Timescale-Separation-Based Source Seeking for USV

Highlights

What are the main findings?

• A source-seeking controller for USVs is developed based on timescale separation, where the task-level reference update and the motion-level tracking are treated on different time scales.
• A steady-state mapping between the reference velocity and the required surge–yaw actuation is established, which enables composite Lyapunov–based convergence analysis of the coupled slow–fast closed-loop system without assuming direct access to the field gradient.

What are the implications of the main findings?

• The controller offers a structured way to incorporate optimization-based reference generation into a dynamic USV model without requiring model reduction to single-integrator or unicycle forms.
• The simulation results indicate that the method can maintain stable seeking behavior in scenarios with typical USV nonlinearities and measurement uncertainties.

Abstract

The primary objective of this study is to enable an unmanned surface vehicle (USV) to autonomously approach the extremum of an unknown scalar field using only real-time field measurements. To this end, a source-seeking method based on timescale separation is developed within a hierarchical control framework that divides the closed-loop system into a slow and a fast subsystem. The slow subsystem governs the gradual evolution of the USV pose and generates reference heading and surge commands from local scalar field information, providing a directional cue toward the field extremum. The fast subsystem applies actuator-level control inputs that ensure these references are tracked with sufficient accuracy through rapid corrective actions. A Lyapunov-based analysis is carried out to study the stability properties of the coupled slow–fast dynamics and to establish conditions under which convergence can be guaranteed in the presence of model nonlinearities and external disturbances. Numerical simulations are conducted to illustrate the resulting system behavior and to verify that the proposed framework maintains stable seeking performance under typical operating conditions.

1. Introduction

Source seeking for unmanned surface vehicles (USVs) [1,2,3,4,5,6] is essential in marine environmental monitoring, plume localization, ecological mapping, and adaptive sampling. In these applications, a USV must navigate toward an extremum of an unknown scalar field using only local measurements while operating under nonlinear hydrodynamics, environmental disturbances, and sensor noise. Although many effective strategies exist for robots modeled by simplified kinematics, extending these approaches to second-order marine vehicles remains challenging. Surge–yaw coupling, nonlinear damping, and constraints on attainable velocities and heading rates all complicate the use of classic source- seeking controllers.

Existing methods fall into two broad categories: non-gradient strategies and gradient-based optimization approaches. Bio-inspired plume-tracking mechanisms [6,7,8,9,10] provide robustness but lack convergence guarantees when applied to nonlinear USV dynamics. Gradient-based algorithms [11,12,13,14,15,16] offer fast convergence on kinematic models such as the single-integrator or unicycle [17,18,19], yet they do not respect the physical limits of a USV. When applied directly, they often generate discontinuous heading changes, oscillatory behaviors, or even loss of convergence due to the mismatch between the desired kinematic update and the feasible surge–yaw motion. Furthermore, Brockett’s necessary condition implies that continuous static feedback cannot asymptotically stabilize nonholonomic surge–yaw dynamics, highlighting the need for dynamically realizable gradient flows.

Time-scale separation and singular perturbation (SP) techniques [20,21,22,23] provide a systematic way to embed high-level decision rules into low-level dynamics. However, applying SP to gradient-based source seeking on nonlinear USV models introduces several open challenges, including how to embed a gradient rule into a dynamically feasible update that respects surge bounds and ensures continuous heading evolution, how to maintain consistency between the gradient-driven reference behavior and the USV’s achievable steady-state velocities, and how to analyze the coupled slow–fast dynamics in the presence of nonlinear damping and field-induced uncertainty.

These considerations motivate a framework in which optimization-layer updates evolve directly within the USV’s physically admissible steady-state velocity space, rather than on a virtual kinematic system. This perspective differs fundamentally from traditional planning–tracking decompositions: the slow subsystem does not produce a geometric reference trajectory but instead defines a physically constrained gradient flow on the steady-state manifold. The fast subsystem ensures accurate tracking of this feasible gradient flow while compensating for hydrodynamic effects, resulting in a consistent closed-loop structure.

Our main contributions are summarized as follows:

• We address the kinematic–dynamic mismatch in USV source seeking by embedding optimization-layer gradient updates directly into the surge–yaw dynamics. This produces a physically realizable time-scale-separated framework that avoids the infeasibility issues inherent in hierarchical kinematic controllers.
• We design a smooth projected-gradient subsystem whose descent direction is restricted to the USV’s attainable surge axis. This ensures continuous heading evolution and bounded yaw-rate commands, and circumvents Brockett’s necessary condition, which rules out asymptotic stabilization via static gradient feedback.
• We introduce a steady-state embedding that maps optimization-layer descent directions to feasible steady-state velocities. This guarantees consistency between the optimization dynamics and the vehicle’s physical constraints, enabling implementable gradient flows for nonholonomic marine vehicles.
• We develop a composite Lyapunov framework that integrates the projected-gradient objective with the attitude and actuator dynamics, accounts for bounded sway-induced drift, and establishes local exponential stability of the full slow–fast closed-loop system.

Overall, the proposed framework provides a unified and physically consistent approach for embedding optimization-based navigation laws into the dynamics of marine vehicles. By treating the gradient update itself as a dynamically feasible flow—rather than a kinematic reference to be tracked—the method offers both theoretical rigor and practical applicability for source seeking with nonlinear USV models.

The structure of this paper is as follows. Section 2 presents the modeling of the USV system. Section 3 introduces the physical system and the problem formulation. Section 4 details the algorithmic design based on time-scale separation. Section 5 provides the stability analysis. Section 6 demonstrates the effectiveness of the proposed framework through numerical simulations.

Notation: For clarity, the following notations are used throughout the paper. $\mathbb{R}$ denotes the set of real numbers, and ${\mathbb{R}}^n$ denotes the n-dimensional Euclidean space. $I_n$ represents the $n\times n$ identity matrix, and $\mathbf{0}$ denotes a zero vector or matrix of appropriate dimension. For vectors $x,y\in {\mathbb{R}}^n$, the Euclidean inner product and norm are denoted by $\langle x,y\rangle =x^{\top }y$ and $\parallel x\parallel =\sqrt{x^{\top }x}$, respectively.

For a matrix A, the spectral norm is written as ${\parallel A\parallel }_2$, and its maximum and minimum eigenvalues are denoted by ${\lambda }_{max}\left(A\right)$ and ${\lambda }_{min}\left(A\right)$. The notation $A\succ 0$ indicates that A is positive definite.

