Abstract
We provide a tutorial and review of the state-of-the-art in robot dynamics algorithms that rely on methods from differential geometry, particularly the theory of Lie groups. After reviewing the underlying Lie group structure of the rigid-body motions and the geometric formulation of the equations of motion for a single rigid body, we show how classical screw-theoretic concepts can be expressed in a reference frame-invariant way using Lie-theoretic concepts and derive recursive algorithms for the forward and inverse dynamics and their differentiation. These algorithms are extended to robots subject to closed-loop and other constraints, joints driven by variable stiffness actuators, and also to the modeling of contact between rigid bodies. We conclude with a demonstration of how the geometric formulations and algorithms can be effectively used for robot motion optimization.
Issue Section:
Review Article
References
1.
Siciliano
, B.
, and Khatib
, O.
, 2016
, Springer Handbook of Robotics
, Springer
, New York
.2.
Featherstone
, R.
, 2014
, Rigid Body Dynamics Algorithms
, Springer
, New York
.3.
Featherstone
, R.
, 1983
, “The Calculation of Robot Dynamics Using Articulated-Body Inertias
,” Int. J. Rob. Res.
, 2
(1
), pp. 13
–30
.4.
Arnold
, V. I.
, 2013
, Mathematical Methods of Classical Mechanics
, Vol. 60, Springer Science & Business Media
, New York
.5.
Brockett
, R. W.
, 1984
, “Robotic Manipulators and the Product of Exponentials Formula
,” Mathematical Theory of Networks and Systems
, Springer
, New York
, pp. 120
–129
.6.
Murray
, R. M.
, Li
, Z.
, and Sastry
, S. S.
, 1994
, A Mathematical Introduction to Robotic Manipulation
, CRC Press
, Boca Raton, FL
.7.
Bloch
, A. M.
, 2015
, Nonholonomic Mechanics and Control
, Vol. 24
, Springer
, New York
.8.
Bullo
, F.
, and Lewis
, A. D.
, 2004
, Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems
, Vol. 49
, Springer Science & Business Media
, New York
.9.
Lewis
, A. D.
, 2007
, “Is It Worth Learning Differential Geometric Methods for Modeling and Control of Mechanical Systems?
,” Robotica
, 25
(6
), pp. 765
–777
.https://pdfs.semanticscholar.org/5fe8/43835ea372bde7235e611ebf784a0e92e8d1.pdf10.
Greenwood
, D. T.
, 2006
, Advanced Dynamics
, Cambridge University Press
, Cambridge, UK
.11.
Belinfante
, J. G.
, and Kolman
, B.
, 1989
, A Survey of Lie Groups and Lie Algebras With Applications and Computational Methods
, SIAM
, Philadelphia, PA.12.
Boothby
, W. M.
, 1986
, An Introduction to Differentiable Manifolds and Riemannian Geometry
, Vol. 120
, Academic Press
, Cambridge, MA
.13.
Curtis
, M. L.
, 2012
, Matrix Groups
, Springer Science & Business Media
, New York
.14.
Chevalley
, C.
, 1999
, Theory of Lie Groups
, Vol. 1, Princeton University Press
, Princeton, NJ
.15.
Lynch
, K. M.
, and Park
, F. C.
, 2017
, Modern Robotics Mechanics, Planning, and Control
, Cambridge University Press
, Cambridge, UK
.16.
McCarthy
, J. M.
, 1990
, Introduction to Theoretical Kinematics
, MIT Press
, Cambridge, MA
.17.
Gantmacher
, F. R.
, 1960
, Theory of Matrices
, Vol. 2
, Chelsea Publishing
, Hartford, VT
.18.
Mladenova
, C. D.
, 1993
, “Group-Theoretical Methods in Manipulator Kinematics and Symbolic Computations
,” J. Intell. Rob. Syst.
, 8
(1
), pp. 21
–34
.19.
Rohmer
, E.
, Singh
, S. P.
, and Freese
, M.
