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Optimal impact-angle guidance / 2025

Optimal Impact Angle Guidance Law Based on Collision Geometry for Axially Maneuvering Targets

Jiedong He, Meng Yu, Yin Wang

A generalized optimal impact-angle guidance framework for head-on and head-pursuit engagements against targets undergoing axial acceleration, drag-like deceleration, and composite axial maneuvers.

DOI: 10.1177/09544100251401024 ↗Browse all figures ↓Run Figures 13–16 in 3D →
Published figures16
Mapped main scripts13
Comparison modules8

01 / Problem and model

Model, frames, and assumptions

This study addresses planar impact-angle-constrained guidance against axially maneuvering targets and unifies head-on and head-pursuit interception strategies through collision-triangle geometry. The code propagates absolute positions and computes relative motion from the target–interceptor position difference.

Coordinate frameTwo-dimensional inertial X–O–Y frame. XI and XT are absolute positions; XT − XI is the relative position. The paper and code use σ for the LOS angle.UnitsPosition: m internally (km in initial tables/trajectory plots) · velocity: m/s · angle: rad internally and deg in plots · acceleration: m/s² internally and g in plotsNotation mappaper/code σ ↔ website λ (LOS); gammaI ↔ gamma_M; gammaT ↔ gamma_T. In the paper, lambdaI and lambdaT remain lead angles.NumericsRK4 state update; adaptive step from 0.01 s to 0.0001 s as range decreases; Newton–Raphson time-to-go estimation.
01

Kinematic guidance model; control dynamics are neglected.

02

Constant interceptor speed throughout each engagement.

03

Planar engagement with a sufficiently fast roll channel.

04

Gravity is omitted from relative-motion kinematics.

05

Small heading-error approximation is used in the derivation.

06

The theoretical guidance law assumes an ideal autopilot without maneuverability constraints.

02 / Nominal conditions

Two nominal interception scenarios

Angles are listed in deg and converted to rad inside the code. Initial positions are listed in km and converted to m inside the simulation.

SCENARIO 1

Scenario 1 · Head-on / decelerating target

  • XI0 = (0, 0) km
  • VI0 = 2500 m/s
  • γI0 = 30 deg
  • XT0 = (50, 0) km
  • VT0 = 3000 m/s
  • γT0 = 150 deg
  • γd = 60 deg
  • uT = −gζVT², ζ = 1.0×10⁻⁶
SCENARIO 2

Scenario 2 · Head-pursuit / accelerating target

  • XI0 = (0, 0) km
  • VI0 = 2500 m/s
  • γI0 = 100 deg
  • XT0 = (50, 0) km
  • VT0 = 3000 m/s
  • γT0 = 150 deg
  • γd = 135 deg
  • uT = 5g

03 / Published evidence

Figures 1–16 and code entry points

Images are extracted from the final composite figures in the paper PDF. Select an image to enlarge it; each card lists the corresponding MATLAB entry point and final result directory.

FIGURE 01ARTICLE P. 3Problem formulation

Engagement geometry between interceptor and target

No MATLAB drawing source

Conceptual geometry; no MATLAB drawing source was found.

FIGURE 02ARTICLE P. 4Guidance formulation

ZEM under two possible interception geometries

No MATLAB drawing source

Conceptual geometry; no MATLAB drawing source was found.

FIGURE 03ARTICLE P. 8Basic characteristics

Different weighting exponents n · Scenario 1

HO_Dece.m

RESULTS0Basic characteristics analysis\Various N\HO_Dece

FIGURE 04ARTICLE P. 9Basic characteristics

Different weighting exponents n · Scenario 2

HP_Acce.m

RESULTS0Basic characteristics analysis\Various N\HP_Acce

FIGURE 05ARTICLE P. 10Basic characteristics

Time-to-go estimation error

HO_Dece.mHP_Acce.m

RESULTS0Basic characteristics analysis\Various N

Generated by the seventh figure window of the n-variation scripts, not by the files named Timetogo_Comparison.