The angle-wrapping operator is defined as

$$wrap\left(\theta \right)=atan2(sin\theta ,cos\theta )\in (-\pi ,\pi ],$$

where $atan2(y,x)$ returns the angle in radians between the vector $(x,y)$ and the positive x-axis. This operator maps any angle to its principal value, ensuring that angular errors remain within $(-\pi ,\pi ]$.

The saturation operator $sat(x,x_{max})$ clips its argument to $[-x_{max},x_{max}]$ elementwise or scalar-wise. When parameter constraints are required, the projection operator ${Proj}_{\Omega }(·)$ is used.

For a function $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$, $\dot{x}$ denotes the time derivative with respect to t, ${\partial }_{x}f$ denotes the partial derivative of f with respect to x, $diag(·)$ constructs diagonal matrices, and $\mathcal{O}(·)$ indicates asymptotic order when relevant.

2. USV Model

2.1. Coordinate Frames and State Variables

We consider a three-degree-of-freedom (3-DOF) underactuated USV operating on the horizontal plane. The position–attitude vector in the inertial frame $O_E-X_E{Y}_E$ is defined as

$$\eta ={[x,y,\psi ]}^{\mathrm{T}},$$

(1)

where x and y denote the Cartesian coordinates of the vessel’s center of mass $o_b$, and $\psi$ denotes the yaw angle. The body-frame velocity vector is

$$\mathit{v}={[u,v,r]}^{\mathrm{T}},$$

where u and v are the surge and sway velocities, and r is the yaw rate. For convenience, the planar position vector is written as

$$p={[x,y]}^{\mathrm{T}}.$$

A schematic diagram of the horizontal-plane motion is shown in Figure 1.

2.2. Kinematic Model

The horizontal-plane kinematics of the USV are given by

$$\dot{\eta }=J\left(\psi \right)\mathit{v},$$

(2)

where

$$J\left(\psi \right)=\left[\begin{array}{ccc}cos\psi & -sin\psi & 0\\ sin\psi & cos\psi & 0\\ 0& 0& 1\end{array}\right],$$

which is the standard body-to-inertial rotation matrix for marine craft.

2.3. Dynamic Model

The rigid-body and hydrodynamic dynamics follow the standard 3-DOF marine craft model:

$$M\dot{\mathit{v}}+C\left(\mathit{v}\right)\mathit{v}+D\mathit{v}=\tau ,$$

(3)

with

$$M=\mathrm{diag}\{m_{11},m_{22},m_{33}\},D=\mathrm{diag}\{d_{11},d_{22},d_{33}\},$$

$$C\left(\mathit{v}\right)=\left[\begin{array}{ccc}0& 0& -m_{22}v\\ 0& 0& m_{11}u\\ m_{22}v& -m_{11}u& 0\end{array}\right],\tau =\left[\begin{array}{c}{\tau }_u\\ 0\\ {\tau }_r\end{array}\right].$$

Here, ${\tau }_u$ and ${\tau }_r$ denote the surge force and yaw moment, and the coefficients $m_{ii},d_{ii}$ arise from hull mass, added mass, and linear hydrodynamic damping.

Remark 1. 

For source-seeking missions, USVs typically operate at low surge speeds (1–5 knots) with small yaw-rate variations to maintain stable sensing. Under these conditions, sway motion remains bounded and weakly excited rather than vanishing, as reported in multiple experimental and theoretical works [3,24,25,26]. The matrix $C\left(\mathit{v}\right)$ used here follows the standard low-speed approximation in Fossen’s marine craft model, specifically the commonly used linearized rigid-body + added-mass Coriolis matrix, which is skew-symmetric and widely adopted in low-speed USV guidance and control [27]. This approximation neglects nonlinear hydrodynamic coupling terms that are insignificant in the low-speed regime considered in this work.

2.4. Low-Speed Modeling Simplifications

In low-speed source-seeking tasks (typically 1–5 knots), hydrodynamic damping strongly suppresses sway, resulting in small lateral motion. The planar kinematics are

$$\dot{p}=R\left(\psi \right)\left[\begin{array}{c}u\\ v\end{array}\right],R\left(\psi \right)=\left[\begin{array}{cc}cos\psi & -sin\psi \\ sin\psi & cos\psi \end{array}\right].$$

(4)

Define the body-frame unit vectors

$$e_1=\left[\begin{array}{c}1\\ 0\end{array}\right],e_2=\left[\begin{array}{c}0\\ 1\end{array}\right],$$

so that (4) becomes

$$\dot{p}=R\left(\psi \right)e_{1}u+R\left(\psi \right)e_{2}v.$$

Let

$$d_p:=R\left(\psi \right)e_{2}v,$$

represent the sway-induced drift, yielding the compact planar form

$$\dot{p}=R\left(\psi \right)e_{1}u+d_p.$$

(5)

The surge-direction unit vector is defined as

$$\mathit{s}(p,\psi ):=R\left(\psi \right)e_1,$$

representing the forward body orientation in the inertial frame.

Assumption 1. 

The sway velocity satisfies

$$\left|v\left(t\right)\right|\le \overline{v},$$

(6)

for some small constant $\overline{v}>0$ determined by low-speed hydrodynamics.

Remark 2. 

Experimental studies on low-speed USV maneuvering [27,28,29] indicate that sway remains below 5–10% of surge speed during tracking, station-keeping, or environmental monitoring. Thus, treating sway as a uniformly bounded disturbance is consistent with standard 3-DOF marine modeling. Since

$$\parallel d_p\parallel =\left|v\right|\le \overline{v},$$

sway appears only as a small bounded drift in the position dynamics.

Remark 3. 

The reduced surge–yaw model

$$\dot{p}=R\left(\psi \right)e_{1}u,$$

(7)

$$\dot{\psi }=r,$$

(8)

is used solely to construct the slow reference dynamics in the proposed time-scale-separated controller. The full 3-DOF dynamic model (3) is always used for the physical USV evolution in simulation and in the fast subsystem. Since $d_p$ satisfies $\parallel d_p\parallel \le \overline{v}$, this reduction introduces no inconsistency; sway is not removed from the physical model but treated as a small perturbation to the slow manifold.

