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Medical-technical robotic systems are typical examples in which external contact forces on a system play an important role in the system dynamics. Mathematical modeling of these systems is challenging due to a variety of reasons. Mathematical models for the described class of systems contain differential equations with an associate algebraic equation, which outlines constrained system dynamics. Such a system is considered to be a singular system of differential equations. In this article, controllability criteria for robotic systems with control tasks are investigated. Contact forces can be characterized by stability and consequently controllability conditions. The controllability conditions are investigated in order to achieve the desired dynamical system behavior. The main objective of this study was to find appropriate mathematical representations and solutions to the singular system in which kinematical constraints are imposed on the motion of the robotic system. The geometric approach to the problem of singular systems with these contact constraints has been presented.