Graphical Abstract Figure
Abstract
The paper presents an approach to perform an instantaneous kinematic analysis of parallel–serial (hybrid) manipulators using screw theory. In this study, we focus on non-kinematically redundant manipulators that include a single parallel mechanism. The proposed systematic procedure allows deriving Jacobian matrices for such manipulators, which provide mathematical relations between the end-effector velocities and speeds in the actuated joints. A generalized structure of the obtained matrices also reflects the constrained motions of the end-effector and the parallel mechanism. To illustrate the developed techniques, we consider three examples where we analyze three well-known parallel–serial manipulators with six, five, and four degrees-of-freedom. Following the proposed method, we determine Jacobian matrices for each manipulator. Next, we apply the presented approach for velocity analysis of a novel parallel–serial manipulator with five degrees-of-freedom. Numerical simulations validate the proposed theoretical techniques. The suggested approach represents the basis for subsequent singularity and performance analysis, and it can be adapted to hybrid manipulators with other architectures.
References
1.
Ye
, W.
, Tang
, T.
, and Li
, Q.
, 2023
, “Robotized Manufacturing Equipment: A Review From the Perspective of Mechanism Topology
,” Sci. China Technol. Sci.
, 66
(6
), pp. 1683
–1697
. 2.
Nouaille
, L.
, Laribi
, M. A.
, Nelson
, C. A.
, Zeghloul
, S.
, and Poisson
, G.
, 2017
, “Review of Kinematics for Minimally Invasive Surgery and Tele-Echography Robots
,” ASME J. Med. Devices
, 11
(4
), p. 040802
. 3.
Kumar
, S.
, Wöhrle
, H.
, de Gea Fernández
, J.
, Müller
, A.
, and Kirchner
, F.
, 2020
, “A Survey on Modularity and Distributivity in Series-Parallel Hybrid Robots
,” Mechatronics
, 68
, p. 102367
. 4.
Müller
, A.
, Zlatanov
, D.
, eds., 2019
, Singular Configurations of Mechanisms and Manipulators
, Springer
, Cham, Switzerland
. 5.
Li
, Q.
, Yang
, C.
, Xu
, L.
, and Ye
, W.
, 2023
, Performance Analysis and Optimization of Parallel Manipulators
, Springer
, Singapore
. 6.
Kelaiaia
, R.
, Chemori
, A.
, Brahmia
, A.
, Kerboua
, A.
, Zaatri
, A.
, and Company
, O.
, 2023
, “Optimal Dimensional Design of Parallel Manipulators With an Illustrative Case Study: A Review
,” Mech. Mach. Theory
, 188
, p. 105390
. 7.
Pisla
, D.
, Szilaghyi
, A.
, Vaida
, C.
, and Plitea
, N.
, 2013
, “Kinematics and Workspace Modeling of a New Hybrid Robot Used in Minimally Invasive Surgery
,” Robot. Comput. Integr. Manuf.
, 29
(2
), pp. 463
–474
. 8.
Deng
, F.
, Liu
, X.
, Zhang
, N.
, and Zhang
, F.
, 2019
, “Dimension Synthesis of a 3T2R Labelling Robot With Hybrid Mechanism
,” J. Eur. des Systèmes Autom.
, 52
(5
), pp. 509
–514
. 9.
Wang
, X.
, Wu
, J.
, and Zhou
, Y.
, 2023
, “Dynamic Modeling and Performance Evaluation of a 5-DOF Hybrid Robot for Composite Material Machining
,” Machines
, 11
(6
), p. 652
. 10.
Yan
, C.
, Gao
, F.
, and Zhang
, Y.
, 2010
, “Kinematic Modeling of a Serial-Parallel Forging Manipulator With Application to Heavy-Duty Manipulations
,” Mech. Based Des. Struct. Mach.
, 38
(1
), pp. 105
–129
. 11.
Singh
, A.
, Singla
, E.
, Soni
, S.
, and Singla
, A.
, 2018
, “Kinematic Modeling of a 7-Degree of Freedom Spatial Hybrid Manipulator for Medical Surgery
,” Proc. Inst. Mech. Eng. Part H: J. Eng. Med.
, 232
(1
), pp. 12
–23
. 12.
Wang
, Z.
, Li
, Y.
, Sun
, P.
, Luo
, Y.
, Chen
, B.
, and Zhu
, W.
, 2021
, “A Multi-objective Approach for the Trajectory Planning of a 7-DOF Serial-Parallel Hybrid Humanoid Arm
,” Mech. Mach. Theory
, 165
, p. 104423
. 13.
Carmichael
, M. G.
, Liu
, D.
, and Waldron
, K. J.
, 2017
, “A Framework for Singularity-Robust Manipulator Control During Physical Human-Robot Interaction
,” Int. J. Rob. Res.
, 36
(5–7
), pp. 861
–876
. 14.
