Trajectory Planning of UAV Group: Pythagorean Hodograph and Bernstein-Bezier Composite Curves in the Plane

Elementary Clothoid-Based Three-Dimensional Curve for Unmanned Aerial Vehicles

“…Smooth spatial paths for autonomous aerial vehicles are usually generated using Bezier curves, splines or Pythagorean Hodographs, among other curves [20][21][22]42]. However, they do not have a clear physical meaning or require to solve an optimization problem in order to take geometric, kinematic or dynamic constraints into account [21,22].…”

“…Smooth spatial paths for autonomous aerial vehicles are usually generated using Bezier curves, splines or Pythagorean Hodographs, among other curves [20][21][22]42]. However, they do not have a clear physical meaning or require to solve an optimization problem in order to take geometric, kinematic or dynamic constraints into account [21,22]. The curve proposed in this Note inherits the properties from the Clothoid-based 3D curves introduced in [24] and, therefore, it has a physical meaning as its design parameters are related to curvature and torsion sharpness.…”

“…Indeed, they proposed the use of Bézier curves as transition curves, but this type of spatial curve has no physical meaning and it is hard to get the relationship between its parameters and vehicle's physics (kinematics and dynamics). Pythagorean Hodographs are also well extended for UAV planning, but they require to solve an optimization problem in order to take constraints into account [21,22]. Among other spatial curves, clothoids (also known as Euler or Cornu Spirals) play an important role, because of their "nice" geometric properties, including a close relation between physical phenomena (normal acceleration and jerks) with the clothoid scaling parameter.…”