Redundancy in the constraining of mechanical systems achieves more stability and larger load capacity for the system, while in actuation it provides better robustness against singularities and higher maneuverability. Few techniques have been developed with the aim to handle redundancy and singularities in dynamics analysis, and further research is still needed in this area. In this paper, we illustrate the concept of actuating and passive constraints. Then, we expand on the existing penalty techniques by incorporating the concept of actuating and passive constraints to present a penalty formulation that is capable of efficiently handling singularities and redundancy in constraining and actuation and can carry out either forward or inverse dynamics analysis of mechanical systems. As such, the proposed approach is referred to as the actuating-passive constraints penalty approach.