Sklar
, M.
, and Tesar
, D.
, 1988
, “Dynamic Analysis of Hybrid Serial Manipulator Systems Containing Parallel Modules
,” ASME J. Mech. Transm. Autom. Des.
, 110
(2
), pp. 109
–115
. 15.
Chung
, G. B.
, Yi
, B. -J.
, Lim
, D. J.
, and Kim
, W.
, 2004
, “An Efficient Dynamic Modeling Methodology for General Type of Hybrid Robotic Systems
,” Proceedings of International Conference on Robotics and Automation (ICRA)
, New Orleans, LA, Apr. 26–May 1, Vol. 2
, pp. 1795
–1802
. 16.
Chablat
, D.
, Wenger
, P.
, and Angeles
, J.
, 1998
, “The Isoconditioning Loci of a Class of Closed-Chain Manipulators
,” Proceedings of IEEE International Conference on Robotics and Automation
, Leuven, Belgium, May 16–20, Vol. 3
, pp. 1970
–1975
. 17.
Wang
, H. B.
, Ishimatsu
, T.
, Schaerer
, C.
, and Huang
, Z.
, 1998
, “Kinematics of a Five Degree-of-Freedom Prosthetic Arm
,” Mech. Mach. Theory
, 33
(7
), pp. 895
–908
. 18.
Guo
, W.
, Li
, R.
, Cao
, C.
, Tong
, X.
, and Gao
, Y.
, 2016
, “A New Methodology for Solving Trajectory Planning and Dynamic Load-Carrying Capacity of a Robot Manipulator
,” Math. Probl. Eng.
, 2016
, p. 1302537
. 19.
Waldron
, K. J.
, Raghavan
, M.
, and Roth
, B.
, 1989
, “Kinematics of a Hybrid Series-Parallel Manipulation System
,” ASME J. Dyn. Syst. Meas. Control
, 111
(2
), pp. 211
–221
. 20.
Yang
, G.
, Chen
, I.-M.
, Yeo
, S. H.
, and Lin
, W.
, 2008
, “Design and Analysis of a Modular Hybrid Parallel-Serial Manipulator for Robotised Deburring Applications,” Smart Devices and Machines for Advanced Manufacturing
, L.
Wang
, J.
Xi
, eds., Springer
, London
, pp. 167
–188
. 21.
He
, J.
, Gao
, F.
, and Sun
, Q.
, 2019
, “Design and Kinematic Analysis of a Novel Hybrid Kinematic Mechanism With Seven-Degrees-of-Freedom and Variable Topology for Operation in Space
,” ASME J. Mech. Rob.
, 11
(1
), p. 011003
. 22.
Xiao
, J.
, Wang
, Y.
, Liu
, S.
, Sun
, Y.
, Liu
, H.
, Huang
, T.
, and Xu
, J.
, 2021
, “Grinding Trajectory Generation of Hybrid Robot Based on Cartesian Direct Teaching Technology
,” Ind. Rob.
, 48
(3
), pp. 341
–351
. 23.
Zhang
, X.
, Wang
, H.
, Rong
, Y.
, Niu
, J.
, Tian
, J.
, and Li
, S.
, 2023
, “Dynamic Modeling of a Class of Parallel-Serial Mechanisms by the Principle of Virtual Work
,” Meccanica
, 58
(1
), pp. 303
–316
. 24.
Davidson
, J. K.
, and Hunt
, K. H.
, 2004
, Robots and Screw Theory: Applications of Kinematics and Statics to Robotics
, Oxford University Press
, Oxford, UK
. 25.
Crane
, III, C. D.
, Griffis
, M.
, and Duffy
, J.
, 2022
, Screw Theory and Its Application to Spatial Robot Manipulators
, Cambridge University Press
, Cambridge, UK
. 26.
Yang
, S.
, and Li
, Y.
, 2020
, “Classification and Analysis of Constraint Singularities for Parallel Mechanisms Using Differential Manifolds
,” Appl. Math. Model.
, 77
(1
), pp. 469
–477
. 27.
Whitney
, D. E.
, 1972
, “The Mathematics of Coordinated Control of Prosthetic Arms and Manipulators
,” ASME J. Dyn. Syst. Meas. Control
, 94
(4
), pp. 303
–309
. 28.
Mohamed
, M. G.
, and Duffy
, J.
, 1985
, “A Direct Determination of the Instantaneous Kinematics of Fully Parallel Robot Manipulators
,” ASME J. Mech. Transm. Autom. Des.
, 107
(2
), pp. 226
–229
. 29.
Kumar
, V.
, 1992
, “Instantaneous Kinematics of Parallel-Chain Robotic Mechanisms
,” ASME J. Mech. Des.
, 114
(3
), pp. 349
–358
. 30.
Wang
, K.
, Dong
, H.
