A new and robust order preserving computational framework which directly deals with index 3 differential algebraic equations (DAE's) that naturally arise in multi-body dynamics systems with constraints is presented. Various existing methods in the current literature have numerous deficiencies in directly treating such index 3 DAE systems leading to unstable and/or unbounded solutions, loss of convergence rate in the unknown variables and Lagrange multipliers, and/or past approaches often resort to index reduction techniques such as looking for stable solutions in the lower index 2 or index 1 DAE counterparts. This leads to other problems and non-physical issues that are unwanted as highlighted in this paper. In contrast, in this paper we provide a simple, stable and robust computational framework in the two-field form of representation of the equation of motion for purposes of illustration, and therein we overcome most major drawbacks in existing approaches. To sharpen the focus of this study, attention is paid to rigid multi-body dynamic systems and simple numerical examples are provided which demonstrate the proposed overall developments which are ideal for subsequent applications to real world problems.