Composite Curve Path following an Underactuated AUV

/is paper addresses the problem of composite curve path following for an underactuated autonomous underwater vehicle by utilizing an adaptive integral line-of-sight (AILOS) guidance and nonlinear iterative sliding mode (NISM) controller. First, the composite curve path is parametrized by a common scalar variable in a continuous way. /en, the kinematics error of an underactuated vehicle is described based on the nonprojection Frenet–Serret frame with a virtual point, which can be eliminated by the virtual point control and AILOS guidance. Meanwhile, the subpath switching algorithm is studied to realize the global path following for the composite curve path. Besides, the NISM controller is cascaded with the AILOS guidance law, and the cascade structure proved to be globally κ-exponentially stable under the influence of slow time-varying currents. Finally, simulations are considered to demonstrate the effectiveness of the proposed composite curve path following control scheme.


Introduction
In recent years, much research has been done in the field of path following for autonomous underwater vehicles (AUVs). In most of the existing literature, the desired geometric path is a single curve [1,2]. However, as is known, a single curve cannot represent complex shapes with a high degree; it may result in a very intractable and impractical path due to Runge's phenomenon [3]. Hence, utilizing the composite curve path as the desired path is more practical since it is composed of multiple subpaths and can meet the demands of varied tasks flexibly in a complex and limited marine environment.
It is known that the shape and the properties of the composite curve path have a great influence on the path following control. Generally, the composite curve path is obtained by a path planning algorithm with two steps. Firstly, utilizing a path search algorithm, the given order of waypoints is obtained based on certain optimization objectives. en, considering the kinematics constraints of the vehicle, multiple curves (such as straight lines, circular arcs, spiral lines, and polynomial curves) can be used to connect all the waypoints to generate a flyable path [4]. In the existing literature, path following of composite curve path such as successive straight lines, straight lines and circles [5,6], and spline interpolation curve [7] has been studied. However, to the best of the authors' knowledge, no general method has been proposed for tracking the composite curve path, which will be studied in this paper.
As the expressions of curve segments vary, and there is no unified parametrization for the composite curve path, they will be inconvenient to calculate and expand. Hence, for better tracking of the composite curve path, it is necessary to choose an appropriate path description method. To solve this problem, all the curve segments are parametrized in a continuous way in this paper. Besides, each curve segment is parametrized by a common scalar variable with the same interval. However, it is usually difficult for the composite curve path to satisfy parametric continuity between the curve segments. Hence, to realize the global path following of the composite curve path, the problem of subpath switching [5,8] is also discussed.
After the composite curve path has been designed and parametrized, an efficient path following system is proposed. For underactuated AUVs, which have no independent control input in the sway and heaven direction, the line-of-sight (LOS) guidance principle is a very suitable and efficient solution for path following. When combined with guidance, the position and heading can be controlled simultaneously just by the heading control. Hence, utilizing LOS guidance, this has been no longer an underactuated problem in terms of the variables to be controlled [9]. Proportional LOS guidance is one of the most widely used methods [10]. However, it cannot handle the unknown environmental disturbance such as wind, wave, or currents. To solve this problem, the integral line-of-sight (ILOS) guidance law is proposed for path following of straight-line paths in the presence of constant and irrotational ocean currents [11]. A modified version of the ILOS algorithm based on the adaptive compensation of the sideslip angle is proposed for path following of a parametrized curve under unknown environmental disturbances [6]. Besides, adaptive integral line-of-sight (AILOS) guidance law is designed for marine craft exposed to ocean currents. e expression of the cross-track kinematics error is reformulated using the concept of relative velocity [12].
Generally, the LOS guidance laws are intuitive as they are formulated at a kinematics level without using the vehicle parameters. rough a cascaded system approach [13], the LOS guidance principle can be interconnected with a heading controller to achieve the path following control. In detail, the LOS guidance principle transforms the position error into the desired heading angle, and the heading control system is responsible for eliminating the heading error. For the cascaded system, the heading controller can be designed independently, so more concise control law can be obtained, such as the PID (proportional integral derivative) control, the fuzzy control, the sliding mode control, and the neural network control. Different control algorithms have their own advantages and disadvantages. In [6], to realize the LOS path following for Dubins paths, the PID controller has been used for the heading control, which is easy to implement, but there are the problems of integral saturation and slow convergence speed. Sliding mode control has strong robustness against external environmental disturbances. In [12], the sliding mode control is constructed for the heading control. However, the control law is based on the yaw dynamics model and can be affected by the uncertainty of model parameters. In [14], disturbance observers and modified terminal sliding mode control are combined to design a robust disturbance rejection control law for the dynamics control of a X-rudder AUV. Fuzzy control [1] and neural network control [7,15] are usually combined with other control methods in order to identify the dynamical uncertainty and time-varying ocean disturbances.
In this paper, a new composite curve path following controller is proposed for an underactuated AUV, based on nonlinear iterative sliding mode (NISM) controller [16] and AILOS guidance. e main contributions are summarized as follows: (1) To eliminate the tracking error, a cascade structure is established based on AILOS guidance law and a NISM controller, which proved to be UGAS (uniformly globally asymptotically stable). (2) Based on the AILOS guidance law and the subpath switching algorithm, the global path following of the composite curve path in the kinematics layer is realized with the unknown ocean currents.
(3) e NISM control is proposed in the dynamics layer, which has the characteristics of fast convergence and strong antidisturbance ability. In addition, the strictly bounded nonlinear hyperbolic tangent function is used to avoid excessive control input caused by the discontinuity at each connection point between two subpaths. Besides, the incremental feedback control law is designed, which is independent of hydrodynamic parameters. e remainder of this paper is organized as follows. Section 2 describes the concept and parametrization of the composite curve path, where the continuity characteristics of the composite curve path are analyzed, and parametric forms of common curves are given. Section 3 presents the kinematics and dynamics expressions of REMUS (remote environmental monitoring units) vehicles and problem formulation. Section 4 addresses the design and proof of the path following control system. Section 5 validates the effectiveness of the previous design for the composite curve path following with several simulation cases and discussions. Section 6 demonstrates the conclusion of this work.

