Abstract
This paper studies the direct geometrico-static problem (DGP) of underconstrained cable-driven parallel robots (CDPRs) with three cables. The task consists in determining the end-effector pose and the cable tensile forces when the cable lengths are assigned. The problem is challenging, because kinematics and statics are coupled, and they must be tackled simultaneously. An effective elimination procedure is proposed and a least-degree univariate polynomial free of spurious factors is obtained in the ideal governing the problem. This is proven to admit 156 solutions in the complex field. Several approaches for the efficient computation of the complete solution set are presented, including an eigenproblem formulation and homotopy continuation.
Issue Section:
Research Papers
References
1.
Roberts
, R. G.
, Graham
, T.
, and Lippitt
, T.
, 1998
, “On the Inverse Kinematics, Statics, and Fault Tolerance of Cable-Suspended Robots
,” J. Rob. Syst.
, 15
(10
), pp. 581
–597
.10.1002/(SICI)1097-4563(199810)15:10<581::AID-ROB4>3.0.CO;2-P2.
Kawamura
, S.
, Kino
, H.
, and Won
, C.
, 2000
, “High-Speed Manipulation by Using Parallel Wire-Driven Robots
,” Robotica
, 18
(1
), pp. 13
–21
.10.1017/S02635747990024773.
Tadokoro
, S.
, Murao
, Y.
, Hiller
, M.
, Murata
, R.
, Kohkawa
, H.
, and Matsushima
, T.
, 2002
, “A Motion Base With 6-DOF by Parallel Cable Drive Architecture
,” IEEE/ASME Trans. Mechatron.
, 7
(2
), pp. 115
–123
.10.1109/TMECH.2002.10112484.
Hiller
, M.
, Fang
, S.
, Mielczarek
, S.
, Verhoeven
, R.
, and Franitza
, D.
, 2005
, “Design, Analysis And Realization of Tendon-Based Parallel Manipulators
,” Mech. Mach. Theory
, 40
(4
), pp. 429
–445
.10.1016/j.mechmachtheory.2004.08.0025.
Alikhani
, A.
, Behzadipour
, S.
, Vanini
, S. A. S.
, and Alasty
, A.
, 2009
, “Workspace Analysis of a Three DOF Cable-Driven Mechanism
,” ASME J. Mech. Rob.
, 1
(4
), p. 041005
.10.1115/1.32042556.
Landsberger
, S. E.
, 1984
, “Design and Construction of a Cable-Controlled, Parallel Link Manipulator
,” Master's thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, Cambridge.7.
Behzadipour
, S.
, and Khajepour
, A.
, 2005
, “A New Cable-Based Parallel Robot With Three Degrees of Freedom
,” Multibody Syst. Dyn.
, 13
(4
), pp. 371
–383
.10.1007/s11044-005-3985-68.
Albus
, J.
, Bostelman
, R.
, and Dagalakis
, N.
, 1993
, “The NIST Robocrane
,” J. Rob. Syst.
, 10
(5
), pp. 709
–724
.10.1002/rob.46201005099.
Su
, Y. X.
, Duan
, B. Y.
, Nan
, R. D.
, and Peng
, B.
, 2001
, “Development of a Large Parallel-Cable Manipulator for the Feed-Supporting System of a Next-Generation Large Radio Telescope
,” J. Rob. Syst.
, 18
(11
), pp. 633
–643
.10.1002/rob.810210.
Bosscher
, P.
, Riechel
, A. T.
, and Ebert-Uphoff
, I.
, 2006
, “Wrench-Feasible Workspace Generation for Cable-Driven Robots
,” IEEE Trans. Rob.
, 22
(5
), pp. 890
–902
.10.1109/TRO.2006.87896711.
Stump
, E.
, and Kumar
, V.
, 2006
, “Workspaces of Cable-Actuated Parallel Manipulators
,” ASME J. Mech. Des.
, 128
(1
), pp. 159
–167
.10.1115/1.212174112.
