Cusp points and assembly changing motions in the PRR-PR-PRR planar parallel manipulator

Abstract

Most parallel manipulators have multiple solutions to the direct kinematic problem. The ability to perform assembly changing motions has received the attention of a few researchers. Cusp points play an important role in the kinematic behavior. This study investigates the cusp points and assembly changing motions in a two degrees of freedom planar parallel manipulator. The direct kinematic problem of the manipulator yields a quartic polynomial equation. Each root in the equation determines the assembly configuration, and four solutions are obtained for a given set of actuated joint coordinates. By regarding the discriminant of the repeated roots of the quartic equation as an implicit function of two actuated joint coordinates, the direct kinematic singularity loci in the joint space are determined by the implicit function. Cusp points are then obtained by the intersection of a quadratic curve and a cubic curve. Two assembly changing motions by encircling different cusp points are highlighted, for each pair of solutions with the same sign of the determinants of the direct Jacobian matrices.

Graphical abstract

Keywords

planar parallel manipulator / assembly changing motions / cusp points / quartic polynomial / discriminant of repeated roots

References

[1] Sun J , Shao L , Fu L F , Han X Y , Li S H . Kinematic analysis and optimal design of a novel parallel pointing mechanism. Aerospace Science and Technology, 2020, 104: 105931

[2] Zhang W X , Zhang W , Ding X L , Sun L . Optimization of the rotational asymmetric parallel mechanism for hip rehabilitation with force transmission factors. Journal of Mechanisms and Robotics, 2020, 12(4): 041006

[3] Merlet J P . Direct kinematics and assembly modes of parallel manipulators. The International Journal of Robotics Research, 1992, 11(2): 150–162

[4] Hunt K H . Structural kinematics of in-parallel-actuated robot-arms. Journal of Mechanisms, Transmissions, and Automation in Design, 1983, 105(4): 705–712

[5] Innocenti C , Parenti-Castelli V . Singularity-free evolution from one configuration to another in serial and fully-parallel manipulators. Journal of Mechanical Design, 1998, 120(1): 73–79

[6] Wenger P, Chablat D. Workspace and assembly modes in fully-parallel manipulators: a descriptive study. In: Lenarčič J, Husty M L, eds. Advances in Robot Kinematics: Analysis and Control. Dordrecht: Springer, 1998, 117–126

[7] Haug E J . Parallel manipulator domains of singularity free functionality. Mechanics Based Design of Structures and Machines, 2021, 49(5): 615–639

[8] Kong X W . Classification of 3-degree-of-freedom 3-UPU translational parallel mechanisms based on constraint singularity loci using Gröbner cover. Journal of Mechanisms and Robotics, 2022, 14(4): 041010

[9] Altuzarra O , Petuya V , Urízar M , Hernández A . Design procedure for cuspidal parallel manipulators. Mechanism and Machine Theory, 2011, 46(2): 97–111

[10] Peidró A , García-Martínez A , Marín J M , Payá L , Gil A , Reinoso O . Design of a mobile binary parallel robot that exploits nonsingular transitions. Mechanism and Machine Theory, 2022, 171: 104733

[11] McAree P R , Daniel R W . An explanation of never-special assembly changing motions for 3-3 parallel manipulators. The International Journal of Robotics Research, 1999, 18(6): 556–574

[12] Zein M , Wenger P , Chablat D . Non-singular assembly-mode changing motions for 3-RPR parallel manipulators. Mechanism and Machine Theory, 2008, 43(4): 480–490

[13] Hernandez A , Altuzarra O , Petuya V , Macho E . Defining conditions for nonsingular transitions between assembly modes. IEEE Transactions on Robotics, 2009, 25(6): 1438–1447

[14] DallaLibera F , Ishiguro H . Non-singular transitions between assembly modes of 2-DOF planar parallel manipulators with a passive leg. Mechanism and Machine Theory, 2014, 77: 182–197

[15] Husty M L. Non-singular assembly mode change in 3-RPR-parallel manipulators. In: Kecskeméthy A, Müller A, eds. Computational Kinematics. Berlin: Springer, 2009, 51–60

[16] Caro S, Wenger P, Chablat D. Non-singular assembly mode changing trajectories of a 6-DOF parallel robot. In: Proceedings of ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Chicago: ASME, 2012, 1245–1254

[17] Husty M, Schadlbauer J, Caro S, Wenger P. The 3-RPS manipulator can have non-singular assembly-mode changes. In: Thomas F, Perez Gracia A, eds. Computational Kinematics. Dordrecht: Springer, 2014, 339–348

[18] Macho E , Petuya V , Altuzarra O , Hernandez A . Planning nonsingular transitions between solutions of the direct kinematic problem from the joint space. Journal of Mechanisms and Robotics, 2012, 4(4): 041005

[19] Bamberger H , Wolf A , Shoham M . Assembly mode changing in parallel mechanisms. IEEE Transactions on Robotics, 2008, 24(4): 765–772

[20] Peidró A , María Marín J , Gil A , Reinoso Ó . Performing nonsingular transitions between assembly modes in analytic parallel manipulators by enclosing quadruple solutions. Journal of Mechanical Design, 2015, 137(12): 122302

