Flatness-Based Quadcopter Trajectory Planning and Tracking With Continuous-Time Safety Guarantees

Constrained PSO-Splines Trajectory Generation for an Indoor Nanodrone

“…We consider the quadcopter model that neglects aerodynamic effects and external disturbances described as in [6]:…”

“…To express all the penalties in function of the control points, the Bspline properties are exploited. Some proofs to formulate the constraints are given in [6]. However, we provide less conservative expressions for the angular and angular velocities constraints (proofs in appendices I and II).…”

“…v) Thrust constraints: (8e) imposes the thrust to stay in the interval [T min , T max ] during the whole trajectory. The expression in function of the control points is obtained in [6] and its associated penalty term is given in (11e). vi) Angular velocity constraints: (8f) forces the angular velocities' magnitudes to stay inferior to ω max .…”

“…While easy to manipulate, having the trajectory as a collection of interlinked segments raises issues of feasibility, e.g., obstacle collision, sudden changes in attitude which cannot be tracked. To overcome those issues, Bézier [4] or B-spline [5], [6], [7] curves can be used as they have many mathematical and geometrical properties that are useful to design a smooth and feasible trajectory. [6] gives some of the properties, formulates constraints in function of the control points defining the trajectory, preprocess the optimization problem with the tool Yalmip [8] and solves it using the solver MOSEK [9].…”

“…To overcome those issues, Bézier [4] or B-spline [5], [6], [7] curves can be used as they have many mathematical and geometrical properties that are useful to design a smooth and feasible trajectory. [6] gives some of the properties, formulates constraints in function of the control points defining the trajectory, preprocess the optimization problem with the tool Yalmip [8] and solves it using the solver MOSEK [9]. Instead of using any solver that has its own requirements on the formulation of the problem (e.g., convexity, linearity and the like), [7] generates a trajectory using Particle Swarm Optimization.…”