Source-linked AI summary
Impact-Time Guidance via Normal Contraction to a Time-to-Go Isochron
Shivam Bajpai, Abhinav Sinha
TL;DR
Impact-time guidance must meet an assigned arrival time while maintaining homing compatibility and respecting bounded lateral acceleration. The paper formulates timing as contraction toward a moving time-to-go isochron, then derives feasibility, calibration, and handover conditions. Under the stated assumptions, the guidance law ensures first capture at the prescribed time, and reported target cases achieve scheduled interception within acceleration bounds.
Problem
Impact-time guidance must ensure that timing-coordinate convergence corresponds to physical capture while remaining compatible with available acceleration and declining timing authority.
Source: §I · p. 1Method
The paper treats the schedule as a moving time-to-go isochron, characterizes homing-compatible maps and predictor mismatch, and regulates normal motion with a robust scalar timing channel.
Source: §I · p. 1, §III · p. 3Results
Under the stated assumptions, terminal calibration and feasible funnel constraints establish first capture at the assigned time; reported stationary and modeled maneuvering target cases achieve scheduled interception within acceleration bounds.
Source: §III · p. 5, §V · pp. 9–10Takeaways & Limitations
Preterminal alignment followed by baseline homing with an exact nominal predictor preserves the assigned arrival time as lateral timing authority diminishes.
Source: §I · p. 2, §V · pp. 9–10
Abstract
from arXiv · showhide
We develop a contraction-based perspective on impact-time guidance that augments a baseline homing command with a timing bias. The proposed perspective treats the prescribed schedule as a moving time-to-go isochron and regulates motion normal to that set through velocity-normal lateral acceleration while the interceptor's speed remains constant. We derive a transport equation that characterizes homing-compatible time-to-go coordinates and define a predictor defect that quantifies the mismatch of approximate maps. We show that the scalar timing channel induces a coordinate-invariant rank-one metric on the normal quotient. To account for bounded lateral acceleration, we formulate a robust scalar filter and derive a necessary and sufficient condition for pointwise feasibility. We then show that terminal calibration and funnel invariance establish first interception at the prescribed time under the stated assumptions. We also develop a preterminal alignment and homing handover that avoids singular inversion as lateral timing authority vanishes near collision-course alignment. The proposed perspective accommodates analytic, numerical, and learned time-to-go maps that satisfy the required calibration and regularity conditions.
I. INTRODUCTION
Impact-time guidance must coordinate arrival while preserving homing compatibility, respecting bounded acceleration, and ensuring timing convergence corresponds to first capture. The paper frames the schedule as a moving time-to-go level set and introduces a contraction-based timing channel with feasibility and handover conditions.
I. INTRODUCTION
- Impact-time scheduling supports simultaneous and cooperative interception and time-constrained pursuit-evasion, alongside related scheduling objectives in rendezvous, docking, and aircraft arrivals.The guidance problem augments a baseline homing command with a timing correction; related approaches include biased PN, Lyapunov-based, and sliding-mode designs.Source: §I · p. 1
- A prescribed arrival time selects a moving time-to-go level set, since states with different ranges, headings, and future paths can share the same time-to-go.Normal perturbations change timing error, while tangential perturbations preserve the timing coordinate to first order; the normal metric leaves tangential homing motion unrestricted.Source: §I · p. 1
- Homing-compatible time-to-go maps are characterized by a transport boundary-value problem, with a predictor defect for approximate maps and a coordinate-invariant rank-one metric on the normal quotient.The formulation treats the prescribed schedule as a moving isochron and measures timing-relevant variations normal to it.Source: §I · p. 1
- A robust scalar projection has a necessary and sufficient feasibility condition under the acceleration bound, while terminal calibration connects funnel invariance to first capture at the assigned instant.The timing coordinate must be calibrated to physical capture and remain positive before the assigned instant to exclude early interception.Source: §I · pp. 1–2
- The proposed formulation combines a homing command with a timing bias and establishes first capture at the assigned instant within the acceleration limit under stated assumptions.It uses a stationary-target PN construction and four alternative predictors to illustrate the analysis under velocity-normal actuation.Source: §I · p. 2
- Declining lateral timing authority near collision-course alignment motivates preterminal alignment and homing handover that preserve the assigned arrival time for an exact nominal predictor.The baseline PN command retains its biased-PN structure.Source: §I · pp. 1–2
II. PROBLEM FORMULATION
The formulation models constant-speed interception with lateral acceleration as the steering input, then defines a baseline time-to-go coordinate and a scheduled moving level set. A timing bias drives motion toward that set, with map compatibility, terminal calibration, and acceleration bounds shaping the problem.
