Hyper-Jerk Analysis of Robot Manipulators

  • Jaime Gallardo-Alvarado


In this work the hyper-jerk analysis of robot manipulators is addressed by means of the theory of screws. The reduced hyper-jerk state of a rigid body as observed from another body or reference frame is obtained as a six-dimensional vector by applying the concept of helicoidal vector field. Moreover, this contribution demonstrates that the reduced hyper-jerk state of a rigid body can be considered, similar to the velocity state, as a twist about a screw. Furthermore, the reduced hyper-jerk state is systematically obtained in pure screw form. Finally, a case study, which is verified with the aid of commercially available software, that consists of solving the kinematics, up to the hyper-jerk analysis, of a zero-torsion parallel manipulator is included in order to show the application of the method of kinematic analysis.


Manipulator Helicoidal vector field Twist about a screw Fourth order analysis Screw theory Kinematics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ball, R.S.: A Treatise on the Theory of Screws. Cambridge University Press, Cambridge (1900, reprinted 1998)MATHGoogle Scholar
  2. 2.
    Brand, L.: Vector and Tensor Analysis. Wiley, New York (1947)MATHGoogle Scholar
  3. 3.
    Rico, J.M., Duffy, J.: An application of screw algebra to the acceleration analysis of serial chains. Mech. Mach. Theory 31, 445–457 (1996)CrossRefGoogle Scholar
  4. 4.
    Rico, J.M., Gallardo, J., Duffy, J.: A determination of singular configurations of serial non-redundant manipulators, and their escapement from singularities using Lie products. In: Merlet, J.-P., Ravani, B. (eds.) Computational Kinematics ’95, pp. 143–152. Kluwer, Dordrecht (1995)Google Scholar
  5. 5.
    Rico, J.M., Gallardo, J.: Acceleration analysis, via screw theory and characterization of singularities of closed chains. In: Lenarčič, J., Parenti-Castelli, V. (eds.) Recent Advances in Robot Kinematics, pp. 139–148. Kluwer, Dordretch (1996)CrossRefGoogle Scholar
  6. 6.
    Gallardo-Alvarado, J., Orozco-Mendoza, H., Rodríguez-Castro, R., Rico-Martínez, J.M.: Kinematics of a class of parallel manipulators which generates structures with three limbs. Multibody Syst. Dyn. 17, 27–46 (2007)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Gallardo, J., Orozco, H., Rico, J.M.: Kinematics of 3-RPS parallel manipulators by means of screw theory. Int. J. Adv. Manuf. Tech. 36, 598–605 (2008)CrossRefGoogle Scholar
  8. 8.
    Gallardo-Alvarado, J., Alici, G., Pérez-González, L.P.: A new family of constrained redundant parallel manipulators. Multibody Syst. Dyn. 23, 57–75 (2010)CrossRefMATHGoogle Scholar
  9. 9.
    Rico, J.M., Gallardo, J. and Duffy, J.: Screw theory and higher order kinematic analysis of open serial and closed chains. Mech. Mach. Theory 34, 559–586 (1999)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Gallardo-Alvarado, J., Rico-Martinez, J.M.: Jerk influence coefficients, via screw theory, of closed chains. Meccanica 36, 213–228 (2001)CrossRefMATHGoogle Scholar
  11. 11.
    Gallardo-Alvarado, J.: Jerk distribution of a 6-3 Gough-Stewart platform. Proc. Instn. Mech. Engrs. Part K: J. Multi-body Dyn. 217, 77–84 (2003)CrossRefGoogle Scholar
  12. 12.
    Gallardo-Alvarado, J.: Jerk analysis of a six-degrees-of-freedom three-legged parallel manipulator. Robot. Cim-Int Manuf. 28, 220–226 (2012)CrossRefGoogle Scholar
  13. 13.
    Lipkin, H.: Time derivatives of screws with applications to dynamic and stiffness. Mech. Mach. Theory 4, 259–273 (2005)CrossRefGoogle Scholar
  14. 14.
    Viviani, P., Schneider, R.: A developmental study of the relationship between geometry and kinematics in drawing movements. J. Exp. Psychol. Hum. Percept. Perform. 17, 198–218 (1991)CrossRefGoogle Scholar
  15. 15.
    Balasubramaniam, R., Wing, A.M., Daffertshofer, A.: Keeping with the beat: movement trajectories contribute to movement timing. Exp. Brain Res. 159, 129–134 (2004)Google Scholar
  16. 16.
    Fradet, L., Lee, G., Dounskaia, N.: Origins of submovements in movements of elderly adults. J. NeuroEngineering Rehabil. 5, 28 (2008)CrossRefGoogle Scholar
  17. 17.
    Safronov, V.A.: Chaotic and ordered processes mediating the knee-jerk reflex. Hum. Physiology 35, 306–315 (2009)CrossRefGoogle Scholar
  18. 18.
    Piazzi, A., Visioli, A.: Global-minimum-jerk trajectory planning of robot manipulators. IEEE Trans. Ind. Electron. 47, 140–149 (2000)CrossRefGoogle Scholar
  19. 19.
    Bruijnen, D., van Helvoort, J., van de Molengraft, R.: Realtime motion path generation using subtargets in a rapidly changing environment. Robot. Auton. Syst. 55, 470–479 (2007)CrossRefGoogle Scholar