, 2013
, “V-Rep: A Versatile and Scalable Robot Simulation Framework
,” IEEE/RSJ International Conference on Intelligent Robots and System (IROS
), Tokyo, Japan, Nov. 3–7, pp. 1321
–1326
.20.
Luh
, J. Y.
, Walker
, M. W.
, and Paul
, R. P.
, 1980
, “On-Line Computational Scheme for Mechanical Manipulators
,” ASME J. Dyn. Syst. Meas. Control
, 102
(2
), pp. 69
–76
.21.
Park
, F. C.
, Bobrow
, J. E.
, and Ploen
, S. R.
, 1995
, “A Lie Group Formulation of Robot Dynamics
,” Int. J. Rob. Res.
, 14
(6
), pp. 609
–618
.22.
Park
, F.
, Choi
, J.
, and Ploen
, S.
, 1999
, “Symbolic Formulation of Closed Chain Dynamics in Independent Coordinates
,” Mech. Mach. Theory
, 34
(5
), pp. 731
–751
.23.
Ploen
, S. R.
, and Park
, F. C.
, 1997
, “A Lie Group Formulation of the Dynamics of Cooperating Robot Systems
,” Rob. Auton. Syst.
, 21
(3
), pp. 279
–287
.24.
Ploen
, S. R.
, 1997
, “Geometric Algorithms for the Dynamics and Control of Multibody Systems,” Ph.D. thesis
, University of California, Irvine, Irvine, CA.http://gram.eng.uci.edu/~sploen/thesis.ps.gz25.
Spong
, M. W.
, Hutchinson
, S.
, and Vidyasagar
, M.
, 2006
, Robot Modeling and Control
, Vol. 3
, Wiley
, New York
.26.
Park
, F.
, and Kim
, M.
, 2000
, “Lie Theory, Riemannian Geometry, and the Dynamics of Coupled Rigid Bodies
,” Z. Angew. Math. Phys.
, 51
(5
), pp. 820
–834
.27.
Walker
, M. W.
, and Orin
, D. E.
, 1982
, “Efficient Dynamic Computer Simulation of Robotic Mechanisms
,” ASME J. Dyn. Syst. Meas. Control
, 104
(3
), pp. 205
–211
.28.
Featherstone
, R.
, and Orin
, D. E.
, 2016
, Dynamics
, Springer
, Cham, Switzerland
, pp. 37
–66
.29.
Lilly
, K. W.
, and Orin
, D. E.
, 1991
, “Alternate Formulations for the Manipulator Inertia Matrix
,” Int. J. Rob. Res.
, 10
(1
), pp. 64
–74
.30.
Rodriguez
, G.
, Jain
, A.
, and Kreutz-Delgado
, K.
, 1991
, “A Spatial Operator Algebra for Manipulator Modeling and Control
,” Int. J. Rob. Res.
, 10
(4
), pp. 371
–381
.31.
Rodriguez
, G.
, Jain
, A.
, and Kreutz-Delgado
, K.
, 1992
, “Spatial Operator Algebra for Multibody System Dynamics
,” J. Astronaut. Sci.
, 40
(1
), pp. 27
–50
.http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.421.9526&rep=rep1&type=pdf32.
Ploen
, S. R.
, and Park
, F. C.
, 1999
, “Coordinate-Invariant Algorithms for Robot Dynamics
,” IEEE Trans. Rob. Autom.
, 15
(6
), pp. 1130
–1135
.33.
Lathrop
, R. H.
, 1985
, “Parallelism in Manipulator Dynamics
,” Int. J. Rob. Res.
, 4
(2
), pp. 80
–102
.34.
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
.35.
Blelloch
, G. E.
, 1990
, “Prefix Sums and Their Applications,” Carnegie Mellon University
, Pittsburgh, PA
, Technical Report No. CMU-CS-90-190
.https://www.cs.cmu.edu/~blelloch/papers/Ble93.pdf36.
Yang
, Y.
, Wu
, Y.
, and Pan
, J.
, 2017
, “Parallel Dynamics Computation Using Prefix Sum Operations
,” IEEE Rob. Autom. Lett.
