Finite-step-integration: An original method for the forward kinematics analysis of parallel manipulators

Abstract

In today’s society, parallel manipulators (PMs) are widely used in industrial production, aerospace, and other fields, where their forward kinematic analyses often serve as the foundation for various tasks, such as design, calibration, and control. In the past few decades, this issue has seemingly been repeatedly addressed using various numerical methods, intelligent algorithms, and algebraic tools. While it is undeniable that solving the equations is easier with current technology, the problem of “how to formulate solvable equations” is often overlooked. This analysis issue typically involves establishing non-linear, multi-parameter, high-dimensional, and strong-coupled mathematical equations, which, from a geometric perspective, is also considered a process of solving a spatial polyhedron. When considering the temporal dimension of motion between two isomorphic polytopes, based on calculus theory, it has been found that this non-linear problem can be transformed into the superposition of multiple iteratively linear equations. Consequently, we propose an original method for the forward kinematic analysis of PMs, namely the finite-step-integration (FSI) method. In this study, the mathematical principles and modeling methods of the FSI method are elucidated, and the modeling and programming processes of the FSI method are demonstrated using general 6-UPS and 3-UPS/S manipulators as examples. Through the analysis of its unique algebraic structure, the methods for singularity determination and motion tracking characteristic analysis are investigated. This method addresses the long-standing challenges in the forward kinematic modeling of PMs, which is applicable for design, calibration, and control, while also offering novel insights for modeling and solving certain non-linear engineering problems.

Graphical abstract

Keywords

finite-step-integration / forward kinematic analysis / parallel manipulator / singularity determination / motion tracking

References

[1] Ye H, Wu J, Huang T. Kinematic calibration of over-constrained robot with geometric error and internal deformation. Mechanism and Machine Theory, 2023, 185: 105345

[2] Latifinavid M, Azizi A. Kinematic modelling and position control of a 3-DOF parallel stabilizing robot manipulator. Journal of Intelligent & Robotic Systems, 2023, 107(2): 17

[3] Han B, Jiang Y, Yang W, Xu Y D, Yao J T, Zhao Y S. Kinematics characteristics analysis of a 3-UPS/S parallel airborne stabilized platform. Aerospace Science and Technology, 2023, 134: 108163

[4] Merlet J P. Solving the forward kinematics of a gough-type parallel manipulator with interval analysis. The International Journal of Robotics Research, 2004, 23(3): 221–235

[5] Geng M C, Zhao T S, Wang C, Chen Y H, Li E W. Forward kinematics analysis of parallel mechanisms with restricted workspace. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2015, 229(14): 2561–2572

[6] Shen C W, Yu J J. Pei X. Cusp points and assembly changing motions in the PRR-PR-PRR planar parallel manipulator. Frontiers of Mechanical Engineering, 2023, 18(2): 27

[7] Liu Y J, Kong M X, Wan N, Ben-Tzvi P. A geometric approach to obtain the closed-form forward kinematics of H4 parallel robot. Journal of Mechanisms and Robotics, 2018, 10(5): 051013

[8] Huang X G, Liao Q Z, Wei S M. Closed-form forward kinematics for a symmetrical 6–6 stewart platform using algebraic elimination. Mechanism and Machine Theory, 2010, 45(2): 327–334

[9] Gallardo-Alvarado J, Garcia-Murillo M A, Alcaraz-Caracheo L A, Torres F J, Sandoval-Castro X Y. Forward kinematics and singularity analyses of an uncoupled parallel manipulator by algebraic screw theory. IEEE Access, 2022, 10: 4513–4522

[10] Zhu G M, Wei S M, Zhang Y, Liao Q Z. Direct kinematic analysis of the spatial parallel mechanism with 3-R(P)S structure based on the point joint relationship. Journal of Mechanisms and Robotics, 2021, 13(6): 061011

[11] Gallardo-Alvarado J, Garcia-Murillo M A, Aguilera-Camacho L D, Alcaraz-Caracheo L A, Sandoval-Castro X Y. An approach to solving the forward kinematics of the 5-RPUR (3T2R) parallel manipulator. Journal of Mechanical Science and Technology, 2023, 37(3): 1443–1453

[12] Cheng G, Li Y, Lodewijks G, Pang Y S, Shan X L. Configuration and singularity analysis of a parallel hip joint simulator based on the forward kinematics. Applied Mathematical Modelling, 2016, 40(15–16): 7281–7292

[13] Gallardo J, Orozco H, Rico J M. Kinematics of 3-RPS parallel manipulators by means of screw theory. The International Journal of Advanced Manufacturing Technology, 2008, 36(5–6): 598–605

[14] Chablat D, Wenger P, Majou F, Merlet J P. An interval analysis based study for the design and the comparison of three-degrees-of-freedom parallel kinematic machines. The International Journal of Robotics Research, 2004, 23(6): 615–624

[15] LiuA X, Yang T L. Kinematic Design of Mechanical Systems. Beijing: China Petrochemical Press, 1999 (in Chinese)

