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Modern robots act in dynamic and partially unknown environments where path replanning can be mandatory if changes in the environment are observed. Task-prioritized control strategies are well known and effective solutions to ensure local adaptation of robot behaviour. The highest priority in a stack of tasks is typically given to the management of correct robot operation or safe interaction with the environment such as obstacles or joint limits avoidance, that we can consider as constraints. If a constraint makes impossible achieving a certain task, such as tracking a Cartesian trajectory, a local control algorithm partially sacrifices the latter which is only accomplished to the best of the robot's ability to generate internal motions. In this control framework, problems may occur in some applications, like in the surgical domain, where it is not safe that some tasks are simply sacrificed without prior notice. The contribution of this work is to introduce a coordinate invariant index, that is used to provide a geometrical interpretation of task conflicts in a task-priority control framework and to develop a method for on-line detection of algorithmic singularities, with the goal of increasing safety and performances during robot operations.

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