The Udwadia-Kalaba equations provide an explicit solution method for complex constrained multibody systems. This method can decouple the generalized accelerations and generalized constraint forces in the system without introducing auxiliary variables, thereby obtaining explicit expressions for both. However, the equations require the mass matrix to be positive definite, which renders them inapplicable to dynamical systems exhibiting singular problems such as singular mass matrices, singular configurations, and redundant constraints. To address this limitation, some researchers have transformed the U-K equations by introducing auxiliary systems, thereby enabling the actual multibody system and the augmented system to share identical dynamic characteristics, which effectively resolves singular problems. Several methods for modifying the fundamental U-K equations through the introduction of auxiliary systems are discussed in detail, and a general procedure for modeling and solving complex constrained multibody systems using the U-K equations is presented. Two illustrative examples featuring singular mass matrices are solved using the U-K equations and their modified variants.