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Three-Dimensional Nonlinear Guidance with Impact Time and Field-of-view Constraints

Ashok R Samrat, Swati Singh, Shashi Ranjan Kumar

TL;DR

This paper tackles time-constrained interception in 3D under bounded seeker FOV by developing nonlinear backstepping guidance laws. It introduces two complementary designs: one using an effective lead angle and another using heading-angle virtual inputs, both ensuring interception at a prescribed time without relying on time-to-go estimates and without decoupling the 3D dynamics. The authors prove convergence properties, derive bounds on achievable impact times, and analyze FOV adherence, supported by extensive simulations including octant control and constant-velocity targets. The results show improved flexibility, reduced sensitivity to large initial heading errors, and the ability to achieve larger feasible impact times compared with existing strategies, making the approach practically appealing for fielded interceptor systems.

Abstract

This paper addresses the time-constrained interception of targets at a predetermined time with bounded field-of-view capability of the seeker-equipped interceptors. We propose guidance laws using the effective lead angle and velocity lead angles of the interceptor to achieve a successful interception of the target. The former scheme extends the existing two-dimensional guidance strategy to a three-dimensional setting. We have shown that such an extension may result in high-frequency switching in the input demand, which may degrade the interceptor's performance. To overcome the potential limitations of such a guidance strategy, we propose an elegant solution using the velocity lead angles and the range error with a backstepping technique. Using the velocity lead angles as virtual inputs, the effective lead angle profile is subsequently regulated to satisfy the seeker's field-of-view bound. Unlike the existing strategies, the proposed guidance strategy does not rely on the time-to-go estimate, which is an appealing feature of the design, as the time-to-go estimate may not always be available with high precision. We provide a theoretical analysis of the error variable and subsequently analytically derive the bounds on achievable impact times. Numerical simulations are performed to support the theoretical findings. The performance of the proposed guidance strategy is compared with that of an existing one, and it has been shown to yield better performance. Finally, a study on different choices of virtual inputs is also provided.

Three-Dimensional Nonlinear Guidance with Impact Time and Field-of-view Constraints

TL;DR

This paper tackles time-constrained interception in 3D under bounded seeker FOV by developing nonlinear backstepping guidance laws. It introduces two complementary designs: one using an effective lead angle and another using heading-angle virtual inputs, both ensuring interception at a prescribed time without relying on time-to-go estimates and without decoupling the 3D dynamics. The authors prove convergence properties, derive bounds on achievable impact times, and analyze FOV adherence, supported by extensive simulations including octant control and constant-velocity targets. The results show improved flexibility, reduced sensitivity to large initial heading errors, and the ability to achieve larger feasible impact times compared with existing strategies, making the approach practically appealing for fielded interceptor systems.

Abstract

This paper addresses the time-constrained interception of targets at a predetermined time with bounded field-of-view capability of the seeker-equipped interceptors. We propose guidance laws using the effective lead angle and velocity lead angles of the interceptor to achieve a successful interception of the target. The former scheme extends the existing two-dimensional guidance strategy to a three-dimensional setting. We have shown that such an extension may result in high-frequency switching in the input demand, which may degrade the interceptor's performance. To overcome the potential limitations of such a guidance strategy, we propose an elegant solution using the velocity lead angles and the range error with a backstepping technique. Using the velocity lead angles as virtual inputs, the effective lead angle profile is subsequently regulated to satisfy the seeker's field-of-view bound. Unlike the existing strategies, the proposed guidance strategy does not rely on the time-to-go estimate, which is an appealing feature of the design, as the time-to-go estimate may not always be available with high precision. We provide a theoretical analysis of the error variable and subsequently analytically derive the bounds on achievable impact times. Numerical simulations are performed to support the theoretical findings. The performance of the proposed guidance strategy is compared with that of an existing one, and it has been shown to yield better performance. Finally, a study on different choices of virtual inputs is also provided.

Paper Structure

This paper contains 20 sections, 10 theorems, 71 equations, 13 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Main Results
  4. Derivation of Guidance Strategy using Effective Lead Angle
  5. Derivation of Guidance Strategy using Heading Angles
  6. Analysis of the Guidance Law
  7. Convergence of Range Error
  8. Lead Angle Analysis
  9. Achievable Impact Time
  10. Choice of Virtual Inputs
  11. Performance Study
  12. Performance of Backstepping using Effective Lead Angle
  13. Octant Of Interceptor's Trajectories
  14. Performance with Different Impact Times
  15. Performance with Different Initial Heading Angles
  16. ...and 5 more sections

Key Result

Lemma 1

Khalil2002Ch4 Let $\pmb{x}=0$ be an equilibrium point for the system $\dot{\pmb{x}}=\pmb{f}(\pmb{x})$. Let $V:\mathbb{R}^n\rightarrow \mathbb{R}$ be a continuously differentiable function such that

Figures (13)

  • Figure 1: 3D engagement Geometry
  • Figure 2: Performance for $t_{\rm f}=70\,$s, initial heading angles $\theta_{\rm m}=-30^\circ,\,\psi_{\rm m}=30^\circ$.
  • Figure 3: Control over the Octant of the Interceptor Trajectories.
  • Figure 4: X-axis view of fig.\ref{['fig:Trajec_Oct_case_e_xyz']}.
  • Figure 5: Performance for $t_{\rm f}=60\,$s, $70\,$s, $80\,$s, initial heading angles $\theta_{\rm m}=-30^\circ,\,\psi_{\rm m}=30^\circ$.
  • ...and 8 more figures

Theorems & Definitions (26)

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