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A spherical amplitude-phase formulation for 3-D adaptive line-of-sight (ALOS) guidance with USGES stability guarantees

Erlend M. Coates, Thor I. Fossen

TL;DR

This paper addresses robust 3-D path following for drift-affected autonomous vehicles by reformulating NED kinematics in a spherical amplitude–phase framework. The spherical representation provides a direct geometric interpretation of velocity orientation and yields error dynamics that are globally well defined, enabling a substantially simpler USGES stability proof for cross- and vertical-track errors. The approach relaxes prior constraints (no constant altitude/depth or zero horizontal crab) and remains valid under fully 3-D motion with nonzero roll, pitch, and flight-path angles. The resulting framework offers a practically impactful, rigorously stable method for drift-compensated 3-D ALOS guidance across marine and aerial platforms.

Abstract

A recently proposed 3-D adaptive line-of-sight (ALOS) path-following algorithm addressed coupled motion dynamics of marine craft, aircraft, and uncrewed vehicles under environmental disturbances such as wind, waves, and ocean currents. Stability analysis established uniform semiglobal exponential stability (USGES) of the cross- and vertical-track errors using a body-velocity-based amplitude-phase representation of the North-East-Down (NED) kinematic differential equations. In this brief paper, we revisit the ALOS framework and introduce a novel spherical amplitude-phase representation. This formulation yields a more geometrically intuitive and physically observable description of the guidance errors and enables a significantly simplified stability proof. Unlike the previous model, which relied on a vertical crab angle derived from body-frame velocities, the new representation uses an alternative vertical crab angle and retains the USGES property. It also removes restrictive assumptions such as constant altitude/depth or zero horizontal crab angle, and remains valid for general 3-D maneuvers with nonzero roll, pitch, and flight-path angles.

A spherical amplitude-phase formulation for 3-D adaptive line-of-sight (ALOS) guidance with USGES stability guarantees

TL;DR

This paper addresses robust 3-D path following for drift-affected autonomous vehicles by reformulating NED kinematics in a spherical amplitude–phase framework. The spherical representation provides a direct geometric interpretation of velocity orientation and yields error dynamics that are globally well defined, enabling a substantially simpler USGES stability proof for cross- and vertical-track errors. The approach relaxes prior constraints (no constant altitude/depth or zero horizontal crab) and remains valid under fully 3-D motion with nonzero roll, pitch, and flight-path angles. The resulting framework offers a practically impactful, rigorously stable method for drift-compensated 3-D ALOS guidance across marine and aerial platforms.

Abstract

A recently proposed 3-D adaptive line-of-sight (ALOS) path-following algorithm addressed coupled motion dynamics of marine craft, aircraft, and uncrewed vehicles under environmental disturbances such as wind, waves, and ocean currents. Stability analysis established uniform semiglobal exponential stability (USGES) of the cross- and vertical-track errors using a body-velocity-based amplitude-phase representation of the North-East-Down (NED) kinematic differential equations. In this brief paper, we revisit the ALOS framework and introduce a novel spherical amplitude-phase representation. This formulation yields a more geometrically intuitive and physically observable description of the guidance errors and enables a significantly simplified stability proof. Unlike the previous model, which relied on a vertical crab angle derived from body-frame velocities, the new representation uses an alternative vertical crab angle and retains the USGES property. It also removes restrictive assumptions such as constant altitude/depth or zero horizontal crab angle, and remains valid for general 3-D maneuvers with nonzero roll, pitch, and flight-path angles.
Paper Structure (12 sections, 3 theorems, 33 equations, 1 figure)

This paper contains 12 sections, 3 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Scope and contributions
  3. Kinematic preliminaries
  4. Along-, cross- and vertical-track errors
  5. NED kinematic differential equations
  6. Amplitude-phase forms of the NED kinematic differential equations
  7. Body-velocity amplitude–phase model
  8. Spherical amplitude–phase model
  9. ALOS guidance law for 3-D path following
  10. Conclusions
  11. Proof of Proposition \ref{['prop:APF2']}
  12. Proof of Theorem 1

Key Result

Proposition 1

(Body-velocity amplitude--phase model) The NED kinematic differential equations eq:kinematics can be expressed in amplitude--phase form as where the phase angles, referred to as the horizontal and vertical crab angles, are:The inverse tangent function, $\tan^{-1}(y/x)$, complies with the definition used by matlab and other programming languages where $x=0$ implies that $\tan^{-1}(y/0) = \mathrm{s

Figures (1)

  • Figure 1: Comparison of vertical planes defined by the body axis and the velocity vector. The operation $\text{proj}(\cdot)$ projects the vector argument into the plane, while $\text{rot}(\cdot)$ rotates about the inertial z-axis to align with the plane.

Theorems & Definitions (7)

  • Proposition 1
  • Proposition 2: Spherical amplitude–phase model
  • Remark 1: Different vertical crab angles
  • Remark 2
  • Theorem 1: USGES of the ALOS guidance law
  • Remark 3: Unrestricted 3-D stability
  • Remark 4: Practical role of projection