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Dynamic analysis is the basic element of mechanical design and control of mechanisms. This work intends to address dynamic methods relevant to constrained robots and mechanisms from a unified analytical point of view, which is based on differential variational principles. A constrained robotic system is a mechanical system, where we need to consider kinematic constraint conditions explicitly in dynamic modeling and analysis. Important classes of constrained robotic systems include, for example, parallel robots and closed-chain mechanisms where the loop closure conditions can be generally expressed by nonlinear holonomic constraint equations, and mobile robots where the system is subjected to linear nonholonomic constraints. Our primary focus is on systems with nonlinear holonomic constraint equations (e.g., parallel robots, robotic systems with closed kinematic chains). However, the approach and formulation discussed are also applicable for nonholonomic systems. In the framework presented, many approaches can be discussed, and new directions can be highlighted that can contribute to the better understanding of dynamic behavior. Two new approaches for the dynamic analysis and for the simulation of constrained robotic systems are introduced and discussed. The paper also points out some areas and methods where further exploration is necessary to shed light on problems and applications related to constrained robotic systems.