Preliminaries
e composite curve path can be defined as a set of curves connected in a specific order. Each curve segment is regarded as a subpath. e parametrization and the continuity of the composite curve path are discussed in this section. Besides, the parametric description is proposed for typical curves.

Path Parametrization.
In complex and limited marine environment, the composite paths can meet the demands of varied tasks flexibly by adjusting the type of subpaths. However, as the expressions of subpaths may vary, they will be inconvenient to calculate and hard to expand for the path following of composite curve path.
To handle this problem, the path parametrization method is adopted to describe the composite curve path. Many parametrization methods can be used to describe a path, which may be continuous, discrete, or even hybrid [17]. In this paper, the composite curve path is parametrized in a continuous way, and every curve segment is parametrized by a common scalar variable s ∈ [0, n] (n is a natural number) with the same interval, which makes it more convenient to calculate considering different kinds of curve segments.
For a continuous parametrization, the composite curve path L with n subpaths can be defined as e path is then simplified by the set 2 Mathematical Problems in Engineering where L i (s) denotes the parametrization of the i th subpath of the composite path L, and p is the set of all the points on L. e coordinates of the point p p (s) in the inertial frame I { } are uniquely determined by a specific value s ∈ [0, n] with the following form: With a continuous parametrization, for any given parameters of the composite curve path, the position of the corresponding point can be determined uniquely.
Regular curves are desired for the subpaths, which means that such paths never degenerate into a point nor do they have corners. Specifically, these curves include both straight lines and circles [18]. Parametrization for the regular curves satisfies where p p ′ (s) is the first derivative of the point p p (s) w.r.t. the path parameters. For each subpath, the first and second derivatives w.r.t. s can be described as e first derivative of s w.r.t. time is _ s � (ds/dt), and the path-tangential speed is calculated as e path-tangential angle (or course angle of the path) is computed as e curvature of the path can be calculated as e angular speed of the path can be described as

