New and recently developed concepts and ideas useful in obtaining efficient computer algorithms for solving the equations of motion of multibody mechanical systems are presented and discussed. These ideas include the use of Euler parameters, Lagrange’s form of d’Alembert’s principle, quasi-coordinates, relative coordinates, and body connection arrays. The mechanical systems considered are linked rigid bodies with adjoining bodies sharing at least one point, and with no “closed loops” permitted. An explicit formulation of the equations of motion is presented.