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Leader-Follower Formation and Tracking Control of Underactuated Surface Vessels

Bo Wang, Antonio Loria

TL;DR

The paper tackles global trajectory tracking and leader–follower formation for underactuated surface vessels with two propellers by exploiting a cascaded control structure that separates kinematics from kinetics. A low-gain, saturated control at the kinematics level is paired with a PD+ velocity controller at the kinetics level, implemented via a left pseudoinverse to accommodate underactuation, and stability is established using ISS, Lyapunov-based analysis, and small-gain arguments under persistently exciting reference trajectories. The main contributions are (i) a novel PD+ formation controller for underactuated Lagrangian vessels within a cascaded-ISS framework and (ii) a scalable formation scheme on directed spanning trees with UGAS guarantees, demonstrated through simulation under disturbances and uncertainties. The results offer a simple, robust, and implementable approach for coordinating multiple underactuated vessels in formation, with practical relevance to rescue, search, and reconnaissance missions.

Abstract

This Technical Note presents a simple control approach for global trajectory tracking and formation control of underactuated surface vessels equipped with only two propellers. The control approach exploits the inherent cascaded structure of the vehicle dynamics and is divided into control designs at the kinematics and kinetics levels. A controller with a low-gain feature is designed at the kinematics level by incorporating the cascaded system method, persistency of excitation, and the small-gain theorem. Furthermore, a PD+ controller is designed to achieve the velocity tracking at the kinetics level. The proposed control laws are partially linear and saturated linear and easy to implement. Based on a leader-follower scheme, our control approach applies to the formation tracking control problem of multi-vehicle systems under a directed spanning tree topology. Our main results guarantee uniform global asymptotic stability for the closed-loop system, which implies robustness with respect to bounded disturbances in the sense of Malkin's total stability, also known as local input-to-state stability.

Leader-Follower Formation and Tracking Control of Underactuated Surface Vessels

TL;DR

The paper tackles global trajectory tracking and leader–follower formation for underactuated surface vessels with two propellers by exploiting a cascaded control structure that separates kinematics from kinetics. A low-gain, saturated control at the kinematics level is paired with a PD+ velocity controller at the kinetics level, implemented via a left pseudoinverse to accommodate underactuation, and stability is established using ISS, Lyapunov-based analysis, and small-gain arguments under persistently exciting reference trajectories. The main contributions are (i) a novel PD+ formation controller for underactuated Lagrangian vessels within a cascaded-ISS framework and (ii) a scalable formation scheme on directed spanning trees with UGAS guarantees, demonstrated through simulation under disturbances and uncertainties. The results offer a simple, robust, and implementable approach for coordinating multiple underactuated vessels in formation, with practical relevance to rescue, search, and reconnaissance missions.

Abstract

This Technical Note presents a simple control approach for global trajectory tracking and formation control of underactuated surface vessels equipped with only two propellers. The control approach exploits the inherent cascaded structure of the vehicle dynamics and is divided into control designs at the kinematics and kinetics levels. A controller with a low-gain feature is designed at the kinematics level by incorporating the cascaded system method, persistency of excitation, and the small-gain theorem. Furthermore, a PD+ controller is designed to achieve the velocity tracking at the kinetics level. The proposed control laws are partially linear and saturated linear and easy to implement. Based on a leader-follower scheme, our control approach applies to the formation tracking control problem of multi-vehicle systems under a directed spanning tree topology. Our main results guarantee uniform global asymptotic stability for the closed-loop system, which implies robustness with respect to bounded disturbances in the sense of Malkin's total stability, also known as local input-to-state stability.
Paper Structure (9 sections, 5 theorems, 64 equations, 3 figures)

This paper contains 9 sections, 5 theorems, 64 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation and Main Results
  3. Cascades-Based Trajectory Tracking Control
  4. Underactuation constraints
  5. Velocity tracking
  6. Control of the error kinematics
  7. Formation Tracking control
  8. Simulation Results
  9. Conclusion

Key Result

Proposition 1

Consider the Lagrangian dynamics (eq:1b) in closed loop with the control law (eq:underactuated-control), where $t\mapsto v^*(t)$ is defined on $[t_\circ, \infty)$ for any $t_\circ \geq 0$ and satisfies (eq:constraint), and $K_d :=\operatorname{diag}\{k_{dx},0,k_{d\omega}\}$, with $k_{dx}\ge {m_{11}^

Figures (3)

  • Figure 1: Small-gain feedback representation of the system (\ref{['eq:26']}).
  • Figure 2: Illustration of the paths in formation tracking.
  • Figure 3: Convergence of the relative errors (in norm) for each leader-follower pair. The shadowed colored regions represent the envelopes containing the trajectories of 100 simulation tests with randomly chosen initial conditions and under the influence of parameter uncertainties and external disturbances.

Theorems & Definitions (11)

  • Proposition 1: Velocity tracking
  • Remark 1
  • Remark 2
  • Remark 3
  • Proposition 2: Nominal system
  • proof
  • Proposition 3: Trajectory tracking
  • proof
  • Proposition 4: Formation tracking
  • proof
  • ...and 1 more