To construct the coordinate system for path following, the following reference frames, including inertial frame I , body-fixed frame B , Serret-Frenet frame F , and resultant velocity frame V (Encarnacao and Pascoal, 2000), are introduced, as shown in Figure 1. The origin of frame B is set to coincide with the vehicle’s center of buoyancy at Q = [ x , y ] ⊤ . In the horizontal plane, the 3 degrees of freedom (DOF) kinematic and dynamic models for the under-actuated marine vehicle are configured as (Fossen, 2011). x ̇ = u ⁡ cos ⁡ ψ − v ⁡ sin ⁡ ψ y ̇ = u ⁡ sin ⁡ ψ + v ⁡ cos ⁡ ψ ψ ̇ = r (1) u ̇ = f u + g u τ u + d u v ̇ = f v + d v r ̇ = f r + g r τ r + d r (2)

In Equations 1, 2, set η = [ x , y , ψ ] ⊤ as the vehicle pose in frame { I } , ν = [ u , v , r ] ⊤ as the vehicle velocity in frame { B } , where the azimuth angle is χ = ψ + β , β = arctan ( v / u ) (Xia et al., 2022). The vector of resultant velocity is defiend as U V = [ U , 0 ] ⊤ in frame { V } , where U = u 2 + v 2 . Additionally, g = [ g u , 0 , g r ] ⊤ , f = [ f u , f v , f r ] ⊤ , d = [ d u , d v , d r ] ⊤ , and τ = [ τ u , 0 , τ r ] ⊤ represent the reciprocal of added mass matrix, corresponding hydrodynamic damping, compound disturbance, and control force and moment (Yuan et al., 2023). Subject to time-varying disturbances, including, uncertain interferences and parameter perturbations, it is denoted that d ̇ ≠ 0 .

In path following, the marine vehicle with length L aims to follow a predefined path continuously parameterized by a time-independent variable ϖ ∈ R as η d = [ x d ( ϖ ) , y d ( ϖ ) , χ d ( ϖ ) ] ⊤ via a virtual target point P = [ x d ( ϖ ) , y d ( ϖ ) ] ⊤ , and the time derivative of position vector is P ̇ d = [ x ̇ d ( ϖ ) ϖ ̇ , y ̇ d ( ϖ ) ϖ ̇ ] ⊤ , where x ̇ d ( ϖ ) = ∂ x d / ∂ ϖ , y ̇ d ( ϖ ) = ∂ y d / ∂ ϖ (Xiang et al., 2017). Thus, the referenced azimuth angle is defined as χ d = arctan ( y ̇ d ( ϖ ) / x ̇ d ( ϖ ) ) . The curvature of referenced path κ p should be limited for the inherent constraints of the vehicle as κ p ≤ 1 / ξ min , where ξ min is the minimum turning radius of the vehicle. Therefore, the problem of path following has converted that the following error η e = η − η d globally converge to a certain neighborhood of zero within a limited time under bounded disturbances.

As stated in reference Breivik and Fossen (2005b), the controlled vehicle aligns with the look-ahead angle χ r in the horizontal plane to reach the desired path. As shown in Figure 2, set s and e represent along-track error and cross-track error, the position errors of path following are P e = [ s , e ] ⊤ = ( R F I ) ⊤ ( Q − P ) , where x e = x − x d and y e = y − y d . According to Yuan et al. (2023), set R V F = [ cos χ r , − sin χ r ; sin χ r , cos χ r ] ⊤ as the rotation matrix of frame { V } with respect to frame { F } , and R F I = [ cos χ d , − sin χ d ; sin χ d , cos χ d ] ⊤ as the rotation matrix of frame { F } with respect to frame { I } . The time derivative of P e are derived in Equation 3 (Yu et al., 2020). P ̇ e = S F ⊤ P e + R V F U V − U d (3) where U d = [ U d , 0 ] ⊤ is the velocity of virtual target point, U d = ϖ ̇ x ̇ d 2 ( ϖ ) + y ̇ d 2 ( ϖ ) , S F = [ 0 , − χ ̇ d ; χ ̇ d , 0 ] ⊤ is skew-symmetric matrix (Breivik and Fossen, 2005a). To stabilize the position errors of the vehicle, the following Lyapunov function candidate is selected as V 11 = 1 2 P e 2 (4)

