Impact Time Control Guidance

Impact-Time-Control Guidance (gnc-autonomy/guidance/impact-time-control-guidance)

Use when the task is computing the impact-time-control guidance (ITCG) command for a salvo or simultaneous-impact engagement against a stationary target: the proportional-navigation (PN) baseline from the closing geometry, the leading-order PNG time-to-go estimate on the collision course, the impact-time error between the commanded remaining time-to-go and the natural PNG estimate, and the time-to-go-error-feedback bias that steers the interceptor group to arrive at the commanded impact time. This is the canonical biased-PN impact-time-control structure of the Jeon-Lee-Tahk family (IEEE Transactions on Aerospace and Electronic Systems 42(2):629-641, 2006, summarized by name and paraphrase only, never reproduced). Pairing leaves: proportional-navigation-guidance owns the unaugmented planar PN law, augmented-proportional-navigation the maneuvering-target augmentation; impact-point-prediction is open-loop ballistic prediction; midcourse-guidance handles waypoint steering and handover.

Domain quick reference

• Planar collision-course intercept against a stationary target: closing speed V_c (m/s), line of sight rate lambda_dot (rad/s), range to target R (m), navigation constant N (dimensionless, > 1), commanded remaining time-to-go t_go_des (s).
• PN baseline acceleration: a_png = N * V_c * lambda_dot.
• PNG time-to-go on the collision course: t_go = R / V_c (exact when the interceptor flies the collision triangle).
• Impact-time error: e_t = t_go_des - t_go.
• Time-to-go-error-feedback bias: a_b = K_it * V_c * e_t / t_go^2, with the feedback gain K_it scheduled on the engagement state (navigation constant by default).
• Total command: a_cmd = a_png + a_b.
• A positive error (later arrival commanded than the natural PNG time) gives a positive bias that lengthens the intercept path; a negative error shortens it.

Workflow

1. Fix the engagement state: closing speed V_c, line of sight rate lambda_dot, range R, navigation constant N and the commanded remaining time-to-go t_go_des. Non-physical inputs are rejected with ValueError (navigation constant at or below 1, non-positive closing speed or range in the estimate, non-positive gain or times in the bias).
2. Compute the PN baseline with png_baseline(nav_constant, closing_speed, los_rate): a_png = N * V_c * lambda_dot.
3. Estimate the natural PNG time-to-go with tgo_estimate_png(closing_speed, range_to_target): t_go = R / V_c.
4. Form the impact-time error e_t = t_go_des - t_go between the commanded remaining time-to-go and the PNG estimate.
5. Compute the time-to-go-error-feedback bias with impact_time_bias(gain, closing_speed, tgo_desired, tgo_actual): a_b = gain * V_c * e_t / t_go^2. The default gain is the navigation constant (K_IT_DEFAULT = 4.0); a custom gain may be passed.
6. Sum the baseline and the bias with itcg_command(nav_constant, closing_speed, los_rate, range_to_target, tgo_desired, gain=None), which returns (a_cmd, a_png, a_b, tgo_png, e_t): the total lateral acceleration command, its PN baseline, the bias term, the time-to-go estimate and the impact-time error.
7. Read the natural (uncontrolled) time to impact with time_to_impact_seconds(range_to_target, closing_speed), the same closed form as the PNG estimate, for the salvo feasibility context.
8. Confirm the deterministic checks with the contract test scripts/test_impact_time_control_guidance.py.

Worked example

All values are REAL outputs of the logic module at the spec anchor state (R = 8000.0 m, V_c = 300.0 m/s, lambda_dot = 0.004 rad/s, N = 4.0, t_go_des = 30.0 s):

• png_baseline(4.0, 300.0, 0.004) = 4.800000000000 m/s^2
• tgo_estimate_png(300.0, 8000.0) = 26.666666666667 s
• impact time error e_t = 30.0 - 26.666666666667 = 3.333333333333 s
• impact_time_bias(4.0, 300.0, 30.0, 26.666666666667) = 5.625000000000 m/s^2
• itcg_command(4.0, 300.0, 0.004, 8000.0, 30.0) = (10.425000000000, 4.800000000000, 5.625000000000, 26.666666666667, 3.333333333333)

The natural PNG time to impact is 26.67 s; commanding a 30 s simultaneous arrival produces a positive bias of 5.625 m/s^2 that lengthens the intercept path. Zero-error identity: at t_go_des equal to the PNG estimate the bias is exactly 0 and the command collapses to the pure PN baseline (4.8 m/s^2). The planar fixed-step Euler probe in the contract test shows the bias lengthens the flight time by about 4.1 s at this command.

Verification

• Deterministic stdlib math only; no RNG, no network, no numeric integration in the leaf (the Euler probe lives in the contract test as a test-side identity check).
• ValueErrors: navigation constant at or below 1.0; negative closing speed in the baseline; non-positive closing speed or negative range in the time-to-go estimate; non-positive gain, actual time-to-go or desired time-to-go in the bias.
• Identities: zero impact-time error gives zero bias and the pure PN command; the natural time to impact equals the PNG time-to-go estimate; doubling the closing speed at constant e_t / t_go^2 doubles the bias; two identical calls return bit-identical results.

Related leaves

• gnc-autonomy/guidance/proportional-navigation-guidance (the unaugmented planar PN law that forms this law's baseline)
• gnc-autonomy/guidance/augmented-proportional-navigation (maneuvering- target augmentation of the same baseline)
• gnc-autonomy/guidance/midcourse-guidance (waypoint steering, handover, trajectory shaping)
• gnc-autonomy/guidance/impact-point-prediction (open-loop unguided ballistic impact prediction)
• gnc-autonomy/control/digital-control-design (sampled-data control of the inner loop that would track this command)

Pitfalls

• Do not claim the proportional-navigation law itself: this leaf forms a PN baseline from given engagement geometry but the standalone law, its capture conditions and its variants belong to the PN siblings.
• Do not add impact-angle constraints: no leaf in the family claims them and they are out of scope for the time-only law.
• The bias formula is deterministic at the given engagement state; this is not a trajectory propagator. Do not present the fixed-step Euler check in the contract test as a leaf capability.
• The t_go = R / V_c estimate is the leading-order collision-course form; for wide heading errors the natural PNG time deviates from it, which is exactly the error the feedback term acts on.
• Keep the commanded time-to-go positive and the navigation constant above 1.0; both are enforced with ValueError.
• Do not use single-word generic tags (guidance, control, navigation, intercept): they would steal corpus tasks from the family router rows.

Behavior contract (gate 3)

The contract test scripts/test_impact_time_control_guidance.py (33 methods, stdlib unittest, offline, deterministic) verifies: the worked-example baseline, time-to-go, error, bias and total command within 1e-9 relative; the zero-error identity (bias zero, command equal to the PN baseline); the ValueError rejections (navigation constant at or below 1, negative closing speed, non-positive closing speed or negative range in the estimate, non-positive gain and non-positive times in the bias); boundary behavior; the linearity of the bias in closing speed; determinism of repeated calls; the natural-time identity; and the planar Euler sign probe showing the bias lengthens the flight time by the anchor margin (between 3.0 and 5.5 s). The test passes under both /usr/bin/python3 and the pyenv 3.13 interpreter; no exact-float equality is asserted on computed sums.

Compliance

STANDARDS-REF, gated false. ARP4754A (reference-only) frames development assurance for guided systems; the ITCG law itself is paraphrased public guidance-theory literature (Jeon, Lee and Tahk 2006) and is never reproduced verbatim.