The explanation of two semi-recursive multibody methods for educational purpose

In this paper, two often-employed alternatives to the semi-recursive method are described and compared in terms of velocity transformation. The text offers a simplified way to understand the theory behind the two semi-recursive approaches. Focusing on planar cases, particular attention is given to reference and joint coordinates. Consequently, a multibody modeling analyst that is familiar with global formulations can easily extend his/her knowledge to the semi-recursive approach. Using semi-recursive methods, the open loops are formulated with a reduced set of coordinates, and the constraint equations are avoided in the dynamic equations of motion. Accordingly, computationally efficient forms of matrices and vectors will be generated to represent the dynamic equations of motion, which leads to enhanced numerical efficiency. The difference between the two studied alternatives is the definition of the reference point (i.e., the origin of the body frame) used to define the reference coordinates of the body. The reference point could be either (1) rigidly attached to the moving body for semi-recursive I or (2) coincident with the origin of the global frame for semi-recursive II. The latter leads to simple expressions of velocity transformation.