Impact Point Prediction

Impact Point Prediction (gnc-autonomy/guidance/impact-point-prediction)

Use when the task is predicting where an unguided ballistic projectile lands from its launch state: range, time of flight, impact coordinates, and how sensitive the landing point is to launch speed and flight path angle errors.

Domain quick reference

Flat earth, vacuum ballistic model, point mass launched from (x0, y0) with speed v0 and flight path angle theta0 above the horizontal, in a constant gravity field g. Angles are radians throughout; units are m, s, m/s, m/s^2.

• Range: R = v0^2 * sin(2 * theta0) / g. Worked anchor: v0 = 100 m/s, theta0 = 45 deg, g = 9.81 m/s^2 gives R = 10000 / 9.81 = 1019.37 m.
• Time of flight: T = 2 * v0 * sin(theta0) / g. Same anchor gives T = 14.42 s.
• Peak height: hp = (v0 * sin(theta0))^2 / (2 * g). Same anchor gives hp = 254.84 m.
• Impact coordinates: xf = x0 + R * cos(heading), yf = y0 + R * sin(heading), heading the azimuth from the +x axis. Anchor: heading = 30 deg gives (882.80, 509.68) m from the origin.
• Range sensitivity: dR/dv0 = 2 * v0 * sin(2 * theta0) / g and dR/dtheta0 = 2 * v0^2 * cos(2 * theta0) / g. At the anchor, dR/dv0 = 20.39 m per (m/s) and dR/dtheta0 = 0: the range is first-order flat in angle at the 45 deg maximum.
• Error propagation: delta_R = dR/dv0 * dv0 + dR/dtheta0 * dtheta, and delta_T = dT/dv0 * dv0 + dT/dtheta0 * dtheta with dT/dv0 = 2 * sin(theta0) / g, dT/dtheta0 = 2 * v0 * cos(theta0) / g. Anchor: dv0 = 1 m/s moves the impact point 20.39 m and shifts the time of flight 0.144 s.
• FAR-25 and CS-25 (reference-only) frame airworthiness certification for transport airplanes; the flat earth range equation itself is common ballistic knowledge.

Workflow

1. Confirm the model applies: an unguided projectile in a flat earth vacuum field, no drag, no curvature. For long ranges or high speeds, treat the result as a first estimate only.
2. Convert theta0 and the heading to radians if given in degrees.
3. Compute range_flat_earth(v0, theta0, g) and time_of_flight(v0, theta0, g); both validate v0 > 0, theta0 in (0, pi/2), and g > 0.
4. Compute impact_point(x0, y0, v0, theta0, heading, g) for the absolute landing coordinates, remembering the launch position offset.
5. Compute range_sensitivity(v0, theta0, g), then impact_error(v0, theta0, dv0, dtheta, g) with the launch condition uncertainties to size the landing point dispersion.
6. Bundle the full prediction with impact_point_prediction(x0, y0, v0, theta0, heading, g), which returns range, time of flight, peak height, and impact coordinates in one dict.
7. State the model limitations with the numbers: drag, curvature, and wind are outside this model and dominate at long range.

Pitfalls

• Confusing impact point prediction with proportional-navigation: PN computes a commanded acceleration from closing velocity and line of sight rate inside an intercept guidance loop; impact point prediction is open-loop ballistic geometry with no guidance law.
• Confusing with pursuit-guidance: pursuit steers the interceptor velocity at the target using an aim heading and capture condition; the ballistic round here is unguided and never steers.
• Confusing with command-to-line-of-sight: CLOS keeps a missile on the tracker to target line with a steering command; impact point prediction has no tracker line and no steering.
• Confusing with rendezvous-phasing: phasing sizes orbital catch-up maneuvers with delta-v around a circular orbit; impact point prediction is a flat earth ground impact problem with no orbit.
• Mixing degrees and radians: theta0 and heading must be radians, or every range and coordinate is wrong by a large factor.
• Treating the vacuum range as the true range: drag shortens the range, and Earth curvature lengthens it at long ranges; the model is a first estimate, not a fire control solution.
• Assuming one error source dominates: at 45 deg the angle sensitivity is exactly zero, so speed errors dominate there, while away from 45 deg angle errors dominate; always check both partial derivatives.
• Forgetting the launch position offset: impact_point returns absolute coordinates from (x0, y0); dropping the offset shifts the whole prediction.

Behavior contract (gate 3)

The range equation, time of flight, peak height, impact coordinates, range sensitivity, first-order error propagation, and the bundled prediction dict are exercised by the gate 3 contract test: scripts/test_impact_point_prediction.py against scripts/impact_point_prediction_logic.py (stdlib unittest, offline). Run: python3 scripts/test_impact_point_prediction.py

Compliance

• FAR-25 (US government work, public domain) and CS-25 (EASA, free-download) are referenced by id only per standards-map.yaml, reference-only: true; no text is copied.
• compliance: STANDARDS-REF, gated: false.