
(1–2
), pp. 199
–215
.28.
Vereshchagin
, A. F.
, 1974
, “Computer Simulation of the Dynamics of Complicated Mechanisms of Robot-Manipulators
,” Eng. Cybern.
, 6
, pp. 65
–70
.29.
Armstrong
, W. W.
, 1979
, “Recursive Solution to the Equations of Motiob of an n-Link Manipulator
,” Fifth World Congress on the Theory of Machines and Mechanisms
(IFToMM), Montreal, Canada, July 8–13, Vol. 2
, pp. 1342
–2346
.30.
Brand
, I. H.
, Johani
, R.
, and Otter
, M.
, 1986
, “A Very Efficient Algorithm for the Simulation of Robots and Similar Multibody Systems Without Inversion of the Mass Matrix
,” IFAC/IFIP/IMACS International Symposium on Theory of Robots
, Vienna, Austria, pp. 95
–100
.31.
Bae
, D. S.
, and Haug
, E. J.
, 1987
, “A Recursive Formulation for Constrained Mechanical Systems—Part I: Open Loop Systems
,” Mech. Struct. Mach.
, 15
(3
), pp. 359
–382
.32.
Rosenthal
, D.
, 1987
, “Order n Formulation for Equations of Motions of Multibody Systems
,” SDIO NASA Workshop on Multibody Simulation
, Jet Propulsion Laboratory, Arcadia, CA, pp. 1122
–1150
.33.
Jain
, A.
, 1991
, “A Unified Formulation of Dynamics for Serial Rigid Multibody Systems
,” J. Guid. Control Dyn.
, 14
(3
), pp. 531
–542
.34.
Stelzle
, W.
, Kecskemethy
, A.
, and Hiller
, M.
, 1995
, “A Comparative Study of Recursive Methods
,” Arch. Study Recursive Methods
, 66
(1
), pp. 9
–19
.35.
Featherstone
, R.
, and Orin
, D. E.
, 2008
, “Dynamics
,” Springer Handbook of Robotics
, B.
Siciliano
and O.
Khatib
, eds., Springer-Verlag
, Berlin
, pp. 37–63.36.
Khalil
, W.
, and Creusot
, D.
, 1997
, “SYMORO+: A System for the Symbolic Modelling of Robots
,” Robotica
, 15
(2
), pp. 153
–161
.37.
Jain
, A.
, and Rodriguez
, G.
, 1993
, “Recursive Dynamics Algorithm for Multibody Systems With Prescribed Motion
,” J. Guid., Control Dyn.
, 16
(4
), pp. 830
–837
.38.
Albu-Schaffer
, A.
, 2008
, “From Torque Feedback-Controlled Lightweight Robots to Intrinsically Compliant Systems
,” IEEE Robotics and Automation Magazine
, pp. 20
–30
.39.
Spong
, M.
, 1987
, “Modelling and Control of Elastic Joint Robots
,” ASME J. Dyn. Syst., Meas., Control
, 109
(4
), pp. 310
–319
.40.
Khalil
, W.
, and Gautier
, M.
, 2000
, “Modeling of Mechanical Systems With Lumped Elasticity
,” IEEE International Conference on Robotics and Automation (ICRA
), San Francisco, CA, Apr. 24–28, pp. 3964
–3969
.41.
Feron
, E.
, and Johnson
, E.
, 2008
, “Aerial Robots
,” Springer Handbook of Robotics
, B.
Siciliano
and O.
Khatib
, eds., Springer-Verlag
, Berlin
, Chap. 44.42.
Khalil
, W.
, and Rongère
, F.
, 2014
, “Dynamic Modeling of Floating Systems: Application to Eel-Like Robot and Rowing System
,” 13th IEEE International Workshop on Advanced Motion Control (AMC
), Yokohama, Japan, Mar. 14–16, pp. 12
–14
.43.
McIsaac
, K. A.
, and Ostrowski
, J. P.
, 1999
, “A Geometric Approach to Anguilliform Locomotion Modelling of an Underwater Eel Robot
,” IEEE International Conference on Robotics and Automation
(ICRA
), Detroit, MI, May 10–15, pp. 2843
–2848
.44.
Ostrowski
, J. P.
, and Burdick
, J. W.
, 1998
, “The Geometric Mechanics of Undulatory Robotics Locomotion
,” Int. J. Rob. Res.
, 17
(7
), pp. 683
–701
.45.
Chevallereau
, C.
, Bessonnet
, G.
, Abba
, G.
, and Aoustin
, Y.
, 2009
, Bipedal Robots
(CAM Control Systems
, Robotics and Manufacturing Series
), ISTE
London
.46.
Mukherjee
, R.
, and Nukamura
, Y.