Take the time derivative of Equation 4 and simplify it as V ̇ 11 = s U d ⁡ cos χ r − U p + e U ⁡ sin χ r (5)

To ensure the Lyapunov function Equation 5 is negative-definite, the update rate of P is set as ϖ ̇ = U d ⁡ cos χ r + k s s x ̇ d 2 ϖ + y ̇ d 2 ϖ (6) with the LOS guidance law is designed as χ r = arctan e Δ (7) where k s > 0 , the look-ahead distance is usually set as Δ ∈ [ 2 L , 5 L ] (Han et al., 2018). Substitute the update velocity Equation 6 and guidance law Equation 7 into Equation 5 as V ̇ 11 = − k s s 2 − U e 2 + Δ 2 e 2 (8)

In Equation 8, the current LOS guidance law features an indeterminate parameter Δ , which limits its applicability across diverse missions and vehicles. For instance, a larger Δ prolongs adjustment time, while a smaller Δ increases oscillations and overshoots. In the basic LOS guidance law, Δ is typically set as a constant.

In this paper, we explore the geometric relationship between the desired path and vehicle, and further find that the cross-track error e and the local path curvature have more effects on Δ . As shown in Figure 3, when Δ = 4 L , the vehicle’s steady-state velocity for a circular path is consistent across different horizontal initial positions, though larger e increases overshoot risk. Figure 4 demonstrates that convergence rate decreases with increasing Δ from a fixed position. Also, it illustrates effect of curvature on errors, revealing Δ does not affect along-track error s . Thus, unlike reference Liu et al. (2017), we disregard the influence of s influence on Δ .

As for the local path curvature κ , the oriented bounding box (OBB) method is used to efficient calculate the point of intersection around the vehicle in green box, as shown in Figure 5. Set OBB(Q,[ x b ⇀ , y b ⇀ , z b ⇀ ],[A,B,C]) is the function to compute the set of points at the boundary of the bounding box (Ding et al., 2004), where [ x b ⇀ , y b ⇀ , z b ⇀ ] are the unit vectors for three axes in frame { B } , and [ A , B , C ] are the length, width and height of the bounding box, as shown in Figure 6.

Combining the cross-track error e and the local path curvature κ into LOS guidance law, a novel GLOS guidance law is proposed to adaptively adjust Δ , and the correction of Δ is designed as Δ = Δ min + Δ max − Δ min e − λ 1 e 2 , n q = 0 Δ min + Δ max − Δ min e − λ 1 e 2 + λ 2 κ , n q ≥ 1 (9) In Equation 9, Δ min = 2 L , Δ max = 5 L (Han et al., 2018), and n q is the number of intersections between OBB(Q,[ x b ⇀ , y b ⇀ , z b ⇀ ],[A,B,C]) and the desired path. λ 1 and λ 2 are adjustable parameters of e and κ , and the selection of λ 2 often needs to consider the length of vehicle L and ξ min .

In the GLOS guidance law, κ differs from the reference Xiang et al. (2015). The latter considers the curvature of the virtual target point on a continuous known desired path, as plotted in Figure 5 as P . By comparison, κ in this paper offers the following advantages:

•The method avoids calculating non-existent curvature at the junction of compound paths.

