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Using the main author's researches on the energies of acceleration and higher order equations of motion, this paper is devoted to new formulations in analytical dynamics of mechanical multibody systems (MBS). Integral parts of these systems are the mechanical robot structures, serial, parallel or mobile on which an application will be presented in order to highlight the importance of the differential motion equations in dynamics behavior. When the components of multibody mechanical systems or in its entirety presents rapid movements or is in transitory motion, are developed higher order variations in respect to time of linear and angular accelerations. According to research of the main author, they are integrated into higher order energies and these in differential equations of motion in higher order, which will lead to variations in time of generalized forces which dominate these types of mechanical systems. The establishing of these differential equations of motion, it is based on a generalization of a principle of analytical differential mechanics, known as the D`Alembert – Lagrange Principle.

This paper is devoted to the presentation of new formulations on the higher order motion energies that are used in the advanced dynamic study of robots. Integral part of these mechanical systems are the mechanical robot structures, on which an application will be presented in order to highlight the importance of the higher order motion energies regarding the dynamic behavior. In current dynamic studies, the kinetic energy is used as a central function in Lagrange - Euler equations. This paper extends the study by developing the acceleration energies of first, second and third order and their implementation in differential equations of motion of third and fourth order, which gives the possibility of applying the initial motion conditions in positions, velocities and accelerations of first and second order. This leads to a more precise control on the transitory motion phases of the multibody systems, in which the robot structures are included.

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