Two Topics from Trajectory Planning and Formation Control

Abstract

This thesis explores two topics: spline trajectory planning and rigidity-based formation control. The analyses presented in this thesis for each topic are self-contained and so this thesis is presented in two parts.

The first chapter focuses on spline trajectory planning. We revisit a popular methodology from the quadrotor trajectory planning literature and address two roadblocks encountered in its practical implementation. Specifically, we consider the computational complexity of prototypical algorithms from the literature as well as their scalability. From this analysis, we propose two new algorithms: (i) Algorithm 1, which generates spline trajectories with linear computational complexity and (ii) Algorithm 2, an iterative algorithm that generates time-optimal spline trajectories with linear computational complexity in each iteration. Both methods are faster and plan larger trajectories than the state-of-the-art. We apply our methods to demonstrate their efficacy by conducting an experimental quadrotor flight and by proposing a novel rapidly-exploring random tree (RRT*) algorithm.

The second chapter considers the use of rigidity theory for the formation control problem. We analyse a non-Euclidean norm and present a rigidity theory for frameworks under the 1-norm. We then use this rigidity theory to derive a distributed control law that, under a reasonable condition, comes with an exponentially stability result. The qualities of our new control law are investigated, particularly in comparison with other, related rigidity-based approaches.