The kinematic analysis and synthesis of parallel mechanisms, particularly those with lower mobility or redundancy, present significant challenges in robotics and mechanical engineering. Traditional approaches often struggle to precisely define the task-relevant motion space when parasitic motions are present. This article introduces a novel method for identifying the "quotient manifold"—the geometric representation of the pure, task-relevant motion of a mechanism—by leveraging Chasles' decomposition models for finite displacements. The proposed methodology provides a systematic approach to characterize the global motion capabilities of "quotient mechanisms," which are designed to produce a specific set of motions while inherent parasitic motions are ignored or decoupled. We outline a theoretical framework rooted in Lie group theory and screw algebra, detailing the algorithm for representing finite displacements as screw motions and subsequently mapping them to identify the quotient manifold. Hypothetical results demonstrate the method's ability to accurately capture the intended motion space and distinguish it from parasitic motions. This approach offers enhanced clarity in kinematic analysis, provides a robust foundation for the type synthesis and optimal design of lower-mobility parallel mechanisms, and contributes to a deeper understanding of complex mechanical system behavior.

Quotient mechanism, kinematic analysis, quotient manifold,

Russo, M., Zhang, D., Liu, X.-J., and Xie, Z., 2024, “A Review of Parallel Kinematic Machine Tools: Design, Modeling, and Applications,” Int. J. Mach. Tool Manuf., 196, p. 104118.

Gosselin, C., and Schreiber, L.-T., 2018, “Redundancy in Parallel Mechanisms: A Review,” ASME Appl. Mech. Rev., 70(1), p. 010802.

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.

Meng, X., Gao, F., Wu, S., and Ge, Q. J., 2014, “Type Synthesis of Parallel Robotic Mechanisms: Framework and Brief Review,” Mech. Mach. Theory, 78, pp. 177–186.

Ye, W., and Li, Q., 2019, “Type Synthesis of Lower Mobility Parallel Mechanisms: A Review,” Chin. J. Mech. Eng., 32(1), p. 38.

Meng, J., Liu, G., and Li, Z., 2007, “A Geometric Theory for Analysis and Synthesis of Sub-6 DoF Parallel Manipulators,” IEEE Trans. Rob., 23(4), pp. 625–649.

Wu, Y., 2010, “Quotient Kinematics Mechanisms: Modeling, Analysis and Synthesis,” Ph.D. thesis, Shanghai Jiao Tong University, Shanghai, China.

Li, Q., Huang, Z., and Herve, J. M., 2004, “Type Synthesis of 3R2T 5-DOF Parallel Mechanisms Using the Lie Group of Displacements,” IEEE Trans. Rob. Autom., 20(2), pp. 173–180.

Li, Q., and Hervé, J. M., 2010, “1T2R Parallel Mechanisms Without Parasitic Motion,” IEEE Trans. Rob., 26(3), pp. 401–410.

Hervé, J. M., 2006, “Uncoupled Actuation of Pan-Tilt Wrists,” IEEE Trans. Rob., 22(1), pp. 56–64.

Hervé, J. M., 1999, “The Lie Group of Rigid Body Displacements, a Fundamental Tool for Mechanism Design,” Mech. Mach. Theory, 34, pp. 719–730.

Wu, Y., Wang, H., and Li, Z., 2011, “Quotient Kinematics Machines: Concept, Analysis, and Synthesis,” ASME J. Mech. Rob., 3(4), p. 041004.

Dai, J. S., 2020, Screw Algebra and Lie Groups and Lie Algebras, Revised Edition ed, Higher Education Press, Beijing, Chap. 5.

Ball, R. S., 1900, A Treatise on the Theory of Screws, Cambridge University Press, London, Chap. 25.

Gogu, G., 2007, Structural Synthesis of Parallel Robots: Part 1: Methodology, Springer, Dordrecht, Chap. 5.

