The core of my research is about planning optimal missions and trajectories for multiple autonomous marine vehicles (AMVs), incorporating e.g. obstacle avoidance, inter-vehicle collision avoidance, or communication constraints. The vehicle dynamics are explicitly incorporated, and special attention is paid to generating energy-minimal trajectories, where the full system is modelled in sufficient detail to actually minimize electrical, not mereley mechanical, power requirements over time.

## PRONTO

At the optimization level, we use the "PRojection Operator based Newton method for Trajectory Optimization", short "PRONTO", as first developed by John Hauser in 2002. The method is based on the notion that the trajectories η of a dynamical system can be viewed as lying on a trajectory manifold (see the figure on the right; shown by courtesy of Alessandro Saccon). By use of the projection operator, a path ξ can be projected onto the manifold.

A point ξi on the trajectory manifold defines a tangent space to the manifold (figure by courtesy of Alessandro Saccon).

By virtue of Newton's method, a descent direction ζi can be found on the tangent space, evaluating an underlying cost functional on the trajectory (figure by courtesy of Alessandro Saccon).

Using e.g. the Armijo line search rule in conjunction with a projection onto the manifold completes the Newton step (figure by courtesy of Alessandro Saccon).

The trajectory for the next iteration is the projection of the Newton step (on the tangent space) onto the trajectory manifold (figure by courtesy of Alessandro Saccon).