Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T17:32:47.318Z Has data issue: false hasContentIssue false
  • English
  • Français

Higher-order Taylor approximation of finite motions in mechanisms

Published online by Cambridge University Press:  29 June 2018

J. J. de Jong ,
A. Müller  and
J. L. Herder
J. J. de Jong*
Affiliation:
University of Twente, Enschede, The Netherlands
A. Müller
Affiliation:
Johannes Kepler University, Linz, Austria. E-mail: a.mueller@jku.at
J. L. Herder
Affiliation:
Delft University of Technology, Delft, The Netherlands. E-mail: j.l.herder@tudelft.nl
*
*Corresponding author. E-mail: j.j.dejong@utwente.nl
Get access Rights & Permissions [Opens in a new window]

Summary

Higher-order derivatives of kinematic mappings give insight into the motion characteristics of complex mechanisms. Screw theory and its associated Lie group theory have been used to find these derivatives of loop closure equations up to an arbitrary order. In this paper, this is extended to the higher-order derivatives of the solution to these loop closure equations to provide an approximation of the finite motion of serial and parallel mechanisms. This recursive algorithm, consisting solely of matrix operations, relies on a simplified representation of the higher-order derivatives of open chains. The method is applied to a serial, a multi-DOF parallel, and an overconstrained mechanism. In all cases, adequate approximation is obtained over a large portion of the workspace.

Keywords

Higher-order kinematicsTaylor approximationScrew theory5-bar mechanismBennet linkage
Type
Articles
Information
Robotica , Volume 37 , Special Issue 7: Computational Kinematics Conference, CK 2017 , July 2019 , pp. 1190 - 1201
Copyright
© Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bartkowiak, R. and Woernle, C., “Necessary and sufficient mobility conditions for single-loop overconstrained nH mechanisms,” Mech. Mach. Theory 103, 65–84 (2016).CrossRefGoogle Scholar
Wohlhart, K., “Degrees of shakiness,” Mech. Mach. Theory 34 (7), 1103–1126 (1999).CrossRefGoogle Scholar
Wohlhart, K., “From higher degrees of shakiness to mobility,” Mech. Mach. Theory 45, 467–476 (Mar. 2010).CrossRefGoogle Scholar
Chen, C., “The order of local mobility of mechanisms,” Mech. Mach. Theory 46 (9), 1251–1264 (2011).CrossRefGoogle Scholar
Rico, J., Gallardo, J. and Duffy, J., “Screw theory and higher order kinematic analysis of open serial and closed chains,” Mech. Mach. Theory 34 (4), 559–586 (1999).CrossRefGoogle Scholar
Müller, A., “Higher derivatives of the kinematic mapping and some applications,” Mech. Mach. Theory 76, 70–85 (Jun. 2014).CrossRefGoogle Scholar
Müller, A., “Recursive higher-order constraints for linkages with lower kinematic pairs,” Mech. Mach. Theory 100, 33–43 (Jun. 2016).CrossRefGoogle Scholar
Van der Wijk, V. and Herder, J. L., “Synthesis method for linkages with center of mass at invariant link point Pantograph based mechanisms,” Mech. Mach. Theory 48, 15–28 (Oct. 2011).Google Scholar
Ma, T. W., “Higher chain formula proved by combinatorics,” Electron. J. Comb. 16 (1), 1–7 (2009).Google Scholar
Brockett, R. W., “Robotic manipulators and the product of exponentials formula,” Math. Theory Netw. Syst. SE - 9 58, 120–129 (1984).CrossRefGoogle Scholar
Vetter, W. J., “Matrix calculus operations and Taylor expansions,” Soc. Ind. Appl. Math. 15 (2), 352–369 (1973).Google Scholar
de Jong, J. J., Müller, A., van Dijk, J. and Herder, J. L., “Differentiation-Free Taylor Approximation of Finite Motion in Closed Loop Kinematics,” In: Computational Kinematics. Mechanisms and Machine Science (Zeghloul, S., Romdhane, L. and Laribi, M., eds.), vol. 50 (2018) pp. 577–584.Google Scholar
Husty, M. L., Pfurner, M. and Schröcker, H. P., “A new and efficient algorithm for the inverse kinematics of a general serial 6R manipulator,” Mech. Mach. Theory 42 (1), 66–81 (2007).CrossRefGoogle Scholar
Bennet, G., “A new mechanism,” Engineering 76 (12), 777–778 (1903).Google Scholar
Baker, J. E., “The Bennett, Goldberg and Myard linkages-in perspective,” Mech. Mach. Theory 14 (4), 239–253 (1979).CrossRefGoogle Scholar