, 2
(3
), pp. 1296
–1303
.37.
Moore
, M.
, and Wilhelms
, J.
, 1988
, “Collision Detection and Response for Computer Animation
,” ACM SIGGRAPH Comput. Graphics
, 22
(4
), pp. 289
–298
.38.
Hahn
, J. K.
, 1988
, “Realistic Animation of Rigid Bodies
,” ACM SIGGRAPH Comput. Graphics
, 22
(4
), pp. 299
–308
.39.
Baraff
, D.
, 1989
, “Analytical Methods for Dynamic Simulation of Non-Penetrating Rigid Bodies
,” ACM SIGGRAPH Comput. Graphics
, 23
(3
), pp. 223
–232
.40.
Mirtich
, B. V.
, 1996
, “Impulse-Based Dynamic Simulation of Rigid Body Systems,” Ph.D. thesis
, University of California at Berkeley, Berkeley, CA.http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.108.783&rep=rep1&type=pdf41.
Lee
, S. H.
, Kim
, J.
, Park
, F. C.
, Kim
, M.
, and Bobrow
, J. E.
, 2005
, “Newton-Type Algorithms for Dynamics-Based Robot Movement Optimization
,” IEEE Trans. Rob.
, 21
(4
), pp. 657
–667
.42.
Wittenburg
, J.
, 2013
, Dynamics of Systems of Rigid Bodies
, Vol. 33
, Springer-Verlag
, New York
.43.
Freeman
, R.
, 1985
, “Kinematic and Dynamic Modeling, Analysis and Control of Robotic Mechanisms,” Ph.D. thesis, University of Florida, Gainesville, FL.44.
Luh
, J.
, and Zheng
, Y. F.
, 1985
, “Computation of Input Generalized Forces for Robots With Closed Kinematic Chain Mechanisms
,” IEEE J. Rob. Autom.
, 1
(2
), pp. 95
–103
.45.
Dietmaier
, P.
, 1998
, “The Stewart-Gough Platform of General Geometry Can Have 40 Real Postures
,” Advances in Robot Kinematics: Analysis and Control
, Springer
, New York
, pp. 7
–16
.46.
Husty
, M. L.
, 1996
, “An Algorithm for Solving the Direct Kinematics of General Stewart-Gough Platforms,” Mech. Mach. Theory
, 31
(4), pp. 365–379.47.
Raghavan
, M.
, 1993
, “The Stewart Platform of General Geometry Has 40 Configurations
,” ASME J. Mech. Des.
, 115
(2
), pp. 277
–277
.48.
Faugère
, J. C.
, and Lazard
, D.
, 1995
, “Combinatorial Classes of Parallel Manipulators
,” Mech. Mach. Theory
, 30
(6
), pp. 765
–776
.49.
Lee
, T. Y.
, and Shim
, J. K.
, 2003
, “Improved Dialytic Elimination Algorithm for the Forward Kinematics of the General Stewart–Gough Platform
,” Mech. Mach. Theory
, 38
(6
), pp. 563
–577
.50.
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
.51.
Do
, W.
, and Yang
, D.
, 1988
, “Inverse Dynamic Analysis and Simulation of a Platform Type of Robot
,” J. Field Rob.
, 5
(3
), pp. 209
–227
.http://onlinelibrary.wiley.com/doi/10.1002/rob.4620050304/pdf52.
Lee
, K. M.
, and Shah
, D. K.
, 1988
, “Dynamic Analysis of a Three-Degrees-of-Freedom In-Parallel Actuated Manipulator
,” IEEE J. Rob. Autom.
, 4
(3
), pp. 361
–367
.53.
Ma
, O.
, 1991
, “Mechanical Analysis of Parallel Manipulators With Simulation, Design, and Control Applications,” Ph.D. thesis
, McGill University, Montreal, QC, Canada.http://digitool.library.mcgill.ca/R/?func=dbin-jump-full&object_id=70192&local_base=GEN01-MCG0254.
Zanganeh
, K. E.
