# Path planning of multiple autonomous vehicles

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Safe and simultaneous arrival of constant speed, constant altitude UAVs on target is solved by design of paths of equal lengths. The starting point of the solution is the well-known Dubins path which is composed of circular arcs and line segments, thus requiring only one simple manoeuvre - constant rate turn. An explicit bound can be imposed on the rate during the design and the resulting paths are the minimum time solution of the problem. However, transition between arc and line segment entails discontinuous changes in lateral accelerations (latax), making this approach impractical for real fixed wing UAVs. Therefore, the Dubins solution is replaced with clothoid and also a novel one, based on quintic Pythagorean Hodograph (PH) curves, whose latax demand is continuous. The clothoid solution is direct as in the case of the Dubins path. The PH path is chosen for its rational functional form. The clothoid and the PH paths are designed to have lengths close to the lengths of the Dubins paths to stay close to the minimum time solution. To derive the clothoid and the PH paths that way, the Dubins solution is first interpreted in terms of Differential Geometry of curves using the path length and curvature as the key parameters. The curvature of a Dubins path is a piecewise constant and discontinuous function of its path length, which is a differential geometric expression of the discontinuous latax demand involved in transitions between the arc and the line segment. By contrast, the curvature of the PH path is a fifth order polynomial of its path length. This is not only continuous, also has enough design parameters (polynomial coefficients) to meet the latax (curvature) constraints (bounds) and to make the PH solution close to the minimum time one. The offset curves of the PH path are used to design a safety region along each path. The solution is simplified by dividing path planning into two phases. The first phase produces flyable paths while the second phase produces safe paths. Three types of paths are used: Dubins, clothoid and Pythagorean Hodograph (PH). The paths are produced both in 2D and 3D. In two dimensions, the Dubins path is generated using Euclidean and Differential geometric principles. It is shown that the principles of Differential geometry are convenient to generalize the path with the curvature. Due to the lack of curvature continuity of the Dubins path, paths with curvature continuity are considered. In this respect, initially the solution with the Dubins path is extended to produce clothoid path. Latter the PH path is produced using interpolation technique. Flyable paths in three dimensions are produced with the spatial Dubins and PH paths. In the second phase, the flyable paths are tuned for simultaneous arrival on target. The simultaneous arrival is achieved by producing the paths of equal lengths. Two safety conditions: (i) minimum separation distance and (ii) non-intersection of paths at equal distance are defined to maneuver in free space. In a cluttered space, an additional condition, threat detection and avoidance is defined to produce safe paths. The tuning is achieved by increasing the curvature of the paths and by creating an intermediate way-point. Instead of imposing safety constraints, the flyable paths are tested for meeting the constraints. The path is replanned either by creating a new way-point or by increasing the curvature between the way-points under consideration. The path lengths are made equal to that of a reference path.