Set the number of intersections n q for the desired path and bounding box as Q = [ Q 1 , Q 2 , … , Q n ] . According to n q , it is divided into n q = 0 , n q = 1 , and n q ≥ 2 , as shown in Figure 7. As for n q = 0 , the vehicle is far from the desired path, and the update of Δ depends on e . When the vehicle approaches the desired path, the desired path and bounding box intersect. The curvature of discrete points is then calculated to correct Δ . As for n q ≥ 2 , the intersections are sequenced according x value in frame { I } . Set κ = ρ ( Q 1 , Q n ) is calculated according to the curvature of discrete point Q 1 , Q , and Q n . If n q = 2 , κ is calculated with Q 1 , Q , and Q 2 , as shown in Figure 5 with blue line. Set Q 1 and Q 2 are the intersections, and ( Q 1 ) x < ( Q 2 ) x . R and O are the radius and center of circumcircle. Theoretically, the desired path and bounding box have infinite intersection points, but due to the curvature constraint in κ p ≤ 1 / ξ min , there are no more than four intersection points, as shown in Figure 7. Therefore, the calculation of κ is efficient. Especially, as for n q = 1 , that means the vehicle is approaching to enter or exit the desired path, set κ = 0 . Above all, κ ∈ [ 0,1 / ξ min ] , and the pseudo-code of GLOS guidance law is present in Algorithm 1.

Algorithm 1

GLOS.

Input:

In Section 3.2, only position error is controlled. According to Section 2.2, the heading error χ e = χ − χ d must also be considered. Define the Lyapunov function candidate as V 12 = ( 1 − cos χ e ) . By combining with Equation 4, the Lyapunov function candidate for position and posture is constructed as V 1 = V 11 + V 12 (10)

The time derivative of Equation 10 are derived as V ̇ 1 = − k s s 2 − U e 2 + Δ 2 e 2 + r − β ̇ − χ ̇ d sin χ e (11)

In order to convert χ e into the error of virtual angular velocity r e = r − r d according to Equation 11, the referenced virtual angular velocity of yaw r d is designed as r d = β ̇ + χ ̇ d − k r ⁡ sin χ e (12) where the control gain is k r > 0 . Substitute the control law Equation 12 in Equation 11 yields Equation 13 as V ̇ 1 = − k s s 2 − U e 2 + Δ 2 e 2 − k r ⁡ sin 2 χ e (13)

In order to stabilize r e , we introduce an ISMC with a novel switching control law to help the sliding mode surface S r related to r e get to zero. The integral sliding surface S r is defined as S r = r e + a r ∫ 0 t r e d t (14) where a r > 0 is constant. Substitute the dynamic model Equation 2 in the time derivative of Equation 14 yields Equation 15 as S ̇ r = f r + b r τ r + d r − r ̇ d + a r r e (15)

Therefore, the yaw DOF controller is designed as τ r = 1 b r r ̇ d − f ̂ r − d ̂ r − a r r e − 1 b r f r e , S s a t r e (16) f r e , S r = μ σ + 1 + r e − m − σ e − n S r (17)

In Equations 16, 17, f ̂ r and d ̂ r are the estimation values, and s a t ( ) is saturation function according to the reference Patre et al. (2018). μ > 0 is coefficient of variational velocity, m > 0 and n ≥ 1 are the coefficients of approach. When r e is far away from S r , r e and S r are bigger, that is lim r e , S r → ∞ f ( r e , S r ) = μ / σ , r e quickly approaches S r . As r e approaches S r , S r → 0 , that is lim S r → 0 f ( r e , S r ) = μ / ( 1 + r e − m ) , r e quickly get to original point with suppressing for the chattering problem. To verify the stability of whole system in the horizontal plane, consider the following Lyapunov function candidate as V 2 = V 1 + 1 2 S r 2 + 1 2 ε f − 1 f ̃ r 2 + 1 2 ε d − 1 d ̃ r 2 (18) where the estimation error of d r and f r are d ̃ r = d r − d ̂ r and f ̃ r = f r − f ̂ r , the time derivative of Equation 18 is drived and simplified as V ̇ 2 = V ̇ 1 − f r e , S r S r + ε f − 1 f ̃ r f ̇ r + ε d − 1 d ̃ r d ̇ r + ε f − 1 f ̃ r ε f S r − f ̂ ̇ r + ε d − 1 d ̃ r ε d S r − d ̂ ̇ r (19)

