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CBF Based Quadratic Program for Trajectory Tracking of Underatuated Marine Vessels

Ji-Hong Li

TL;DR

This work tackles trajectory tracking for underactuated marine vessels by leveraging two polar-coordinate transformations that introduce singularities. It replaces traditional AMO/EMO-based safeguards with a control barrier function (CBF) framework embedded in a quadratic program (QP) to enforce singularity-avoidance constraints (CC-1 and CC-2) while tracking a reference via a backstepping-based τ^{ref}. The method yields ECBF conditions that translate into linear QP constraints, enabling the actual control input τ to deviate from τ^{ref} only when necessary to bypass singularities, and proves exponential stability of the closed-loop system under CCs. Numerical simulations demonstrate the approach's ability to maintain tracking across singular points, with surge-speed treated as a constraint rather than a fixed assumption, and highlight considerations for actuator limits and potential future work on input saturation.

Abstract

By introducing two polar coordinates transformations, the marine vessel's original two-input-three-output second-order tracking model can be reduced to a two-input-two-output feedback form. However, the resulting system does not confirm to the strict-feedback structure, leading to potential singularity when designing the stabilizing function for the virtual input in the recursive controller design. Moreover, the polar coordinate transformation itself inherently introduces singularities. To address these singularity issues, this paper employs a control barrier function (CBF) based approach and formulates the trajectory tracking problem as a quadratic program (QP) solved via a QP optimizer. Numerical simulations are carried out to demonstrate the effectiveness of the proposed method.

CBF Based Quadratic Program for Trajectory Tracking of Underatuated Marine Vessels

TL;DR

This work tackles trajectory tracking for underactuated marine vessels by leveraging two polar-coordinate transformations that introduce singularities. It replaces traditional AMO/EMO-based safeguards with a control barrier function (CBF) framework embedded in a quadratic program (QP) to enforce singularity-avoidance constraints (CC-1 and CC-2) while tracking a reference via a backstepping-based τ^{ref}. The method yields ECBF conditions that translate into linear QP constraints, enabling the actual control input τ to deviate from τ^{ref} only when necessary to bypass singularities, and proves exponential stability of the closed-loop system under CCs. Numerical simulations demonstrate the approach's ability to maintain tracking across singular points, with surge-speed treated as a constraint rather than a fixed assumption, and highlight considerations for actuator limits and potential future work on input saturation.

Abstract

By introducing two polar coordinates transformations, the marine vessel's original two-input-three-output second-order tracking model can be reduced to a two-input-two-output feedback form. However, the resulting system does not confirm to the strict-feedback structure, leading to potential singularity when designing the stabilizing function for the virtual input in the recursive controller design. Moreover, the polar coordinate transformation itself inherently introduces singularities. To address these singularity issues, this paper employs a control barrier function (CBF) based approach and formulates the trajectory tracking problem as a quadratic program (QP) solved via a QP optimizer. Numerical simulations are carried out to demonstrate the effectiveness of the proposed method.

Paper Structure

This paper contains 16 sections, 39 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Vessel Model
  4. Two Polar Coordinates Transformations
  5. System Singularities
  6. Problem Formulation
  7. Main Results
  8. Control Barrier Functions for Singularities Avoidance
  9. CBF for CC-1
  10. CBF for CC-2
  11. Design of Reference Controller $\bm{\tau}^{r\!e\!f}$
  12. Kinematic Tracking
  13. Dynamic Tracking
  14. QP-Based Tracking Controller Design
  15. Numerical Study
  16. ...and 1 more sections

Figures (7)

  • Figure 1: Reference trajectory and its tracking using $\bm{\tau}=\bm{\tau}^{r\!e\!f}$.
  • Figure 2: Variation trends of all possible causes associated with SP-1 to SP-4.
  • Figure 3: Reference trajectory and its tracking using the proposed method in (\ref{['eq37']}).
  • Figure 4: Tracking error convergence with the proposed method (\ref{['eq37']}).
  • Figure 5: Variation trends of all possible causes associated with SP-1 to SP-4 in the case of the proposed method (\ref{['eq37']}).
  • ...and 2 more figures