2.5. Model Properties

(i)

Bounded inertia and damping. The inertia and damping matrices satisfy

$$0<m_{min}I\le M\le m_{max}I,0<d_{min}I\le d_{max}I,$$

for some constants $m_{min},m_{max},d_{min},d_{max}>0$.

(ii)

Skew symmetry of the Coriolis matrix. The Coriolis–centrifugal matrix satisfies

$$C\left(\mathit{v}\right)+C{\left(\mathit{v}\right)}^{\mathrm{T}}=0,{\mathit{v}}^{\mathrm{T}}C\left(\mathit{v}\right)\mathit{v}=0.$$

Remark 4. 

The positive definiteness of M and D reflects basic physical constraints on mass distribution and hydrodynamic dissipation.

3. Problem Formulation

3.1. Scalar Field and Source Location

The USV is assumed to measure or estimate a scalar field represented by a smooth function

$$\Phi :{\mathbb{R}}^2\to \mathbb{R},$$

defined on the position $p={[x,y]}^{\mathrm{T}}$. The goal of source seeking is to drive the USV toward the (unknown) minimizer

$$p^{*}=arg\underset{p\in {\mathbb{R}}^2}{min}\Phi \left(p\right).$$

(9)

3.2. Assumptions on the Field

Assumption 2. 

The scalar field Φ satisfies the following:

1. 

Φ is twice continuously differentiable, and $\nabla \Phi$ is globally Lipschitz; i.e., there exists $L>0$ such that

$$\parallel \nabla \Phi \left(p_1\right)-\nabla \Phi \left(p_2\right)\parallel \le L\parallel p_1-p_2\parallel ,\forall p_1,p_2\in {\mathbb{R}}^2.$$

2. 

Near the minimizer $p^{*}$, the Hessian is uniformly positive definite:

$${\nabla }^2\Phi \left(p\right)⪰{\mu }_{\Phi }I,{\mu }_{\Phi }>0.$$

Remark 5. 

Assumption 2 ensures regularity of the scalar field and enables Lyapunov-based analysis of the closed-loop convergence properties, consistent with smooth environmental fields encountered in practical source-seeking scenarios.

3.3. Control Objective

The control objective is to design a feedback controller using the model (7)–(8) and dynamics (3) such that the USV trajectory converges to the neighborhood of the minimizer $p^{*}$:

$$\underset{t\to \infty }{lim\; sup}\parallel p\left(t\right)-p^{*}\parallel \le \epsilon ,$$

(10)

for any prescribed accuracy $\epsilon >0$ (in Figure 2).

Remark 6. 

A key difficulty is that the surge–yaw kinematics of an underactuated USV form a nonholonomic system, so Brockett’s necessary condition [30] precludes any continuous, time-invariant static feedback from asymptotically stabilizing the closed loop to a point. Consequently, the ideal descent law $\dot{p}=-\nabla \Phi \left(p\right)$ cannot be directly realized by the feasible model (7), whose input vector fields do not span a neighborhood of the origin; any static attempt to track $-\nabla \Phi \left(p\right)$ would induce heading discontinuities or chattering. This motivates the dynamic, timescale-separated design in Section 4, which embeds the gradient direction into the USV’s steady-state feasible velocity manifold and thereby circumvents the constraint imposed by Brockett’s condition.

4. Source-Seeking Algorithm Based on Timescale Separation

This section develops a composite controller for the USV source-seeking problem using a timescale-separated architecture. The slow subsystem generates a reference motion based on the gradient field, while the fast subsystem ensures that the actual control input $\tau$ rapidly tracks its steady-state value. The slow subsystem evolves under the reduced kinematics (7) and (8), whereas the fast subsystem follows the full dynamic model (3).

4.1. Slow Subsystem: Reference Heading and Velocity Design

Given the scalar field $\Phi$ satisfying Assumption 2, define the negative-gradient direction

$${\psi }_{\nabla }\left(p\right)=atan2\left(-{\partial }_y\Phi \left(p\right),-{\partial }_x\Phi \left(p\right)\right).$$

(11)

Directly setting the desired heading equal to ${\psi }_{\nabla }$ may introduce discontinuities when the field is nonuniform.

To characterize the angular mismatch between the vessel’s heading and the gradient-descent direction, we define the raw heading error

$$e_g={\psi }_{\nabla }-\psi ,$$

(12)

representing the unwrapped angular difference.

To ensure continuity under angle wrap-around, a wrapped version is used:

$$e_{\psi }=wrap\left(e_g\right)\in (-\pi ,\pi ],$$

(13)

producing a smooth and bounded heading-error signal.

A saturated proportional correction is then introduced:

$$\Delta \psi =k_{\psi }{sat}_{{\theta }_{max}}\left(e_{\psi }\right),$$

(14)

where $k_{\psi }$ is the heading correction gain and ${\theta }_{max}$ specifies the maximum allowable adjustment.

The desired heading is finally obtained through

$${\psi }_d=\psi +\Delta \psi ,$$

(15)

ensuring that heading commands evolve smoothly and within prescribed bounds.

The desired surge velocity is defined as the projection of the normalized gradient direction onto the vessel’s forward axis:

$$u^{★}(p,\psi )=k_u{\displaystyle \frac{{(-\nabla \Phi \left(p\right))}^{\top }\left(R\left(\psi \right){\mathit{e}}_1\right)}{1+\parallel \nabla \Phi \left(p\right)\parallel }}.$$

(16)

The desired yaw rate is selected as a stabilizing feedback law:

$$r^{★}(p,\psi )=k_{r}wrap({\psi }_d-\psi ).$$

(17)

Thus, the reference velocity vector for the slow subsystem is

$${\mathit{v}}^{★}(p,\psi )=\left[\begin{array}{c}{u}^{★}\\ 0\\ r^{★}\end{array}\right].$$

(18)

Assumption 3. 

There exists $\overline{V}>0$ such that $\parallel {\mathit{v}}^{★}(p,\psi )\parallel \le \overline{V}$ for all $(p,\psi )$.

Remark 7. 

The design (15)–(18) produces smooth reference trajectories even in spatially varying fields. The heading update enforces bounded rotational motion, while the projected gradient direction guarantees motion toward decreasing values of $\Phi \left(p\right)$.

