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Parametric curves are extensively used in engineering. The most commonly used parametric curves are, Bézier, B-splines, (NURBSs), and rational Bézier. Each and every one of them has special features, being the main difference between them the complexity of their mathematical definition. While Bézier curves are the simplest ones, B-splines or NURBSs are more complex. In mobile robotics, two main problems have been addressed with parametric curves. The first one is the definition of an initial trajectory for a mobile robot from a start location to a goal. The path has to be a continuous curve, smooth and easy to manipulate, and the properties of the parametric curves meet these requirements. The second one is the modification of the initial trajectory in real time attending to the dynamic properties of the environment. Parametric curves are capable of enhancing the trajectories produced by path planning algorithms adapting them to the kinematic properties of the robot. In order to avoid obstacles, the shape modification of parametric curves is required. In this chapter, an algorithm is proposed for computing an initial Bézier trajectory of a mobile robot and subsequently modifies it in real time in order to avoid obstacles in a dynamic environment.

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