Ghasemi
, A.
, Eghtesad
, M.
, and Farid
, M.
, 2009
, “Workspace Analysis for Planar and Spatial Redundant Cable Robots
,” ASME J. Mech. Rob.
, 1
(4
), p. 044502
.10.1115/1.321102613.
Diao
, X.
, and Ma
, O.
, 2009
, “Force-Closure Analysis of 6-DOF Cable Manipulators With Seven or More Cables
,” Robotica
, 27
(2
), pp. 209
–215
.10.1017/S026357470800459114.
Bouchard
, S.
, Gosselin
, C.
, and Moore
, B.
, 2010
, “On the Ability of a Cable-Driven Robot to Generate a Prescribed Set of Wrenches
,” ASME J. Mech. Rob.
, 2
(1
), p. 011010
.10.1115/1.400055815.
Gouttefarde
, M.
, Daney
, D.
, and Merlet
, J.-P.
, 2011
, “Interval-Analysis-Based Determination of the Wrench-Feasible Workspace of Parallel Cable-Driven Robots
,” IEEE Trans. Rob.
, 27
(1
), pp. 1
–13
.10.1109/TRO.2010.209006416.
Lau
, D.
, Oetomo
, D.
, and Halgamuge
, S.
, 2011
, “Wrench-Closure Workspace Generation for Cable Driven Parallel Manipulators Using a Hybrid Analytical-Numerical Approach
,” ASME J. Mech. Des.
, 133
(7
), p. 071004
.10.1115/1.400422217.
Azizian
, K.
, and Cardou
, P.
, 2012
, “The Dimensional Synthesis of Planar Parallel Cable-Driven Mechanisms Through Convex Relaxations
,” ASME J. Mech. Rob.
, 4
(3
), p. 031011
.10.1115/1.400695218.
Behzadipour
, S.
, and Khajepour
, A.
, 2006
, “Stiffness of Cable-Based Parallel Manipulators With Application to Stability Analysis
,” ASME J. Mech. Des.
, 128
(1
), pp. 303
–310
.10.1115/1.211489019.
Surdilovic
, D.
, Radojicic
, J.
, and Krüger
, J.
, 2013
, “Geometric Stiffness Analysis of Wire Robots: A Mechanical Approach
,” Cable-Driven Parallel Robots
, T.
Bruckmann
and A.
Pott
, eds., Springer-Verlag
, Berlin Heidelberg
, pp. 389
–404
.20.
Merlet
, J.-P.
, 2004
, “Analysis of the Influence of Wires Interference on the Workspace of Wire Robots
,” On Advances in Robot Kinematics
, J.
Lenarčič
and C.
Galletti
, eds., Kluwer Academic Publishers
, Dordrecht
, pp. 211
–218
.21.
Perreault
, S.
, Cardou
, P.
, Gosselin
, C. M.
, and Otis
, M. J.-D.
, 2010
, “Geometric Determination of the Interference-Free Constant-Orientation Workspace of Parallel Cable-Driven Mechanisms
,” ASME J. Mech. Rob.
, 2
(3
), p. 031016
.10.1115/1.400178022.
Pusey
, J.
, Fattah
, A.
, Agrawal
, S.
, and Messina
, E.
, 2004
, “Design and Workspace Analysis of a 6-6 Cable-Suspended Parallel Robot
,” Mech. Mach. Theory
, 39
(7
), pp. 761
–778
.10.1016/j.mechmachtheory.2004.02.01023.
Rosati
, G.
, Zanotto
, D.
, and Agrawal
, S. K.
, 2011
, “On the Design of Adaptive Cable-Driven Systems
,” ASME J. Mech. Rob.
, 3
(2
), p. 021004
.10.1115/1.400358024.
Yamamoto
, M.
, Yanai
, N.
, and Mohri
, A.
, 2004
, “Trajectory Control of Incompletely Restrained Parallel-Wire-Suspended Mechanism Based on Inverse Dynamics
,” IEEE Trans. Rob.