, Sinatra
, R.
, and Angeles
, J.
, 1997
, “Kinematics and Dynamics of a Six-Degree-of-Freedom Parallel Manipulator With Revolute Legs
,” Robotica
, 15
(4
), pp. 385
–394
.55.
Negrut
, D.
, Haug
, E.
, and Iancu
, M.
, 1997
, “Variable Step Implicit Numerical Integration of Stiff Multibody Systems
,” NATO Advanced Study Institute on Computational Methods in Mechanisms
, Varna, Bulgarie, June 16–28, pp. 157
–166
.https://pdfs.semanticscholar.org/e016/35ec401970a6deba8f869e7a88fa13d13ed0.pdf56.
Haug
, E.
, and Yen
, J.
, 1992
, “Implicit Numerical Integration of Constrained Equations of Motion Via Generalized Coordinate Partitioning
,” ASME J. Mech. Des.
, 114
(2
), pp. 296
–304
.57.
Lilly
, K.
, and Orin
, D.
, 1994
, “Efficient Dynamic Simulation of Multiple Chain Robotic Mechanisms
,” ASME J. Dyn. Syst. Meas. Control
, 116
(2
), pp. 223
–223
.58.
Wehage
, R.
, and Haug
, E.
, 1982
, “Generalized Coordinate Partitioning for Dimension Reduction in Analysis of Constrained Dynamic Systems
,” ASME J. Mech. Des.
, 104
(1
), pp. 247
–255
.59.
Roberson
, R. E.
, and Schwertassek
, R.
, 2012
, Dynamics of Multibody Systems
, Springer Science & Business Media
, New York
.60.
Baumgarte
, J.
, 1972
, “Stabilization of Constraints and Integrals of Motion in Dynamical Systems
,” Comput. Methods Appl. Mech. Eng.
, 1
(1
), pp. 1
–16
.61.
Nakamura
, Y.
, and Ghodoussi
, M.
, 1989
, “Dynamics Computation of Closed-Link Robot Mechanisms With Nonredundant and Redundant Actuators
,” IEEE Trans. Rob. Autom.
, 5
(3
), pp. 294
–302
.62.
Park
, F.
, and Kim
, J. W.
, 1999
, “Singularity Analysis of Closed Kinematic Chains
,” ASME J. Mech. Des.
, 121
(1
), pp. 32
–38
.63.
Park
, J.
, Haan
, J.
, and Park
, F. C.
, 2007
, “Convex Optimization Algorithms for Active Balancing of Humanoid Robots
,” IEEE Trans. Rob.
, 23
(4
), pp. 817
–822
.64.
Collette
, C.
, Micaelli
, A.
, Andriot
, C.
, and Lemerle
, P.
, 2008
, “Robust Balance Optimization Control of Humanoid Robots With Multiple Non Coplanar Grasps and Frictional Contacts
,” IEEE International Conference on Robotics and Automation (ICRA
), Pasadena, CA, May 19–23, pp. 3187
–3193
.65.
Wensing
, P. M.
, and Orin
, D. E.
, 2013
, “Generation of Dynamic Humanoid Behaviors Through Task-Space Control With Conic Optimization
,” IEEE International Conference on Robotics and Automation (ICRA
), Karlsruhe, Germany, May 6–10, pp. 3103
–3109
.66.
Caron
, S.
, Pham
, Q. C.
, and Nakamura
, Y.
, 2015
, “Stability of Surface Contacts for Humanoid Robots: Closed-Form Formulae of the Contact Wrench Cone for Rectangular Support Areas
,” IEEE International Conference on Robotics and Automation (ICRA
), Seattle, WA, May 26–30, pp. 5107
–5112
.67.
Gilardi
, G.
, and Sharf
, I.
, 2002
, “Literature Survey of Contact Dynamics Modelling
,” Mech. Mach. Theory
, 37
(10
), pp. 1213
–1239
.68.
Anitescu
, M.
, and Potra
, F. A.
, 1997
, “Formulating Dynamic Multi-Rigid-Body Contact Problems With Friction as Solvable Linear Complementarity Problems
,” Nonlinear Dyn.
