Research Papers

Forward Kinematics Solution Distribution and Analytic Singularity-Free Workspace of Linear-Actuated Symmetrical Spherical Parallel Manipulators

[+] Author and Article Information
Dongming Gan

Robotics Institute,
Khalifa University of Science,
Technology & Research,
Abu Dhabi 127788, UAE
e-mail: dongming.gan@kustar.ac.ae

Jian S. Dai

School of Natural and Mathematical Sciences,
King's College London,
University of London,
London WC2R2LS, UK

Jorge Dias

Robotics Institute,
Khalifa University of Science,
Technology & Research,
Abu Dhabi 127788, UAE
Faculty of Science and Technology,
University of Coimbra,
Coimbra 3030-790, Portugal

Lakmal Seneviratne

Robotics Institute,
Khalifa University of Science,
Technology & Research,
Abu Dhabi 127788, UAE
School of Natural and Mathematical Sciences,
King's College London,
University of London,
London WC2R2LS, UK

1Corresponding author.

Manuscript received September 27, 2013; final manuscript received February 9, 2015; published online March 23, 2015. Assoc. Editor: Qiaode Jeffrey Ge.

J. Mechanisms Robotics 7(4), 041007 (Nov 01, 2015) (8 pages) Paper No: JMR-13-1194; doi: 10.1115/1.4029808 History: Received September 27, 2013; Revised February 09, 2015; Online March 23, 2015

This paper presents a new kinematics model for linear-actuated symmetrical spherical parallel manipulators (LASSPMs) which are commonly used considering their symmetrical kinematics and dynamics properties. The model has significant advantages in solving the forward kinematic equations, and in analytically obtaining singularity loci and the singularity-free workspace. The Cayley formula, including the three Rodriguez–Hamilton parameters from a general rotation matrix, is provided and used in describing the rotation motion and geometric constraints of LASSPMs. Analytical solutions of the forward kinematic equations are obtained. Then singularity loci are derived, and represented in a new coordinate system with the three Rodriguez–Hamilton parameters assigned in three perpendicular directions. Limb-actuation singularity loci are illustrated and forward kinematics (FK) solution distribution in the singularity-free zones is discussed. Based on this analysis, unique forward kinematic solutions of LASSPMs can be determined. By using Cayley formula, analytical workspace boundaries are expressed, based on a given mechanism structure and input actuation limits. The singularity-free workspace is demonstrated in the proposed coordinate system. The work gives a systematic method in modeling kinematics, singularity and workspace analysis which provides new optimization design index and a simpler kinematics model for dynamics and control of LASSPMs.

