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The equation of Motion by Gibbs - Appell formulation has been used the least for deriving the dynamic equations of manipulator robots. So, in this paper a new systematic method for deriving the equation of motion of n - rigid robotic manipulators with revolute - prismatic joints (R - P - J) is considered. The equation of motion for this robotic system is obtained based on (G - A) formulation. All the mathematical operations are done by only 3×3 and 3×1 matrices. Also, all dynamic characteristics of a link are expressed in the same link local coordinate system. Based on the developed formulation, an algorithm is proposed that recursively and systematically derives the equation of motion. Finally, a computational simulation for a manipulator with three (R - P - J) is presented to show the ability of this algorithm in deriving and solving high degree of freedom of robotic system.

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The main purpose of this paper is to derive the inverse dynamic equation of motion of n-rigid robotic manipulator that mounted on a mobile platform, systematically. To avoid the Lagrange multipliers associated with the nonholonomic constraints the approach of Gibbs-Appell formulation in recursive form is adopted. For modeling the system completely and precisely the dynamic interactions between the manipulator and the mobile platform as well as both nonholonomic constraints associated with the no-slipping and the no-skidding conditions are also included. In order to reduce the computational complexity, all the mathematical operations are done by only 3×3 and 3×1 matrices. Also, all dynamic characteristics of a link are expressed in the same link local coordinate system. Finally, a computational simulation for a manipulator with five revolute joints that mounted on a mobile platform is presented to show the ability of this algorithm in generating the equation of motion of mobile robotic manipulators with high degree of freedom.

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This paper presents a research into the progress of modeling of N-viscoelastic robotic manipulators. The governing equations of the system is obtained by using Gibbs-Appell (G-A) formulation and Assumed Mode Method (AMM). When the beam is short in length direction, shear deformation is a factor that may have substantial effects on the dynamics of the system. So, in modeling the assumption of Timoshenko Beam Theory (TBT) and its associated mode shapes has been considered. Although considering the effects of damping in continuous systems makes the formulations more complex, two important damping mechanisms, namely, Kelvin-Voigt damping as internal damping and the viscous air damping as external damping have been considered. Finally, to validate the proposed formulation a comparative assessment between the results achieved from experiment and simulation is presented in time domain.

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This paper presents the investigation of general formulation and numerical solution of the dynamic load carrying capacity (DLCC) of flexible link manipulator. The proposed method is based on open loop optimal control problem. A two point boundary value problem (TPBVP) is provided, extracted from the Pontryagin's minimum principle. The indirect approach is employed to derive optimality conditions. The system’s dynamics equation of motion is obtained from Gibbs-Appell (G-A) formulation and assumed mode method (AMM). Elastic properties of the links are modeled according to the assumption of Timoshenko beam theory (TBT) and its associated mode shapes. As TBT is more accurate compared with the Euler-Bernoulli beam theory, it is exploited for mathematical modeling of flexible links. The main contribution of the paper is to calculate the maximum allowable load of a flexible link robot while an optimal trajectory is provided. Finally, the result of the simulation and experimental platform are compared for a two-link flexible arm to verify the introduced technique. The efficiency of the proposed method is illustrated by performing some simulation studies on the IUST flexible link manipulator. Simulation and experimental results confirm the validity of the claimed capability for controlling point-to-point motion of the proposed method and its application toward DLCC calculation.

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In this article, the dynamic equations of multiple flexible links robotic manipulators fabricated of functionally graded materials (FGM), whose properties vary continuously along the axial direction and also along the thickness, are examined. Gibbs-Appell methodology and Timoshenko Beam Theory according to the Assumed Mode Method are utilized to obtain the equations of motion and to model the flexible characteristics of links, respectively. Subsequently, the influence of power law index on the vibration response of a two-link functionally graded robotic manipulator is studied for two cases in which the mechanical properties of links vary once along the axial direction and again along the thickness direction of each link. By introducing a parameter called signal energy, it is shown that the power law index has a substantial effect on the vibrational behaviors of the mentioned system; and that by choosing a proper power law index, system vibrations can be reduced considerably in a passive way.