, 14
(3
), pp. 231
–247
.69.
Trinkle
, J. C.
, Pang
, J. S.
, Sudarsky
, S.
, and Lo
, G.
, 1997
, “On Dynamic Multi-Rigid-Body Contact Problems With Coulomb Friction
,” Z. Angew. Math. Mech.
, 77
(4
), pp. 267
–279
.70.
Montana
, D. J.
, 1988
, “The Kinematics of Contact and Grasp
,” Int. J. Rob. Res.
, 7
(3
), pp. 17
–32
.71.
Lemke
, C. E.
, 1965
, “Bimatrix Equilibrium Points and Mathematical Programming
,” Manage. Sci.
, 11
(7
), pp. 681
–689
.72.
Cottle
, R. W.
, and Dantzig
, G. B.
, 1968
, “Complementary Pivot Theory of Mathematical Programming
,” Linear Algebra Appl.
, 1
(1
), pp. 103
–125
.73.
Murty
, K. G.
, and Yu
, F. T.
, 1988
, Linear Complementarity, Linear and Nonlinear Programming
, Vol. 3, Heldermann Verlag, Berlin.74.
Cottle
, R. W.
, Pang
, J. S.
, and Stone
, R. E.
, 2009
, The Linear Complementarity Problem
, SIAM
, Philadelphia, PA.75.
Dwivedy
, S. K.
, and Eberhard
, P.
, 2006
, “Dynamic Analysis of Flexible Manipulators: A Literature Review
,” Mech. Mach. Theory
, 41
(7
), pp. 749
–777
.76.
Gasparetto
, A.
, 2004
, “On the Modeling of Flexible-Link Planar Mechanisms: Experimental Validation of an Accurate Dynamic Model
,” ASME J. Dyn. Syst. Meas. Control
, 126
(2
), pp. 365
–375
.77.
Yoshikawa
, T.
, Ohta
, A.
, and Kanaoka
, K.
, 2001
, “State Estimation and Parameter Identification of Flexible Manipulators Based on Visual Sensor and Virtual Joint Model
,” IEEE International Conference on Robotics and Automation (ICRA
), Seoul, South Korea, May 21–26, pp. 2840
–2845.
78.
Kalra
, P.
, and Sharan
, A. M.
, 1991
, “Accurate Modelling of Flexible Manipulators Using Finite Element Analysis
,” Mech. Mach. Theory
, 26
(3
), pp. 299
–313
.79.
Dubowsky
, S.
, Gu
, P. Y.
, and Deck
, J. F.
, 1991
, “The Dynamic Analysis of Flexibility in Mobile Robotic Manipulator Systems
,” VIII World Congress on the Theory of Machines and Mechanics
, Prague, Czechoslavakia, Aug. 26–31, pp. 26
–31
.http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.33.9388&rep=rep1&type=pdf80.
Naganathan
, G.
, and Soni
, A. H.
, 1988
, “Nonlinear Modeling of Kinematic and Flexibility Effects in Manipulator Design
,” J. Mech. Trans. Autom.
, 110
(3
), pp. 243
–254
.81.
Theodore
, R. J.
, and Ghosal
, A.
, 1995
, “Comparison of the Assumed Modes and Finite Element Models for Flexible Multilink Manipulators
,” Int. J. Rob. Res.
, 14
(2
), pp. 91
–111
.82.
Book
, W. J.
, 1984
, “Recursive Lagrangian Dynamics of Flexible Manipulator Arms
,” Int. J. Rob. Res.
, 3
(3
), pp. 87
–101
.83.
Siciliano
, B.
, and Book
, W. J.
, 1988
, “A Singular Perturbation Approach to Control of Lightweight Flexible Manipulators
,” Int. J. Rob. Res.
, 7
(4
), pp. 79
–90
.84.
Martins
, J.
, Botto
, M. A.
, and da Costa
, J. S.
, 2002
, “Modeling of Flexible Beams for Robotic Manipulators
,” Multibody Syst. Dyn.
