A cuspidal serial robot can change inverse kinematic solutions (IKS) without crossing singularities because it has multiple IKS in a singularity-free region. This property of robots has been researched for over thirty years but has not been taken seriously when designing new robots. The presented work points out issues related to nonsingular change of IKS and path planning specific to the cuspidal robots present in existing commercial robots used in various applications. The multiple IKS at the initial end-effector pose allows the user to choose an initial IKS that may lead to a continuous and repeatable path. We analyze in detail how the initial IKS choice affects the prescribed path’s feasibility and repeatability. Cuspidal robots can be used safely if the workspace is analyzed, considering the cuspidality property. For these reasons, we propose a path testing and planning methodology that considers different path scenarios. Given the rise of unconventional designs in 6R robots, the identification of cuspidal properties in the design phase of a robot is of paramount importance. We recall all the known criteria for cuspidality and propose new methods to decide if a given 6R robot is cuspidal. Accordingly, a practical guideline is proposed for deciding the cuspidality of a generic 6R robot.
AnderssonJAEGillisJHornG, et al. (2019) CasADi - a software framework for nonlinear optimization and optimal control. Mathematical Programming Computation.
2.
AngererAHofbaurMW (2013) Industrial versatility of inverse kinematics algorithms for general 6r manipulators. In: 16th International Conference on Advanced Robotics, ICAR 2013, 25-29 November 2013. Montevideo, Uruguay: IEEE, 1–7. DOI: 10.1109/ICAR.2013.6766464.
3.
AstudilloAGillisJDiehlM, et al. (2022) Position and orientation tunnel-following nmpc of robot manipulators based on symbolic linearization in sequential convex quadratic programming. IEEE Robotics and Automation Letters7: 2867–2874. DOI: 10.1109/LRA.2022.3142396.
4.
BailiMWengerPChablatD (2004) A classification of 3R orthogonal manipulators by the topology of their workspace. In: A Classification of 3R Orthogonal Manipulators by the Topology of Their Workspace. New Orleans, LA, USA: IEEE, Vol. 2, 1933–1938.
5.
BockHPlittK (1984) A multiple shooting algorithm for direct solution of optimal control problems. IFAC Proceedings Volumes.
6.
BorrelPLiegeoisA (1986) A study of multiple manipulator inverse kinematic solutions with applications to trajectory planning and workspace determination. In: Proceedings. 1986 IEEE International Conference on Robotics and Automation. San Francisco, CA, USA: Institute of Electrical and Electronics Engineers, 3, 1180–1185.
7.
BurdickJW (1989) On the inverse kinematics of redundant manipulators: characterization of the self-motion manifolds. In: 1989 International Conference on Advanced Robotics. Ohio: IEEE, 4, 10.
8.
CapcoJManongsongSM (2019) Implementing hupf algorithm for the inverse kinematics of general 6r/p manipulators. In: EnglandMKoepfWSadykovTM, et al. (eds.) Computer Algebra in Scientific Computing. Cham: Springer International Publishing, 78–90. ISBN 978-3-030-26831-2.
9.
CapcoJSafey El DinMSchichoJ (2020) Robots, computer algebra and eight connected components. In: ISSAC ’20: International Symposium on Symbolic and Algebraic Computation, ISSAC’20: Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation. Kalamata / Virtual, Greece: ACM, 62–69.
10.
ChablatDPrébetRSafey El DinM, et al. (2022) Deciding cuspidality of manipulators through computer algebra and algorithms in real algebraic geometry. In: Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation, ISSAC ’22. Villeneuve-d’Ascq, France: Association for Computing Machinery, 439–448, ISBN 9781450386883. DOI:10.1145/3476446.3535477.
11.
DenavitJHartenbergRS (1955) A kinematic notation for lower-pair mechanisms based on matrices. Journal of Applied Mechanics22(2): 215–221. DOI: 10.1115/1.4011045.
12.
DotyKLMelchiorriCSchwartzEM, et al. (1995) Robot manipulability. In: IEEE Transactions on Robotics and Automation. Piscataway: IEEE.
13.
El OmriJWengerP (1995) How to recognize simply a non-singular posture changing 3-dof manipulator. In: Proceedings of the 7th International Conference on Advanced Robotics. Barcelona: Universitat Politècnica de Catalunya, 215–222.
14.
GorlaBRenaudM (1984) Modèles des robots manipulateurs: application à leur commande. Toulouse: Cepadues-Éditions.
15.
GosselinCLiuH (2014) Polynomial inverse kinematic solution of the Jaco robot. In: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Volume Volume 5B: 38th Mechanisms and Robotics Conference, V05BT08A055. DOI: 10.1115/DETC2014-34152.
16.
GutierrezAGudaVKMugishaS, et al. (2022) Trajectory planning in dynamics environment: application for haptic perception in safe human-robot interaction. In: DuffyVG (ed) Digital Human Modeling and Applications in Health, Safety, Ergonomics and Risk Management. Health, Operations Management, and Design. Cham: Springer International Publishing, 313–328. ISBN 978-3-031-06018-2.
17.
HustyMLPfurnerMSchröckerHP (2007) A new and efficient algorithm for the inverse kinematics of a general serial 6R manipulator. Mechanism and Machine Theory42(1): 66–81. DOI: 10.1016/j.mechmachtheory.2006.02.001.
18.
InnocentiCParenti-CastelliV (1988) Position analysis of robot manipulators: Regions and sub-regions. In Proceedings of Int. Conf. on Advances in Robot Kinematics150–158.
19.
KhalilWDombreE (2004) Modeling, Identification and Control of Robots. New York: Springer. DOI: 10.1016/B978-1-903996-66-9.X5000-3.