, Spyrakos-Papastavridis
, E.
, Qiu
, C.
, and Dai
, J. S.
, 2022
, “A Repelling-Screw-Based Approach for the Construction of Generalized Jacobian Matrices for Nonredundant Parallel Manipulators
,” Mech. Mach. Theory
, 176
, p. 105009
. 31.
Joshi
, S. A.
, and Tsai
, L. -W.
, 2002
, “Jacobian Analysis of Limited-DOF Parallel Manipulators
,” ASME J. Mech. Des.
, 124
(2
), pp. 254
–258
. 32.
Huang
, T.
, Liu
, H. T.
, and Chetwynd
, D. G.
, 2011
, “Generalized Jacobian Analysis of Lower Mobility Manipulators
,” Mech. Mach. Theory
, 46
(6
), pp. 831
–844
. 33.
Huang
, T.
, Yang
, S.
, Wang
, M.
, Sun
, T.
, and Chetwynd
, D. G.
, 2015
, “An Approach to Determining the Unknown Twist/Wrench Subspaces of Lower Mobility Serial Kinematic Chains
,” ASME J. Mech. Rob.
, 7
(3
), p. 031003
. 34.
Etemadi-Zanganeh
, K.
, and Angeles
, J.
, 1995
, “Instantaneous Kinematics of General Hybrid Parallel Manipulators
,” ASME J. Mech. Des.
, 117
(4
), pp. 581
–588
. 35.
Monsarrat
, B.
, and Gosselin
, C. M.
, 2002
, “Jacobian Matrix of General Parallel and Hybrid Mechanisms With Rigid and Flexible Links: A Software-Oriented Approach
,” Proceedings of ASME 2002 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
, Montreal, Quebec, Canada, Sept. 29–Oct. 2, Vol. 5
, pp. 461
–470
. 36.
Sun
, P.
, Li
, Y.
, Chen
, K.
, Zhu
, W.
, Zhong
, Q.
, and Chen
, B.
, 2021
, “Generalized Kinematics Analysis of Hybrid Mechanisms Based on Screw Theory and Lie Groups Lie Algebras
,” Chin. J. Mech. Eng.
, 34
(1
), p. 98
. 37.
Lee
, S.
, and Kim
, S.
, 1993
, “Efficient Inverse Kinematics for Serial Connections of Serial and Parallel Manipulators
,” Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)
, Yokohama, Japan, July 26–30, Vol. 3
, pp. 1635
–1641
. 38.
Huang
, M. Z.
, Ling
, S. -H.
, and Sheng
, Y.
, 1993
, “A Study of Velocity Kinematics for Hybrid Manipulators With Parallel-Series Configurations
,” Proceedings of IEEE International Conference on Robotics and Automation (ICRA)
, Atlanta, GA, May 2–6, Vol. 1
, pp. 456
–461
. 39.
Lee
, M. K.
, Park
, K. W.
, and Choi
, B. O.
, 1999
, “Kinematic and Dynamic Models of Hybrid Robot Manipulator for Propeller Grinding
,” J. Robot. Syst.
, 16
(3
), pp. 137
–150
. 40.
Gallardo-Alvarado
, J.
, 2023
, “Unified Infinitesimal Kinematics of a 3-RRR/PRR Six-Degree-of-Freedom Parallel-Serial Manipulator
,” Meccanica
, 58
(4
), pp. 795
–811
. 41.
Wu
, Y.
, and Carricato
, M.
, 2018
, “Design of a Novel 3-DOF Serial-Parallel Robotic Wrist: A Symmetric Space Approach,” Robotics Research
, A.
Bicchi
, W.
Burgard
, eds., Springer
, Cham, Switzerland
, pp. 389
–404
. 42.
Hu
, B.
, Shi
, Y.
, Xu
, L.
, and Bai
, P.
, 2020
, “Reconsideration of Terminal Constraint/Mobility and Kinematics of 5-DOF Hybrid Manipulators Formed by One 2R1T PM and One RR SM
,” Mech. Mach. Theory
, 149
, p. 103837
. 43.
Laryushkin
, P.
, Antonov
, A.
, Fomin
, A.
, and Essomba
, T.
, 2022
, “Velocity and Singularity Analysis of a 5-DOF (3T2R) Parallel-Serial (Hybrid) Manipulator
,” Machines
, 10
(4
), p. 276
. 44.
Antonov
, A.
, and Fomin
, A.
, 2023
, “Velocity Analysis of a 5-DOF Hybrid Manipulator,” New Advances in Mechanisms, Transmissions and Applications
, M. A.
Laribi
, C. A.
Nelson
, M.
Ceccarelli
, and S.
Zeghloul
, eds., Springer
, Cham, Switzerland
, pp. 161
–170
. 45.
Gosselin
, C.
, and Schreiber
, L.-T.
