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Research Papers

Reconfiguration Analysis of Multimode Single-Loop Spatial Mechanisms Using Dual Quaternions

[+] Author and Article Information
Xianwen Kong

School of Engineering and Physical Sciences,
Heriot-Watt University,
Edinburgh EH14 4AS, UK
e-mail: X.Kong@hw.ac.uk

Manuscript received September 17, 2016; final manuscript received May 28, 2017; published online August 4, 2017. Assoc. Editor: Venkat Krovi.

J. Mechanisms Robotics 9(5), 051002 (Aug 04, 2017) (8 pages) Paper No: JMR-16-1273; doi: 10.1115/1.4037111 History: Received September 17, 2016; Revised May 28, 2017

Although kinematic analysis of conventional mechanisms is a well-documented fundamental issue in mechanisms and robotics, the emerging reconfigurable mechanisms and robots pose new challenges in kinematics. One of the challenges is the reconfiguration analysis of multimode mechanisms, which refers to finding all the motion modes and the transition configurations of the multimode mechanisms. Recent advances in mathematics, especially algebraic geometry and numerical algebraic geometry, make it possible to develop an efficient method for the reconfiguration analysis of reconfigurable mechanisms and robots. This paper first presents a method for formulating a set of kinematic loop equations for mechanisms using dual quaternions. Using this approach, a set of kinematic loop equations of spatial mechanisms is composed of six polynomial equations. Then the reconfiguration analysis of a novel multimode single-degree-of-freedom (1DOF) 7R spatial mechanism is dealt with by solving the set of loop equations using tools from algebraic geometry. It is found that the 7R multimode mechanism has three motion modes, including a planar 4R mode, an orthogonal Bricard 6R mode, and a plane symmetric 6R mode. Three (or one) R (revolute) joints of the 7R multimode mechanism lose their DOF in its 4R (or 6R) motion modes. Unlike the 7R multimode mechanisms in the literature, the 7R multimode mechanism presented in this paper does not have a 7R mode in which all the seven R joints can move simultaneously.

