Automatic formulation of falling multiple flexible-link robotic manipulators using 3×3 rotational matrices

Document Type : Research Article

Author

Assistant Professor, Department of Mechanical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran

10.22064/tava.2017.35085.1032

Abstract

In this paper, the effect of normal impact on the mathematical modeling of flexible multiple links is investigated. The response of such a system can be fully determined by two distinct solution procedures. Highly nonlinear differential equations are exploited to model the falling phase of the system prior to normal impact; and algebraic equations are used to model the normal collision of this open-chain robotic system. To avoid employing the Lagrangian method which suffers from too many differentiations, the governing equations of such complicated system are acquired via the Gibbs-Appell (G-A) methodology. The main contribution of the present work is the use of an automatic algorithm according to 3×3 rotational matrices to obtain the system’s motion equations more efficiently. Accordingly, all mathematical formulations are completed by the use of 3×3 matrices and 3×1 vectors only. The dynamic responses of this system are greatly reliant on the step sizes. Therefore, as well as solving the obtained differential equations by using several ODE solvers, a computer program according to the Runge-Kutta method was also developed. Finally, the computational counts of both algorithms i.e., 3×3 rotational matrices and 4×4 transformation matrices are compared to prove the efficiency of the former in deriving the motion equations.

Highlights

  • Modelling of finite and impulsive motions for a multi-flexible-link system is presented.
  • The recursive Gibbs-Appell formulation is used to derive the motion equations.
  • An algorithm based on 3×3 rotational matrices is applied to derive dynamic equations.
  • Joints’ pre- and post-collision velocities are found related using Newton’s kinematic impact law.
  • Simulation results for single and double elastic links are presented in time domain.

Keywords

Main Subjects

References

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Journal of Theoretical and Applied Vibration and Acoustics
Volume 3, Issue 1 - Serial Number 1
January 2017
Pages 15-40

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