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Gosselin, C., St. Pierre, E., and Gagné, M., 1996, “On the Development of the Agile Eye: Mechanical Design, Control Issues and Experimentation,” IEEE Rob. Autom. Mag., 3(4), pp. 29–37. [CrossRef]
Gosselin, C., and Angeles, J., 1989, “The Optimum Kinematic Design of a Spherical Three-Degree-of-Freedom Parallel Manipulator,” ASME J. Mech. Des., 111(2), pp. 202–207. [CrossRef]
Vischer, P., and Clavel, R., 2000, “Argos: A Novel 3-DOF Parallel Wrist Mechanism,” Int. J. Robot. Res., 19(1), pp. 5–11. [CrossRef]
Gregorio, R. D., 2001, “A New Parallel Wrist Using Only Revolute Pairs: The 3-RUU Wrist,” Robotica, 19(3), pp. 305–309. [CrossRef]
Cox, D., and Tesar, D., 1989, “The Dynamic Model of a Three-Degree-of-Freedom Parallel Robotic Shoulder Module,” Fourth International Conference on Advanced Robotics, Columbus, OH, June 13–15, pp. 475–487. [CrossRef]
Hofschulte, J., Seebode, M., and Gerth, W., 2004, “Parallel Manipulator Hip Joint for a Bipedal Robot,” Climbing and Walking Robots, Springer, New York, pp. 601–609.
Cui, L., and Dai, J. S., 2012, “Reciprocity-Based Singular Value Decomposition for Inverse Kinematic Analysis of the Metamorphic Multifingered Hand,” ASME J. Mech. Rob., 4(3), p. 034502. [CrossRef]
Cui, L., and Dai, J. S., 2011, “Posture, Workspace, and Manipulability of the Metamorphic Multifingered Hand With an Articulated Palm,” ASME J. Mech. Rob., 3(2), p. 021001. [CrossRef]
Li, T., and Payandeh, S., 2002, “Design of Spherical Parallel Mechanisms for Application to Laparoscopic Surgery,” Robotica, 20(2), pp. 133–138. [CrossRef]
Dai, J., Zhao, T., and Nester, C., 2004, “Sprained Ankle Physiotherapy Based Mechanism Synthesis and Stiffness Analysis of Rehabilitation Robotic Devices,” Autonom. Rob., 16(2), pp. 207–218. [CrossRef]
Gogu, G., 2012, “Parallel Wrists With Three Degrees of Freedom, Structural Synthesis of Parallel Robots,” Solid Mechanics and Its Applications, Vol. 183, Springer, Dordrecht, The Netherlands, pp. 483–552.
Innocenti, C., and Parenti-Castelli, V., 1993, “Echelon Form Solution of Direct Kinematics for the General Fully-Parallel Spherical Wrist,” Mech. Mach. Theory, 28(4), pp. 553–561. [CrossRef]
Kong, X.-W., 1998, “Forward Displacement Analysis of Three New Classes of Analytic Spherical Parallel Manipulators,” ASME Paper No. DETC98/MECH-5953. [CrossRef]
Gan, D. M., Seneviratne, L. D., and Dias, J., 2012, “Design and Analytical Kinematics of a Robot Wrist Based on a Parallel Mechanism,” 14th International Symposium on Robotics and Applications, Puerto Vallarta, Mexico, June 24–28, pp. 1–6.
Karouia, M., and Hervé, J. M., 2000, “A Three-DOF Tripod for Generating Spherical Rotation,” Advances in Robot Kinematics, J.Lenarcic, and M. M.Stanisic, eds., Kluwer Academic Publishers, Amsterdam, The Netherlands, pp. 395–402.
Tsai, L. W., and Sameer, J., 2000, “Kinematics and Optimization of a Spatial 3-UPU Parallel Manipulator,” ASME J. Mech. Des., 122(4), pp. 439–446. [CrossRef]
Di Gregorio, R., 2003, “Kinematics of the 3-UPU Wrist,” Mech. Mach. Theory, 38(3), pp. 253–263. [CrossRef]
Zhang, K. T., Dai, J. S., and Fang, Y. F., 2012, “Geometric Constraint and Mobility Variation of Two 3SvPSv Metamorphic Parallel Mechanisms,” ASME J. Mech. Des., 135(1), p. 011001. [CrossRef]
Gosselin, C. M., Sefrioui, J., and Richard, M. J., 1994, “On the Direct Kinematics of Spherical Three-Degree-of-Freedom Parallel Manipulators of General Architecture,” ASME J. Mech. Des., 116(2), pp. 594–598. [CrossRef]
Huang, Z., and Yao, Y. L., 1999, “A New Closed-Form Kinematics of the Generalized 3-DOF Spherical Parallel Manipulator,” Robotica, 17(5), pp. 475–485. [CrossRef]
Alizade, R. I., Tagiyiev, N. R., and Duffy, J., 1994, “A Forward and Reverse Displacement Analysis of an In-Parallel Spherical Manipulator,” Mech. Mach. Theory, 29(1), pp. 125–137. [CrossRef]
Ji, P., and Wu, H., 2001, “Algebraic Solution to Forward Kinematics of 3-DOF Spherical Parallel Manipulator,” J. Rob. Syst., 18(5), pp. 251–257. [CrossRef]
Bai, S. P., Hansen, M. R., and Angeles, J., 2009, “A Robust Forward-Displacement Analysis of Spherical Parallel Robots,” Mech. Mach. Theory, 44(12), pp. 2204–2216. [CrossRef]
Bonev, I. A., Chablat, D., and Wenger, P., 2006, “Working and Assembly Modes of the Agile Eye,” IEEE International Conference on Robotics and Automation (ICRA 2006), Orlando, FL, May 15–19, pp. 2317–2322. [CrossRef]
Kong, X., and Gosselin, C. M., 2010, “A Formula That Produces a Unique Solution to the Forward Displacement Analysis of a Quadratic Spherical Parallel Manipulator: The Agile Eye,” ASME J. Mech. Rob., 2(4), p. 044501. [CrossRef]
Kong, X., Gosselin, C. M., and Ritchie, J. M., 2011, “Forward Displacement Analysis of a Linearly Actuated Quadratic Spherical Parallel Manipulator,” ASME J. Mech. Rob., 3(1), p. 011007. [CrossRef]
Bottema, O., and Roth, B., 1979, Theoretical Kinematics, North-Holland, New York, pp. 9–11.
Gan, D. M., Liao, Q. Z., Dai, J. S., and Wei, S. M., 2010, “Design and Kinematics Analysis of a New 3CCC Parallel Mechanism,” Robotica, 28(7), pp. 1065–1072. [CrossRef]
Lee, T.-Y., and Shim, J.-K., 2003, “Improved Dialytic Elimination Algorithm for the Forward Kinematics of the General Stewart–Gough Platform,” Mechanism and Machine Theory, 38(6), pp. 563–577. [CrossRef]
Gan, D. M., Liao, Q. Z., Dai, J. S., Wei, S. M., and Seneviratne, L. D., 2009, “Forward Displacement Analysis of a New 1CCC-5SPS Parallel Mechanism Using Grobner Theory,” Proc. Inst. Mech. Eng., Part C, 223(C5), pp. 1233–1241. [CrossRef]
Husty, M. L., 1996, “An Algorithm for Solving the Direct Kinematics of General Stewart-Gough Platforms,” Mech. Mach. Theory, 31(4), pp. 365–380. [CrossRef]
Schröcker, H.-P., and Husty, M. L., 2007, “Kinematic Mapping Based Assembly Mode Evaluation of Spherical Four-Bar Mechanisms,” 12th IFToMM World Congress, Besançon, France, June 18–21. Available at http://www.iftomm.org/iftomm/proceedings/proceedings_WorldCongress/WorldCongress07/articles/sessions/papers/A7.pdf
Hayes, M. J. D., Husty, M. L., and Zsombor-Murray, P., 1999, “Kinematic Mapping of Planar Stewart–Gough Platforms,” 17th Canadian Congress of Applied Mechanics (CanCAM '99), Hamilton, ON, Canada, May 30–June 3, pp. 319–320.
Gosselin, C. M., and Angeles, J., 1990, “Singularity Analysis of Closed-Loop Kinematic Chains,” IEEE Trans. Rob. Autom., 6(3), pp. 281–290. [CrossRef]
Wei, G., and Dai, J. S., 2010, “Geometric and Kinematic Analysis of a Seven-Bar Three-Fixed-Pivoted Compound-Joint Mechanism,” Mech. Mach. Theory, 45(2), pp. 170–184. [CrossRef]
Sefrioui, J., and Gosselin, C. M., 1994, “Étude et Représentation des Lieux de Singularité des Manipulateurs Parallèles Sphériques à Trois Degrés de Liberté Avec Actionneurs Prismatiques,” Mech. Mach. Theory, 29(4), pp. 559–579. [CrossRef]
Alici, G., and Shirinzadeh, B., 2004, “Topology Optimization and Singularity Analysis of a 3-SPS Parallel Manipulator With a Passive Constraining Spherical Joint,” Mech. Mach. Theory, 39(2), pp. 215–235. [CrossRef]
Gosselin, C. M., and Wang, J., 2002, “Singularity Loci of A Special Class of Spherical Three-Degree-of-Freedom Parallel Mechanisms With Revolute Actuators,” Int. J. Rob. Res., 21(7), pp. 649–659. [CrossRef]
Bonev, I. A., and Gosselin, C. M., 2005, “Singularity Loci of Spherical Parallel Mechanisms,” IEEE International Conference on Robotics and Automation, (ICRA 2005), Barcelona, Spain, Apr. 18–22, pp. 2968–2973. [CrossRef]
Bonev, I. A., and Gosselin, C. M., 2006, “Analytical Determination of the Workspace of Symmetrical Spherical Parallel Mechanisms,” IEEE Trans. Rob., 22(5), pp. 1011–1017. [CrossRef]
Yang, G., and Chen, I.-M., 2006, “Equivolumetric Partition of Solid Spheres With Applications to Orientation Workspace Analysis of Robot Manipulators,” IEEE Trans. Rob., 22(5), pp. 869–879. [CrossRef]
Merlet, J., 1995, “Determination of the Orientation Workspace of Parallel Manipulators,” J. Intell. Rob. Syst., 13(2), pp. 143–160. [CrossRef]
Chen, C., and Jackson, D., 2011, “Parameterization and Evaluation of Robotic Orientation Workspace: A Geometric Treatment,” IEEE Trans. Rob., 27(4), pp. 656–663. [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]


