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

Document Type: Full Length Article


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



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.


  • 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.


Main Subjects

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