, 20
(5
), pp. 840
–850
.10.1109/TRO.2004.82950125.
Fattah
, A.
, and Agrawal
, S. K.
, 2006
, “On the Design of Cable-Suspended Planar Parallel Robots
,” ASME J. Mech. Des.
, 127
(5
), pp. 1021
–1028
.10.1115/1.190300126.
Heyden
, T.
, and Woernle
, C.
, 2006
, “Dynamics and Flatness-Based Control of a Kinematically Undetermined Cable Suspension Manipulator
,” Multibody Syst. Dyn.
, 16
(2
), pp. 155
–177
.10.1007/s11044-006-9023-527.
Michael
, N.
, Kim
, S.
, Fink
, J.
, and Kumar
, V.
, 2009
, “Kinematics and Statics of Cooperative Multi-Robot Aerial Manipulation With Cables
,” ASME 2009 International Design Engineering Technical Conferences, San Diego, Vol. 7, pp. 83
–91
, Paper No. DETC2009–87677.
28.
Jiang
, Q.
, and Kumar
, V.
, 2010
, “The Inverse Kinematics of 3-D Towing
,” Advances in Robot Kinematics: Motion in Man and Machine
, J.
Lenarčič
and M. M.
Stanišic`
, eds., Springer
, Dordrecht
, pp. 321
–328
.29.
Jiang
, Q.
, and Kumar
, V.
, 2010
, “The Direct Kinematics of Objects Suspended From Cables
,” ASME 2010 International Design Engineering Technical Conferences, Montreal, Canada, Vol. 2, pp. 193
–
202
, Paper No. DETC2010–28036.
30.
Collard
, J.-F.
, and Cardou
, P.
, 2013
, “Computing the Lowest Equilibrium Pose of a Cable-Suspended Rigid Body
,” Optim. Eng.
, (in press).10.1007/s11081-012-9191-531.
Morizono
, T.
, Kurahashi
, K.
, and Kawamura
, S.
, 1998
, “Analysis and Control of a Force Display System Driven by Parallel Wire Mechanism
,” Robotica
, 16
(5
), pp. 551
–563
.10.1017/S026357479800055132.
Surdilovic
, D.
, Zhang
, J.
, and Bernhardt
, R.
, 2007
, “STRING-MAN: Wire-Robot Technology for Safe, Flexible and Human-Friendly Gait Rehabilitation
,” Proceedings of the 2007 IEEE International Conference on Rehabilitation Robotics, Noordwijk, The Netherlands, pp. 446
–453
.33.
Rosati
, G.
, Gallina
, P.
, and Masiero
, S.
, 2007
, “Design, Implementation and Clinical Tests of a Wire-Based Robot for Neurorehabilitation
,” IEEE Trans. Neural Syst. Rehabil. Eng.
, 15
(4
), pp. 560
–569
.10.1109/TNSRE.2007.90856034.
Merlet
, J.-P.
, and Daney
, D.
, 2010
, “A Portable, Modular Parallel Wire Crane for Rescue Operations
,” Proceedings of the 2010 IEEE International Conference on Robotics and Automation, Anchorage, pp. 2834
–2839
.35.
Gobbi
, M.
, Mastinu
, G.
, and Previati
, G.
, 2011
, “A Method for Measuring the Inertia Properties of Rigid Bodies
,” Mech. Syst. Signal Process.
, 25
(1
), pp. 305
–318
.10.1016/j.ymssp.2010.09.00436.
Carricato
, M.
, and Merlet
, J.-P.
, 2010
, “Geometrico-Static Analysis of Under-Constrained Cable-Driven Parallel Robots
,” Advances in Robot Kinematics: Motion in Man and Machine
, J.
Lenarčič
and M. M.
Stanišic`
, eds., Springer
, Dordrecht
, pp. 309
–319
.37.
Carricato
, M.
, and Merlet
, J.-P.
, 2013
, “Stability Analysis of Underconstrained Cable-Driven Parallel Robots
,” IEEE Trans. Rob.
