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There are many robot system configurations where external contact forces on a robot play an important role in the system dynamics. Such contact forces arise due to constraints on the motion of the robot. Mathematical models of such robot system naturally give rise to a mathematical system of differential equations and algebraic equations, that can be viewed as a singular system of differential equations. Such singular models are developed for several robot system configurations. A general reduction approach is presented that can, in principle, form the basis for the development of planning and feedback control schemes for such robot system configurations.