Flatness-based Quadcopter Trajectory Planning and Tracking with Continuous-time Safety Guarantees

Abstract

This work presents a convex optimization framework for the planning and tracking of quadcopter trajectories with continuous-time safety guarantees. Using B-spline basis functions and the differential flatness property of quadcopters, a second-order cone program is formulated to generate optimal trajectories that respect safe state and input constraints in the continuous-time sense. A quadratic program (QP) based on control barrier functions is proposed to guarantee bounded trajectory tracking in continuous time by filtering a nominal controller, where the QP is shown to be always feasible. Furthermore, conditions that ensure the safe tracking controller respects thrust, roll angle, and pitch angle constraints are also proposed. The effectiveness of the proposed framework is demonstrated by real-world experiments using a Crazyflie2.1 nano quadcopter.The video for the experiments of Section VI is available at https://xu.me.wisc.edu/wp-content/uploads/sites/1196/2021/10/ continuous-safety.mp4.

Optimal Control for Kinematic Bicycle Model with Continuous-time Safety Guarantees: A Sequential Second-order Cone Programming Approach

“…In the case of system (1), the flat trajectory y(t) needs to be at least twice-differentiable. B. B-Spline Curves B-splines are common in trajectory generation [4], [16], [17]. A d-th degree B-spline basis λ i,d (t) with d ∈ Z >0 is defined over a given knot vector τ = (τ 0 , .…”

“…, and b T r,j is the j-th row of a time-invariant matrix +r+1) are defined in [17], [19]. Definition 1.…”

“…Definition 1. [17] The columns of P (r) P B r are called the r-th order virtual control points (VCPs) of s (r) (t) and denoted as p (r) i where i = 0, 1, . .…”

“…B-splines have some nice properties such as continuity, convexity, and local support. The following result from [17] ensures continuous-time set inclusion by using such properties.…”

“…While the obstacle-free space D is generally nonconvex, the convexity assumption is easily relaxed by considering the concept of "safe corridor" (union of convex sets) and enforcing the conditions of Proposition 4 segmentwise instead of globally. The reader is referred to our previous work [17] and to [26], [27] for more information.…”