Computer-Aided Design 75–76 (2016) 47–60 Contents lists available at ScienceDirect Computer-Aided Design journal homepage: www.elsevier.com/locate/cad Path planning with obstacle avoidance by G 1 PH quintic splines Carlotta Giannelli a,∗ , Duccio Mugnaini b , Alessandra Sestini a a Dipartimento di Matematica e Informatica ‘‘U. Dini’’, Università di Firenze, Viale Morgagni 67/A, I-50134 Firenze, Italy b Dipartimento di Scienze e Alta Tecnologia, Università degli Studi dell’Insubria, Via Valleggio 11, I-22100 Como, Italy article info Article history: Received 12 July 2015 Accepted 5 February 2016 Keywords: Path planning Obstacle avoidance Spline Pythagorean-hodograph curves Tension parameters abstract We propose a two-step approach for the construction of planar smooth collision-free navigation paths. Obstacle avoidance techniques that rely on classical data structures are initially considered for the identification of piecewise linear paths having no intersection with the obstacles of a given scenario. Variations of the shortest piecewise linear path with angle-based criteria are proposed and discussed. In the second part of the scheme we rely on spline interpolation algorithms with tension parameters to provide a smooth planar control strategy. In particular, we consider the class of curves with Pythagorean structures, because they provide an exact computation of fundamental geometric quantities. A selection of test cases demonstrates the quality of the new motion planning scheme. © 2016 Elsevier Ltd. All rights reserved. 1. Introduction The design of motion planning strategies plays a fundamental role in modern computer applications with focus on different kinds of simulation environments naturally related to robotics, as well as to scientific visualization and interactive navigation [1,2]. The issue of finding an optimal trajectory for a given path should properly combine the geometric part of the motion, usually identified by a path planning scheme, with a suitable time law. The path planning problem includes the identification of paths that do not intersect any obstacle. In order to avoid forbidden configurations related to a given scenario, several graph-like structures may be considered, see for example [3] for a recent survey related to possible collision-free piecewise linear solutions. Using a standard graph search algorithm, a graph with non- negative edge weights can be exploited to compute the path with lowest total cost between any two vertices of the graph. In particular, the output of the algorithm may return an optimal path with respect to a distance (shortest path) criterion. In order to provide an optimal trade-off between the accuracy of the prescribed trajectory and the flexibility of interactive navigations, the information concerning the collision-free piecewise linear path may be subsequently combined with spline interpolation techniques that provide a smooth planar control strategy, see ∗ Corresponding author. E-mail addresses: carlotta.giannelli@unifi.it (C. Giannelli), dmugnaini@uninsubria.it (D. Mugnaini), alessandra.sestini@unifi.it (A. Sestini). e.g., [4]. Previous attempts in this direction usually considered solutions related to classical spline methods [5,6]. By considering interpolation schemes with tension control – see, e.g., [7,8] and the references therein – as a control tool on the shape of the interpolating curve, we present a two-step approach for smooth path planning with obstacle avoidance. In the first step, algorithms for the modification of the shortest piecewise linear path associated to the trapezoidal map and the visibility graph according to simple angle-based criteria are proposed and discussed. In the second step, we consider the class of curves with Pythagorean structures, because they usually provide paths with fair shape and always guarantee exact computation of fundamental geometric quantities like curvature and arc length [9]. This can also facilitate the physical part of the motion which requires accurate arc length and curvature computations [10]. The structure of the paper is as follows. Section 2 provides the preliminary material that introduces the problem setting and the two graph structures considered in the subsequent algorithms, namely the trapezoidal map and the visibility graph. The design of a piecewise linear collision-free path is addressed in Section 3. Different algorithms that rely on the information provided by the above mentioned data structures are presented and discussed. Section 4 provides a brief overview of Pythagorean- hodograph (PH) curves by focusing on G 1 PH quintic Hermite spline interpolants with tension parameters. An asymptotic analysis that can be exploited for choosing the free parameters involved in the interpolation scheme is developed in Section 5. A final illustrative example in a non-trivial obstructed scenario is presented in Section 6. Finally, Section 7 concludes the paper. http://dx.doi.org/10.1016/j.cad.2016.02.004 0010-4485/© 2016 Elsevier Ltd. All rights reserved.