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References

Uicker, J. J. , Denavit, J. , and Hartenberg, R. S. , 1964, “ An Iterative Method for the Displacement Analysis of Spatial Mechanisms,” ASME J. Appl. Mech., 31(2), pp. 309–314. [CrossRef]
Yang, A. T. , 1969, “ Displacement Analysis of Spatial Five-Link Mechanisms Using (3×3) Matrices With Dual-Number Elements,” ASME J. Eng. Ind., 91(1), pp. 152–157. [CrossRef]
Lee, H. Y. , and Liang, C. G. , 1988, “ Displacement Analysis of the General Spatial 7-Link 7R Mechanism,” Mech. Mach. Theory, 23(3), pp. 219–226. [CrossRef]
Raghavan, M. , and Roth, B. , 1993, “ Inverse Kinematics of the General 6R Manipulator and Related Linkages,” ASME J. Mech. Des., 115(3), pp. 502–508. [CrossRef]
Wang, Y. , Hang, L. , and Yang, T. , 2006, “ Inverse Kinematics Analysis of General 6R Serial Robot Mechanism Based on Groebner Base,” Front. Mech. Eng. China, 1(1), pp. 115–124.
Husty, M. L. , Pfurner, M. , and Schröcker, H.-P. , 2007, “ A New and Efficient Algorithm for the Inverse Kinematics of a General Serial 6R Manipulator,” Mech. Mach. Theory, 42(1), pp. 66–81. [CrossRef]
Selig, J. M. , 2005, Geometric Fundamentals of Robotics, Springer, New York.
McCarthy, J. M. , 2000, Geometric Design of Linkages, Springer-Verlag, New York.
Gervasi, P. , Karakusevic, V. , and Zsombor-Murray, P. J. , 1998, “ An Algorithm for Solving the Inverse Kinematics of a 6R Serial Manipulator Using Dual Quaternions and Grassmannians,” Advances in Robot Kinematics: Analysis and Control, J. Lenarčič and M. L. Husty , eds., Springer, Dordrecht, The Netherlands, pp. 383–392. [CrossRef]
Wohlhart, K. , 1996, “ Kinematotropic Linkages,” Recent Advances in Robot Kinematics, J. Lenarcic and V. Parenti-Castelli , eds., Kluwer Academic, Dordrecht, The Netherlands, pp. 359–368. [CrossRef]
Galletti, C. , and Fanghella, P. , 2001, “ Single-Loop Kinematotropic Mechanisms,” Mech. Mach. Theory, 36(6), pp. 743–761. [CrossRef]
Fanghella, P. , Galletti, C. , and Gianotti, E. , 2006, “ Parallel Robots That Change Their Group of Motion,” Advances in Robot Kinematics, Springer, Dordrecht, The Netherlands, pp. 49–56. [CrossRef]
Lee, C. C. , and Hervé, J. M. , 2005, “ Discontinuously Movable Seven-Link Mechanisms Via Group-Algebraic Approach,” Proc. Inst. Mech. Eng., Part C, 219(6), pp. 577–587. [CrossRef]
Kong, X. , and Huang, C. , 2009, “ Type Synthesis of Single-DOF Single-Loop Mechanisms With Two Operation Modes,” Reconfigurable Mechanisms and Robots, KC Edizioni, Genova, Italy, pp. 141–146.
Huang, C. , Kong, X. , and Ou, T. , 2009, “ Position Analysis of a Bennett-Based Multiple-Mode 7R Linkage,” ASME Paper No. DETC2009-87241.
Wohlhart, K. , 2010, “ Multifunctional 7R Linkages,” International Symposium on Mechanisms and Machine Theory, AzCIFToMM, Izmir, Turkey, Oct. 5–8, pp. 85–91.
Song, C. Y. , Chen, Y. , and Chen, I.-M. , 2013, “ A 6R Linkage Reconfigurable Between the Line-Symmetric Bricard Linkage and the Bennett Linkage,” Mech. Mach. Theory, 70, pp. 278–292. [CrossRef]
He, X. , Kong, X. , Chablat, D. , Caro, S. , and Hao, G. , 2014, “ Kinematic Analysis of a Single-Loop Reconfigurable 7R Mechanism With Multiple Operation Modes,” Robotica, 32(07), pp. 1171–1188. [CrossRef]
Kong, X. , and Pfurner, M. , 2015, “ Type Synthesis and Reconfiguration Analysis of a Class of Variable-DOF Single-Loop Mechanisms,” Mech. Mach. Theory, 85, pp. 116–128. [CrossRef]
Zhang, K. T. , and Dai, J. S. , 2014, “ A Kirigami-Inspired 8R Linkage and Its Evolved Overconstrained 6R Linkages With the Rotational Symmetry of Order Two,” ASME J. Mech. Rob., 6(2), p. 021007. [CrossRef]
Zhang, K. , Müller, A. , and Dai, J. S. , 2015, “ A Novel Reconfigurable 7R Linkage With Multifurcation,” Advances in Reconfigurable Mechanisms and Robots II, X. Ding , X. Kong , and J. S. Dai , eds., Springer, Cham, Switzerland, pp. 15–25. [CrossRef]
He, X. , Kong, X. , Hao, G. , and Ritchie, J. M. , 2015, “ Design and Analysis of a New 7R Single-Loop Mechanism With 4R, 6R and 7R Operation Modes,” Advances Reconfigurable Mechanisms and Robots II, X. Ding , X. Kong , and J. S. Dai , eds., Springer, Cham, Switzerland, pp. 27–37. [CrossRef]
Lopez-Custodio, P. C. , Rico, J. M. , Cervantes-Snchez, J. J. , and Prez-Soto, G. I. , 2016, “ Reconfigurable Mechanisms From the Intersection of Surfaces,” ASME J. Mech. Rob., 8(2), p. 021029. [CrossRef]
Müller, A. , and Piipponen, S. , 2015, “ On Regular Kinematotropies,” 14th World Congress in Mechanism and Machine Science, Taipei, Taiwan, Oct. 25–30, Paper No. IMD-123. https://www.researchgate.net/publication/283149233_On_Regular_Kinematotropies
Cox, D. A. , Little, J. B. , and O'Shea, D. , 2007, Ideals, Varieties, and Algorithms, Springer, New York. [CrossRef]
Walter, D. R. , Husty, M. L. , and Pfurner, M. , 2009, “ A Complete Kinematic Analysis of the SNU 3-UPU Parallel Manipulator,” Contemporary Mathematics, Vol. 496, American Mathematical Society, Providence, RI, pp. 331–346.
Husty, M. L. , and Schröcker, H.-P. , 2013, “ Kinematics and Algebraic Geometry,” 21st Century Kinematics, J. M. McCarthy , ed., Springer, London, pp. 85–123.
Wampler, C. W. , and Sommese, A. J. , 2013, “ Applying Numerical Algebraic Geometry to Kinematics,” 21st Century Kinematics, J. M. McCarthy , ed., Springer, London, pp. 125–159. [CrossRef]
Decker, W. , and Pfister, G. , 2013, A First Course in Computational Algebraic Geometry, Cambridge University Press, Cambridge, UK. [CrossRef]
Kong, X. , 2014, “ Reconfiguration Analysis of a 3-DOF Parallel Mechanism Using Euler Parameter Quaternions and Algebraic Geometry Method,” Mech. Mach. Theory, 74, pp. 188–201. [CrossRef]
Kong, X. , Yu, J. , and Li, D. , 2015, “ Reconfiguration Analysis of a Two Degrees-of-Freedom 3-4R Parallel Manipulator With Planar Base and Platform,” ASME J. Mech. Rob., 8(1), p. 011019. [CrossRef]
Carbonari, L. , Callegari, M. , Palmieri, G. , and Palpacelli, M.-C. , 2014, “ A New Class of Reconfigurable Parallel Kinematic Machines,” Mech. Mach. Theory, 79, pp. 173–183. [CrossRef]
Nurahmi, L. , Schadlbauer, J. , Caro, S. , Husty, M. , and Wenger, Ph. , 2015, “ Kinematic Analysis of the 3-RPS Cube Parallel Manipulator,” ASME J. Mech. Rob., 7(1), p. 011008. [CrossRef]
Kong, X. , 2016, “ Reconfiguration Analysis of a 4-DOF 3-RER Parallel Manipulator With Equilateral Triangular Base and Moving Platform,” Mech. Mach. Theory, 98, pp. 180–189. [CrossRef]
Kong, X. , 2016, “ Reconfiguration Analysis of a Variable Degrees-of-Freedom Parallel Manipulator With Both 3-DOF Planar and 4-DOF 3T1R Operation Modes,” ASME Paper No. DETC2016-59203.
Coste, M. , and Demdah, K. M. , 2015, “ Extra Modes of Operation and Self Motions in Manipulators Designed for Schoenflies Motion,” ASME J. Mech. Rob., 7(4), p. 041020. [CrossRef]
Nurahmi, L. , Caro, S. , Wenger, P. , Schadlbauer, J. , and Husty, M. , 2016, “ Reconfiguration Analysis of a 4-RUU Parallel Manipulator,” Mech. Mach. Theory, 96(Pt. 2), pp. 269–289. [CrossRef]
Arponen, T. , Piipponen, S. , and Tuomela, J. , 2013, “ Kinematical Analysis of Wunderlich Mechanism,” Mech. Mach. Theory, 70, pp. 16–31. [CrossRef]
Radavelli, L. , Simoni, R. , De Pieri, E. , and Martins, D. , 2012, “ A Comparative Study of the Kinematics of Robots Manipulators by Denavit–Hartenberg and Dual Quaternion,” Mec. Comput., 31, pp. 2833–2848. http://www.cimec.org.ar/ojs/index.php/mc/article/viewFile/4224/4150
Gan, D. , Liao, Q. , Wei, S. , Dai, J. S. , and Qiao, S. , 2008, “ Dual Quaternion-Based Inverse Kinematics of the General Spatial 7R Mechanism,” Proc. Inst. Mech. Eng., Part C, 222(8), pp. 1593–1598. [CrossRef]
Qiao, S. , Liao, Q. , Wei, S. , and Su, H. J. , 2009, “ Inverse Kinematic Analysis of the General 6R Serial Manipulators Based on Double Quaternions,” Mech. Mach. Theory, 45(2), pp. 193–199. [CrossRef]
Li, Z. , and Schicho, J. , 2015, “ A Technique for Deriving Equational Conditions on the Denavit–Hartenberg Parameters of 6R Linkages That Are Necessary for Movability,” Mech. Mach. Theory, 94, pp. 1–8. [CrossRef]
Perez, A. , and McCarthy, J. M. , 2004, “ Dual Quaternion Synthesis of Constrained Robotic Systems,” ASME J. Mech. Des., 126(3), pp. 425–435. [CrossRef]
Ge, Q. J. , Varshney, A. , Menon, J. , and Chang, C. , 2004, “ On the Use of Quaternions, Dual Quaternions, and Double Quaternions for Freeform Motion Synthesis,” Mach. Des. Res., 20(Z1), pp. 147–150. http://d.wanfangdata.com.cn/periodical/jxsjyyj2004z1044
Thomas, F. , 2014, “ Approaching Dual Quaternions From Matrix Algebra,” IEEE Trans. Rob., 30(5), pp. 1037–1048. [CrossRef]
Kong, X. , 2016, “ Kinematic Analysis of Conventional and Multi-Mode Spatial Mechanisms Using Dual Quaternions,” ASME Paper No. DETC2016-59194.

