The significance of particular computational algorithm by design within the context of a generalized single step single solve order preserving computational framework encompassing the class of LMS methods which directly deals with index 3 differential algebraic equations (DAE's) that naturally arise in multi-body dynamics systems with constraints is presented. For deriving the mathematical model for constraint dynamical systems such as multi-body dynamics, the the resulting governing equations naturally arise as index 3 DAE systems involving a position level constraint. there exist numerous computational difficulties with solving the index 3 systems and most existing techniques either loose their expected order of accuracy or fail to complete the analysis, and consequently index reduction methods are employed which in turn cause other problems such as drift in solutions. The present computational framework of a family of algorithms for index 3 DAE systems described here, have been derived from the GSSSS family of algorithms for linear dynamical non-constraint systems, based on the algorithmic time level consistency theorem. The algorithmic framework is capable of providing stable, robust, and accurate integration of index 3 DAE's directly for both rigid and rigid/flexible multi-body dynamics applications. A challenging numerical example is employed and numerical illustrations demonstrate the significance and importance of the presented technique, namely a novel midpoint rule formulation with midpoint based acceleration in resolving the numerical difficulties in contrast to the classical midpoint rule with endpoint acceleration and the classical Newmark method.