Forward kinematics analysis of parallel manipulator using modified global Newton-Raphson method

Abstract

In order to obtain direct solutions of parallel manipulator without divergence in real time, a modified global Newton-Raphson (MGNR) algorithm was proposed for forward kinematics analysis of six-degree-of-freedom (DOF) parallel manipulator. Based on geometrical frame of parallel manipulator, the highly nonlinear equations of kinematics were derived using analytical approach. The MGNR algorithm was developed for the nonlinear equations based on Tailor expansion and Newton-Raphson iteration. The procedure of MGNR algorithm was programmed in Matlab/Simulink and compiled to a real-time computer with Microsoft visual studio.NET for implementation. The performance of the MGNR algorithms for 6-DOF parallel manipulator was analyzed and confirmed. Applying the MGNR algorithm, the real generalized pose of moving platform is solved by using the set of given positions of actuators. The theoretical analysis and numerical results indicate that the presented method can achieve the numerical convergent solution in less than 1 ms with high accuracy (1×10⁻⁹ m in linear motion and 1×10⁻⁹ rad in angular motion), even the initial guess value is far from the root.

Keywords

parallel manipulator / forward kinematics / global Newton-Raphson / real-time system

References

[1] MerletJ. P.Parallel manipulator [M], 2000, Norwell, MA, Kluwer Academic Publishers: 6-12

[2] ChinJ. H., SunY. H., ChengY. M.. Force computation and continuous path tracking for hydraulic parallel manipulators [J]. Control Engineering Practice, 2008, 16(6): 697-709

[3] HeQ.-h., HeX.-y., ZhuJ.-xin.. Fault detection of excavator’s hydraulic system based on dynamic principal component analysis [J]. Journal of Central South University of Technology, 2008, 15(5): 700-705

[4] BandeP., SeibtM., UhlmannE., SahaS. K., RaoP. V. M.. Kinematics analysis of Dodekapod [J]. Mechanism and Machine Theory, 2005, 40(6): 740-756

[5] JaimeG. A., JoseM. R. M., GurselA.. Kinematics and singularity analyses of a 4-DOF parallel manipulator using screw theory [J]. Mechanism and Machine Theory, 2006, 41(9): 1048-1061

[6] PetuyaV., GutierrezJ. M., AlonsoA., AltuzarraO., HernandezA.. A numerical procedure to solve non-linear kinematic problems in spatial mechanism [J]. International Journal for Numerical Methods in Engineering, 2008, 73(6): 825-843

[7] JinY., ChenI. M., YangG. L.. Kinematic design of a 6-DOF parallel manipulator with decoupled translation and rotation [J]. IEEE Transactions on Robotics, 2006, 22(3): 545-551

[8] BaronL., AngelesJ.. The direct kinematics of parallel manipulator under join-sensor redundancy [J]. IEEE Transactions on Robotics and Automation, 2000, 16(1): 12-19

[9] LeeT. Y., ShimJ. K.. Improved dialytic elimination algorithms for the forward kinematics of the general Stewart-Gough platform [J]. Mechanism and Machine Theory, 2003, 38(6): 563-577

[10] SelfridgeR. G., MatthewG. K.. Forward analysis of some special Stewart platforms [J]. Journal of Robotic System, 2000, 17(10): 517-526

[11] RaghavanM.. The Stewart platform of general geometry has 40 configurations [J]. ASME Journal of Mechanical Design, 1993, 115(2): 227-282

[12] KamraR., KohliD., DhingraA. K.. Forward displacement analysis of a six-DOF parallel manipulator actuated by 3R3P and 4R2P chains [J]. Mechanism and Machine Theory, 2002, 37(6): 619-637

[13] DidritO., PetitotM., WalterE.. Guaranteed solution of direct kinematic problems for general configuration of parallel manipulator [J]. IEEE Transactions on Robotics and Automation, 1998, 14(2): 259-265

[14] MerletJ. P.. Solving the forward kinematics of a Gough-type parallel manipulator with interval analysis [J]. International Journal of Robotics Research, 2004, 23(3): 221-235

[15] JiP., WuH.-tao.. A closed-form forward kinematics solution for the 6-6P Stewart platform [J]. IEEE Transactions on Robotics and Automation, 2001, 17(4): 522-526

[16] QuarteroniA., SaccoR., SaleriF.Numerical mathematics [M], 2000, Berlin, Springer

[17] KELLEY C T. Solving nonlinear equations with Newton’s method [M]. Philadelphia: SIAM, 2003: 158-165.

[18] ParikhP. J., LamS. S. Y.. A hybrid strategy to solve the forward kinematics problem in parallel manipulators [J]. IEEE Transactions on Robotics, 2005, 21(1): 18-25

[19] RoccoV., VincenzoP. C.. Accurate and fast body pose estimation by three point position data [J]. Mechanism and Machine Theory, 2007, 42(9): 1170-1183

[20] KimH. S., ChoY. M., LeeK.. Robust nonlinear task space control for 6 DOF parallel manipulator [J]. Automatic, 2005, 41(9): 1591-1600