Safe and simultaneous arrival of constant speed, constant altitude unmanned air vehicles (UAVs) on target is solved by design of paths of equal lengths. The starting point for our solution is the well-known Dubins path, which is composed of circular arc and line segments, thus requiring only one simple maneuver—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 segments entails discontinuous changes in lateral acceleration (latax), making this approach impractical for real fixed wing UAVs. Therefore, we replace the Dubins solution with a novel one, based on quintic Pythagorean hodograph (PH) curves, whose latax demand is continuous. 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 PH paths, 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 but also has enough design parameters (polynomial coefficients) to meet the latax (curvature) constraints (bounds) and make the PH solution close to the minimum time one. The solution involves the design of paths meeting the curvature constraint and is followed by producing multiple paths of equal length by increasing the lengths of the shorter paths to match the longest one. The safety constraint, intercollision avoidance is achieved by satisfying two conditions: minimum separation distance and nonintersection at equal distance. The offset curves of the PH path are used to design a safety region along each path.
Differential Geometric Path Planning of Multiple UAVs
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Shanmugavel, M., Tsourdos, A., White, B. A., and Żbikowski, R. (January 16, 2007). "Differential Geometric Path Planning of Multiple UAVs." ASME. J. Dyn. Sys., Meas., Control. September 2007; 129(5): 620–632. https://doi.org/10.1115/1.2767657
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