Path
Smoothness. e smoothness of the desired path has an essential impact on the motion control of the underactuated vehicle. Moreover, two notions can be used to describe the path smoothness, namely, the geometric continuity (GC) and the parametric continuity (PC) [19].
GC is denoted by G n , with n specifying the degree of smoothness. e brief definition of GC up to the second degree can be given as follows: (i) G 0 : the only requirement is that all subpaths are connected (ii) G 1 : the path-tangential angle in the connection point is continuous (iii) G 2 : the tangential angle and the curvature of the path are continuous Similarly, PC is denoted by C n , with n specifying the degree of smoothness. C n up to the second degree can be defined as follows: (i) C 0 : the definition is the same as that of G 0 (ii) C 1 : the velocity vector orientation and magnitude are continuous (iii) C 2 : the acceleration is continuous Compared with GC, PC is a stricter form of continuity which imposes constraints on how the parameter propagates along the path. Moreover, PC is a measure of smoothness for parametrizations.
From (1), it can be concluded that the composite curve path can satisfy at least the C 0 (or G 0 ) continuity. However, higher-order PC is usually hard to realize for the composite curve path with different types of curve segments. e derivative of the parameter in the connection point is discontinuous. Relatively, G 1 and G 2 are easy to build and can be used for path following of the composite curve path.
Some common composite curve paths can be used as examples (see Table 1).

Parametric Description of Typical
Curves. Based on [19], the parametric descriptions of straight lines, circular arcs, and Fermat's spiral with a standard scalar variable are given. e parametric description of the line segment is as follows: where (x 0 , y 0 ) is the starting point, L is the length of the line segment, and χ l is the path-tangential angle of the line segment. Furthermore, the parametrization of circular arcs can be expressed as

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Here, (c x0 , c y0 ) is the center of the circle, and R is the radius. Besides, α 0 and α 1 are the heading angles of vectors from the center of the circle to the starting point and the endpoint.
To avoid singularity, the parametrization of Fermat's spiral can be described as For the mirrored curve of Fermat's spiral, the following parametrization is proposed: where (x 0 , y 0 ) is the starting point, (x end , y end ) is the endpoint, θ is the polar angle, θ end is the polar angle of the endpoint, χ 0 and χ end are the path-tangential angles of Fermat's spiral at the starting and the endpoint, and ρ determines the direction of spiral rotation.

Problem Statement
REMUS vehicles are low-cost AUVs designed by the Woods Hole Oceanographic Institution serving in a range of oceanographic applications, such as surveying and mapping. e vehicles are torpedo-shaped and underactuated without lateral thrust; a propeller and fins are used for steering and diving. Besides, the mathematical model of REMUS vehicles has been well researched, which can be used in motion control simulation of underactuated AUVs [20].
is section describes the kinematics and dynamics expressions of the REMUS vehicles and problem formulation of the composite curve path following.

AUV Model considering Currents.
Considering the influences of currents, the kinematics model in the horizontal plane can be expressed in terms of the relative surge and sway velocities [20,21].
As depicted in Figure 1, (x) and y are the coordinates of the center of mass of the vehicle expressed in the inertial frame I { }. ψ and r define its heading angle and yaw velocity. e pair (V x , V y ) denotes the northeast current velocities in I { }. Hence, the body-fixed current velocities in surge and sway directions (u c , v c ) are given by e relative surge and sway velocity can be defined as where u and v are the surge and sway velocity relative to the Earth. Besides, the relative resultant velocity can be expressed as e REMUS vehicle considered in this paper is based on the following assumptions.

Assumption 1.
e vehicle has two axial planes of symmetry, top-bottom and port-starboard symmetry, respectively.

Assumption 2.
e vehicle center of gravity is the same as the vehicle center of buoyancy, and the origin of the vehicle body-fixed coordinate system is located at the vehicle center of buoyancy.
Neglecting the motions in heave, roll, and pitch directions, the 3-DOF dynamics model of REMUS vehicle in the horizontal plane can be simplified as Path GC Comments Piecewise linear path G 0 Generated directly through waypoints but is not suitable for path following of underactuated AUV Circular smoothing G 1 Generated by approximating methods, and curvature between straight and circular segments is discontinuous Dubins path G 1 Generated by interpolating methods, and curvature between straight and circular segments is discontinuous Clothoid smoothing G 2 Generated by approximating methods, linear varying curvature with an increased computational cost Fermat's spiral smoothing G 2 Generated by interpolating methods, curvature-continuous paths with a very low computational cost compared to clothoid smoothing [19] where ΣX and ΣY define the nonacceleration terms along the x-axis and y-axis, and ΣN defines the nonacceleration terms of rotation along the z-axis. e description of ΣX, ΣY, and ΣNcan be shown as where u r , v r , and r are relative surge, sway, and yaw velocity of the vehicle, which constitute the kinematics status of the vehicle. e thrust X T and the rudder angle δ r denote the control input. e symbols similar to X (·) , Y (·) , and N (·) represent the hydrodynamic parameters of the model, which are used for the calculation of hydrodynamic forces and moments. m represents the mass of the vehicle, and I zz is the moments of inertia. e values of these parameters are presented in Table 2. Figure 2, the composite curve path L is composed of two subpaths, L 1 and L 2 . Q is the origin of the body-fixed coordinate system of the AUV and P is the virtual point moving along the path. e kinematics error is described in the Frenet-Serret frame F { } [22] attached to the point P. However, different from the traditional Frenet-Serret frame F { }, P is not the closest point on the path to the vehicle, but a point which evolves according to the designed control law. In this way, the singularity caused by parametrization with arc length can also be avoided [23]. e tracking error expressed in F { } can be described as

