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Nonlinear Optimal Guidance for Intercepting Moving Targets

Han Wang, Zheng Chen

TL;DR

This work addresses real-time nonlinear optimal guidance for intercepting moving targets by deriving PMP-based necessary and sufficient conditions that reveal extremals depend on two scalar parameters, then constructing a parameterized extremal family to generate a locally optimal trajectory dataset. A lightweight neural network learns the mapping from state and target motion to the optimal lateral command, enabling real-time guidance with guaranteed local optimality under studied conditions. The method is extended to maneuvering targets through augmentation terms inspired by established guidance laws, and numerical simulations show substantial reductions in control effort compared to traditional PN and PEOG schemes, including robustness against target maneuvers. The proposed framework offers a practical pathway to near-optimal, real-time intercept guidance in 2D engagements with both nonmaneuvering and maneuvering targets.

Abstract

This paper introduces a nonlinear optimal guidance framework for guiding a pursuer to intercept a moving target, with an emphasis on real-time generation of optimal feedback control for a nonlinear optimal control problem. Initially, considering the target moves without maneuvering, we derive the necessary optimality conditions using Pontryagin's Maximum Principle. These conditions reveal that each extremal trajectory is uniquely determined by two scalar parameters. Analyzing the geometric property of the parameterized extremal trajectories not only leads to an additional necessary condition but also allows to establish a sufficient condition for local optimality. This enables the generation of a dataset containing at least locally optimal trajectories. By studying the properties of the optimal feedback control, the size of the dataset is reduced significantly, allowing training a lightweight neural network to predict the optimal guidance command in real time. Furthermore, the performance of the neural network is enhanced by incorporating the target's acceleration, making it suitable for intercepting both uniformly moving and maneuvering targets. Finally, numerical simulations validate the proposed nonlinear optimal guidance framework, demonstrating its better performance over existing guidance laws.

Nonlinear Optimal Guidance for Intercepting Moving Targets

TL;DR

This work addresses real-time nonlinear optimal guidance for intercepting moving targets by deriving PMP-based necessary and sufficient conditions that reveal extremals depend on two scalar parameters, then constructing a parameterized extremal family to generate a locally optimal trajectory dataset. A lightweight neural network learns the mapping from state and target motion to the optimal lateral command, enabling real-time guidance with guaranteed local optimality under studied conditions. The method is extended to maneuvering targets through augmentation terms inspired by established guidance laws, and numerical simulations show substantial reductions in control effort compared to traditional PN and PEOG schemes, including robustness against target maneuvers. The proposed framework offers a practical pathway to near-optimal, real-time intercept guidance in 2D engagements with both nonmaneuvering and maneuvering targets.

Abstract

This paper introduces a nonlinear optimal guidance framework for guiding a pursuer to intercept a moving target, with an emphasis on real-time generation of optimal feedback control for a nonlinear optimal control problem. Initially, considering the target moves without maneuvering, we derive the necessary optimality conditions using Pontryagin's Maximum Principle. These conditions reveal that each extremal trajectory is uniquely determined by two scalar parameters. Analyzing the geometric property of the parameterized extremal trajectories not only leads to an additional necessary condition but also allows to establish a sufficient condition for local optimality. This enables the generation of a dataset containing at least locally optimal trajectories. By studying the properties of the optimal feedback control, the size of the dataset is reduced significantly, allowing training a lightweight neural network to predict the optimal guidance command in real time. Furthermore, the performance of the neural network is enhanced by incorporating the target's acceleration, making it suitable for intercepting both uniformly moving and maneuvering targets. Finally, numerical simulations validate the proposed nonlinear optimal guidance framework, demonstrating its better performance over existing guidance laws.

Paper Structure

This paper contains 21 sections, 5 theorems, 44 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Characterization of Optimal Trajectories
  4. Necessary Conditions
  5. Parametrization of Extremal Trajectories
  6. Supplementary Optimality Conditions
  7. NOG for Intercepting Nonmaneuvering Targets
  8. Guidance Architecture
  9. Dataset Generation for Training NN
  10. Guidance for Intercepting Maneuvering Targets
  11. Numerical Simulations
  12. Simulations for Intercepting Nonmaneuvering Target
  13. Engagement I
  14. Engagement II
  15. Simulations for Intercepting Constant Maneuvering Target
  16. ...and 6 more sections

Key Result

Lemma 1

Given a pursue and a target, let the heading angle of the target be 0, i.e., $\theta_T = 0$. Then, for any speed ratio $\mu \in (0,1)$ and any initial condition $(r_0,\lambda_0,\theta_{P0})$ for Problem pro1, there exists $\tau>0$, $\Theta_0 \in [0,2\pi)$, and $\Lambda_0\in [0,2\pi)$ so that Conversely, given any $\mu \in (0,1)$ and any $\Theta_0,\Lambda_0 \in [0,2\pi)$, there exists an initial c

Figures (11)

  • Figure 1: Engagement geometry.
  • Figure 2: Guidance framework for intercepting a nonmaneuvering target.
  • Figure 3: The geometry for relative motion.
  • Figure 4: Trajectories, profiles of guidance command and pursuer's heading angle of Engagement I & II.
  • Figure 5: Comparison of trajectories related to different guidance laws for intercepting a constant maneuvering target.
  • ...and 6 more figures

Theorems & Definitions (5)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5