, 2018
, “Redundancy in Parallel Mechanisms: A Review
,” ASME Appl. Mech. Rev.
, 70
(1
), p. 010802
. 46.
Huang
, Z.
, Liu
, J.
, and Zeng
, D.
, 2009
, “A General Methodology for Mobility Analysis of Mechanisms Based on Constraint Screw Theory
,” Sci. China Ser. E Technol. Sci.
, 52
(5
), pp. 1337
–1347
. 47.
Song
, Y.
, Kang
, X.
, and Dai
, J. S.
, 2020
, “Instantaneous Mobility Analysis Using the Twist Space Intersection Approach for Parallel Mechanisms
,” Mech. Mach. Theory
, 151
, p. 103866
. 48.
Zhao
, J.
, Li
, B.
, Yang
, X.
, and Yu
, H.
, 2009
, “Geometrical Method to Determine the Reciprocal Screws and Applications to Parallel Manipulators
,” Robotica
, 27
(6
), pp. 929
–940
. 49.
Yang
, G.
, Chen
, W.
, and Ho
, E. H. L.
, 2002
, “Design and Kinematic Analysis of a Modular Hybrid Parallel-Serial Manipulator
,” Proceedings of International Conference on Control, Automation, Robotics and Vision (ICARCV)
, Singapore, Dec. 2–5, Vol. 1
, pp. 45
–50
. 50.
Carricato
, M.
, and Zlatanov
, D.
, 2014
, “Persistent Screw Systems
,” Mech. Mach. Theory
, 73
, pp. 296
–313
. 51.
Dong
, C.
, Liu
, H.
, Liu
, Q.
, Sun
, T.
, Huang
, T.
, and Chetwynd
, D. G.
, 2018
, “An Approach for Type Synthesis of Overconstrained 1T2R Parallel Mechanisms,” Computational Kinematics
, S.
Zeghloul
, L.
Romdhane
, and M.
Laribi
, eds., Springer
, Cham, Switzerland
, pp. 274
–281
. 52.
Liu
, Q.
, Liu
, H.
, Xiao
, J.
, Tian
, W.
, Ma
, Y.
, and Li
, B.
, 2023
, “Open-Architecture of CNC System and Mirror Milling Technology for a 5-Axis Hybrid Robot
,” Robot. Comput. Integr. Manuf.
, 81
, p. 102504
. 53.
Saafi
, H.
, Laribi
, M. A.
, Zeghloul
, S.
, and Arsicault
, M.
, 2018
, “On the Development of a New Master Device Used for Medical Tasks
,” ASME J. Mech. Rob.
, 10
(4
), p. 044501
. 54.
Saafi
, H.
, Laribi
, M. A.
, and Zeghloul
, S.
, 2020
, “Forward Kinematic Model Resolution of a Special Spherical Parallel Manipulator: Comparison and Real-Time Validation
,” Robotics
, 9
(3
), p. 62
. 55.
Antonov
, A.
, and Fomin
, A.
, 2023
, “Mechanism Design and Inverse Kinematics of a 5-DOF Medical Assistive Manipulator,” New Trends in Medical and Service Robotics
, D.
Tarnita
, N.
Dumitru
, D.
Pisla
, G.
Carbone
, and I.
Geonea
, eds., Springer
, Cham, Switzerland
, pp. 334
–342
. 56.
Bottema
, O.
, and Roth
, B.
, 1990
, Theoretical Kinematics
, Dover Publications
, New York
. https://scholar.google.com/scholar?cluster=772944265291167393557.
Gallardo-Alvarado
, J.
, 2005
, “Kinematics of a Hybrid Manipulator by Means of Screw Theory
,” Multibody Syst. Dyn.
, 14
(3–4
), pp. 345
–366
. 58.
Gallardo-Alvarado
, J.
, and Posadas-García
, J.
, 2013
, “Mobility Analysis and Kinematics of the Semi-General 2(3-RPS) Series-Parallel Manipulator
,” Robot. Comput. Integr. Manuf.
, 29
(6
), pp. 463
–472
. 59.
Nayak
, A.
, Caro
, S.
, and Wenger
, P.
, 2019
, “Kinematic Analysis of the 3-RPS-3-SPR Series-Parallel Manipulator
,” Robotica
, 37
(7
), pp. 1240
–1266
. 60.
Zoppi
, M.
, Zlatanov
, D.
, and Molfino
, R.
, 2006
, “On the Velocity Analysis of Interconnected Chains Mechanisms
,” Mech. Mach. Theory
, 41
(11
), pp. 1346
–1358
. 61.
Ding
, H.
, Cao
, W.
, Chen
, Z.
, and Kecskeméthy
, A.
, 2015
, “Structural Synthesis of Two-Layer and Two-Loop Spatial Mechanisms With Coupling Chains
,” Mech. Mach. Theory
, 92
, pp. 289
–313
. 