Kinematics Error Description. As depicted in
where x e is the along-track error and y e is the cross-track error, (x, y) and (x p , y p ) are the coordinates of the AUV and the virtual point in the inertial coordinate system, and χ p is the path-tangential angle of the desired path. Differentiating (20) yields the error dynamics build in where r p � _ χ p is the angular speed of the path and β r � a tan 2(v r , u r ) is the drift angle.
Generally, the problem of the composite curve path following for underactuated AUV can be formulated as follows: Given a constant thrust X T and desired composite curve path L, select an appropriate way to parametrize the path (1), design guidance and virtual point control laws to generate the desired heading angle ψ d and realize the global path following, and then develop an active heading controller to Mathematical Problems in Engineering achieve the expected heading angle, so that the path tracking error can be eliminated.

Path following Control Design
To eliminate the tracking error under the influence of unknown static currents, a new path following controller is proposed with cascade structure, as shown in Figure 3. First, the designed composite curve path is parametrized by a common scalar variable in a continuous way, and a point (P) is selected as the virtual target to be followed. en, the kinematics error between the coordinates of the AUV and the virtual target point can be calculated including the along-track error and the cross-track error. To eliminate the along-track error, the speed of the virtual point ( _ s) is introduced as a control input. Besides, utilizing the adaptive ILOS guidance with a current observer, the desired heading angle is given to eliminate the cross-track error. Next, the heading control system (an improved NISM controller) is designed to realize the desired heading angle (ψ e � 0) by controlling the rudder (δ r ) under constant thrust (X T ). Finally, the cascade structure proved to be globally k-exponentially stable under the influence of static currents.

e AILOS Guidance.
e path-tangential speed of the virtual point can be used as a control input [18,23]: where k > 0 is a gain parameter; the virtual point will move toward the direct projection of the vehicle onto the x-axis of F { }, whose purpose is to reduce the along-track error to zero.
Simultaneously, adaptive ILOS guidance law is used to derive a desired heading angle to eliminate the cross-track error. As illustrated in Figure 1, the desired heading angle is designed as [3] with where Δ > 0 is the look-ahead distance along the tangential path in F { }, which is given in meters and usually takes values between 1.5 and 2.5 times the length of a vehicle. Besides, the parameter α is a virtual control input used to compensate for the disturbance of unknown currents.
As the heading error can be expressed as ψ e � ψ − ψ d , the expression of _ y e in (21) can be rewritten as Substituting (25) into (26) gives en, (27) can be rewritten as where To eliminate the influence of the currents, α can be designed as where θ n � −sin χ p V x + cos χ p V y U r .
As the currents considered in this paper are much smaller than the speed of the vehicle, it is easy to conclude Theorem 1. Assume that ψ e � 0, U r > 0, Δ > 0, and V x and V y are known. Furthermore, assume that U p is computed using (22) and ψ d is calculated as (24). en, the equilibrium point (x e , y e ) � (0, 0) of the system (32) is globally k-exponentially stable.
Proof. By substituting ψ e � 0 into (32), the Lyapunov Function Candidate (LFC) V 1 � (1/2)x 2 e + (1/2)y 2 e has the time derivative which is negative since k > 0 and U r > 0. Hence, the equilibrium point [x e , y e ] 2 � 0 is uniformly globally asymptotically stable (UGAS) and uniformly locally exponentially stable (ULES) or globally k-exponentially stable [24]. However, currents are generally difficult to measure. To eliminate the influence of the unknown currents, the current observer is designed to predict the currents. e currents can be described asV cn � (V x , V y ) T , and V cn is the estimated value of V cn . e position of AUV in I { } can be express as η 1 � (x, y) T , and η 1 is the estimated value of η 1 . According to (14), the derivative of η 1 can be expressed as Here, ] 1r � (u r , v r ), and R(ψ) � cos(ψ) −sin(ψ) sin(ψ) cos(ψ) . Mathematical Problems in Engineering e current observer can be designed as [25] _ Assume that the currents are changing slowly relative to time, which means _ V cn � 0. Substituting (34) to (35) gives where k ob1 > 0, k ob2 > 0.
Consider the following Lyapunov function candidate: where X � [V cn , η 1 ]; P � P T > 0 is given by where q > 0. Substitution of (36) into (38) gives When k ob1 > 0 and k ob2 > 0, it can be seen that the leading principal minors of P are positive. Hence, V 2 is positive definite. e time derivative of V 2 is Substituting of (36) and (38) into (40) gives Consequently, the equilibrium point [V cn , η 1 ] T � 0 is GES, according to eorem 4.10 in [26].

