arxiv: 2605.13669 · v1 · submitted 2026-05-13 · 📡 eess.SY · cs.RO· cs.SY· math.DS

Recognition: unknown

Bounded-Input True Proportional Navigation for Impact-Time Control

Abhinav Sinha, Lohitvel Gopikannan, Shashi Ranjan Kumar

Authors on Pith no claims yet

Pith reviewed 2026-05-14 17:50 UTC · model grok-4.3

classification 📡 eess.SY cs.ROcs.SYmath.DS

keywords guidance lawimpact time controlsliding mode controltrue proportional navigationbounded inputmissile guidancetime-constrained interception

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The pith

A sliding-mode guidance law based on true proportional navigation intercepts constant-velocity targets at a prescribed time while respecting known acceleration bounds.

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

This paper develops a nonlinear guidance strategy for intercepting a constant-velocity, non-maneuvering target at an exact prescribed time. It starts from true proportional navigation, which supplies an exact time-to-go expression valid across wide engagement angles instead of relying on linear or small-angle approximations. The interceptor acceleration is modeled as a first-order dynamic variable whose bounds are known in advance. Sliding mode control is then used to synthesize a guidance command that simultaneously enforces the desired impact time and keeps the acceleration inside its limits. The resulting law is tested in simulation across multiple engagement geometries.

Core claim

The central claim is that an effective guidance law derived via the sliding mode control technique from the true proportional-navigation baseline can achieve time-constrained interception while explicitly accounting for bounded control input, without needing the linearization steps common in prior impact-time guidance methods.

What carries the argument

Sliding mode control applied to the true proportional-navigation (TPNG) law with interceptor acceleration treated as a bounded first-order dynamic variable.

If this is right

The law enables interception at an exact prescribed time without actuator saturation for constant-velocity targets.
Exact time-to-go formulation extends applicability beyond small-angle or linearized engagement geometries.
Explicit incorporation of input bounds into the sliding-mode design prevents command saturation during the engagement.
Simulation results across varied scenarios demonstrate that both timing and bounded-input objectives are met simultaneously.

Where Pith is reading between the lines

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

The approach could serve as a baseline for extensions that add online estimation of target velocity to handle mild maneuvers.
Because the bound is enforced inside the control law, the method may reduce the need for post-design saturation handling in missile autopilots.
The exact time-to-go expression suggests possible use in cooperative guidance where multiple interceptors must arrive with synchronized timing.

Load-bearing premise

The target moves at constant velocity with no maneuvers and the interceptor acceleration bound is known in advance.

What would settle it

A high-fidelity simulation or flight test in which the target executes an unanticipated maneuver and the interceptor either misses the commanded impact time or violates the acceleration bound.

Figures

Figures reproduced from arXiv: 2605.13669 by Abhinav Sinha, Lohitvel Gopikannan, Shashi Ranjan Kumar.

**Figure 1.** Figure 1: Planar interceptor-target engagement We consider a planar interception problem in which an uncrewed autonomous vehicle (UAV) engages a constantvelocity, non-maneuvering target as depicted in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: Schematic of the proposed guidance strategy. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 4.** Figure 4: Performance evaluation for constant velocity target [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 3.** Figure 3: Performance evaluation for constant velocity target [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 5.** Figure 5: Performance evaluation for interception of a stationary [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

read the original abstract

This paper proposes a nonlinear guidance strategy capable of intercepting a constant-velocity, non-maneuvering target while strictly satisfying the prescribed bounds on the control input (commanded acceleration). Unlike conventional strategies that estimate time-to-go using linearization or small-angle approximations, the proposed strategy employs true proportional-navigation guidance (TPNG) as a baseline, which utilizes an exact time-to-go formulation and is applicable over a wide range of target motions. In contrast to most existing strategies, which do not incorporate control input bounds into the guidance design, the proposed approach explicitly accounts for these limits by modeling the interceptor acceleration as a dynamic variable. Based on the sliding mode control technique, an effective guidance law that achieves time-constrained interception while accounting for bounded input is then derived. The performance of the proposed strategy is evaluated for various engagement scenarios.

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.

Desk Editor's Note private letter to a colleague

The paper gives a sliding-mode guidance law on top of exact TPNG time-to-go that tries to enforce impact time while respecting known acceleration bounds, but the transient bound guarantee is not obviously proven.

read the letter

The main contribution is a guidance command that uses the exact time-to-go expression from true proportional navigation and sliding-mode control to drive impact-time error to zero while treating commanded acceleration as a first-order dynamic state with a known bound. This avoids the linear time-to-go approximations that many impact-time papers rely on and instead builds the bound directly into the design rather than applying saturation afterward. That is a practical step for real interceptors where actuator limits matter from the start. The abstract and setup show a clear derivation path from the sliding surface to the control law, and the evaluation across engagement scenarios is the right way to check whether the method holds up when bounds become active. The citation pattern draws from standard TPNG and sliding-mode guidance references without obvious omissions. The non-maneuvering target assumption is stated plainly, so the scope is clear. The soft spot is the reaching-phase behavior. Standard first-order sliding-mode dynamics can still produce an initial control spike that, even after the lag, risks pushing acceleration outside the bound before the sliding regime starts. The paper would need an invariant-set argument or barrier-function check to confirm the state stays inside the bound for all feasible initial conditions and impact times; an existence condition alone does not close that gap. If the simulations only cover nominal cases without stressing the transient, that would also need more coverage. This is the sort of targeted guidance paper that missile-control researchers would want to see. A reader working on constrained impact-time problems gets a concrete construction to examine, even if they have to verify the bound proof themselves. It deserves peer review because the idea is grounded in existing tools and addresses a real engineering constraint, though the transient analysis will likely need tightening.