, 29
(1
), pp. 288
–296
.10.1109/TRO.2012.221779538.
Merlet
, J.-P.
, 2013
, “Wire-Driven Parallel Robot: Open Issues
,” Romansy 19 − Robot Design, Dynamics and Control
, V.
Padois
, P.
Bidaud
, and O.
Khatib
, eds., Springer
, Vienna
, pp. 3
–10
.39.
McCarthy
, J. M.
, 2011
, “21st Century Kinematics: Synthesis, Compliance, and Tensegrity
,” ASME J. Mech. Rob.
, 3
(2
), p. 020201
.10.1115/1.400318140.
Carricato
, M.
, 2013
, “Inverse Geometrico-Static Problem of Under-Constrained Cable-Driven Parallel Robots With Three Cables
,” ASME J. Mech. Rob.
, 5
(4
), p. 041002
.10.1115/1.402429141.
Carricato
, M.
, Abbasnejad
, G.
, and Walter
, D.
, 2012
, “Inverse Geometrico-Static Analysis of Under-Constrained Cable-Driven Parallel Robots With Four Cables
,” Latest Advances in Robot Kinematics
, J.
Lenarčič
and M.
Husty
, eds., Springer
, Dordrecht
, pp. 365
–372
.42.
Carricato
, M.
, and Abbasnejad
, G.
, 2013
, “Direct Geometrico-Static Analysis of Under-Constrained Cable-Driven Parallel Robots With 4 Cables
,” Cable-Driven Parallel Robots
, T.
Bruckmann
and A.
Pott
, eds., Springer-Verlag
, Berlin Heidelberg
, pp. 269
–285
.43.
Abbasnejad
, G.
, and Carricato
, M.
, 2013
, “Direct Geometrico-Static Problem of Underconstrained Cable-Driven Parallel Robots With 5 Cables
,” Proceedings of the 6th International Workshop on Computational Kinematics, Barcelona, Spain.44.
Raghavan
, M.
, and Roth
, B.
, 1995
, “Solving Polynomial Systems for the Kinematic Analysis and Synthesis of Mechanisms and Robot Manipulators
,” ASME J. Mech. Des.
, 117
(2B
), pp. 71
–79
.10.1115/1.283647345.
Carricato
, M.
, and Merlet
, J.-P.
, 2011
, “Direct Geometrico-Static Problem of Under-Constrained Cable-Driven Parallel Robots With Three Cables
,” 2011 IEEE International Conference on Robotics and Automation, Shanghai, China, pp. 3011
–3017
.46.
Bottema
, O.
, and Roth
, B.
, 1990
, Theoretical Kinematics, Dover Publications
, New York
.47.
Morgan
, A.
, and Sommese
, A.
, 1987
, “A Homotopy for Solving General Polynomial Systems That Respects m-Homogeneous Structures
,” Appl. Math. Comput.
, 24
(2
), pp. 101
–113
.10.1016/0096-3003(87)90063-448.
Morgan
, A.
, and Sommese
, A.
, 1987
, “Computing all Solutions to Polynomial Systems Using Homotopy Continuation
,” Appl. Math. Comput.
, 24
(2
), pp. 115
–138
.10.1016/0096-3003(87)90064-649.
Wampler
, C. W.
, 1992
, “Bezout Number Calculations for Multi-Homogeneous Polynomial Systems
,” Appl. Math. Comput.
, 51
(2–3
), pp. 143
–157
.10.1016/0096-3003(92)90070-H50.
Merlet
, J.-P.
, 2006
. Parallel Robots
, Springer
, Dordrecht
.51.
Dhingra
, A. K.
, Almadi
, A. N.
, and Kohli
, D.
, 2000
, “A Gröbner-Sylvester Hybrid Method for Closed-Form Displacement Analysis of Mechanisms
,” ASME J. Mech. Des.
, 122
(4
), pp. 431
–438
.10.1115/1.129039552.