Subpath Switching Algorithm.
e path-tangential speed U P of the virtual point is used as a control input to stabilize the along-track error. Moreover, according to (6), the first derivative of the parameter s of the virtual point w.r.t. time can be expressed as It can be seen that _ s is singularity-free for all regular paths. Besides, the value of _ s is dependent on U P and the parametrization of curves. As the composite curve path usually does not satisfy C 1 continuity, and each curve segment is parametrized differently, _ s is discontinuous at the waypoint. Besides, as each curve segment with different characteristics is parametrized with the same parameter interval, the length of the curve segments will have a significant impact on the value of _ s. Relatively, the equal amount of _ s will have a different meaning for different curve segments. Hence, the switching algorithm has to be considered for path following of the composite curve path, especially when the length of subpaths varies a lot. e parameter s of the virtual point can be obtained from the numerical integration of _ s. It can be expressed in Euler's method: When s k + Δs k > � i, the reference point will switch to the L i+1 subpath from the L i curve segments. Considering the different effect of _ s for two curve segments, at this time, s k+1 cannot be calculated by (43) directly. e problem can be solved by setting s k+1 � i, which is the parameter of the starting point of L i+1 subpath, as the displacement of the reference point in a period is small and has little impact on the path following. Ultimately, the global path following for composite curve path can be achieved.
Besides, an additional path switching mechanism can be introduced to achieve better tracking effect at path switching, especially for the piecewise linear path with G 0 continuity. As shown in Figure 4, it is suggested that a socalled circle of acceptance is associated with each waypoint connecting two subpaths [8]. e switching criterion can be defined as where R k is the radius of the circle of acceptance. e magnitude of R k can be determined by considering the turning ability of the AUV. Generally, the parameter R k can be selected as R k ≤ Δ. After the criterion has been satisfied, the next curve segment will be followed.
Generally, for composite curve path following, each subpath switching can be regarded as the beginning of tracking a new path initialized by the current state of the AUV.

Heading Controller Design.
To obtain the desired heading angle given by the guidance law, the heading control system is designed to make the heading error converge to zero.
A non-model-based NISM control algorithm is utilized to realize the heading control. e nonlinear sliding surfaces are designed as σ 1 ψ e � k 1 tanh k 2 ψ e + _ ψ e , where k i > 0(i � 0, 1, . . . , 4) are the control parameters, tanh(x) is a strictly bounded nonlinear hyperbolic tangent function, and σ 1 and σ 2 are designed slide mode surfaces.