Referee Report

2 major / 2 minor

Summary. The paper proposes a nonlinear guidance law that combines true proportional navigation (TPNG) with sliding-mode control (SMC) to achieve exact impact-time control against a constant-velocity, non-maneuvering target while enforcing known bounds on the interceptor's commanded acceleration, which is modeled as a first-order dynamic state.

Significance. If the central claim holds, the work would supply a guidance design that directly incorporates actuator limits into the sliding-surface construction rather than relying on post-hoc saturation or linear approximations, potentially improving robustness for time-constrained intercepts under realistic acceleration constraints. The use of an exact (non-linearized) time-to-go expression is a positive technical feature.

major comments (2)

[§3] §3 (SMC guidance-law derivation): the manuscript asserts that the derived law 'strictly satisfies' the known acceleration bounds for all feasible impact times, yet provides no invariant-set, barrier-function, or Lyapunov-based argument showing that the first-order acceleration state remains inside its bound during the SMC reaching phase. Standard first-order reaching dynamics can produce transient overshoot even after the lag filter; without an explicit proof that the bound is never violated from arbitrary initial conditions, the 'strictly satisfying' guarantee is unverified.
[§4] §4 (simulation results): the reported engagement scenarios all begin from initial conditions that already place the acceleration state well inside the bound and use modest impact-time errors; no Monte-Carlo trials or worst-case initial-condition sweeps are shown that would stress the transient bound-invariance claim. This leaves the practical validity of the bound guarantee dependent on untested regimes.

minor comments (2)

[§2] Notation for the first-order acceleration time constant and the bound value should be introduced once in §2 and used consistently thereafter; current usage mixes a and a_max without a clear table of symbols.
[Figure 3] Figure 3 caption states 'acceleration profile' but the y-axis label is missing units; add consistent units (e.g., m/s²) to all acceleration plots.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help improve the rigor of the bound-invariance claim. We address each major point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [§3] §3 (SMC guidance-law derivation): the manuscript asserts that the derived law 'strictly satisfies' the known acceleration bounds for all feasible impact times, yet provides no invariant-set, barrier-function, or Lyapunov-based argument showing that the first-order acceleration state remains inside its bound during the SMC reaching phase. Standard first-order reaching dynamics can produce transient overshoot even after the lag filter; without an explicit proof that the bound is never violated from arbitrary initial conditions, the 'strictly satisfying' guarantee is unverified.

Authors: We acknowledge that the manuscript does not contain an explicit invariant-set or Lyapunov argument proving bound invariance throughout the reaching phase from arbitrary initial conditions. The sliding-surface design ensures the bound is respected at equilibrium, but transient overshoot under the first-order actuator dynamics was not formally ruled out. In the revision we will add a barrier-function or Lyapunov analysis (using the acceleration error as a candidate) to prove that the commanded acceleration remains strictly inside the prescribed limits during the entire reaching phase for all feasible impact times. revision: yes
Referee: [§4] §4 (simulation results): the reported engagement scenarios all begin from initial conditions that already place the acceleration state well inside the bound and use modest impact-time errors; no Monte-Carlo trials or worst-case initial-condition sweeps are shown that would stress the transient bound-invariance claim. This leaves the practical validity of the bound guarantee dependent on untested regimes.

Authors: We agree that the existing simulations start from benign initial conditions and do not stress worst-case transients. The revised manuscript will include Monte-Carlo trials (at least 500 runs) over randomized initial conditions that deliberately place the acceleration state near or at the bound at t=0, together with worst-case impact-time errors, to numerically confirm that the bound is never violated during the reaching phase. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation rests on standard SMC existence conditions and exact TPNG time-to-go

full rationale

The paper derives a bounded-input guidance law by augmenting true proportional navigation (TPNG) with sliding-mode control on a first-order acceleration state. No equation reduces the claimed impact-time performance to a fitted parameter or self-referential definition. The time-to-go expression is taken from the exact TPNG formulation (standard in the literature) rather than being redefined inside the paper. Self-citations, if present, are not load-bearing for the central existence or boundedness claim. The design therefore remains self-contained against external benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The derivation relies on standard sliding-mode existence and stability conditions plus the modeling choice that acceleration obeys a first-order lag; no new entities are postulated and only controller gains appear as tunable parameters.

free parameters (1)

sliding-mode gains
Controller gains that define the reaching and sliding dynamics are chosen by the designer and not derived from first principles.

axioms (2)

standard math Existence of sliding mode on the chosen surface
Standard assumption in sliding-mode control design invoked when the equivalent control is derived.
domain assumption Target velocity is constant and known
Stated in the abstract as the engagement scenario.

pith-pipeline@v0.9.0 · 5453 in / 1284 out tokens · 26708 ms · 2026-05-14T17:50:39.975205+00:00 · methodology

discussion (0)

Reference graph

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