An effective solution to the problem of Hermite G 1 interpolation with a clothoid curve is provided. At the beginning the problem is naturally formulated as a system of nonlinear equations with multiple solutions that is generally difficult to solve numerically. All the solutions of this nonlinear system are reduced to the computation of the zeros of a single nonlinear equation. A simple strategy, together with the use of a good and simple guess function, permits to solve the single nonlinear equation with a few iterations of the Newton-Raphson method.The computation of the clothoid curve requires the computation of Fresnel and Fresnel related integrals. Such integrals need asymptotic expansions near critical values to avoid loss of precision. This is necessary when, for example, the solution of interpolation problem is close to a straight line or an arc of circle. Moreover, some special recurrences are deduced for the efficient computation of asymptotic expansion.The reduction of the problem to a single nonlinear function in one variable and the use of asymptotic expansions make the solution algorithm fast and robust.

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We consider a vehicle consisting of a robotic walking assistant pushed by a user. The robot can guide the person along a path and suggest a velocity by various means. The vehicle moves in a crowded environment and can detect the different pedestrian in the surroundings. We propose a reactive planner that modifies the path in order to avoid the pedestrian in the surroundings. The algorithm relies on a very accurate model to predict the motion of each pedestrian, i.e. the Headed Social Force Model (HSFM). The possible trajectories for both the vehicle and the pedestrians are modelled as clothoid curves, which are efficient to manage from the numeric point of view and are very comfortable to follow for the user. Probabilistic techniques are used to account for the variability of the motion of each pedestrian. The path is efficient to generate, is collision free (up to a certain probability) and is comfortable to follow. Simulations and comparisons with a state of the art planner using real data as well as experiments are reported to prove the effectiveness of the method.

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