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Higher-order Taylor approximation of finite motions in mechanisms

  • J. J. de Jong (a1), A. Müller (a2) and J. L. Herder (a3)
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.

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*Corresponding author. E-mail: j.j.dejong@utwente.nl
References
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1. Bartkowiak, R. and Woernle, C., “Necessary and sufficient mobility conditions for single-loop overconstrained nH mechanisms,” Mech. Mach. Theory 103, 6584 (2016).
2. Wohlhart, K., “Degrees of shakiness,” Mech. Mach. Theory 34 (7), 11031126 (1999).
3. Wohlhart, K., “From higher degrees of shakiness to mobility,” Mech. Mach. Theory 45, 467476 (Mar. 2010).
4. Chen, C., “The order of local mobility of mechanisms,” Mech. Mach. Theory 46 (9), 12511264 (2011).
5. 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), 559586 (1999).
6. Müller, A., “Higher derivatives of the kinematic mapping and some applications,” Mech. Mach. Theory 76, 7085 (Jun. 2014).
7. Müller, A., “Recursive higher-order constraints for linkages with lower kinematic pairs,” Mech. Mach. Theory 100, 3343 (Jun. 2016).
8. 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, 1528 (Oct. 2011).
9. Ma, T. W., “Higher chain formula proved by combinatorics,” Electron. J. Comb. 16 (1), 17 (2009).
10. Brockett, R. W., “Robotic manipulators and the product of exponentials formula,” Math. Theory Netw. Syst. SE - 9 58, 120129 (1984).
11. Vetter, W. J., “Matrix calculus operations and Taylor expansions,” Soc. Ind. Appl. Math. 15 (2), 352369 (1973).
12. 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.
13. 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), 6681 (2007).
14. Bennet, G., “A new mechanism,” Engineering 76 (12), 777778 (1903).
15. Baker, J. E., “The Bennett, Goldberg and Myard linkages-in perspective,” Mech. Mach. Theory 14 (4), 239253 (1979).
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Robotica
  • ISSN: 0263-5747
  • EISSN: 1469-8668
  • URL: /core/journals/robotica
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