Sommese
, A. J.
, and Wampler
, C. W.
, 2005
, The Numerical Solution of Systems of Polynomials Arising in Engineering and Science
, World Scientific Publishing
, Singapore
.53.
Cox
, D.
, Little
, J.
, and O'Shea
, D.
, 2007
, Ideals, Varieties, and Algorithms
, Springer
, New York
.54.
Faugère
, J. C.
, Gianni
, P.
, Lazard
, D.
, and Mora
, T.
, 1993
, “Efficient Computation of Zero-Ddimensional Gröbner Bases by Change of Ordering
,” J. Symb. Comput.
, 16
(4
), pp. 329
–344
.10.1006/jsco.1993.105155.
Möller
, H. M.
, 1998
, “Gröbner Bases and Numerical Analysis
,” Gröbner Bases and Applications
, Vol. 251 of London Mathematical Society Lecture Note Series, B.
Buchberger
and F.
Winkler
, eds., Cambridge University Press
, Cambridge
, pp. 159
–178
.56.
Corless
, R. M.
, 1996
, “Gröbner Bases and Matrix Eigenproblems
,” ACM SIGSAM Bull.
, 30
(4
), pp. 26
–32
.10.1145/242961.24296857.
Manocha
, D.
, and Krishnan
, S.
, 1996
, “Solving Algebraic Systems Using Matrix Computations
,” ACM SIGSAM Bull.
, 30
(4
), pp. 4
–21
.10.1145/242961.24296558.
Higham
, N.
, Mackey
, D.
, and Tisseur
, F.
, 2006
, “The Conditioning of Linearizations of Matrix Polynomials
,” SIAM J. Matrix Anal. Appl.
, 28
(4
), pp. 1005
–1028
.10.1137/05062828359.
Higham
, N.
, Li
, R.-C.
, and Tisseur
, F.
, 2007
, “Backward Error of Polynomial Eigenproblems Solved by Linearization
,” SIAM J. Matrix Anal. Appl.
, 29
(4
), pp. 1218
–1241
.10.1137/06066373860.
Bates
, D. J.
, Hauenstein
, J. D.
, Sommese
, A. J.
, and Wampler
, C. W.
, “bertini: Software for Numerical Algebraic Geometry
,” Available at: http://bertini.nd.edu61.
Morgan
, A. P.
, and Sommese
, A. J.
, 1989
, “Coefficient-Parameter Polynomial Continuation
,” Appl. Math. Comput.
, 29
(2
), pp. 123
–160
.10.1016/0096-3003(89)90099-462.
Abbasnejad
, G.
, and Carricato
, M.
, 2012
, “Real Solutions of the Direct Geometrico-Static Problem of Under-Constrained Cable-Driven Parallel Robots With 3 Cables: A Numerical Investigation
,” Meccanica
, 47
(7
), pp. 1761
–1773
.10.1007/s11012-012-9552-363.
Merlet
, J.-P.
, 2009
, “Interval Analysis for Certified Numerical Solution of Problems in Robotics
,” Int. J. Appl. Math. Comput. Sci.
, 19
(3
), pp. 399
–412
.10.2478/v10006-009-0033-364.
Berti
, A.
, Merlet
, J.-P.
, and Carricato
, M.
, 2013
, “Solving the Direct Geometrico-Static Problem of 3-3 Cable-Driven Parallel Robots by Interval Analysis: Preliminary Results
,” Cable-Driven Parallel Robots
, T.
Bruckmann
and A.
Pott
, eds., Springer-Verlag
, Berlin Heidelberg
, pp. 251
–268
.65.
Dietmaier
, P.
, 1998
, “The Stewart-Gough Platform of General Geometry can have 40 Real Postures
,” Advances in Robot Kinematics: Analysis and Control
, J.
Lenarčič
and M. L.
Husty
, eds., Kluwer Academic Publishers
, Dordrecht
, pp. 7
–16
.Copyright © 2013 by ASME
You do not currently have access to this content.