FIGURE 06ARTICLE P. 11Basic characteristics

Varying initial flight-path angles · Scenario 1

HO_Dece_Initial_Cdt.m

RESULTS0Basic characteristics analysis\Various Initial Cdt\HO_Dece

FIGURE 07ARTICLE P. 12Basic characteristics

Varying initial flight-path angles · Scenario 2

HP_Acce_Initial_Cdt.m

RESULTS0Basic characteristics analysis\Various Initial Cdt\HP_Acce

FIGURE 08ARTICLE P. 13Basic characteristics

Varying terminal impact angles · Scenario 1

HO_Dece_Terminal_Cdt.m

RESULTS0Basic characteristics analysis\Various Terminal Cdt\HO_Dece

FIGURE 09ARTICLE P. 14Basic characteristics

Varying terminal impact angles · Scenario 2

HP_Acce_Terminal_Cdt.m

RESULTS0Basic characteristics analysis\Various Terminal Cdt\HP_Acce

FIGURE 10ARTICLE P. 15Basic characteristics

Complex composite axial target maneuvers

Against_Complex_TM.m

RESULTS0Basic characteristics analysis\Complex TM

Uses the published switching thresholds R1 = 0.8R0 and R2 = 0.05R0. test_Complex_TM.m is only a parameter-search utility.

FIGURE 11ARTICLE P. 16Robustness analysis

Target-parameter estimation errors · Scenario 1

HO_Dece_Robustness.m

RESULTS1Robustness analysis\HO_Dece

FIGURE 12ARTICLE P. 17Robustness analysis

Target-parameter estimation errors · Scenario 2

HP_Acce_Robustness.m

RESULTS1Robustness analysis\HP_Acce

FIGURE 13ARTICLE P. 18Comparison analysis

Energy-optimal comparison · Scenario 1

HO_EnergyOptimal_Comparison.mHO_Dece_{Proposed,TWOIAG,DBOIAG,RWOIAG}.m

RESULTS2Comparison analysis\Energy optimal\HO_Dece

FIGURE 14ARTICLE P. 19Comparison analysis

Energy-optimal comparison · Scenario 2

HP_EnergyOptimal_Comparison.mHP_Acce_{Proposed,TWOIAG,DBOIAG,RWOIAG}.m

RESULTS2Comparison analysis\Energy optimal\HP_Acce

FIGURE 15ARTICLE P. 20Comparison analysis

Time-to-go weighted comparison · Scenario 1

HO_Timetogo_Comparison.mHO_Dece_{Proposed,TWOIAG,DBOIAG,RWOIAG}.m

RESULTS2Comparison analysis\Time to go weighted\HO_Dece

The published legend says ODIAG; the actual function and manuscript text identify the method as DBOIAG.

FIGURE 16ARTICLE P. 21Comparison analysis

Time-to-go weighted comparison · Scenario 2

HP_Timetogo_Comparison.mHP_Acce_{Proposed,TWOIAG,DBOIAG,RWOIAG}.m

RESULTS2Comparison analysis\Time to go weighted\HP_Acce

04 / Evidence notes

Differences between paper and code
that must not be silently ignored

These items are not automatically “corrected.” The site preserves both the published description and actual code behavior until numerical reruns further establish their effects.

01

CONFIRMED

Monte Carlo distribution

The manuscript states N(0, 0.3²) with a ±30% limit. The MATLAB scripts use σ = 0.3/3 = 0.1 and truncate samples to ±0.3.

02

CONFIRMED

Random seed

The robustness scripts call randn without rng(seed); published scatter locations cannot be reproduced point-for-point without an added seed.

03

CONFIRMED

Figure 15 legend

The figure displays ODIAG, while the called function and manuscript text identify DBOIAG.

04

REQUIRES RERUN

Command clipping

The eight comparison functions clip commands to ±200g although the theoretical derivation assumes an unconstrained ideal autopilot. Whether clipping is active should be logged during reruns.

05

CONFIRMED

Terminal threshold

Scripts use R < 1 m plus either tgo < 10⁻⁴ s or 10⁻³ s; the exact threshold must be recorded per run.

05 / Local source map

Code archive location

LOCAL CODE ROOTPaper_1 · author-maintained MATLAB archive

Final main scripts

  • HO_Dece.m
  • HP_Acce.m
  • HO_Dece_Initial_Cdt.m
  • HP_Acce_Initial_Cdt.m
  • HO_Dece_Terminal_Cdt.m
  • HP_Acce_Terminal_Cdt.m
  • Against_Complex_TM.m
  • HO_Dece_Robustness.m
  • HP_Acce_Robustness.m
  • HO_EnergyOptimal_Comparison.m
  • HP_EnergyOptimal_Comparison.m
  • HO_Timetogo_Comparison.m
  • HP_Timetogo_Comparison.m

Not final figure entries

  • HO_Const.m — constant-speed illustration
  • HP_Const.m — constant-speed illustration
  • test_Const.m — non-paper test parameters
  • test_Complex_TM.m — parameter-search utility
  • 5.1 / 5.2 / 5.3 — earlier result folders