II. PROBLEM FORMULATION
- The interceptor maintains constant speed while lateral acceleration changes its velocity direction; the model assumes a sufficiently fast autopilot tracks that command.The lateral input is the sole steering command, with no independent radial channel.Source: §II · p. 2
- The baseline first-hitting-time map gives the remaining time until capture under a homing law, and the exact map decreases at one second per second along nominal trajectories.Its characteristic curves are the baseline homing trajectories; the map can be solved analytically, integrated along trajectories, or tabulated offline.Source: §II · p. 2
- A predictor defect measures departure of a candidate time-to-go map from the exact baseline map, and its accumulation along the nominal trajectory bounds their difference at capture.The defect is zero when the candidate map evolves as the exact map under nominal homing.Source: §II · pp. 2–3
- The time-to-go map must be smooth, vanish on the capture set, have regular level sets with bounded gradient on considered timing tubes, and satisfy terminal calibration conditions.The internal dynamics must remain in the engagement domain; other engagement coordinates, including seeker field of view and terminal angle, need their own constraints.Source: §II · pp. 2–3
- The prescribed impact time defines a moving zero-error level set; with the exact baseline map, every state on that set reaches capture at the scheduled absolute time.States with different ranges, headings, and future paths can share a baseline time-to-go.Source: §II · p. 3
- A timing bias augments the baseline homing command to contract toward the scheduled level set, while requiring zero bias on that set when the predictor defect and disturbance vanish.The commanded lateral acceleration must remain within its bound.Source: §II · p. 3
- The acceleration limit constrains attainable impact times; a bounded-curvature path inequality screens stationary-target cases, while moving-target reachability needs further analysis.The online feasibility condition accounts for timing sensitivity along the engagement.Source: §II · p. 3
III. MAIN RESULTS
The section treats the assigned schedule as a moving time-to-go isochron and regulates timing error through normal steering. It develops bounded-input feasibility and terminal conditions for first capture, then addresses loss of timing authority near alignment.
III. MAIN RESULTS
- The timing channel induces a coordinate-invariant rank-one metric that is positive on the normal quotient and vanishes on tangent perturbations to the scheduled isochron.In a local tube, timing-error magnitude is comparable to distance from the moving isochron; tangent perturbations remain governed by homing dynamics and other constraints.Source: §III · p. 4
- The scalar filter is feasible exactly when the required timing-action interval intersects the authority available from bounded lateral acceleration.At zero timing authority, feasibility requires 0 to lie within the required interval; excessive uncertainty or rapid funnel shrinkage can make that interval empty even with large acceleration limits.Source: §III · p. 5
- The scalar timing-channel analysis extends to vector actuation by replacing scalar authority with the input-sensitivity norm times the input bound.The extension includes three-dimensional point-mass guidance with two lateral steering components.Source: §III · p. 5
- A predictor defect measures a map’s mismatch with the baseline homing flow, while local authority and feasibility tests assess whether that map supports timing regulation.The predictor coordinates are terminally calibrated on compact subsets of their domains, and loss-of-authority surfaces enter the nonsingularity analysis.Source: §III · pp. 5–6
- Under the stated assumptions and pointwise feasibility, the timing funnel is robustly invariant and first capture occurs at the prescribed absolute time.The coordinate remains positive before the assigned time, while terminal calibration links convergence to physical capture.Source: §III · p. 5
- For a nontrivial stationary-target PN trajectory with N > 1 approaching collision-course capture, bounded nominal timing correction requires c ≥ N + 1; c > N + 1 makes it vanish at capture.Source: §III · p. 6
- The guarantees require a target model, sufficient control authority, and terminally compatible prediction error; arbitrary unknown target motion is outside the results.The analysis takes achieved lateral acceleration as input; first-order acceleration or turn-rate dynamics require a new dynamic contraction and filter design.Source: §III · p. 6
- A preterminal handover avoids continued inversion as authority declines: align the exact time-to-go coordinate, then return to baseline homing for first capture at tf.With c > 1 and authority bounded away from zero before handover, the correction is bounded and tends to zero at handover.Source: §III · pp. 6–7
IV. SIMULATIONS
The simulations examine timing regulation with a stationary-target large-heading PN approximation and a maneuvering-target deviated-pursuit predictor. The maneuvering-target case reports interception at the prescribed time, while both cases track timing behavior and predictor mismatch.
IV. SIMULATIONS
- The stationary-target case uses a large-heading PN approximation, while the maneuvering-target case evaluates a deviated-pursuit predictor.The reported histories include trajectories, range, time-to-go, timing error, lateral acceleration, timing effectiveness, and predictor defect.Source: §IV · p. 7
- The maneuvering-target trajectories reach interception at the prescribed time, with timing error inside the funnel and approaching zero by handover.The deviated-pursuit predictor initially differs from the exact coordinate and approaches its terminal profile as the engagement evolves.Source: §IV · p. 8
- In the deviated-pursuit case, the preterminal bias stays within the acceleration bounds and vanishes at handover, while the predictor defect measures disagreement with the baseline homing flow.The exact coordinate has reference defect εT,H = 0.Source: §IV · p. 9
V. CONCLUSIONS AND FUTURE WORK
The paper presents normal-contraction impact-time guidance with a scalar timing correction to baseline homing. Under the stated assumptions, calibrated timing convergence and feasible funnel constraints yield first capture at the assigned time; the reported cases achieve scheduled interception within acceleration bounds.
V. CONCLUSIONS AND FUTURE WORK
- Terminal calibration and feasible funnel constraints link timing convergence to first capture at the assigned impact time under the stated engagement assumptions.As lateral timing authority declines, preterminal alignment followed by an exact nominal predictor allows baseline homing to preserve the assigned arrival time.Source: §V · p. 9
- The reported stationary and modeled maneuvering-target cases achieve scheduled interception within acceleration bounds, while approximate predictors show geometry-varying mismatch.Source: §V · p. 10
- Future work will incorporate actuator dynamics and sensing constraints and examine three-dimensional cooperative interception.Source: §V · p. 10