Gosselin, C. M., Vollmer, F., Cote, G., and Wu, Y. N., 2004, “Synthesis and Design of Reactionless Three-Degree-of-Freedom Parallel Mechanisms,” IEEE Trans. Rob. Autom., 20, pp. 191–199.

Kong, X. W., and Gosselin, C. M., 2008, “Type Synthesis of Six-DOF Wrist-Partitioned Parallel Manipulations,” ASME J. Mech. Des., 130(6), p. 062302.

Huang, Z., and Li, Q. C., 2003, “Type Synthesis of Symmetrical Lower-Mobility Parallel Mechanisms Using the Constraint-Synthesis Method,” Int. J. Rob. Res., 22, pp. 59–79.

Xie, F., Liu, X.-J., You, Z., and Wang, J., 2014, “Type Synthesis of 2T1R-Type Parallel Kinematic Mechanisms and the Application in Manufacturing,” Rob. Comput.-Integr. Manuf., 30(1), pp. 1–10.

Hunt, K. H., 1978, Kinematie Geometry of Mechanisms, Oxford University Press, New York, Chap. 12.

Tosi, D., Legnani, G., Pedrocchi, N., Righettini, P., and Giberti, H., 2010, “Cheope: A New Reconfigurable Redundant Manipulator,” Mech. Mach. Theory, 45(4), pp. 611–626.

Borras, J., Thomas, F., and Torras, C., 2014, “New Geometric Approaches to the Analysis and Design of Stewart–Gough Platforms,” IEEE/ASME Trans. Mechatron., 19(2), pp. 445–455.

Cirillo, A., Cirillo, P., De Maria, G., Marino, A., Natale, C., and Pirozzi, S., 2017, “Optimal Custom Design of Both Symmetric and Unsymmetrical Hexapod Robots for Aeronautics Applications,” Rob. Comput.-Integr. Manuf., 44, pp. 1–16.

Kim, Y.-S., Shi, H., Dagalakis, N., Marvel, J., and Cheok, G., 2019, “Design of a Six-DOF Motion Tracking System Based on a Stewart Platform and Ball-and-Socket Joints,” Mech. Mach. Theory, 133, pp. 84–94.

Selig, J. M., and Di Paola, V., 2024, “Some Five Dimensional Persistent Submanifolds of the Study Quadric,” Mech. Mach. Theory, 196, p. 105632.

Carricato, M., and Zlatanov, D., 2014, “Persistent Screw Systems,” Mech. Mach. Theory, 73, pp. 296–313.

Wu, Y., and Carricato, M., 2017, “Identification and Geometric Characterization of Lie Triple Screw Systems and Their Exponential Images,” Mech. Mach. Theory, 107, pp. 305–323.

Wu, Y., and Carricato, M., 2020, “Persistent Manifolds of the Special Euclidean Group SE(3): A Review,” Comput. Aided Geom. Des., 79, p. 101872.

Wu, Y., and Carricato, M., 2018, “Symmetric Subspace Motion Generators,” IEEE Trans. Rob., 34(3), pp. 716–735.

Lee, C.-C., and Hervé, J. M., 2006, “Translational Parallel Manipulators With Doubly Planar Limbs,” Mech. Mach. Theory, 41(4), pp. 433–455.

Dai, J. S., 2012, “Finite Displacement Screw Operators With Embedded Chasles’ Motion,” ASME J. Mech. Rob., 4(4), p. 041002.

Boothby, W., 2007, An Introduction to Diferentiable Manifolds and Riemannian Geometry, 2nd ed., Academic, Amsterdam, Chap. 6.

Dai, J. S., 2014, Geometrical Foundations and Screw Algebra for Mechanisms and Robotics, Higher Education Press, Beijing, Chap. 5.

Samuel, A. E., McAree, P. R., and Hunt, K. H., 1991, “Unifying Screw Geometry and Matrix Transformations,” Int. J. Rob. Res., 10(5), pp. 454–472.