4.2. Fast Subsystem: Control Input Tracking

For the full dynamics (3), the steady-state input required to achieve the reference velocity ${\mathit{v}}^{★}$ is

$${\tau }^{★}(p,\psi ,\mathit{v})=C\left(\mathit{v}\right){\mathit{v}}^{★}+D{\mathit{v}}^{★}.$$

(19)

To enforce timescale separation, the actuator input evolves according to

$$\epsilon \dot{\tau }=-\left(\tau -{\tau }^{★}\right),0<\epsilon \ll 1.$$

(20)

Remark 8. 

The reference dynamics evolve on the nominal $O\left(1\right)$ time scale, whereas the input-update law (20) drives τ to its quasi–steady value ${\tau }^{★}$ on the fast $O\left(\epsilon \right)$ time scale. Because $\tau \left(t\right)$ converges exponentially to ${\tau }^{★}(p,\psi ,\mathit{v})$ with rate $1/\epsilon$, the closed-loop system exhibits a standard singular perturbation structure: the fast subsystem contracts rapidly to the slow manifold $\tau ={\tau }^{★}$, while the slow variables $(p,\psi )$ evolve independently on the outer time scale. This establishes the required slow–fast decomposition for the composite Lyapunov analysis.

4.3. Composite Control Law

The complete controller consists of the slow-layer reference generator and the fast-layer tracking dynamics. The resulting closed-loop control law is

$$\begin{array}{cc}\hfill {\psi }_d& =\psi +k_{\psi }{sat}_{{\theta }_{max}}\left(wrap({\psi }_{\nabla }-\psi )\right),\hfill \end{array}$$

(21a)

$$\begin{array}{cc}\hfill u^{★}& =k_u{\displaystyle \frac{{(-\nabla \Phi \left(p\right))}^{\top }\left(R\left(\psi \right){\mathit{e}}_1\right)}{1+\parallel \nabla \Phi \left(p\right)\parallel }},\hfill \end{array}$$

(21b)

$$\begin{array}{cc}\hfill r^{★}& =k_{r}wrap({\psi }_d-\psi ),\hfill \end{array}$$

(21c)

$$\begin{array}{cc}\hfill {\mathit{v}}^{★}& ={[u^{★},0,r^{★}]}^{\mathrm{T}},\hfill \end{array}$$

(21d)

$$\begin{array}{cc}\hfill {\tau }^{★}& =C\left(\mathit{v}\right){\mathit{v}}^{★}+D{\mathit{v}}^{★},\hfill \end{array}$$

(21e)

$$\begin{array}{cc}\hfill \epsilon \dot{\tau }& =-(\tau -{\tau }^{★}).\hfill \end{array}$$

(21f)

A schematic of the controller architecture is shown in Figure 3.

In Algorithm 1, we present the source-seeking algorithm based on time-scale separation. The approach decomposes the control task into a slow subsystem, which governs the reference kinematics and drives the vessel toward the gradient of the unknown scalar field, and a fast subsystem, which stabilizes the full surge–yaw dynamics to track the reference motion. The algorithm is implemented in a recursive manner, updating the reference velocities and heading continuously based on local measurements of the scalar field.

Algorithm 1: Composite Control Algorithm for USV Source Seeking

Require:   Initial conditions $\left(p\right(0),\psi (0),\mathit{v}(0\left)\right)$; parameters $k_u,k_r,k_{\psi },{\theta }_{max},\epsilon$   1: procedure USV_SOURCE_SEEKING($p,\psi ,\mathit{v}$)   2:     while $\parallel p\left(t\right)-p^{*}\parallel >\epsilon$ do   3:          Measure $p,\psi ,\mathit{v}$ and compute $\nabla \Phi \left(p\right)$   4:          Compute gradient direction: ${\psi }_{\nabla }=atan2(-{\partial }_y\Phi ,-{\partial }_x\Phi )$   5:          Compute desired heading ${\psi }_d$ using (15)   6:          Compute desired surge velocity $u^{★}$ using (16)   7:          Compute desired yaw rate $r^{★}$ using (17)   8:          Form reference velocity ${\mathit{v}}^{★}$ using (18)   9:          Compute steady-state control ${\tau }^{★}$ using (19) 10:          Update input using $\epsilon \dot{\tau }=-(\tau -{\tau }^{★})$ 11:          Update USV motion using dynamics (3) and kinematics (7)–(8) 12:     end while 13: end procedure

5. Main Results and Stability Analysis

5.1. Fast Subsystem

The fast layer enforces actuator tracking of the steady-state value (3) where the reference velocity ${\mathit{v}}^{★}(p,\psi )$ is defined in (18). The fast tracking law is given by the first-order system (20). Define the velocity and torque errors

$$w:=\mathit{v}-{\mathit{v}}^{★},\tilde{\tau }:=\tau -{\tau }^{★}.$$

(22)

Assume the following:

(A1)

$m_{min}I⪯M⪯m_{max}I$ and $d_{min}I⪯D⪯d_{max}I$.

(A2)

The reference satisfies ${sup}_{t\ge 0}\parallel {\dot{\mathit{v}}}^{★}\left(t\right)\parallel \le \sigma <\infty$.

(A3)

The Coriolis matrix is skew-symmetric: $C\left(\mathit{v}\right)+C{\left(\mathit{v}\right)}^{\top }=0$.

Lemma 1 

(Fast-layer exponential tracking). Under (A1)–(A3), the control error satisfies

$$\parallel \tilde{\tau }\left(t\right)\parallel \le \parallel \tilde{\tau }\left(0\right)\parallel e^{-t/\epsilon }+\epsilon \underset{s\in [0,t]}{sup}\parallel {\dot{\tau }}^{★}\left(s\right)\parallel .$$

(23)

Furthermore, there exist constants $c_1,c_2,\lambda >0$ and ${\epsilon }^{★}>0$ such that, for all $0<\epsilon <{\epsilon }^{★}$,

$$\parallel w\left(t\right)\parallel \le c_1{e}^{-\lambda t}\parallel w\left(0\right)\parallel +c_2{e}^{-\lambda t}\parallel \tilde{\tau }\left(0\right)\parallel +\mathcal{O}\left(\epsilon \sigma \right).$$

(24)

Hence $w\left(t\right)\to 0$ exponentially as $\epsilon \to 0$.

Proof. 