In order to set V 2 negative semi-definite, the adaptive interference laws are designed as f ̂ ̇ r = ε f S r d ̂ ̇ r = ε d S r (20)

Substitute the adaptive interference law Equation 20 in Equation 19 as V ̇ 2 ≤ − k s s 2 − U e 2 + Δ 2 e 2 − k r ⁡ sin 2 χ e − f r e , S r S r + ε f − 1 f ̃ r f ̇ r + ε d − 1 d ̃ r d ̇ r (21)

In Equation 21, according to the reference Yuan et al. (2022), d ̃ r d ̇ r ≤ 0 and f ̃ r f ̇ r ≤ 0 . Also, f ( r e , S r ) > 0 , and V ̇ 2 ≤ 0 . If and only if s = 0 , e = 0 , χ e = 0 , and S r = 0 , that V ̇ 2 = 0 . The control system converges asymptotically according to the Lyapunov stability theorem. For the surge velocity u , the PID controller or the dynamic controller in Equation 16 stabilize the error of surge velocity u e . Therefore, this paper will not do too much elaboration.

Case I employs a compound straight-line and curve desired path to verify the effectiveness of the proposed method. Four different methods track the compound desired path starting from [ x , y , ψ ] ⊤ = [ 5 , − 5 , π / 2 ] ⊤ : (1) The proposed method, which is the GLOS guidance law with an improved exponent switching law based on the SMC; (2) Method 1, the LOS guidance law with a traditional switching law based on the SMC; (3) Method 2, the LOS guidance law with an improved exponent switching law based on the SMC and a fixed Δ = 2 L ; (4) Method 3, the LOS guidance law with an improved exponent switching law based on the SMC and a fixed Δ = 5 L . The desired path is set as Equation 21 y d = 4 3 x d , 0 ≤ x d < 30 1600 − x d − 70 2 1 2 + 40 , 30 ≤ x d ≤ 110 4 3 100 − x d + 40 , 110 < x d ≤ 140 (22)

As shown in Figure 8, for Methods 2 and Methods 3, a smaller Δ leads to slower convergence to the desired path, while a larger Δ causes overshoots in the initial phase under the same control law. In contrast, the proposed method and Method one utilize the GLOS guidance law, which helps the vehicle avoid the slow convergence and overshoot issues mentioned above, particularly at the initial position and turning points. Figure 9 displays more details about the following errors of different methods. As shown in Figure 10, compared with Method 1, the improved exponent switching law significantly suppresses τ r oscillations under the same parameters and quickly stabilizes r e with strong anti-disturbance performance.

Case II uses the sinusoidal desired path, which is y d = 20 ⁡ sin ⁡ 0.03 x d , to verify the robustness of proposed method. The desired path is tracked by proposed method from different initial position as: (1) e = 0 , [ x , y , ψ ] ⊤ = [ 0,0 , π / 2 ] ⊤ ; (2) e = 5 , [ x , y , ψ ] ⊤ = [ 0 , − 5 , π / 2 ] ⊤ ; (3) e = 10 , [ x , y , ψ ] ⊤ = [ 5 , − 10 , π / 2 ] ⊤ ; (4) e = 15 , [ x , y , ψ ] ⊤ = [ 10 , − 15 , π / 2 ] ⊤ .

Figure 11 demonstrates the proposed method applied under different initial positions. The GLOS guidance efficiently directs the vehicle regardless of the initial cross-track error distance e . As shown in Figure 12, even when following a desired path with variable curvature, the following errors stabilize at all positions, particularly at the start position and corners with larger curvature. As shown in Figure 13, at the beginning of the following, the change of Δ is mainly affected by cross-track error e , Δ increase with the decrease of e . After following up the desired path, the change of Δ is mainly affected by κ . Δ decrease with the increase of κ . In general, the change of Δ better helps the vehicle to realize path following control.