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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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Figures

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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