, 7
(1
), pp. 79
–100
.85.
Asada
, H.
, Ma
, Z. D.
, and Tokumaru
, H.
, 1990
, “Inverse Dynamics of Flexible Robot Arms: Modeling and Computation for Trajectory Control
,” ASME J. Dyn. Syst. Meas. Control
, 112
(2
), pp. 177
–185
.86.
Boyer
, F.
, and Coiffet
, P.
, 1996
, “Generalization of Newton-Euler Model for Flexible Manipulators
,” J. Rob. Syst.
, 13
(1
), pp. 11
–24
.87.
Spong
, M. W.
, 1987
, “Modeling and Control of Elastic Joint Robots
,” ASME J. Dyn. Syst. Meas. Control
, 109
(4
), pp. 310
–319
.88.
Buondonno
, G.
, and De Luca
, A.
, 2015
, “A Recursive Newton-Euler Algorithm for Robots With Elastic Joints and Its Application to Control
,” IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS
), Hamburg, Germany, Sept. 28–Oct. 2, pp. 5526
–5532
.89.
Betts
, J. T.
, 1998
, “Survey of Numerical Methods for Trajectory Optimization
,” J. Guid. Control Dyn.
, 21
(2
), pp. 193
–207
.90.
Pontryagin
, L. S.
, 1987
, Mathematical Theory of Optimal Processes
, CRC Press
, Boca Raton, FL
.91.
Bryson
, A. E.
, 1975
, Applied Optimal Control: Optimization, Estimation and Control
, CRC Press
, Boca Raton, FL
.92.
De Boor
, C.
, 1978
, A Practical Guide to Splines
, Vol. 27
, Springer-Verlag
, New York
.93.
Bobrow
, J. E.
, Martin
, B.
, Sohl
, G.
, Wang
, E. C.
, Park
, F. C.
, and Kim
, J.
, 2001
, “Optimal Robot Motions for Physical Criteria
,” J. Rob. Syst.
, 18
(12
), pp. 785
–795
.94.
Martin
, B. J.
, and Bobrow
, J. E.
, 1999
, “Minimum-Effort Motions for Open-Chain Manipulators With Task-Dependent End-Effector Constraints
,” Int. J. Rob. Res.
, 18
(2
), pp. 213
–224
.95.
Ata
, A. A.
, 2007
, “Optimal Trajectory Planning of Manipulators: A Review
,” J. Eng. Sci. Technol.
, 2
(1
), pp. 32
–54
.https://www.researchgate.net/publication/49596198_OPTIMAL_TRAJECTORY_PLANNING_OF_MANIPULATORS_A_REVIEW96.
Gill
, P. E.
, and Murray
, W.
, 1972
, “Quasi-Newton Methods for Unconstrained Optimization
,” IMA J. Appl. Math.
, 9
(1
), pp. 91
–108
.97.
Fletcher
, R.
, 2013
, Practical Methods of Optimization
, Wiley
, Hoboken, NJ
.98.
Sideris
, A.
, and Bobrow
, J. E.
, 2005
, “An Efficient Sequential Linear Quadratic Algorithm for Solving Nonlinear Optimal Control Problems
,” IEEE Trans. Autom. Control
, 50
(12
), pp. 2043
–2047
.99.
Bobrow
, J. E.
, Park
, F. C.
, and Sideris
, A.
, 2006
, “Progress on the Algorithmic Optimization of Robot Motion
,” Fast Motions in Biomechanics and Robotics: Optimization and Feedback Control
(Lecture Notes in Control and Information Sciences, Vol. 340), Springer, Berlin.100.
Li
, W.
, and Todorov
, E.
, 2004
, “Iterative Linear Quadratic Regulator Design for Nonlinear Biological Movement Systems,” 1st International Conference on Informatics in Control, Automation and Robotics (ICINCO
), Setúbal, Portugal, Aug. 25–28, pp. 222
–229
.https://homes.cs.washington.edu/~todorov/papers/LiICINCO04.pdf101.