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To stabilize σ 1 and σ 2 , the incremental feedback control law is designed as where _ δ r is the derivative of the rudder angle. Different from the standard NISM controller [27], tanh(x) is used for the incremental feedback calculation of _ δ r . By considering the discontinuity of the connection point between two curve segments, the value of σ 2 may become very large suddenly. is will cause excessive rudder speed which is not practical and may cause damage to the rudder system. Hence, in (46), k p can be used to limit the maximum rudder speed. When there is time-varying interference, k s can be set to a smaller value to ensure the stability of the control system. e signs of k p and k s are selected relating to the definition of the rudder. For the model of the REMUS vehicle, the rudder angle is defined as positive, when the resulting force causes the vehicle to turn left. Hence, k p and k s are selected to be positive. In addition, k 5 is also selected to be positive, which is used to adjust the range of coordinate where _ δ r is varying at an exponential rate. According to (45), when σ 2 � 0, one can obtain Hence,σ 1 ⟶ 0. Similarly, when σ 1 � 0, one can obtain Hence, ψ e ⟶ 0. is means that the heading error will converge to zero when σ 2 � 0. According to (47) and (48) and the properties of the hyperbolic function, it can be known that when k 4 σ 1 and k 2 ψ e are large, σ 1 and ψ e will tend to zero at a fixed rate. Moreover, when k 4 σ 1 and k 2 ψ e are small, σ 1 and ψ e will exponentially converge to zero. Besides, k 3 and k 1 determine the maximum attenuation rate of σ 1 and ψ e , and k 4 and k 2 are used to adjust the range of coordinate where σ 1 and ψ e are decaying at an exponential rate. Theorem 3. Assume that the sliding mode surfaces (45) and the incremental feedback control law (46) are used for the heading autopilot system, the definition of the sign of the rudder is known ((z _ r/zδ r ) < 0), and k p > 0, k s ≥ 0; then, the equilibrium point ψ e � 0 is globally k-exponentially stable. Proof.
e Lyapunov Function Candidate (LFC) According to (45), where r e � r − r d . According to (18) and (19) and the parameters of the REMUS AUV, ignoring the variables independent of δ r , one can obtain According to (45), it can be known that Hence, it can be derived that Depending on the properties of the hyperbolic function and Lyapunov stability theory, it can be concluded that σ 2 � 0 is UGAS and ULES or globally k-exponentially stable. Similarly, according to (47) and (48), ψ e � 0 is globally k-exponentially stable. x pt � f 1 t, x pt + g t, x pt , y pt y pt , System (54) can be regarded as which is perturbed by the output of the system 2pt : _ y pt � f 2 t, y pt .
Besides, the theorems proposed by [13] can be used to prove the stability of the cascade system. To be convenient, the theorems are presented here.
Composite curve path P a th -t a n g e n ti a l li n e P k-1 P (x p , y p ) Figure 4: e subpath switching algorithm of the composite curve path.

Theorem 4. Cascaded system (54) is GUAS if the following three assumptions hold:
(i) Assumption on 1pt : the system f 1 (t, x pt ) is GUAS and there exists a continuously differentiable function V(t, x pt ): IR ≥0 × IR n ⟶ IR that satisfies where W x is a positive definite proper function and c pt > 0 and η pt > 0 are constants. (ii) Assumption on the interconnection: the function g(t, x pt , y pt ) satisfies for all t ≥ t 0 where θ 1 , θ 2 : IR ≥0 ⟶ IR ≥0 are continuous functions. (iii) Assumption on 2pt : the system 2pt is GUAS and where the function κ(·) is a class κ function [24].

Theorem 5.
In addition to the assumptions in eorem 4, if both 1pt and 2pt are globally κ-exponentially stable, then cascaded system (54) is globally κ-exponentially stable. For path following under the disturbance of unknown static currents, the stability of cascade system can be proved in two steps. First, without considering the currents, the cascade system described in Theorem 6 can prove to be globally κ-exponentially stable. en, based on eorem 6, the cascade system including a current observer can prove to be globally κ-exponentially stable under the influence of unknown static currents. Theorem 6. Without considering the currents, the guidance system expressed as (32) and the heading control system described by (45) and (56) can be analyzed as a cascade structure, which is globally κ-exponentially stable at (x e , y e , ψ e ) � (0, 0, 0).
Besides, the interconnection term satisfies |ϕ(y e , ψ)| < � c, and c � 1.73 is the upper bound [12]. Hence, the assumption on the interconnection is also satisfied.
As Σ 1 2 is globallyk-exponentially stable, the last assumption is also satisfied. Hence, all conditions of eorem 4 and eorem 5 are satisfied, and the cascade system is globally k-exponentially stable.
e nominal system (Σ 2 1 system with V cn � 0 ) has proved to be globally k-exponentially stable in eorem 6. Hence, according to converse Lyapunov theory in [26], there exists Lyapunov function V 4 � (x e , y e , ψ e ) that satisfies the assumption on the nominal system. Besides, as the trigonometric functions are strictly bounded and the V cn is bounded, it can be concluded that ϕ 2 (V cn ) and ϕ 3 (V cn ) are bounded. Moreover, the current observer is GES. Hence, all conditions of eorem 4 and eorem 5 are satisfied, and the cascade system is globally k-exponentially stable.