[21] Coste M, Chablat D, Wenger P. Nonsingular change of assembly mode without any cusp. In: Lenarčič J, Khatib O, eds. Advances in Robot Kinematics. Cham: Springer, 2014, 105–112

[22] Coste M, Wenger P, Chablat D. Hidden cusps. In: Lenarčič J, Merlet J P, eds. Advances in Robot Kinematics 2016. Cham: Springer, 2018, 129–138

[23] Moroz G , Rouiller F , Chablat D , Wenger P . On the determination of cusp points of 3-RPR parallel manipulators. Mechanism and Machine Theory, 2010, 45(11): 1555–1567

[24] Manubens M , Moroz G , Chablat D , Wenger P , Rouillier F . Cusp points in the parameter space of degenerate 3-RPR planar parallel manipulators. Journal of Mechanisms and Robotics, 2012, 4(4): 041003

[25] Thomas F . A distance geometry approach to the singularity analysis of 3R robots. Journal of Mechanisms and Robotics, 2016, 8(1): 011001

[26] Salunkhe D H , Spartalis C , Capco J , Chablat D , Wenger P . Necessary and sufficient condition for a generic 3R serial manipulator to be cuspidal. Mechanism and Machine Theory, 2022, 171: 104729

[27] Salunkhe D, Capco J, Chablat D, Wenger P. Geometry based analysis of 3R serial robots. In: Altuzarra O, Kecskeméthy A, eds. Advances in Robot Kinematics 2022. Cham: Springer, 2022, 65–72

[28] Kohli D , Spanos J . Workspace analysis of mechanical manipulators using polynomial discriminants. Journal of Mechanisms, Transmissions, and Automation in Design, 1985, 107(2): 209–215

[29] Waldron K J , Hunt K H . Series-parallel dualities in actively coordinated mechanisms. The International Journal of Robotics Research, 1991, 10(5): 473–480

[30] Iqbal H , Khan M U A , Yi B J . Analysis of duality-based interconnected kinematics of planar serial and parallel manipulators using screw theory. Intelligent Service Robotics, 2020, 13(1): 47–62

[31] Macho E, Altuzarra O, Pinto C, Hernandez A. Transitions between multiple solutions of the direct kinematic problem. In: Lenarčič J, Wenger P, eds. Advances in Robot Kinematics: Analysis and Design. Dordrecht: Springer, 2008, 301–310

[32] Bohigas O , Henderson M E , Ros L , Manubens M , Porta J M . Planning singularity-free paths on closed-chain manipulators. IEEE Transactions on Robotics, 2013, 29(4): 888–898

[33] Urízar M , Petuya V , Altuzarra O , Macho E , Hernández A . Computing the configuration space for tracing paths between assembly modes. Journal of Mechanisms and Robotics, 2010, 2(3): 031002

[34] Thomas F, Wenger P. On the topological characterization of robot singularity loci. A catastrophe-theoretic approach. In: Proceedings of 2011 IEEE International Conference on Robotics and Automation. Shanghai: IEEE, 2011, 3940–3945

[35] Burnside W S, Panton A W. The Theory of Equations: With an Introduction to the Theory of Binary Algebraic Forms. 3rd ed. Dublin: Hodges, Figgis and Co., Ltd., 1892

[36] Blinn J F . Quartic discriminants and tensor invariants. IEEE Computer Graphics and Applications, 2002, 22(2): 86–91

[37] Arikawa K . Kinematic analysis of mechanisms based on parametric polynomial system: basic concept of a method using Gröbner cover and its application to planar mechanisms. Journal of Mechanisms and Robotics, 2019, 11(2): 020906

[38] Gosselin C , Angeles J . Singularity analysis of closed-loop kinematic chains. IEEE Transactions on Robotics and Automation, 1990, 6(3): 281–290

Nomenclature

b1, b2, l1, l2	Geometric parameters of the manipulator under study
Ci	Coefficient of the quartic polynomial equation
fi	Constraint equation of a non-redundant manipulator
g	Reduced configuration space of a non-redundant manipulator
h, φ	Pose coordinates of the moving platform of the manipulator under study
I, J	Redefined variables of the quartic polynomial equation
JDKP	Direct Jacobian matrix
JIKP	Inverse Jacobian matrix
p, α	Actuated joint coordinates of the manipulator under study
P, λ	Redefined unknown variables
q	Direct kinematic singularity loci in the joint space of a non-redundant manipulator
t1, t2	Output variables for a non-redundant manipulator
u	Tangent-half-angle of φ
ρ1, ρ2	Input variables for a non-redundant manipulator
\mathrm{\Delta }	Quartic discriminant

Acknowledgements

The author’s contributions are as follows: Xu Pei was in charge of the whole trial; Chengwei Shen wrote the manuscript; Jingjun Yu assisted with sampling. The first author thanks Prof. Lubin Hang of Shanghai University of Engineering Science for his previous introduction on cusp points and singularity-free transitions between DKP solutions in parallel manipulators, which stimulated interest in this study. This research was partially supported by the National Natural Science Foundation of China (Grant Nos. U1813221 and 52075015).

Conflict of Interests

The authors declare no competing financial interests.

RIGHTS & PERMISSIONS

2023 Higher Education Press