Integral Line-of-Sight Guidance and Control of Underactuated Marine Vehicles: Theory, Simulations and Experiments Walter Caharija, Kristin Y. Pettersen, Marco Bibuli, Pedro Calado, Enrica Zereik, José Braga, Jan T. Gravdahl, Asgeir J. Sørensen, Milan Milovanovi´ c, Gabriele Bruzzone Abstract—This paper presents an extensive analysis of the integral line-of-sight (ILOS) guidance method for path following tasks of under- actuated marine vehicles, operating on and below the sea surface. It is shown that due to the embedded integral action, the guidance law makes the vessels follow straight lines by compensating for the drift effect of environmental disturbances such as currents, wind and waves. The ILOS guidance is first applied to a 2D model of surface vessels that includes the underactauted sway dynamics of the vehicle as well as disturbances in the form of constant irrotational ocean currents and constant dynamic, attitude dependent, forces. The actuated dynamics are not taken into account at this point. A Lyapunov closed loop analysis yields explicit bounds on the guidance law gains to guarantee uniform global asymptotic stability (UGAS) and uniform local exponential stability (ULES). The complete kinematic and dynamic closed loop system of the 3D ILOS guidance law is analyzed next, hence extending the analysis to underactuated AUVs for 3D straight-line path following applications in the presence of constant irrotational ocean currents. The actuated surge, pitch and yaw dynamics are included in the analysis where the closed loop system forms a cascade, and the properties of UGAS and ULES are shown. The 3D ILOS control system is a generalization of the 2D ILOS guidance. Finally, results from simulations and experiments are presented to validate and illustrate the theoretical results, where the 2D ILOS guidance is applied to the CART and the LAUV vehicles. Index Terms—Path following, LOS guidance, Nonlinear control, Un- deractuated vessels, Experiments, CART USSV, LAUV I. I NTRODUCTION Environmental forces and disturbances such as ocean currents, wind and waves are often referred to as sea loads [1], and their effect can significantly undermine maritime activities and pose serious threats to the people involved. The unavoidable occurrence of dealing with heavy seas and the need to guarantee ship maneuverability as well as safety of the crew on board has lead to improved vessel hulls, smarter navigation techniques and better meteorological forecasts. Unmanned marine vehicles such as autonomous underwater ve- hicles (AUVs), remotely operated vehicles (ROVs) and unmanned surface vehicles (USV) make it possible to operate in otherwise hazardous and inaccessible areas for humans (deep waters or under the ice). In particular, AUVs are becoming more popular and are starting to replace ROVs in activities such as search and rescue, surveying and pipeline inspection [2]. The unmanned USVs are also experiencing a significant development phase: [3] demonstrates that cooperating USVs can perform emergency towing operations, while in [4] a USV is used to retrieve overboard personnel. Supported by the Research Council of Norway through the Centres of Ex- cellence funding scheme, project number 223254 and the Strategic University Program, project number 192427. W. Caharija, K. Y. Pettersen and A. J. Sørensen are with the Centre for Autonomous Marine Operations and Systems, NTNU, Trondheim, Norway. {walter.caharija,kristin.y.pettersen}@itk.ntnu.no, asgeir.sorensen@ntnu.no J. T. Gravdahl is with the Dept. of Engineering Cybernetics, NTNU, Trondheim, Norway. jan.tommy.gravdahl@itk.ntnu.no M. Bibuli, E. Zereik and G. Bruzzone are with ISSIA-CNR, Genova, Italy. {marco.bibuli,enrica.zereik,gabriele.bruzzone}@ ge.issia.cnr.it P. Calado and J. Braga are with LSTS, University of Porto, Portugal. {pdcalado,jose.braga}@fe.up.pt M. Milovanovi´ c is with Rolls-Royce Marine AS, Bergen, Norway. milan.milovanovic@rolls-royce.com Most marine surface vessels are underactuated since they are equipped with fixed stern propellers and steering rudders, or alter- natively with azimuth thrusters only. Even when tunnel thrusters are installed, such actuators are effective exclusively at low maneuvering speeds [5]. Similar arguments apply to underwater vehicles: most existing AUVs are torpedo shaped and equipped with stern propellers, steering rudders and diving rudders only [6]. As a result, the absence of actuation in sway/heave poses significant challenges on the control system design side in path following and trajectory tracking scenarios, especially when the vessel is subject to disturbances acting in the underactuated transverse directions. Whether on the surface or under the surface, many offshore oil and gas activities involve path following tasks of marine vessels. Path following is a motion control scenario where a vessel or underwater vehicle has to follow a predefined path without any time constraints. For a detailed discussion on the fundamental differences between different motion control scenarios the reader is referred to [7]–[10]. A review of different approaches to path following and other control problems of marine vehicles and vessels is given in [11], [12] where both linear and nonlinear control strategies are used. In particular, nonlinear control approaches have become popular since they take into account the dominating nonlinear behavior, and reduced-state stabilization techniques are often used to address nonlinear control problems involving underactuated marine vessels. For instance, [13] proposes a nonlinear controller for 2D path following tasks of 3 degrees-of-freedom (DOFs) underactuated marine vehicles. The work of [13] is further developed in [14] and [15], where 3D and 2D path following is considered. Stabilizing all the DOFs of an underactuated vehicle using a single controller is an ambitious and powerful approach since it gives complete control over the vehicle. The work of [16] presents one of the first solutions to the full-state stabilization problem of underactuated 3-DOFs surface vessels. In [16], the controllers are designed to make the vessel follow a 2D path and to stabilize the heading dynamics. These results are improved in [17] and extended to trajectory tracking in [18]. Motivated by [16], [19] presents a path following control solution for a 3-DOFs underactuated marine vehicle required to follow a straight line. These results are extended to underactuated underwater vehicles for path following of 3D curves in [20]. The Lyapunov direct method and backstepping techniques are exploited for full-state stabilization of underactuated 3-DOFs surface vessels for tracking and path following scenarios in [21], [22]. This paper focuses on the nonlinear line-of-sight (LOS) guidance principle. The nonlinear LOS law is widely used to solve practical path following problems of marine vehicles due to its simplicity and intuitiveness: it imitates a helmsman steering the vessel toward a point lying at a constant distance ahead of the ship along the desired path. In particular, it is used in [23]–[27] for path following control in 2D of fully actuated as well as underactuated ships. In [25] the LOS guidance law for 3-DOFs underactuated surface vessels is tested on a model ship but the zero dynamics and the cross-track error dynamics are not analyzed. The work of [25] is further developed in [28], [29]. The complete kinematic/dynamic closed loop behavior of a LOS guidance system is analyzed in detail with a full state approach