, 1992
, “Formulation and Efficient Computation of Inverse Dynamics of Space Robots
,” IEEE Trans. Rob. Autom.
, 8
(3
), pp. 400
–406
.47.
Jain
, A.
, and Rodriguez
, G.
, 1995
, “Base-Invariant Symmetric Dynamics of Free-Flying Manipulators
,” IEEE Trans. Rob. Autom.
, 11
(4
), pp. 585
–597
.48.
Khalil
, W.
, 2010
, “Dynamic Modeling of Robots Using Newton-Euler Formulation
,” Informatics in Control, Automation and Robotics
(Lecture Notes in Electrical Engineering), Vol. 89
, J. A.
Cetto
, J.-L.
Ferrier
, and J.
Filipe
, eds., Springer
, Berlin
, pp. 3
–20
.49.
Boyer
, F.
, Porez
, M.
, and Khalil
, W.
, 2006
, “Macro-Continuous Torque Algorithm for a Three-Dimensional Eel-Like Robot
,” IEEE Rob. Trans.
, 22
(4
), pp. 763
–775
.50.
Boyer
, F.
, Shaukat
, A.
, 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
.51.
Boyer
, F.
, and Porez
, M.
, 2015
, “Multibody System Dynamics for Bio-Inspired Locomotion: From Geometric Structures to Computational Aspects
,” Bioinspiration Biomimetics
, 10
(2
), p. 025007
.52.
Khalil
, W.
, Vijayalingam
, A.
, Khomutenko
, B.
, Mukhanov
, I.
, Lemoine
, P.
, and Ecorchard
, G.
, 2014
, “OpenSYMORO: An Open-Source Software Package for Symbolic Modelling of Robots
,” IEEE/ASME
International Conference on Advanced Intelligent Mechatronics, Besancon, France, July, pp. 1206
–1211
.53.
De Luca
, A.
, and Siciliano
, B.
, 1991
, “Recursive Lagrangian Dynamics of Flexible Manipulator Arms
,” IEEE Trans. Syst., Man Cybern.
, 21
(4
), pp. 826
–839
.54.
Damaren
, C.
, and Sharf
, I.
, 1995
, “Simulation of Flexible-Link Manipulators With Inertial and Geometric Nonlinearities
,” ASME J. Dyn. Syst., Meas., Control
, 117
(1
), pp. 74
–87
.55.
Porez
, M.
, Boyer
, F.
, and Belkhiri
, A.
, 2014
, “A Hybrid Dynamic Model for Bio-Inspired Soft Robots: Application to a Flapping-Wing Micro Air Vehicle
,” International Conference on Robotics and Automation
(ICRA
), Hong Kong, China, May 31–June 7, pp. 3556
–3563
.56.
Boyer
, F.
, and Coiffet
, P.
, 1996
, “Generalization of Newton-Euler Model for Flexible Manipulators
,” J. Rob. Syst.
, 13
(1
), pp. 11
–24
.57.
Boyer
, F.
, Glandais
, N.
, and Khalil
, W.
, 2002
, “Flexible Multibody Dynamics Based on a Non-Linear Euler-Bernoulli Kinematics
,” Int. J. Numer. Methods Eng.
, 54
(1
), pp. 27
–59
.58.
Sane
, S. P.
, and Dickinson
, M. H.
, 2002
, “The Aerodynamic Effects of Wing Rotation and a Revised Quasi-Steady Model of Flapping Flight
,” J. Exp. Biol.
, 205
, pp. 1087
–1096
.59.
McMichael
, J. M.
, and Francis
, M. S.
, 1997
, “Micro Air Vehicles–Toward a New Dimension in Flight
,” DARPA Tactical Technology Office, Arlington, VA, http://www.uadrones.net/military/research/1997/0807.htm60.
Boyer
, F.
, Porez
, M.
, Ferhat
, M.
, and Morel
, Y.
, 2016
, “Locomotion Dynamics for Bio-Inspired Robots With Soft Appendages: Application to Flapping Flight and Passive Swimming
,” J. Nonlinear Sci.
, (epub).61.
Khalil
, W.
, Gallot
, G.
, and Boyer
, F.
, 2007
, “Dynamic Modeling and Simulation of a 3-D Serial Eel-Like Robot
,” IEEE Trans. Syst., Man Cybern., Part C
, 37
(6
), pp. 1259
–1268
.62.
Porez
, M.
, Boyer
, F.
, and Ijspeert
, A. J.
, 2014
, “Improved Lighthill Fish Swimming Model for Bio-Inspired Robots: Modeling, Computational Aspects and Experimental Comparisons
,” Int. J. Rob. Res.
, 33
(10
), pp. 1322
–1341
.Copyright © 2017 by ASME
You do not currently have access to this content.