From (20),

$$\epsilon \dot{\tilde{\tau }}=-\tilde{\tau }-\epsilon {\dot{\tau }}^{★}.$$

Solving this linear ODE gives the exact expression

$$\tilde{\tau }\left(t\right)=\tilde{\tau }\left(0\right)e^{-t/\epsilon }-{\int }_0^t{e}^{-(t-s)/\epsilon }{\dot{\tau }}^{★}\left(s\right)ds.$$

Bounding the convolution term by ${\int }_0^t{e}^{-(t-s)/\epsilon }ds\le \epsilon$ yields (23).

Subtracting the dynamics of ${\mathit{v}}^{★}$ gives

$$M\dot{w}+C\left(\mathit{v}\right)w+Dw=\tilde{\tau }-M{\dot{\mathit{v}}}^{★}.$$

With the quadratic Lyapunov function

$$V_f=\frac{1}{2}{\mathit{w}}^{\top }\mathit{M}\mathit{w},$$

(25)

skew-symmetry of $C\left(\mathit{v}\right)$ implies

$${\dot{V}}_f\le -{\lambda }_{min}{\left(D\right)\parallel w\parallel }^2+\parallel w\parallel \parallel \tilde{\tau }\parallel +\parallel M\parallel \parallel w\parallel \parallel {\dot{\mathit{v}}}^{★}\parallel .$$

Applying Young’s inequality to the two coupling terms,

$${\dot{V}}_f\le -\mu V_f+{\displaystyle \frac{1}{2{\eta }_1}}\parallel \tilde{\tau }{\parallel }^2+{\displaystyle \frac{{\parallel M\parallel }^2}{2{\eta }_2}}{\parallel {\dot{\mathit{v}}}^{★}\parallel }^2,$$

for some $\mu >0$ when ${\eta }_1,{\eta }_2$ are sufficiently small.

Insert the estimate (23) into the inequality above and apply the comparison lemma. Since both forcing terms decay as $e^{-t/\epsilon }$, one obtains constants $c_1,c_2,\lambda >0$ such that

$$V_f\left(t\right)\le c_1{e}^{-2\lambda t}{V}_f\left(0\right)+c_2{e}^{-2\lambda t}{\parallel \tilde{\tau }\left(0\right)\parallel }^2+\mathcal{O}\left({\epsilon }^2{\sigma }^2\right).$$

Using ${\displaystyle \frac{1}{2}}{m}_{max}{\parallel w\parallel }^2\ge V_f\ge {\displaystyle \frac{1}{2}}{m}_{min}{\parallel w\parallel }^2$ produces (24). □

5.2. Slow Subsystem

The reference kinematics are

$$\dot{p}=R\left(\psi \right)e_1{u}^{★}(p,\psi ),\dot{\psi }=r^{★}(p,\psi ),$$

(26)

where

$$\begin{array}{cc}\hfill u^{★}& =k_u{\displaystyle \frac{-{(\nabla \Phi \left(p\right))}^{\top }s(p,\psi )}{1+\parallel \nabla \Phi \left(p\right)\parallel }},\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill r^{★}& =k_r\mathrm{wrap}({\psi }_d-\psi ),\hfill \end{array}$$

(28)

and

$$\begin{array}{cc}\hfill {\psi }_d& =\psi +k_{\psi }{\mathrm{sat}}_{{\theta }_{max}}\left(\mathrm{wrap}({\psi }_{\nabla }-\psi )\right),\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill {\psi }_{\nabla }& =\mathrm{atan}2\left(-{\partial }_y\Phi \left(p\right),-{\partial }_x\Phi \left(p\right)\right).\hfill \end{array}$$

(30)

Near the minimizer $(p^{★},{\psi }^{★})$, assume the following:

(A1)

$\Phi$ satisfies Assumption 2;

(A2)

${\psi }_{\nabla }$ is $C^1$ with $|{\partial }_p{\psi }_{\nabla }|\le C_{\psi }$;

(A3)

angular errors are small so that $\mathrm{wrap}$ and $\mathrm{sat}$ act linearly.

Define the gradient-based angle errors

$$e_g:={\psi }_{\nabla }-\psi ,e_{\psi }:=\psi -{\psi }_d,$$

(31)

where $e_g$ is the raw heading error and $e_{\psi }$ is its linearized equivalent used for local analysis (12)–(13). In the small-angle region relevant to the stability analysis, the wrap operator satisfies $\mathrm{wrap}({\psi }_{\nabla }-\psi )={\psi }_{\nabla }-\psi$, and the desired-heading update ${\psi }_d=\psi +k_{\psi }({\psi }_{\nabla }-\psi )$ is locally smooth. Thus, the two error variables satisfy

$$e_{\psi }=-k_{\psi }{e}_g,r^{★}=k_r{k}_{\psi }{e}_g.$$

(32)

This shows that $e_{\psi }$ and $e_g$ represent the same physical heading deviation, with the former adopted as the linearized form for Lyapunov analysis.

The Lyapunov function for the slow subsystem is

$$V_s(p,e_{\psi })=\Phi \left(p\right)-\Phi \left(p^{*}\right)+\frac{1}{2}{e}_{\psi }^2,$$

(33)

which is locally positive definite.

Lemma 2 

(Slow-layer dissipation). Under (L1), the Lyapunov derivative of the slow subsystem satisfies

$${\dot{V}}_s\le -{\alpha }_s{V}_s+{\Gamma }_s,{\alpha }_s=min\left\{{\displaystyle \frac{2c_1}{{\mu }_{\Phi }}},c_2\right\},$$

(34)

for some constants $c_1,c_2>0$ and bounded residual ${\Gamma }_s\ge 0$.

Proof. 

