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The paper deals with the problem of positioning a manipulator in a cluttered environment while avoiding collision with obstacles. Recently a control strategy based on invariant sets has been introduced by some of the authors: it consists of covering the configuration space by means of a connected family of polyhedral regions which can be rendered controlled-invariant. Each of these regions includes some crossing points to the confining (and partially overlapping) regions. The control is hierarchically structured: a high-level controller establishes a proper sequence of regions to be crossed to reach the one in which the target configuration is included. A low-level controller solves the problem of tracking, within a region, the crossing point to the next confining region and, eventually, tracking the reference whenever it is included in the current one. Here we focus on the low-level controller, providing two novel contributions: first we extend the previous results, based on a vertex representation of the polyhedral sets, to the face representation which is more natural and offers significant computational advantages for on-line implementation; second, we provide a new low-level speed-saturated controller in order to improve the performance of the previous one in terms of convergence speed. We also investigate the robustness of the proposed controller. Experimental results on a Cartesian robot are provided.