  20. 20.
    Gasparetto, A., Zanotto, V.: A new method for smooth trajectory planning of robot manipulators. Mech. Mach. Theory 42, 455–471 (2007)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Bagci, C.: Synthesis of double-crank (drag-link) driven mechanisms with adjustable motion and dwell time ratios. Mech. Mach. Theory 12, 619–638 (1977)CrossRefGoogle Scholar
  22. 22.
    Barre, P.-J, Bearee, R., Borne, P., Dumetz, E.: Influence of a jerk controlled movement law on the vibratory behaviour of high-dynamics systems. J. Intell. Robot. Syst. 42, 275–293 (2005)CrossRefGoogle Scholar
  23. 23.
    Naskar, T.K., Acharyya, S.: Measuring cam-follower performance. Mech. Mach. Theory 45, 678–691 (2010)CrossRefMATHGoogle Scholar
  24. 24.
    Macfarlane, S., Croft, E.: Jerk-bounded manipulator trajectory planning: design for real-time applications. IEEE Trans. Robot. Autom. 19, 42–52 (2003)CrossRefGoogle Scholar
  25. 25.
    Dumetz, E., Dieulot, J.-Y., Barre, P.-J., Colas, F., Delplace, T.: Control of an industrial robot using acceleration feedback. J. Intell. Robot. Syst. 46, 111–128 (2006)CrossRefGoogle Scholar
  26. 26.
    Boryga, M., Grabos, A.: Planning of manipulator motion trajectory with higher-degree polynomials use. Mech. Mach. Theory 44, 1400–1419 (2009)CrossRefMATHGoogle Scholar
  27. 27.
    Bearee, R., Barre, P.-J., Bloch, S.: Influence of high-speed machine tool control parameters on the contouring accuracy. Application to linear and circular interpolation. J. Intell. Robot. Syst. 40, 321–341 (2004)Google Scholar
  28. 28.
    Xu, R.Z., Xie, L., Li, C.X., Du, D.S.: Adaptive parametric interpolation scheme with limited acceleration and jerk values for NC machining. Int. J. Adv. Manuf. Tech. 36, 343–354 (2008)Google Scholar
  29. 29.
    Wan, D., Wang, S.L., Zhu, C.C., Meng, F.: Feedrate scheduling and jerk control algorithm for high-speed CNC machining. Int. J. Adv. Manuf. Manag. 17, 216–231 (2009)Google Scholar
  30. 30.
    Sugimoto, K., Duffy, J.: Application of linear algebra to screw systems. Mech. Mach. Theory 17, 73–83 (1982)CrossRefGoogle Scholar
  31. 31.
    Hunt, K.H.: Structural kinematics of in-parallel-actuated robot arms. ASME J. Mech. Trans. Aut. Des. 105, 705–712 (1983)CrossRefGoogle Scholar
  32. 32.
    Lee, K.M., Shah, D.K.: Kinematic analysis of a three-degree-of-freedom in-parallel actuated manipulator. IEEE J. Robot. Autom. 4, 354–360 (1988)CrossRefGoogle Scholar
  33. 33.
    Huang, Z., Fang, Y.F.: Kinematic characteristics analysis of 3 DOF in-parallel actuated pyramid mechanism. Mech. Mach. Theory 31, 1009–1018 (1996)CrossRefGoogle Scholar
  34. 34.
    Kim, H.S., Tsai, L.-W.: Kinematic synthesis of a spatial 3-RPS parallel manipulator. ASME J. Mech. Des. 125, 92–97 (2003)CrossRefGoogle Scholar
  35. 35.
    Liu, C.H., Cheng, S.: Direct singular positions of 3RPS parallel manipulators. ASME J. Mech. Des. 126, 1006–1016 (2004)CrossRefGoogle Scholar
  36. 36.
    Rao, N.M., Rao, K.M.: Multi-position dimensional synthesis of a spatial 3-RPS parallel manipulator. ASME J. Mech. Des., Technical Brief 128, 815–819 (2006)CrossRefGoogle Scholar
  37. 37.
    Sokolov, A., Xirouchakis, P.: Singularity analysis of a 3-DOF parallel manipulator with R-P-S joint structure. Robotica 24, 131–142 (2006)CrossRefGoogle Scholar
  38. 38.
    Huang, Z., Mu, D., Zeng, D.: The screw motion simulation on 3-RPS parallel pyramid mechanism. In: Proceedings of the 2007 IEEE International Conference on Mechatronics and Automation, pp. 2860–2864. Harbin, China (2007)Google Scholar
  39. 39.
    Bonev, I.A.: Direct kinematics of zero-torsion parallel mechanisms. In: Proceedings of the IEEE International Conference on Robotics and Automation, pp. 3851–3856. Pasadena, California (2008)Google Scholar
  40. 40.
    Rad, C.-R, Manic, M., Balan, R., Stan, S.-D.: Real time evaluation of inverse kinematics for a 3-RPS medical parallel robot usind dSpace platform. In: IEEE Proceedins 3rd Conference on Human System Interactions (HSI), pp. 48–53 (2010)Google Scholar
  41. 41.
    Yu, L., Zhang, L., Zhang, N., Yang, S., Wang, D.: Kinematics simulation and Analysis of 3-RPS parallel robot on SimMechanics. In: Proceedings of the 2010 IEEE International Conference on Information and Automation, pp. 2363–2367. Harbin, China (2010)Google Scholar
  42. 42.
    Gallardo, J., Lesso, R., Rico, J.M., Alici, G.: The kinematics of modular spatial hyper-redundant manipulators formed from RPS-type limbs. Robot. Autonomous Syst. 59, 12–21 (2011)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringInstituto Tecnológico de CelayaCelaya, Gto.México

Personalised recommendations