[16] PathakP, Barnwal A K, SriwastavN, SinghR, SinghM. An algorithm based on homotopy perturbation theory and its mathematical analysis for singular nonlinear system of boundary value problems. Mathematical Methods in the Applied Sciences, 2023 (in press)

[17] Gallardo-Alvarado J. An application of the Newton-homotopy continuation method for solving the forward kinematic problem of the 3-RRS Parallel Manipulator. Mathematical Problems in Engineering, 2019, 2019(1): 3123808

[18] Bates D J, Newell A J, Niemerg M. BertiniLab: A MATLAB interface for solving systems of polynomial equations. Numerical Algorithms, 2016, 71(1): 229–244

[19] Shen H P, Zhao Y N, Li J, Wu G L, Chablat D. A novel partially-decoupled translational parallel manipulator with symbolic kinematics, singularity identification and workspace determination. Mechanism and Machine Theory, 2021, 164: 104388

[20] Koupaei J A, Hosseini S M M. A new hybrid algorithm based on chaotic maps for solving systems of nonlinear equations. Chaos, Solitons and Fractals, 2015, 81: 233–245

[21] Liu J B, Yang L, Yang K Q. Nonlinear transform and jacobi elliptic function solutions of nonlinear equations. Chaos, Solitons and Fractals, 2004, 20(5): 1157–1164

[22] Omran A, Kassem A. Optimal task space control design of a Stewart manipulator for aircraft stall recovery. Aerospace Science and Technology, 2011, 15(5): 353–365

[23] Wu X Y, Wu S L, Wang C Z, Luo J F, Wang Y J. Kinematics analysis of a novel hybrid manipulator for optomechanical modules assembly. Fusion Engineering and Design, 2024, 202: 114316

[24] Zhu Q D, Zhang Z. An efficient numerical method for forward kinematics of parallel robots. IEEE Access, 2019, 7: 128758–128766

[25] Wang X S, Hao M L, Cheng Y H. On the use of differential evolution for forward kinematics of parallel manipulators. Applied Mathematics and Computation, 2008, 205(2): 760–769

[26] Mao B Y, Xie Z J, Wang Y B, Handroos H, Wu H P. A hybrid strategy of differential evolution and modified particle swarm optimization for numerical solution of a parallel manipulator. Mathematical Problems in Engineering, 2018, 2018(1): 9815469

[27] Wang Y, Yu J J, Pei X. PEI X. Fast forward kinematics algorithm for real-time and high-precision control of the 3-RPS parallel mechanism. Frontiers of Mechanical Engineering, 2018, 13(3): 368–375

[28] Siciliano B. The tricept robot: Inverse kinematics, manipulability analysis and closed-loop direct kinematics algorithm. Robotica, 1999, 17(4): 437–445

[29] Carmichael M G, Liu D K, Waldron K J. A framework for singularity-robust manipulator control during physical human-robot interaction. The International Journal of Robotics Research, 2017, 36(5–7): 861–876

[30] Dantam N T. Robust and efficient forward, differential, and inverse kinematics using dual quaternions. The International Journal of Robotics Research, 2021, 40(10-11): 1087–1105

[31] Wu X Y, Bai S P. Analytical determination of shape singularities for three types of parallel manipulators. Mechanism and Machine Theory, 2020, 149: 103812

[32] Merlet J P. A formal-numerical approach for robust in-workspace singularity detection. IEEE Transactions on Robotics, 2007, 23(3): 393–402

[33] Ben-Horin P, Shoham M. Application of grassmann-cayley algebra to geometrical interpretation of parallel robot singularities. The International Journal of Robotics Research, 2009, 28(1): 127–141

[34] Kanaan D, Wenger P, Caro S, Chablat D. Singularity analysis of lower mobility parallel manipulators using grassmann-cayley algebra. IEEE Transactions on Robotics, 2009, 25(5): 995–1004

[35] Yang S F, Li Y M. Classification and analysis of constraint singularities for parallel mechanisms using differential manifolds. Applied Mathematical Modelling, 2020, 77: 469–477

[36] Voglewede P A, Ebert-Uphoff I. Overarching framework for measuring closeness to singularities of parallel manipulators. IEEE Transactions on Robotics, 2005, 21(6): 1037–1045

[37] Yao H J, Li Q C, Chen Q H, Chai X X. Measuring the closeness to singularities of a planar parallel manipulator using geometric algebra. Applied Mathematical Modelling, 2018, 57: 192–205

[38] Arian A, Danaei B, Abdi H, Nahavandi S. Kinematic and dynamic analysis of the Gantry-Tau, a 3-DOF translational parallel manipulator. Applied Mathematical Modelling, 2017, 51: 217–231

Conflict of Interest

The authors declare no conflict of interest.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 52105035 and 62203094), the Special Central Funds for Guiding Local Scientific and Technological Development, China (Grant No. 236Z1801G), the Higher Education Youth Top Talent Project of Hebei Province, China (Grant No. BJK2024042), the Natural Science Foundation of Hebei Province, China (Grant Nos. E2021203109 and F2023501021), and the Graduate Student Innovation Capability Training and Support Project of Hebei Province, China (Grant No. CXZZBS2024053).

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