Wang
, C. Y.
, Timoszyk
, W. K.
, and Bobrow
, J. E.
, 2001
, “Payload Maximization for Open Chained Manipulators: Finding Weightlifting Motions for a Puma 762 Robot
,” IEEE Trans. Rob. Autom.
, 17
(2
), pp. 218
–224
.102.
Albro
, J. V.
, Sohl
, G. A.
, Bobrow
, J. E.
, and Park
, F. C.
, 2000
, “On the Computation of Optimal High-Dives
,” IEEE International Conference on Robotics and Automation (ICRA
), San Francisco, CA, Apr. 24–28, pp. 3958
–3963
.103.
Sohl
, G. A.
, and Bobrow
, J. E.
, 1999
, “Optimal Motions for Underactuated Manipulators,” ASME Design Engineering Technical Conferences, Las Vegas, NV, Sept. 12–15, pp. 519–528.104.
Wang
, C.-Y. E.
, Bobrow
, J. E.
, and Reinkensmeyer
, D. J.
, 2001
, “Swinging From the Hip: Use of Dynamic Motion Optimization in the Design of Robotic Gait Rehabilitation
,” IEEE International Conference on Robotics and Automation (ICRA
), Seoul, South Korea, May 21–26, pp. 1433
–1438
.105.
Kuindersma
, S.
, Deits
, R.
, Fallon
, M.
, Valenzuela
, A.
, Dai
, H.
, Permenter
, F.
, Koolen
, T.
, Marion
, P.
, and Tedrake
, R.
, 2016
, “Optimization-Based Locomotion Planning, Estimation, and Control Design for the Atlas Humanoid Robot
,” Auton. Rob.
, 40
(3
), pp. 429
–455
.106.
Lo
, J.
, Huang
, G.
, and Metaxas
, D.
, 2002
, “Human Motion Planning Based on Recursive Dynamics and Optimal Control Techniques
,” Multibody Sys. Dyn.
, 8
(4
), pp. 433
–458
.107.
Chesse
, S.
, and Bessonnet
, G.
, 2001
, “Optimal Dynamics of Constrained Multibody Systems. Application to Bipedal Walking Synthesis
,” IEEE International Conference on Robotics and Automation (ICRA
), Seoul, South Korea, May 21–26, pp. 2499
–2505
.108.
Goswami
, A.
, Espiau
, B.
, and Keramane
, A.
, 1997
, “Limit Cycles in a Passive Compass Gait Biped and Passivity-Mimicking Control Laws
,” Auton. Rob.
, 4
(3
), pp. 273
–286
.109.
Berkemeier
, M. D.
, 1998
, “Modeling the Dynamics of Quadrupedal Running
,” Int. J. Rob. Res.
, 17
(9
), pp. 971
–985
.110.
Fukuda
, T.
, and Saito
, F.
, 1996
, “Motion Control of a Brachiation Robot
,” Rob. Auton. Syst.
, 18
(1–2
), pp. 83
–93
.111.
Spong
, M. W.
, 1995
, “The Swing Up Control Problem for the Acrobot
,” IEEE Control Syst.
, 15
(1
), pp. 49
–55
.112.
Koon
, W.
, and Marsden
, J.
, 1997
, “Optimal Control for Holonomic and Nonholonomic Mechanical Systems With Symmetry and Lagrangian Reduction
,” SIAM J. Control Optim.
, 35
(3
), pp. 901
–929
.113.
Cortes
, J.
, and Martinez
, S.
, 2000
, “Optimal Control for Nonholonomic Systems With Symmetry
,” 39th IEEE Conference on Decision and Control (CDC
), Sydney, Australia, Dec. 12–15, pp. 5216
–5218
.114.
Hussein
, I. I.
, and Bloch
, A. M.
, 2008
, “Optimal Control of Underactuated Nonholonomic Mechanical Systems
,” IEEE Trans. Autom. Control
, 53
(3
), pp. 668
–682
.115.
Mombaur
, K.
, Laumond
, J. P.
