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.

References

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.pdf
10.
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.gz
25.
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=pdf
32.
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.pdf
36.
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=pdf
41.
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/pdf
52.
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-MCG02
54.
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.pdf
56.
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=pdf
80.
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_REVIEW
96.
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.pdf
101.
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.pdf
124.
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
.
You do not currently have access to this content.