Simulation Results
To verify the effectiveness of the control system for the composite curve path following, three simulation cases are carried out. Case 1 is implemented to verify the performance of the proposed path following controller for the composite curve path without currents. Besides, the PID control and the conventional sliding mode control are introduced as the heading controller to be compared with the improved NISM controller. Based on Case 1, Case 2 is carried out to prove the effectiveness of the additional subpath switching algorithm. In Case 3, unknown static currents are introduced to test the anti-interference ability of the controller based on the current observer. e composite curve path is designed to contain G0, G1, and G2 continuity at the same time and is composed of seven subpaths, including straight lines, circular arcs, and Fermat's spirals. e parameters of the composite curve path and the controller are shown in Tables 3 and 4. As most parameters of the guidance and heading control system have a clear physical meaning, the parameters are based on hand tuning. e values of the look-ahead distance Δ and the heading control parameter k p have obvious influence on the control effect and can also be adjusted automatically. Relevant methods can be found in [3,16].

Case 1: Path following without Currents.
For path following with no currents, V x � 0 and V y � 0. Due to the cascade structure design, the adaptive integral guidance law can be flexibly combined with heading controllers to achieve the desired path tracking. e PID control and conventional sliding mode control [12] are introduced as the heading controller to be compared with the improved NISM controller. e desired composite path and the trajectory of path following are shown in Figure 5. It can be concluded that, by the path parametrization according to (1) and the control of virtual point as (24), the global path planning for composite curve path can be realized.
As shown in Figure 6, compared with the PID control and conventional sliding mode control, the NISM control has faster convergence speed and smaller steady-state error. Besides, the NISM control adopts the incremental feedback control law, which is not based on the model parameters. Hence, it is not easily affected by the uncertainty of the model parameters. In Figure 7, after modifying the value of the hydrodynamic parameter N uuδ from −6.15 to −4.15, the NISM control can still maintain the heading control, while the conventional sliding mode control produces control error. e degree of the GC has a high impact on the control of path following. Comparatively, G 1 and G 2 paths have less effect on the path following. However, for G 0 paths, there will be an apparent deviation due to the sudden change of the desired heading angle ψ d . Hence, G 0 paths are not suitable for path following of underactuated vehicle, and an additional path switching mechanism (44) is required to handle this problem. e discontinuity of the connection point between two curve segments will cause excessive rudder speed which is not practical and may cause damage to the rudder system. Strictly bounded nonlinear hyperbolic tangent function is used in the NISM control, and the parameter k p can be used to limit the maximum rudder speed. In Figure 8, the rudder              speed is limited to 15 o /s. Hence, the proposed NISM control is more suitable for the composite curve path following.

Case 2: Subpath Switching
Algorithm. From Case 1, it is known that the tracking trajectory deviates from the desired path obviously at the switching point due to the sudden change of the heading angle for G 0 paths. Hence, in Case 2, an additional path switching mechanism (44) is used to handle this problem.
As shown in Figures 9 and 10, after using the additional path switching algorithm, the vehicle will track the next subpath early near the connection point when the switching criterion is satisfied. In addition, in this way, the tracking error between two subpaths can be reduced obviously.

Case 3: Path following Unknown Static
Currents. For path following with static currents, V x and V y are constant values, where V x is 0.1 (m/s) and V y is 0.2 (m/s). e simulation results are as follows.
From Figure 11, we can see that, without the current observer, the path following controller will produce certain tracking error under the disturbance of unknown current especially for the curve paths. Besides, from Figure 12, it can be concluded that the currents can be well estimated by the designed observer. en, the influence of currents can be eliminated at the kinematics level with U p and α calculated as (22) and (30). Finally, by the cascade structure composed of AILOS guidance and the improved NISM controller, the vehicle can converge to and move along the desired composite curve path.

Conclusion
is paper addresses the problems of composite curve path following for an underactuated AUV in the horizontal plane. e global path following of the composite curve path is realized by the virtual point control and the subpath switching algorithm after the parametrization of the composite curve and description of the kinematics error. Besides, the cascade structure composed of AILOS guidance and the improved NISM control proved to be UGAS under the influence of constant currents. Future work will expand the application of the composite curve path following to threedimensional space.

Data Availability
Partial data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.