## Collision Avoidance and Path Following for Underactuated Marine Vehicles

##### Abstract

In this thesis we propose and investigate a collision avoidance algorithm for underactuated marine vehicles, both in two and three dimensions. We also investigate the stability properties of a path following algorithm for such vehicles, this too in two and three dimensions.
The first part of this thesis establishes some mathematical preliminaries, and provides a mathematical model of the vehicles in question. The underactuation of the vehicles become apparent in the vehicle model in that a part of the vehicle's velocity cannot be directly controlled, but is induced when the vehicle is maneuvering. Specifically, when a 2D vehicle turns, the momentum of the vehicle will transform some of its forward speed to a sideways (sway) speed. A similar phenomenon occurs when a 3D vehicle pitches, which will induce a heave speed along the normal axis of the vehicle. This e ect is important to consider both during path following and during collision avoidance.
In the second part of the thesis we examine the integral line of sight (ILOS) guidance law for straight-line path following. The guidance law imitates the way an experienced helmsman steers a ship by aiming the vehicle a certain distance, called the lookahead distance, ahead of it on the path. Integral effect is added to compensate for an ocean current, which is modeled as a kinematic disturbance uniform in time and space.
The ILOS guidance law has been in successful use for several years, for example on the Hugin series of autonomous underwater vehicles (AUVs). In this thesis, we will examine the stability properties of the guidance law, and give conditions under which it can be shown to provide uniform semiglobal exponential stability (USGES) of the closed-loop error dynamics. Furthermore, we propose a new variant of the guidance law, where the lookahead distance is designed to increase with the vehicle speed, and we provide USGES conditions for the speed dependence. This design choice is motivated both by intuition and by practical considerations, and will make the vehicle avoid overshoot and oscillations caused by slow convergence or saturations in the underlying controllers. Finally, we will examine 3D path following, using the ILOS guidance law to steer both heading and pitch. We remove a common assumption that the vehicle is neutrally buoyant in water, and provide conditions of USGES in this scenario as well.
In the third part of the thesis, we focus on collision avoidance in two dimensions. We propose a collision avoidance algorithm, the constant avoidance angle (CAA) algorithm, which employs a novel mechanism for compensation of the obstacle's velocity. The intuition behind the algorithm is simple; the vehicle measures the direction to the edges of the obstacle, and adds an avoidance angle to each of these edges in the direction away from the obstacle. Thus, the algorithm creates two safe directions, one on the port side of the obstacle, and one on the starboard side.
If the obstacle is moving, each of these edges are rotated to compensate for the obstacle's velocity. The resulting directions are safe at a given vehicle speed, which can thus be used as an input to the CAA algorithm. Unlike for algorithms which specify both the direction and magnitude of the velocity required to avoid an obstacle, the CAA algorithm thus provides flexibility in the design of the desired speed trajectory.
We will show how this flexibility can be utilized by applying the algorithm to a unicycle restricted to keep a constant speed, thus demonstrating how the algorithm is suitable for vehicles with a limited speed envelope. This includes both fixed-wing aircraft, which must avoid stalling, and many marine vehicles, which may have a high acceleration cost, and which can lose controllability at low speeds. In the unicycle case, we provide conditions under which safe avoidance of a moving obstacle is guaranteed. Specifcally, we derive an upper bound on the required yaw rate during the maneuver, as well as a lower bound on the required distance from the obstacle at which, at the latest, the vehicle must start the avoidance maneuver in order to turn away safely.
We next apply the algorithm to an underactuated marine vehicle required to keep a constant forward speed. The underactuated sway components of the vehicle makes the vehicle's heading point in a di erent direction than the vehicle's course (i.e. the direction of the vehicle's velocity vector). Thus, even though the vehicle points in a safe direction, it may still move towards a collision. We solve this by the use of a course controller, where we employ a model of the underactuated dynamics in order to steer the vehicle course. We provide conditions under which safe avoidance is still guaranteed, and under which all the control signals in the system remain well de ned. We furthermore combine the collision avoidance law both with a target reaching guidance law, and with a path following guidance law. The results are veri ed both in simulations and through full-scale experiments. In the experiments, we also also demonstrate how multiple obstacles can be handled by the algorithm.
In the fourth and final part of the thesis, we extend the CAA algorithm to 3D. The vehicle now measures a three-dimensional cone to the outline of the obstacle, and each ray of this cone is rotated an avoidance angle away from it, creating an extended vision cone. Obstacle motion is compensated for by a transformation of this vision cone, using the same technique as in two dimensions. The resulting collection of rays constitute a compensated vision cone, where each ray is a provably safe direction. As in the 2D case, the vehicle speed is used as an input to the algorithm, and we first demonstrate this by applying the algorithm to a 3D vehicle with nonholonomic constraints in sway and heave, and with a constant forward speed. We utilize the flexibility offered by operating in 3D by choosing a safe direction which seeks to move behind the obstacle, while minimizing the required pitch and yaw rate. This enables us to build on the results from the analysis of the 2D algorithm in deriving upper bounds on the required pitch and yaw rate during the avoidance maneuver. Furthermore, we are able to limit the vehicle pitch during the maneuver, thus showing how the algorithm can make the vehicle comply with operational constraints often present in practice. Finally, we derive conditions under which a safe avoidance maneuver can be guaranteed.
The 3D CAA algorithm is applied to an underactuated underwater vehicle with underactuation in sway and heave and with a constant desired forward speed. To steer the vehicle's velocity direction, we propose a Flow frame controller, where the Flow frame is defined as a frame aligned with the vehicle's velocity vector. Through the use of this frame, we derive conditions on the controller and on the CAA algorithm under which obstacle avoidance is ensured. The sway and heave speeds are furthermore guaranteed to be bounded during the maneuver, and bounds on the pitch of the vehicle's velocity vector are upheld. The results are verified through several simulations, as well as through full-scale experiments on the Hugin autonomous underwater vehicle, and it is demonstrated how the algorithm can be applied to a multi-obstacle scenario.