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Three-dimensional Nonlinear Path-following Guidance with Bounded Input Constraints

Saurabh Kumar, Shashi Ranjan Kumar, Abhinav Sinha

TL;DR

This work tackles 3D UAV path-following with bounded actuator authority by formulating the path as a moving pseudo-target and designing a nonlinear, saturation-aware guidance law. The method jointly commands the linear speed $V_U$ and pitch/yaw rates $\omega_U^y,\omega_U^z$ through bounded-input laws, ensuring the relative range $r$ and lead angles $(\psi_U,\theta_U)$ converge to zero in a fixed time while staying within safe actuation sets. Auxiliary variables and smooth saturation models guarantee global fixed-time convergence and avoid singularities, with proofs provided in appendices. The approach is geometry-agnostic, applicable to both multi-rotor and fixed-wing UAVs, and validated via extensive simulations that cover diverse 3D paths and variable path-curvature or pseudo-target speed, demonstrating robust, bounded-input path-following in realistic scenarios.

Abstract

In this paper, we consider the tracking of arbitrary curvilinear geometric paths in three-dimensional output spaces of unmanned aerial vehicles (UAVs) without pre-specified timing requirements, commonly referred to as path-following problems, subjected to bounded inputs. Specifically, we propose a novel nonlinear path-following guidance law for a UAV that enables it to follow any smooth curvilinear path in three dimensions while accounting for the bounded control authority in the design. The proposed solution offers a general treatment of the path-following problem by removing the dependency on the path's geometry, which makes it applicable to paths with varying levels of complexity and smooth curvatures. Additionally, the proposed strategy draws inspiration from the pursuit guidance approach, which is known for its simplicity and ease of implementation. Theoretical analysis guarantees that the UAV converges to its desired path within a fixed time and remains on it irrespective of its initial configuration with respect to the path. Finally, the simulations demonstrate the merits and effectiveness of the proposed guidance strategy through a wide range of engagement scenarios, showcasing the UAV's ability to follow diverse curvilinear paths accurately.

Three-dimensional Nonlinear Path-following Guidance with Bounded Input Constraints

TL;DR

This work tackles 3D UAV path-following with bounded actuator authority by formulating the path as a moving pseudo-target and designing a nonlinear, saturation-aware guidance law. The method jointly commands the linear speed and pitch/yaw rates through bounded-input laws, ensuring the relative range and lead angles converge to zero in a fixed time while staying within safe actuation sets. Auxiliary variables and smooth saturation models guarantee global fixed-time convergence and avoid singularities, with proofs provided in appendices. The approach is geometry-agnostic, applicable to both multi-rotor and fixed-wing UAVs, and validated via extensive simulations that cover diverse 3D paths and variable path-curvature or pseudo-target speed, demonstrating robust, bounded-input path-following in realistic scenarios.

Abstract

In this paper, we consider the tracking of arbitrary curvilinear geometric paths in three-dimensional output spaces of unmanned aerial vehicles (UAVs) without pre-specified timing requirements, commonly referred to as path-following problems, subjected to bounded inputs. Specifically, we propose a novel nonlinear path-following guidance law for a UAV that enables it to follow any smooth curvilinear path in three dimensions while accounting for the bounded control authority in the design. The proposed solution offers a general treatment of the path-following problem by removing the dependency on the path's geometry, which makes it applicable to paths with varying levels of complexity and smooth curvatures. Additionally, the proposed strategy draws inspiration from the pursuit guidance approach, which is known for its simplicity and ease of implementation. Theoretical analysis guarantees that the UAV converges to its desired path within a fixed time and remains on it irrespective of its initial configuration with respect to the path. Finally, the simulations demonstrate the merits and effectiveness of the proposed guidance strategy through a wide range of engagement scenarios, showcasing the UAV's ability to follow diverse curvilinear paths accurately.
Paper Structure (5 sections, 7 theorems, 65 equations, 8 figures, 1 table, 1 algorithm)

This paper contains 5 sections, 7 theorems, 65 equations, 8 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Dynamical Model
  3. Derivation of the Guidance Strategy
  4. Performance Evaluations
  5. Conclusions

Key Result

Lemma 1

(6104367) If there exists a continuous radially unbounded function, $\mathcal{V}(\varpi):\mathbb R^n\to \mathbb{R}_+ \cup \{0\}$ such that (1) $\mathcal{V}(\varpi)=0$$\implies$$\varpi \in \mathfrak{R}$; (2) any solution $\varpi(t,\gamma_0)$ of the system eqn:dyn_system satisfies for some $\kappa_1,\kappa_2,\alpha,\beta>0$ such that $\alpha>1$ and $0<\beta<1,$ then the origin is fixed-time stable

Figures (8)

  • Figure 1: UAV-target path following scenario.
  • Figure 2: Block-diagram representation of the proposed strategy.
  • Figure 3: UAV following a helix-like curvilinear path with $V\textsubscript{0}=3$ m/s.
  • Figure 4: UAV following a helix-like curvilinear path with $V\textsubscript{0}=0$ m/s.
  • Figure 5: UAV following a helix-like curvilinear path with different $T\textsubscript{2}$ and $T\textsubscript{3}$.
  • ...and 3 more figures

Theorems & Definitions (26)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 1
  • Lemma 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • ...and 16 more