Differentiating $V_s$ gives

$${\dot{V}}_s={(\nabla \Phi )}^{\top }\dot{p}+e_{\psi }(r^{★}-{\dot{\psi }}_d)={(\nabla \Phi )}^{\top }s{u}^{★}+e_{\psi }(r^{★}-{\dot{\psi }}_d).$$

Using (27),

$${(\nabla \Phi )}^{\top }s{u}^{★}=-k_u{\displaystyle \frac{{\left({(\nabla \Phi )}^{\top }s\right)}^2}{1+\parallel \nabla \Phi \parallel }}\le 0.$$

Differentiating ${\psi }_d$ and substituting (32) yields

$$e_{\psi }(r^{★}-{\dot{\psi }}_d)=-k_r{k}_{\psi }{e}_{\psi }^2+k_{\psi }^2{e}_g\left({\partial }_p{\psi }_{\nabla }\right)s{u}^{★}.$$

Since $|{\partial }_p{\psi }_{\nabla }|\le C_{\psi }$ and $|u^{★}|\le k_u$,

$$\left|k_{\psi }^2{e}_g\left({\partial }_p{\psi }_{\nabla }\right)s{u}^{★}\right|\le k_{\psi }{C}_{\psi }{k}_u\left|e_{\psi }\right|.$$

The position dynamics with sway drift satisfy

$${(\nabla \Phi )}^{\top }{d}_p\le \parallel \nabla \Phi \parallel \overline{v}:={\Gamma }_s.$$

Combining the terms gives

$${\dot{V}}_s\le -k_u{\displaystyle \frac{{\left({(\nabla \Phi )}^{\top }s\right)}^2}{1+\parallel \nabla \Phi \parallel }}-k_r{k}_{\psi }{e}_{\psi }^2+k_{\psi }{C}_{\psi }{k}_u\left|e_{\psi }\right|+{\Gamma }_s.$$

Near $p^{★}$,

$${(\nabla \Phi )}^{\top }s\ge c_g\parallel p-p^{★}\parallel ,$$

so the descent term dominates:

$$k_u{\displaystyle \frac{{\left({(\nabla \Phi )}^{\top }s\right)}^2}{1+\parallel \nabla \Phi \parallel }}\ge c_1{\parallel p-p^{★}\parallel }^2.$$

Applying Young’s inequality to the coupling term,

$$k_{\psi }{C}_{\psi }{k}_u\left|e_{\psi }\right|\le \frac{c_2}{2}{e}_{\psi }^2+\frac{k_{\psi }^2{C}_{\psi }^2{k}_u^2}{2c_2}.$$

Substituting these bounds yields

$${\dot{V}}_s\le -c_1{\parallel p-p^{★}\parallel }^2-\frac{c_2}{2}{e}_{\psi }^2+{\Gamma }_s.$$

Since

$$V_s\ge \frac{{\mu }_{\Phi }}{2}{\parallel p-p^{★}\parallel }^2+\frac{1}{2}{e}_{\psi }^2,$$

we obtain the desired inequality (34). □

5.3. Composite Stability

To analyze the full coupled slow–fast closed-loop system, consider the composite Lyapunov function

$$\Psi (w,p,\psi )=(1-\delta )V_f\left(w\right)+\delta V_s(p,e_{\psi }),\delta \in (0,1),$$

(35)

where $V_f$ (25) and $V_s$ (33).

Define

$${\alpha }_f:={\displaystyle \frac{2d_{min}}{m_{max}}},{\alpha }_s:=min\{k_u{\mu }_{\Phi },k_r\}.$$

(36)

Introduce the coupling constants (precisely those appearing in the cross-term bounds of $\dot{\Psi }$):

$$\begin{array}{cc}\hfill {\overline{c}}_1\left(\delta \right)& :=(1-\delta )\left(\parallel D\parallel k_u+\parallel M\parallel k_{u}L(\overline{V}+1)\right)\hfill \\ & +\delta \left(1+{\kappa }_M\parallel D\parallel k_u+{\kappa }_M\parallel M\parallel k_{u}L(\overline{V}+1)\right),\hfill \\ \hfill {\overline{c}}_2\left(\delta \right)& :=(1-\delta )\left(\parallel D\parallel k_r+\parallel M\parallel \left(k_r{C}_{\psi }+k_{u}L(\overline{V}+1)\right)\right)\hfill \\ & +\delta \left({\kappa }_M\parallel D\parallel k_r+{\kappa }_M\parallel M\parallel \left(k_r{C}_{\psi }+k_{u}L(\overline{V}+1)\right)\right).\hfill \end{array}$$

Theorem 1

(Composite (local) exponential stability). Suppose the model and field assumptions of Section 2, Section 3 and Section 4 hold, and assume the reference satisfies $\parallel {\dot{v}}^{★}\left(t\right)\parallel \le \sigma$ for some finite σ. Let $\delta \in (0,1)$ and define ${\alpha }_f,{\alpha }_s,{\overline{c}}_1,{\overline{c}}_2$ as above. If the singular perturbation parameter ε satisfies

$$\begin{array}{|c|}\hline {\displaystyle \frac{{\overline{c}}_2\left(\delta \right)}{{\alpha }_s^2}}<\epsilon <{\displaystyle \frac{1}{\sqrt{{\alpha }_s{\overline{c}}_1\left(\delta \right)}}}\\ \hline\end{array}$$

(37)

(and the slow-layer local gain condition used in Section 5.2 holds), then there exists a neighborhood $\mathcal{N}$ of the equilibrium $(p^{*},{\psi }^{*},w=0,\tilde{\tau }=0)$ and positive constants $c,\lambda$ such that

$$\Psi \left(t\right)\le c\Psi \left(0\right)e^{-\lambda t},$$

i.e., the equilibrium is locally exponentially stable. Furthermore, the interval (37) is constructive.

Proof. 

From $\Psi =(1-\delta )V_f+\delta V_s$ we obtain

$$\dot{\Psi }=(1-\delta ){\dot{V}}_f+\delta {\dot{V}}_s.$$

Fast subsystem. For suitable ${\eta }_1,{\eta }_2>0$,

$${\dot{V}}_f\le -{\alpha }_f{V}_f+{\displaystyle \frac{1}{2{\eta }_1}}\parallel \tilde{\tau }{\parallel }^2+{\displaystyle \frac{{\parallel M\parallel }^2}{2{\eta }_2}}{\parallel {\dot{v}}^{★}\parallel }^2.$$

(38)

Slow subsystem. There exist constants ${\gamma }_1,{\gamma }_2>0$ and a bounded residual $\Gamma$ such that

$${\dot{V}}_s\le -{\alpha }_s{V}_s+{\gamma }_1\parallel w\parallel \parallel \nabla \Phi \parallel +{\gamma }_2\parallel w\parallel \left|e_{\psi }\right|+\Gamma .$$

(39)

