An Integral Line-of-Sight Guidance Law with a Speed-dependent Lookahead Distance Martin S. Wiig 1,2 , Kristin Y. Pettersen 1,2 , Else-Line M. Ruud 2 and Thomas R. Krogstad 2 Abstract— This paper presents an algorithm that makes an underactuated marine vehicle follow a straight line path while in the presence of a constant ocean current. When following the path, the vehicle maintains a desired surge speed which is measured relative to the water, and which may be constant or time-varying. The algorithm is an integral line-of-sight guidance law where the lookahead distance is designed to depend linearly on the desired relative surge speed of the vehicle. This dependency makes it possible to keep the maneuvering demands of the vehicle limited, even when the vehicle surge speed is large. It is shown that if the desired relative surge speed is constant along the path, the resulting error dynamics has a uniformly semiglobally exponentially stable equilibrium at the origin, thus achieving the path following and velocity control objectives. Furthermore, in the case of a general, time- varying desired speed trajectory, it is shown that the solutions of the system remain bounded. The results are supported by simulations, as well as experiments with an unmanned surface vehicle. I. I NTRODUCTION Precise path following is a requirement for several kinds of marine operations, such as sea bed surveying, underwater pipeline inspection and sub sea photography. To achieve such tasks, the vehicles will rely on a guidance system that steers the vehicle onto the path. These tasks can often be achieved by following a set of straight line segments. However, the desired speed along the path will vary. For example, during a transit task the vehicle may drive as fast as possible, while during a task involving underwater photography the speed needs to be quite low to avoid blurry images. Furthermore, many marine vehicles are underactu- ated and can be modeled as vehicles equipped with stern propellers and steering rudders only. This gives a control force in the forward direction (surge), and a control moment for orientation (yaw), but no sideways (sway) control force. In this paper we investigate a guidance law for straight-line path following for underactuated vehicles at varying desired forward speed. Path following for underactuated marine vehicles has been considered for instance in [1]–[5]. The line-of-sight (LOS) family of guidance laws steers the vehicle towards a point on the path ahead of the it, and have proven well suited This work was partly supported by the Research Council of Norway through the Centres of Excellence funding scheme, project no. 223254 - NTNU AMOS 1 Centre for Autonomous Marine Operations and Systems (NTNU AMOS), Department of Engineering Cybernetics, Norwegian University of Science and Technology, 7491 Trondheim, Norway. Martin.Wiig@itk.ntnu.no 2 Norwegian Defence Research Establishment (FFI), P.O. Box 25, N-2027 Kjeller, Norway. for underactuated vehicles. The algorithm was presented [6], and it was shown in [7] that the algorithm provided uniform global κ-exponential stability (i.e. uniform global asymptotic stability (UGAS) and uniform local exponential stability (ULES) [8]) of the path error and the state errors of a simple vehicle model in 3 degrees of freedom (3-DOF). More complete models of the vehicles were analyzed in [9] and [10], while [11] proved uniform semiglobal exponential stability (USGES) of the LOS guidance. Integral action was added in the integral line-of-sight (ILOS) guidance law in [12], where global stability was proved in the presence of a constant ocean current. By considering the vehicle velocity measured relative to the water, it was possible to extend this result to global κ- exponential stability in [13], and further to USGES in [14]. This is as close to uniform global exponential stability (UGES) which it is possible to get with LOS and ILOS guidance laws, as there is a trigonometric saturation in the kinematic representation. An important design parameter for (I)LOS guidance laws is the lookahead distance Δ. In [15], the speed dependency of the optimal lookahead distance for a given vessel employing the LOS guidance law was investigated. It was shown that the optimal Δ increases with increasing surge speed of the vehicle. This matches with intuition, as a longer lookahead distance will give smoother turns at higher speed. In particu- lar, it is to be expected that the overshoot of the system will be reduced, even in the presence of slow heading controllers. Furthermore, this also matches with how an experienced helmsman would steer a ship; the faster the ship goes, the further ahead the helmsman will look. Hence, in this paper, we propose a lookahead distance that increases linearly with the desired relative surge speed, u rd . Most of the previous work on (I)LOS guidance laws has assumed a constant u rd . However, when the lookahead distance varies with u rd , it is natural to investigate the case when u rd is time-varying. Such a scenario occurs, for example, when an (I)LOS guidance law is combined with a desired surge speed trajectory to obtain trajectory tracking, as in [16] and [17], or formation control, as in [18]–[20]. In [16] and [17], a LOS guidance law is used to steer the vehicle heading, while a surge speed law is used to obtain trajectory following along the path. Straight-line paths and a constant lookahead distance are considered in [16], which proves that the system, including the dynamics, is κ-exponentially stable. More general curved paths, and a lookahead distance that varies with the desired trajectory following speed are considered in [17], which shows local