Figures

Grahic Jump Location
Fig. 2

A multimode 7R mechanism

Grahic Jump Location
Fig. 3

Motion modes of the multimode 7R mechanism: (a) motion mode 1: planar 4R mode in circuit 1, (b) motion mode 1: planar 4R mode in circuit 2, (c) motion mode 2: orthogonal bricard 6R mode in circuit 1, (d) motion mode 2: orthogonal bricard 6R mode in circuit 2, (e) motion mode 3: plane symmetric 6R mode in circuit 1, and (f) motion mode 3: plane symmetric 6R mode in circuit 2

Grahic Jump Location
Fig. 4

Transition configurations between the planar 4R mode and the orthogonal bricard 6R mode of the multimode 7R mechanism: (a) transition configuration 1 between motion modes 1 and 2 in circuit 1 and (b) transition configuration 2 between motion modes 1 and 2 in circuit 2

Grahic Jump Location
Fig. 5

Transition configurations between the planar 4R mode and the plane symmetric 6R mode of the multimode 7R mechanism: (a) transition configuration 3 between motion modes 1 and 3 in circuit 1 and (b) transition configuration 4 between motion modes 1 and 3 in circuit 2

Grahic Jump Location
Fig. 6

Reconfiguration of the multimode 7R mechanism: (a) motion modes 1, 2, and 3 in circuit 1 and (b) motion modes 1, 2, and 3 in circuit 2

Grahic Jump Location
Fig. 7

A 3D-printed prototype of the multimode 7R mechanism in transition configuration 2

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