Using the composite constants ${\overline{c}}_1\left(\delta \right),{\overline{c}}_2\left(\delta \right)$ and Young’s inequality,

$${\overline{c}}_1\left(\delta \right)\parallel w\parallel \parallel \nabla \Phi \parallel \le {\displaystyle \frac{{\alpha }_s}{4}}{\parallel \nabla \Phi \parallel }^2+{\displaystyle \frac{{\overline{c}}_1{\left(\delta \right)}^2}{{\alpha }_s}}{\parallel w\parallel }^2,$$

$${\overline{c}}_2\left(\delta \right)\parallel w\parallel \left|e_{\psi }\right|\le {\displaystyle \frac{{\alpha }_s}{4}}{e}_{\psi }^2+{\displaystyle \frac{{\overline{c}}_2{\left(\delta \right)}^2}{{\alpha }_s}}{\parallel w\parallel }^2.$$

Using norm equivalence $c_f{\parallel w\parallel }^2\le V_f\le C_f{\parallel w\parallel }^2$ and $c_s{(\parallel \nabla \Phi \parallel }^2+e_{\psi }^2)\le V_s\le C_s{(\parallel \nabla \Phi \parallel }^2+e_{\psi }^2)$ on $\mathcal{N}$, substituting into $(1-\delta ){\dot{V}}_f+\delta {\dot{V}}_s$ yields

$$\dot{\Psi }\le -(1-\delta ){\alpha }_f{V}_f-\delta {\alpha }_s{V}_s+{\displaystyle \frac{{\overline{c}}_1{\left(\delta \right)}^2+{\overline{c}}_2{\left(\delta \right)}^2}{{\alpha }_s}}{V}_f+(1-\delta )\frac{1}{2{\eta }_1}{\parallel \tilde{\tau }\parallel }^2+\tilde{\mathrm{\Xi }},$$

where

$$\tilde{\mathrm{\Xi }}:=\Gamma +\delta {\displaystyle \frac{{\parallel M\parallel }^2}{2{\eta }_2}}{\parallel {\dot{v}}^{★}\parallel }^2\le C_v{\sigma }^2+C_{\Gamma }.$$

Define

$$A_w(\epsilon ,\delta )=(1-\delta ){\alpha }_f-{\displaystyle \frac{{\overline{c}}_1{\left(\delta \right)}^2+{\overline{c}}_2{\left(\delta \right)}^2}{{\alpha }_s}},A_s\left(\delta \right)={\displaystyle \frac{\delta {\alpha }_s}{2}}.$$

Because ${\alpha }_f=\mathsf{\Theta }(1/\epsilon )$, the positivity condition $A_w(\epsilon ,\delta )>0$ yields the explicit interval (37). The term $\parallel \tilde{\tau }{\left(t\right)\parallel }^2$ decays exponentially fast since

$$\parallel \tilde{\tau }\left(t\right)\parallel \le \parallel \tilde{\tau }\left(0\right)\parallel e^{-t/\epsilon }+\mathcal{O}\left(\epsilon \sigma \right),$$

and is absorbed into the dissipative part on a sufficiently small neighborhood $\mathcal{N}$.

Thus, for any $\delta \in (0,1)$ and any $\epsilon$ in the constructive interval (37), there exist $c,\lambda >0$ such that $\dot{\Psi }\le -\lambda \Psi$ on $\mathcal{N}$, proving local exponential stability. □

6. Numerical Simulation

6.1. Simulation Setup

6.1.1. Simulation Setup of the Scalar Field

The source is positioned at

$$p^{*}={[1,1]}^{\top },$$

and generates a strongly convex scalar field

$$\Phi \left(p\right)=0.5{(x-1)}^2+0.5{(y-1)}^2,$$

with curvature bounds ${\mu }_{\Phi }=L=2$. The USV seeks the minimizer of this field.

6.1.2. USV Dynamic Model

The Unmanned Surface Vehicle (USV) is modeled using the standard 3-DOF surge–sway–yaw dynamics

$$M\dot{\mathit{v}}+C\left(\mathit{v}\right)\mathit{v}+D\mathit{v}=\tau ,$$

with the inertia and damping matrices defined as

$$M=\mathrm{diag}(5.0,5.0,2.0),D=\mathrm{diag}(3.0,3.0,1.0).$$

These satisfy the assumed physical bounds

$$m_{min}=2,m_{max}=5,d_{min}=1.$$

The initial state is

$$(x,y,\psi )=(8.0,5.0,0.0),(u,v,r)=(0,0,0),$$

and the workspace is defined as $x,y\in [-5,25]$ m.

6.1.3. Controller Parameters

The proposed controller uses the following gains:

$$k_u=8.0,k_r=5.0,k_{\psi }=0.7,$$

with surge saturation $u_{max}=1.0$ m/s.

The timescale separation parameter is chosen as

$$\epsilon =0.2,$$

which satisfies the theoretical stability interval

$${\displaystyle \frac{{\overline{c}}_2\left(\delta \right)}{{\alpha }_s^2}}<\epsilon <{\displaystyle \frac{1}{\sqrt{{\alpha }_s{\overline{c}}_1\left(\delta \right)}}}.$$

6.1.4. Numerical Integration and Convergence Criterion

Simulations are performed over a 50 s horizon using a maximum integration step of $0.05$ s. A run is considered converged when the position and heading errors satisfy

$$\parallel p\left(t\right)-p^{*}\parallel <0.05\mathrm{m},|\psi \left(t\right)-{\psi }^{*}|<1^{\circ }.$$

6.2. Simulation Settings for Baseline Algorithms

To provide a fair comparison with the proposed timescale-separated controller, two widely used source-seeking methods are implemented as baselines:

• Gradient-based Source-Seeking (Gradient).
• Extremum Seeking Control (ESC).

All algorithms share the same USV model, noise realizations, disturbances, and Monte Carlo sampling conditions.

6.2.1. Common Dynamic Model and Initial Conditions

Each controller operates on the same 3-DOF surge–sway–yaw dynamics:

$$M\dot{\mathit{v}}+C\left(\mathit{v}\right)\mathit{v}+D\mathit{v}=\tau ,$$

with inertia and damping matrices

$$M=\mathrm{diag}(3.0,3.0,1.0),D=\mathrm{diag}(2.0,2.0,0.5).$$

The initial state is

$$(x_0,y_0,{\psi }_0)=(x_0,y_0,0),(u_0,v_0,r_0)=(0,0,0),$$

where $(x_0,y_0)$ is sampled independently for each Monte Carlo run.