, and Yoshida
, E.
, 2008
, “An Optimal Control Model Unifying Holonomic and Nonholonomic Walking
,” Eighth IEEE-RAS International Conference on Humanoid Robots (ICHR
), Daejeon, South Korea, Dec. 1–3, pp. 646
–653
.116.
Ostrowski
, J. P.
, Desai
, J. P.
, and Kumar
, V.
, 2000
, “Optimal Gait Selection for Nonholonomic Locomotion Systems
,” Int. J. Rob. Res.
, 19
(3
), pp. 225
–237
.117.
Reuter
, J.
, 1998
, “Mobile Robots Trajectories With Continuously Differentiable Curvature: An Optimal Control Approach
,” IEEE/RSJ International Conference on Intelligence Robots and Systems (IROS
), Victoria, BC, Canada, Oct. 13–17, pp. 38
–43.
118.
Duleba
, I.
, and Sasiadek
, J. Z.
, 2003
, “Nonholonomic Motion Planning Based on Newton Algorithm With Energy Optimization
,” IEEE Trans. Control Syst. Technol.
, 11
(3
), pp. 355
–363
.119.
Korayem
, M. H.
, Nikoobin
, A.
, and Azimirad
, V.
, 2009
, “Maximum Load Carrying Capacity of Mobile Manipulators: Optimal Control Approach
,” Robotica
, 27
(1
), pp. 147
–159
.120.
Korayem
, M. H.
, Nazemizadeh
, M.
, and Rahimi
, H. N.
, 2013
, “Trajectory Optimization of Nonholonomic Mobile Manipulators Departing to a Moving Target Amidst Moving Obstacles
,” Acta Mech.
, 224
(5
), pp. 995
–1008
.121.
Ge
, X. S.
, and Zhang
, Q. Z.
, 2006
, “Optimal Control of Nonholonomic Motion Planning for a Free-Falling Cat
,” First International Conference on Innovative Computing, Information, and Control (ICICIC'06
), Beijing, China, Aug. 30–Sept. 1, pp. 599
–602
.122.
Crawford
, L. S.
, and Sastry
, S. S.
, 1995
, “Biological Motor Control Approaches for a Planar Diver
,” 34th IEEE Conference on Decision and Control (CDC
), New Orleans, LA, Dec. 13–15, pp. 3881
–3886
.123.
Braun
, D. J.
, Howard
, M.
, and Vijayakumar
, S.
, 2011
, “Exploiting Variable Stiffness in Explosive Movement Tasks
,” Robotics: Science and Systems VII
, Los Angeles, CA, June 27–June 30, pp. 25–32.http://www.roboticsproceedings.org/rss07/p04.pdf124.
Braun
, D.
, Howard
, M.
, and Vijayakumar
, S.
, 2012
, “Optimal Variable Stiffness Control: Formulation and Application to Explosive Movement Tasks
,” Auton. Rob.
, 33
(3
), pp. 237
–253
.125.
Braun
, D. J.
, Petit
, F.
, Huber
, F.
, Haddadin
, S.
, Van Der Smagt
, P.
, Albu-Schäffer
, A.
, and Vijayakumar
, S.
, 2013
, “Robots Driven by Compliant Actuators: Optimal Control Under Actuation Constraints
,” IEEE Trans. Rob.
, 29
(5
), pp. 1085
–1101
.126.
Garabini
, M.
, Passaglia
, A.
, Belo
, F.
, Salaris
, P.
, and Bicchi
, A.
, 2011
, “Optimality Principles in Variable Stiffness Control: The VSA Hammer
,” IEEE/RSJ International Conference on Intelligence Robots and System (IROS
), San Francisco, CA, Sept. 25–30, pp. 3770
–3775
.127.
Haddadin
, S.
, Weis
, M.
, Wolf
, S.
, and Albu-Schäffer
, A.
, 2011
, “Optimal Control for Maximizing Link Velocity of Robotic Variable Stiffness Joints
,” IFAC Proc. Vol.
, 44
(1
), pp. 6863
–6871
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