Path planning on riemannian manifolds with applications to quadrotor load transportation

Resumen

In the last century, the field of geometric mechanics has proven itself to be invaluable in applications related to physics, aerospace engineering, and robotics. At its core, geometric mechanics seeks to study mechanical systems from a geometric perspective. Often, this amounts to determining the properties of some dynamical system by taking advantage of the geometric properties of its configuration manifold. One particular application that has become increasingly pervasive—and technologically feasible—in recent years is path planning in robotic systems, from robotic manipulators in assembly lines to teams of quadrotor UAVs carrying a payload. The first step in such a problem is to design a control scheme which causes the system to converge to some desired trajectory. Once a trajectory tracking control scheme has been developed, the natural next step is to design the trajectories themselves. In most cases, these trajectories must interpolate some set of knot points, and it is desirable that they minimize some quantity while doing so. A quantity of particular interest is the energy consumption, which in many applications is related to the magnitude of the controls. The magnitude of the controls, in turn, can often be related to the norm of the covariant acceleration of the state variables with respect to the Riemannian metric induced by the kinetic energy—a fundamentally geometric quantity. Thus, such path-planning problems have a natural correspondence with a certain variational problem on a Riemannian manifold, and can thus be studied using the tools of geometric mechanics. This particular variational problem gives rise to curves called Riemannian cubic polynomials, which have studied extensively in the literature over the last 40 years. In this thesis, we are motivated by a slightly different problem. Specifically, one where, in addition to finding optimal interpolants, the resulting trajectories avoid certain prescribed obstacles in phase space, or avoid inter-agent collisions in the case of multi-agent systems. A strategy implemented in recent years to formulate this task as a variational problem (where the tools of geometric mechanics can be applied) is to augment the action functional which we wish to minimize with an artificial potential function which grows large near obstacles. In that sense, it is expected that minimizers of the resulting variational problem will naturally seek to avoid collisions. In this thesis, we study this variational problem—deriving necessary and sufficient conditions for optimality, reducing said equations by taking advantage of the symmetry of the configuration manifold and the associated Riemannian metric, and designing artificial potentials which guarantee obstacle or collision avoidance. We then apply these techniques to the application of teams of quadrotor UAVs transporting a payload via inflexible elastic cables. That is, we design a trajectory tracking control scheme, and then write energy-optimal trajectories in terms of the proposed variational problem