6.2.2. Baseline 1: Gradient-Based Source Seeking

The gradient-based controller drives the USV in the negative gradient direction of the field:

$$\nabla \Phi \left(p\right)=\left[\begin{array}{c}x-1\\ y-1\end{array}\right].$$

The desired heading and surge commands are

$${\psi }_{\mathrm{grad}}=\mathrm{atan}2(-g_y,-g_x),r\left(t\right)=k_r\mathrm{wrap}({\psi }_{\mathrm{grad}}-\psi ),$$

$$u\left(t\right)=k_u{\left(-{\displaystyle \frac{\nabla \Phi \left(p\right)}{\parallel \nabla \Phi \left(p\right)\parallel }}\right)}^{\top }{e}_1,e_1={[cos\psi ,sin\psi ]}^{\top },$$

with gains

$$k_u=3.0,k_r=2.0.$$

This method provides a direct but kinematically mismatched gradient-descent update, and is therefore included as a representative baseline.

6.2.3. Baseline 2: Extremum Seeking Control (ESC)

ESC perturbs the heading with a small sinusoidal excitation:

$${\psi }_{\mathrm{esc}}\left(t\right)=\mathrm{atan}2(-g_y,-g_x)+0.2sin\left(0.5t\right),$$

and computes the control inputs as

$$r\left(t\right)=k_r{k}_{\psi }\mathrm{wrap}({\psi }_{\mathrm{esc}}-\psi ),u\left(t\right)=k_u{\left(-{\displaystyle \frac{\nabla \Phi }{1+\parallel \nabla \Phi \parallel }}\right)}^{\top }{e}_1,$$

where

$$k_u=3.0,k_r=2.0,k_{\psi }=0.7,{\theta }_{max}={15}^{\circ }.$$

6.2.4. Noise and Disturbance Model

All three algorithms use the exact same realizations of environmental noise:

• Position noise: $\tilde{p}=p+\mathcal{N}(0,{0.03}^2)$
• Field-value noise: $\tilde{\Phi }=\Phi \left(p\right)+\mathcal{N}(0,{0.02}^2)$
• Environmental velocity disturbance: $w_d\sim \mathcal{N}(0,{0.01}^2)\mathrm{m}/\mathrm{s}$

This ensures that performance differences arise from the controller design rather than stochastic variability.

6.2.5. Monte Carlo Simulation Setup

A Monte Carlo study is conducted for all algorithms:

• Number of trials: $N=30$.
• Initial positions: $(x_0,y_0)\sim U(-20,20)\times U(-20,20)$.
• Simulation horizon: $T=150\mathrm{s}$ with $\Delta t\le 0.02\mathrm{s}$.
• Convergence threshold: $\parallel p\left(t\right)-p^{★}\parallel <0.5\mathrm{m}$.

All trajectories, convergence times, and steady-state errors are aggregated for quantitative comparison among the proposed controller, Gradient, and ESC.

6.3. Simulation Results

6.3.1. Single-Trajectory Behavior

As can be seen from Figure 4, the USV trajectories, starting from different initial positions, all converge effectively to the target point, showing a clear convergence trend. This indicates that the adopted control strategy is capable of guiding the USV to stably approach the target source.

6.3.2. Trajectory Performance Under Noise-Free Conditions

Figure 5 illustrates the navigation trajectories of the proposed method in three distinct non-uniform field environments (background contour lines represent field distributions, with denser lines indicating steeper gradients). From left to right, we see the quadratic field (circular contours, uniform gradient variation), elliptical field (x-direction steeper gradient), and multi-well field (dual potential wells with local minima). The blue square denotes the initial position, and the green star denotes the target source. In all three fields, the proposed method generates trajectories that adapt to the field gradient characteristics: it converges steadily in the quadratic field, adjusts its path to match the x-direction-dominated gradient in the elliptical field, and avoids local minima to reach the global target in the multi-well field. This demonstrates the strong adaptability of the proposed method to diverse non-uniform field environments.

Figure 6 presents the motion trajectories of the three controllers in the scalar field: the proposed controller (blue) exhibits the smoothest and most directionally focused trajectory with no obvious oscillations; the gradient-based controller (red) shows significant oscillations and path reversals, deviating notably from the target direction; the ESC controller (yellow) performs moderately, with only slight jitter and a less curved path than the gradient method.

The position error curves in Figure 7 demonstrate that the proposed controller converges the fastest, approaching near-zero error in approximately 25 s; the ESC controller follows with a steady error decrease; the gradient-based controller, due to persistent oscillations, requires significantly longer convergence time and exhibits ongoing error fluctuations in the later stage. In summary, the proposed timescale-separated controller outperforms the two baseline algorithms in both trajectory smoothness and convergence speed.

6.3.3. Error Convergence Characteristics

Figure 8 and Figure 9 together show that the proposed method achieves faster, smoother, and more reliable convergence than the gradient-based and ESC approaches. The Monte Carlo trajectories indicate that our controller generates more directed and stable paths toward the source, while the gradient method suffers from strong oscillations and ESC shows slower, higher-frequency exploratory motion. These behaviors are reflected in the convergence-time statistics: our method has the smallest average time and tightest distribution, whereas the gradient method exhibits large variance and several slow outliers, and ESC performs moderately but remains consistently slower. Overall, the results demonstrate that the proposed strategy provides superior efficiency and robustness under noisy conditions.

6.3.4. Effect of Measurement Noise

Figure 10 compares the three algorithms under different noise levels, with three trajectories shown for each case. The proposed method (ours) maintains smooth and consistent paths across all noise strengths, showing only minor perturbations and reliably converging to the target. The gradient method is highly sensitive to noise: its trajectories exhibit large oscillations and drift significantly as noise increases, often deviating from the target region. The ESC method performs more stably than the gradient method but still develops noticeable lateral swings and longer paths under higher noise.

7. Conclusions

In this paper, we proposed a timescale-separated controller that embeds a gradient-consistent steady-state mapping into a slow subsystem and enforces actuator tracking through a fast subsystem, providing a dynamically feasible source-seeking strategy for underactuated USVs. Using the $\Psi$-function and $\Lambda$-matrix framework from autonomous optimization, we derived explicit conditions that guarantee local exponential convergence under realistic surge–yaw coupling and nonlinear damping, offering the first stability result for source seeking with second-order USV dynamics. Simulations across multiple scalar fields, noise levels, and Monte Carlo trials show that the method yields smoother trajectories, faster convergence, and smaller steady-state errors than gradient baseline, demonstrating its robustness and practical applicability.