## 2 Literature Review  
An overview of LOS guidance principles for marine craft is offered in [19]. Path following is usually divided into two stages by the cross-track error: convergence and guidance. In convergence, the look-ahead distance is so important that it affects the tracking performance. Large look-ahead distance reduces overshoot and results in smooth, but slow convergence. Small look-ahead distance corresponds to fast convergence to the path, but with a large overshoot [20]. An ALOS method considering a lateral error to solve the above problems is derived in [21; 22]. Simulation results show the effectiveness of this method, so that it is widely used in the research of path following. In addition, there is a strong correlation between the curvature of the desired path and the look-ahead distance if the reference path is curve. The comparisons between look-ahead and lateral error is summarized in [23], and guidelines for more accurate vehicle operations are provided for curve path. However, there is no specific formula for the relationship between look-ahead distance and curvature. Furthermore, the reference path cannot be followed by an underactuated AUV subjected to the effect of a sideslip angle [24]. To solve this problem, scholars have devoted many efforts to the field in recent years. The integral LOS (ILOS) is presented in [25; 26] to counteract the disturbance of ocean currents by using an integral, which may cause the reduced stability margin due to phase lag. A proportional LOS (PLOS) considering the effect of drift angle is presented in [27; 28]. The path tracking deviation under PLOS has been significantly improved. A novel extended state observer-based LOS (ESO-LOS) is presented in [29] to estimate the drift angle, which also has obvious improvements with complicated parameters. A model predictive control scheme with LOS (MPCLOS) is proposed in [30] to formulate the path generation capability and has a good tracking performance. These control methods have been theoretically verified; however, they are rarely used to test their reliability in practical. Therefore, the actual path following performance of these methods is not clear. On the premise of feasibility and parameter simplicity, this manuscript presents a 3 DOFALOS guidance method. The adaptive method mainly considers three degrees of freedom: the cross-track error, the curvature of reference path and the forward speed of AUV. The real-time look-ahead distance is obtained under the established rules. Then, a proportional guidance method related to drift angle is designed to obtain the desired heading. Finally, the fractional order proportion integration differentiation (FOPID) controller is introduced to realize the heading control. Simulation results show the feasibility and effectiveness of the proposed method. Furthermore, the AUV's path following performance is greatly improved in practical applications. The major contributions of the paper are as follows:  
* Firstly, the appropriate rules are deduced, i.e., the larger the cross-track error, the smaller the look-ahead distance. The larger the curvature, the smaller the look-ahead distance. The smaller the forward speed, the smaller the look-ahead distance. In addition, the specific relationship between these 3 DOF (the cross-track error, the path curvature and the forward speed) and the look-ahead is introduced in detail. The change of the intensity of look-ahead distance under three degrees of freedom is depicted in a four-dimensional graph. * Secondly, the proportional guidance method is derived in detail to eliminate the influence of drift angle on path following, which makes the resultant speed of AUV on the desired path. Hence, the 3 DOF adaptive proportional guidance method is constructed based on the above methods. _Lyapunov_ theory proves the stability of the proposed closed-loop system. * Finally, FOPID is employed as a heading controller to realize heading tracking.