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

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.