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arxiv: 2606.02872 · v1 · pith:BNM6WA2Onew · submitted 2026-06-01 · 📡 eess.SY · cs.MA· cs.RO· cs.SY

Terminal Time and Angle-Constrained Nonlinear Intercept Guidance

Shivam Bajpai , Abhinav Sinha This is my paper

Pith reviewed 2026-06-28 13:03 UTC · model grok-4.3

classification 📡 eess.SY cs.MAcs.ROcs.SY
keywords impact timeimpact anglesliding mode controlguidance lawadaptive controlnonlinear kinematicsinterceptionunderactuated systems
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The pith

A hierarchical sliding mode guidance law with a two-layer manifold regulates both impact time and angle using only lateral acceleration.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a method to control an interceptor's impact time and impact angle simultaneously with a single control input, the lateral acceleration, despite the underactuated nonlinear kinematics. It does this by designing a two-layer sliding manifold where the first layer handles individual error dynamics for time and angle, and the second combines them into a composite surface. A variable-gain adaptive law is then derived to drive the system to the desired terminal conditions for stationary targets and extended to constant-velocity ones. This matters because many interception scenarios require precise timing and approach angle, which single-constraint guidance laws cannot guarantee together.

Core claim

The central claim is that a two-layer sliding manifold can be used to construct a hierarchical sliding mode-based guidance law that concurrently satisfies time and angle constraints in nonlinear engagement kinematics with a single input, first for stationary targets via a variable-gain adaptive law and then extended to constant velocity targets.

What carries the argument

The two-layer sliding manifold, with sub-surfaces for impact time and impact angle errors in the first layer and a composite manifold in the second.

If this is right

  • The guidance law ensures time and angle-constrained interception against a stationary target.
  • It extends to intercept a constant velocity target.
  • Simulations for various engagement scenarios attest to the efficacy.
  • The approach handles the underactuated system without conflict in constraints.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This method could potentially be tested in scenarios with maneuvering targets by adapting the sliding surfaces.
  • Integration with estimation techniques might allow application when target velocity is uncertain.
  • The variable-gain adaptation might reduce sensitivity to initial condition variations in real engagements.

Load-bearing premise

That the two-layer sliding manifold can be constructed to concurrently regulate both terminal constraints without conflict in the underactuated nonlinear engagement kinematics.

What would settle it

A simulation or flight test where the achieved impact time or angle deviates from the commanded values by more than the tolerance allowed by the sliding mode dynamics would falsify the claim.

read the original abstract

This paper considers the problem of simultaneously controlling an interceptor's impact time and impact angle using its lateral acceleration as the sole control input. With a single control input, the nonlinear engagement kinematics is inherently underactuated, which complicates guidance law synthesis. To overcome this challenge, a hierarchical sliding mode-based guidance law is developed to concurrently regulate the two terminal constraints. The proposed architecture consists of a two-layer sliding manifold. The first layer comprises two sub-sliding surfaces corresponding to the impact time and impact angle error dynamics, respectively, while the second layer introduces a composite sliding manifold that combines the two individual sub-surfaces. Then, a variable-gain adaptive guidance law is designed to ensure time and angle-constrained interception against a stationary target, which is further extended to intercept a constant velocity target. Simulations are conducted for various engagement scenarios to attest to the efficacy of the proposed approach.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a hierarchical sliding-mode guidance law for simultaneous terminal time and impact angle constraints in nonlinear intercept problems. With lateral acceleration as the sole input, the underactuated engagement kinematics are addressed via a two-layer sliding manifold (first layer with separate impact-time and impact-angle sub-surfaces; second layer a composite manifold). A variable-gain adaptive law is derived for stationary targets and extended to constant-velocity targets; efficacy is asserted via simulations across engagement scenarios.

Significance. If the stability analysis and convergence properties are rigorously established, the work offers a systematic sliding-mode construction for multi-constraint guidance on underactuated systems. The variable-gain adaptation and explicit extension to moving targets are constructive features that could be useful in missile guidance applications. The approach follows standard hierarchical sliding-mode techniques but applies them to a practically relevant pair of terminal constraints.

major comments (2)
  1. [Abstract / guidance law derivation] The central claim that the two-layer manifold concurrently enforces both constraints without conflict rests on the underactuated kinematics; however, the abstract supplies no Lyapunov function, finite-time convergence argument, or controllability condition for the composite surface. A load-bearing stability step is therefore missing from the provided description.
  2. [Simulation section] Simulations are stated to attest to efficacy, yet no quantitative metrics (e.g., impact-time error, angle error, control effort, or robustness to initial-condition variation) or comparison against existing impact-time-and-angle laws appear in the abstract. This leaves the performance claim difficult to evaluate.
minor comments (2)
  1. [Guidance law section] Notation for the sliding surfaces and the variable-gain adaptation law should be introduced with explicit definitions before the stability discussion.
  2. [Extension to moving targets] The extension from stationary to constant-velocity targets would benefit from a short paragraph clarifying which terms in the manifold are modified and why the same adaptive structure remains valid.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below, with references to the full text where the abstract provides only a summary.

read point-by-point responses

  1. Referee: [Abstract / guidance law derivation] The central claim that the two-layer manifold concurrently enforces both constraints without conflict rests on the underactuated kinematics; however, the abstract supplies no Lyapunov function, finite-time convergence argument, or controllability condition for the composite surface. A load-bearing stability step is therefore missing from the provided description.

    Authors: The abstract is a concise overview and does not contain the full technical details. The manuscript develops the two-layer sliding manifold in Section III and provides the Lyapunov stability analysis, finite-time convergence arguments, and conditions ensuring the composite surface enforces both constraints without conflict in Section IV, including handling of the underactuated kinematics. We will revise the abstract to briefly reference these stability results. revision: yes

  2. Referee: [Simulation section] Simulations are stated to attest to efficacy, yet no quantitative metrics (e.g., impact-time error, angle error, control effort, or robustness to initial-condition variation) or comparison against existing impact-time-and-angle laws appear in the abstract. This leaves the performance claim difficult to evaluate.

    Authors: Space constraints limit the abstract to a high-level statement. Section V contains the simulation results with the requested quantitative metrics (impact-time and angle errors, control effort, robustness to initial conditions) and comparisons to existing impact-time-and-angle guidance laws. We will revise the abstract to include key performance figures from these simulations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is a direct constructive design

full rationale

The paper constructs a hierarchical sliding-mode guidance law with a two-layer manifold to enforce impact time and angle constraints on an underactuated engagement model. This is presented as an explicit design choice using standard sliding-mode techniques, with the variable-gain adaptive law derived directly from the manifold definition and extended to constant-velocity targets. No fitted parameters are renamed as predictions, no self-citation chains support load-bearing uniqueness claims, and no result reduces to its inputs by construction. The approach remains self-contained as a synthesis method without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract does not provide sufficient detail to identify specific free parameters, axioms, or invented entities.

pith-pipeline@v0.9.1-grok · 5682 in / 1006 out tokens · 51711 ms · 2026-06-28T13:03:34.475126+00:00 · methodology

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Reference graph

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