Grahic Jump Location
Fig. 1

LASSPMs and the representative kinematics model. (a) 3SPS-1S, (b) 3UPU wrist, (c) 3RRR, and (d) representative kinematics model.

Grahic Jump Location
Fig. 2

Singularity Loci of LASSPMs. (a) α = π/4, (b) α = sin-1(2/3) ≈ 0.95 with orthogonal base and platform, (c) α = π/3, and (d) variable α and the singularity loci evolution.

Grahic Jump Location
Fig. 3

Limb actuation singularity loci. (a) Leg 1(J1 = 0) and the mechanism singularity configurations, (b) leg 2 (J2 = 0), and (c) leg 3 (J3 = 0).

Grahic Jump Location
Fig. 4

Limb actuation singularity loci in the mechanism singularity loci. (a) α = π/4 and (b) α = sin-1(2/3) with orthogonal base and platform.

Grahic Jump Location
Fig. 5

Assembly zones of the LASSPM with orthogonal base and platform. (a) Connection at infinity, (b) assembly zones, and (c) FK solution distribution.

Grahic Jump Location
Fig. 6

FK solution distribution of the LASSPM with α = π/4

Grahic Jump Location
Fig. 7

Workspace representation and the physical meaning. (a) Workspace representation, (b) corresponding mechanism configuration, and (c) trajectory.

Grahic Jump Location
Fig. 8

Workspace of the LASSPM with α = π/4, ϕimin = 0.6, ϕimax = 2.1. (a) Workspace boundaries of limb 1, (b) mechanism workspace boundaries, and (c) workspace with singularity.

Grahic Jump Location
Fig. 9

Workspace of the LASSPM with orthogonal base and platform with ϕimin = 0.6, ϕimax = 2.1. (a) Mechanism workspace boundaries and (b) workspace with singularity.

Grahic Jump Location
Fig. 10

Workspace of the LASSPM with orthogonal base and platform with ϕimin = 0.8, ϕimax = 1.5. (a) Workspace in z7 and z8, (b) workspace in all assembly zones, and (c) workspace in z1 and z4.




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