Dynamic modeling of tree-type robotic systems by combining 3 × 3 $3\times3$ rotation and 4 × 4 $4\times4$ transformation matrices

Abstract

The objective of this article is to present a systematic method of deriving the mathematical formulation for the oblique impact of a tree-type robotic manipulator. The dynamic response of this system (confined within a curved-wall environment) is expressed by two diverse models. A set of differential equations is employed to obtain the dynamic behavior of the system when it has no contact with any object in its environment (flying phase), and a set of algebraic equations is used to describe the collision of the system with the curved walls (impact phase). The Gibbs–Appell formulation in recursive form and the Newton’s impact law are utilized to derive the governing equations of this robotic system for the flying and impact phases, respectively. The main innovation of this article is the development of an automatic approach based on the combination of $3\times3$ rotation and $4\times4$ transformation matrices. In fact, this is the first time the merits of $3\times3$ rotation matrices (i.e., improving the computational efficiency of the developed algorithm) have been merged with the capabilities of $4\times4$ transformation matrices (i.e., deriving more compact motion equations by combining rotations with translations). Finally, a case study involving a tree-type robotic system with 12 degrees of freedom has been simulated to show the efficiency of the proposed dynamic modeling.