20.
KohliDSpanosJ (1985) Workspace analysis of mechanical manipulators using polynomial discriminants. Journal of Mechanisms, Transmissions, and Automation in Design107(2): 209–215. DOI: 10.1115/1.3258710.
21.
MachoEPetuyaVAltuzarraO, et al. (2012) Planning nonsingular transitions between solutions of the direct kinematic problem from the joint space. Journal of Mechanisms and Robotics4(4): 041005, URL. DOI: 10.1115/1.4007306.
22.
MaciejewskiAAKleinCA (1989) The singular value decomposition: computation and applications to robotics. The International Journal of Robotics Research8(6): 63–79, URL. DOI: 10.1177/027836498900800605.
23.
MakarychevKMakarychevY (2020) Certified algorithms: worst-case analysis and beyond. In: VidickT (ed) 11th Innovations in Theoretical Computer Science Conference (ITCS 2020), Leibniz International Proceedings in Informatics (LIPIcs). Dagstuhl, Germany: Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Vol. 151, 49:1–49:14.
24.
MarauliTDurgeshSHGattringerH, et al. (2023) Time-optimal point-to-point motion planning and assembly mode change of cuspidal manipulators: application to 3r and 6r robot. In: 2023 IEEE/RSJ International Conference on Intelligent Robots and Systems, Detroit, USA: IEEE, 6.
25.
NguyenDQBriotSWengerP (2012) Analysis of the dynamic performance of serial 3r orthogonal manipulators. In: Volume 3: Advanced Composite Materials and Processing; Robotics; Information Management and PLM; Design Engineering, Engineering Systems Design and Analysis. Hoboken: John Wiley & Sons, 175–184. DOI: 10.1115/ESDA2012-82208.
26.
PaganelliD (2008) Topological Analysis of Singularity Loci for Serial and Parallel Manipulators. Bologna, Italy: Universita di Bologna. PhD Thesis.
27.
PaiDLeuM (1992) Genericity and singularities of robot manipulators. IEEE Transactions on Robotics and Automation8(5): 545–559.
28.
PieperDL (1968) The Kinematics of Manipulators under Computer Control. USA: Stanford University. PhD Thesis.
29.
PrimroseE (1986) On the input-output equation of the general 7r-mechanism. Mechanism and Machine Theory21(6): 509–510. DOI: 10.1016/0094-114X(86)90134-5.
30.
SalunkheDH (2023) Cuspidal robots: theoretical study, classification, and application to commercial robots. Nantes: École centrale de Nantes. PhD Thesis.
31.
SalunkheDCapcoJChablatD, et al. (2022a) Geometry based analysis of 3r serial robots. In: AltuzarraOKecskeméthyA (eds) Advances in Robot Kinematics 2022. Cham: Springer International Publishing, 65–72. ISBN 978-3-031-08140-8.
32.
SalunkheDHSpartalisCCapcoJ, et al. (2022b) Necessary and sufficient condition for a generic 3r serial manipulator to be cuspidal. Mechanism and Machine Theory171: 104729.
33.
SalunkheDHChablatDWengerP (2023) Trajectory planning issues in cuspidal commercial robots. In: 2023 IEEE International Conference on Robotics and Automation (ICRA). Piscataway: IEEE, 7426–7432. DOI: 10.1109/ICRA48891.2023.10161444.
34.
SmithDLipkinH (1990) Analysis of fourth order manipulator kinematics using conic sections. In: Proceedings., IEEE International Conference on Robotics and Automation. Cincinnati, OH, USA: IEEE Computer Society Press, 274–278. ISBN 978-0-8186-9061-7. DOI: 10.1109/ROBOT.1990.125986.
35.
ThomasF (2015) A distance geometry approach to the singularity analysis of 3R robots. Journal of Mechanisms and Robotics8(1): 011001. DOI: 10.1115/1.4029500.
36.
TrinhCZlatanovDZoppiM, et al. (2015) A geometrical approach to the inverse kinematics of 6R serial robots with offset wrists. In: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Volume Volume 5C: 39th Mechanisms and Robotics Conference, V05CT08A016.
WächterABieglerLT (2006) On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Mathematical Programming.
39.
WamplerCW (1986) Manipulator inverse kinematic solutions based on vector formulations and damped least-squares methods. IEEE Transactions on Systems, Man, and Cybernetics16(1): 93–101. DOI: 10.1109/TSMC.1986.289285.
40.
WengerP (1992) A new general formalism for the kinematic analysis of all non-redundant manipulators. In: Proceedings of the 1992 IEEE International Conference on Robotics and Automation. France: Nice, 442–447.
41.
WengerP (1997) Design of cuspidal and non-cuspidal robot manipulators. In: Proceedings of International Conference on Robotics and Automation, 3, 2172–2177. DOI:10.1109/ROBOT.1997.619284.
42.
WengerP (1998) Classification of 3R positioning manipulators. Journal of Mechanical Design120(2): 327–332. DOI: 10.1115/1.2826976.
43.
WengerP (2004) Uniqueness domains and regions of feasible paths for cuspidal manipulators. IEEE Transactions on Robotics20(4): 745–750.
44.
WengerPChablatD (2022) A review of cuspidal serial and parallel manipulators. Journal of Mechanisms and Robotics15(4): 040801. DOI: 10.1115/1.4055677.
45.
ZeinMWengerPChablatD (2008) Non-singular assembly-mode changing motions for 3-rpr parallel manipulators. Mechanism and Machine Theory43(4): 480–490. DOI: 10.1016/j.mechmachtheory.2007.03.011.
Please find the following supplemental material available below.
For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.
For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.
