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Nonlinear Optimal Guidance for Impact Time Control with Field-of-View Constraint

Fangmin Lu, Zheng Chen, Kun Wang

TL;DR

This work tackles impact time control with Field-of-View constraints by converting the inequality-constrained nonlinear OCP into an equality-constrained problem using a saturation function, then deriving necessary optimality conditions via Pontryagin's Maximum Principle. A parameterized extremal system is constructed so that varying a few trajectory parameters generates rich, optimal-like data, which is used to train a neural network that maps the current polar state $\left(r,\sigma\right)$ and time-to-go $t_g$ to an optimal guidance command $u$ in real time. The neural network, with a compact two-hidden-layer architecture, delivers guidance commands in about $1\times 10^{-2}$ seconds and respects the FOV bound while achieving nearly optimal control effort, as demonstrated across multiple simulations and comparisons with existing PN- and SMC-based ITCG methods. This approach enables real-time, FOV-compliant, nonlinear optimal guidance and provides a scalable framework for extending to higher dimensions. $|\,\sigma\,|\leq\sigma_M$ and related PMP conditions underpin the core methodology.

Abstract

An optimal guidance law for impact time control with field-of-view constraint is presented. The guidance law is derived by first converting the inequality-constrained nonlinear optimal control problem into an equality-constrained one through a saturation function. Based on Pontryagin's maximum principle, a parameterized system satisfying the necessary optimality conditions is established. By propagating this system, a large number of extremal trajectories can be efficiently generated. These trajectories are then used to train a neural network that maps the current state and time-to-go to the optimal guidance command. The trained neural network can generate optimal commands within 0.1 milliseconds while satisfying the field-of-view constraint. Numerical simulations demonstrate that the proposed guidance law outperforms existing methods and achieves nearly optimal performance in terms of control effort.

Nonlinear Optimal Guidance for Impact Time Control with Field-of-View Constraint

TL;DR

This work tackles impact time control with Field-of-View constraints by converting the inequality-constrained nonlinear OCP into an equality-constrained problem using a saturation function, then deriving necessary optimality conditions via Pontryagin's Maximum Principle. A parameterized extremal system is constructed so that varying a few trajectory parameters generates rich, optimal-like data, which is used to train a neural network that maps the current polar state and time-to-go to an optimal guidance command in real time. The neural network, with a compact two-hidden-layer architecture, delivers guidance commands in about seconds and respects the FOV bound while achieving nearly optimal control effort, as demonstrated across multiple simulations and comparisons with existing PN- and SMC-based ITCG methods. This approach enables real-time, FOV-compliant, nonlinear optimal guidance and provides a scalable framework for extending to higher dimensions. and related PMP conditions underpin the core methodology.

Abstract

An optimal guidance law for impact time control with field-of-view constraint is presented. The guidance law is derived by first converting the inequality-constrained nonlinear optimal control problem into an equality-constrained one through a saturation function. Based on Pontryagin's maximum principle, a parameterized system satisfying the necessary optimality conditions is established. By propagating this system, a large number of extremal trajectories can be efficiently generated. These trajectories are then used to train a neural network that maps the current state and time-to-go to the optimal guidance command. The trained neural network can generate optimal commands within 0.1 milliseconds while satisfying the field-of-view constraint. Numerical simulations demonstrate that the proposed guidance law outperforms existing methods and achieves nearly optimal performance in terms of control effort.

Paper Structure

This paper contains 14 sections, 4 theorems, 33 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Nonlinear Kinematics
  4. Optimal Control Problem
  5. Parameterization of the Optimal Solution
  6. Necessary Conditions
  7. Additional Necessary Conditions
  8. Parameterization of Extremal Trajectories
  9. Real-Time Solution via NNs
  10. Dataset Generation
  11. Optimal Guidance via NNs
  12. Simulations
  13. Conclusions
  14. Proof of Lemmas

Key Result

Lemma 1

Given any extremal trajectory $(x(\cdot),y(\cdot),\theta(\cdot))$ on $[0,t_f]$, if there exists a time ${\hat{t}}\in (0,t_f)$ so that the velocity vector is collinear with the LOS, i.e., then the extremal trajectory $(x(\cdot),y(\cdot),\theta(\cdot))$ on $[0,t_f]$ is not an optimal trajectory.

Figures (5)

  • Figure 1: Geometry of a two-dimensional interception.
  • Figure 2: A schematic view of the optimal guidance.
  • Figure 3: Optimality validation with FOV limits of 30, 45 and 60 deg.
  • Figure 4: Optimality validation with FOV limits of 60 deg and tg of 50, 55, and 60 s.
  • Figure 5: Performance comparison with the PN-based method Dong2022article and SMC-based method 2018Nonsingular.

Theorems & Definitions (6)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • proof : Proof of Lemma \ref{['le:align']}:
  • proof : Proof of Lemma \ref{['lemma:jacobian']}: