Impact Angle Control Guidance

Impact-Angle-Control Guidance (gnc-autonomy/guidance/impact-angle-control-guidance)

Use when the task is computing the impact-angle-control guidance command for a planar terminal intercept against a stationary target with a commanded terminal flight path angle: the proportional-navigation-like collision-course nulling baseline from the crossrange offset, the crossrange velocity and the time-to-go, the terminal crossrange velocity that realizes the commanded impact angle, the impact-angle error between the commanded terminal flight path angle and the current flight path angle, and the impact-angle-error-feedback bias that reshapes the intercept path so the interceptor meets the target at the commanded impact angle. This is the energy-optimal impact-angle-constrained guidance structure of the Ryoo-Cho-Tahk family (Ryoo, Cho and Tahk, "Optimal Guidance Laws with Terminal Impact Angle Constraint," Journal of Guidance, Control, and Dynamics 28(4):724-732, 2005, summarized by name and paraphrase only, never reproduced). Pairing leaves: proportional-navigation owns the unaugmented planar PN law, augmented-proportional-navigation the maneuvering-target augmentation, impact-time-control-guidance the time-constrained member of the same terminal-law family; midcourse-guidance handles waypoint steering and handover; impact-point-prediction is open-loop ballistic prediction.

Domain quick reference

• Planar terminal intercept against a stationary target: speed V (m/s), current flight path angle gamma (rad), commanded terminal flight path angle gamma_f (rad), crossrange offset y (m), crossrange velocity component v_perp (m/s), time-to-go t_go (s).
• Crossrange velocity: v = V * sin(gamma); the terminal crossrange velocity v_f = V * sin(gamma_f) is the crossrange velocity that realizes the commanded impact angle.
• Time-to-go: t_go = range_to_target / closing_speed (exact on the collision course).
• Impact-angle error: e_g = gamma_f - gamma.
• Collision-course nulling baseline: a_base = -W_Y * (y + v_perp * t_go) / t_go^2, zero when the interceptor is already on the collision course (y
  • v_perp * t_go = 0). W_Y = 6.0.
• Impact-angle-error-feedback bias: a_bias = 2.0 * (v - v_f) / t_go, positive when the current crossrange velocity exceeds the terminal one (a steeper terminal dive must first be set up), negative when the commanded course is shallower.
• Total command: a_cmd = a_base + a_bias.

Workflow

1. Fix the engagement state: speed V, current flight path angle gamma, commanded terminal flight path angle gamma_f, crossrange offset y, crossrange velocity v_perp and time-to-go t_go. Non-physical inputs are rejected with ValueError (non-positive speed, non-positive closing speed or negative range in the time-to-go estimate, non-positive time-to-go in the bias and baseline).
2. Compute the crossrange velocity with crossrange_velocity(speed, gamma): v = speed * sin(gamma). The same function gives the terminal crossrange velocity v_f from the commanded terminal flight path angle.
3. Estimate the time-to-go with tgo_estimate(closing_speed, range_to_target): t_go = range_to_target / closing_speed.
4. Form the impact-angle error with impact_angle_error(commanded_gamma, gamma): e_g = commanded_gamma - gamma.
5. Compute the impact-angle-error-feedback bias with impact_angle_bias(speed, tgo, gamma, commanded_gamma): a_bias = 2.0 * (v - v_f) / tgo.
6. Compute the collision-course nulling baseline with collision_nulling_baseline(tgo, crossrange_offset, v_perp): a_base = -W_Y * (crossrange_offset + v_perp * tgo) / tgo^2.
7. Sum the baseline and the bias with impact_angle_guidance_command(tgo, crossrange_offset, v_perp, speed, gamma, commanded_gamma), which returns (a_cmd, a_base, a_bias, t_go, e_g): the total lateral acceleration command, its collision-course nulling baseline, the impact-angle-error bias, the time-to-go estimate and the impact-angle error.
8. Confirm the deterministic checks with the contract test scripts/test_impact_angle_control_guidance.py.

Worked example

All values are REAL outputs of the logic module at the spec anchor state (stationary target at the origin, interceptor at range R = 10000.0 m, line of sight 30 deg below the horizontal, V = 300.0 m/s, gamma_0 = -30 deg, commanded terminal flight path angle gamma_f = -60 deg, crossrange offset y_0 = +5000.0 m, closing speed V_c = 300.0 m/s on the collision course):

• tgo_estimate(300.0, 10000.0) = 33.333333333333 s
• crossrange_velocity(300.0, -30 deg) = -150.000000000000 m/s
• crossrange_velocity(300.0, -60 deg) = -259.807621135332 m/s
• impact_angle_error(-60 deg, -30 deg) = -0.523598775598 rad (-30.000000000000 deg)
• collision_nulling_baseline(33.333333333333, 5000.0, -150.0) = -0.000000000000 m/s^2 (the interceptor starts on the collision course: y_0 + v_0 * t_go0 = 0)
• impact_angle_bias(300.0, 33.333333333333, -30 deg, -60 deg) = 6.588457268120 m/s^2
• impact_angle_guidance_command(33.333333333333, 5000.0, -150.0, 300.0, -30 deg, -60 deg) = (6.588457268120, -0.000000000000, 6.588457268120, 33.333333333333, -0.523598775598)

Zero-error identity: at gamma equal to the commanded terminal flight path angle the bias is exactly 0 and the command collapses to the collision-course nulling baseline alone. Terminal straight course: at t_go = 10 s with v = v_f and the crossrange offset set to -v_f * t_go (2598.076211353 m), both the baseline and the bias vanish and the total command is 0.000000000000 m/s^2. Bias linearity: doubling the speed at the worked-example geometry doubles the bias magnitude to 13.176914536240 m/s^2. Shallowing-command bias (gamma = -60 deg, commanded -10 deg) = -12.462790070115 m/s^2.

Verification

• Deterministic stdlib math only; no RNG, no network, no numeric integration in the leaf (the fixed-step Euler checks live in the contract test as test-side identity probes, not as a leaf capability).
• ValueErrors: non-positive speed in crossrange_velocity, impact_angle_bias and impact_angle_guidance_command; non-positive closing speed or negative range in tgo_estimate; non-positive time-to-go in impact_angle_bias, collision_nulling_baseline and impact_angle_guidance_command.
• Identities: zero impact-angle error gives zero bias and the command equal to the collision-course baseline alone; the collision-course offset gives a zero baseline; the terminal straight course gives a zero total command; doubling the speed at fixed geometry and angles doubles the bias magnitude; two identical calls return bit-identical results.
• Linear-model closed-loop check: fixed-step Euler integration of the linearized crossrange kinematics under the closed-form law drives the crossrange velocity to within 0.05 m/s of the terminal crossrange velocity and the offset to within 1.0 m of zero as the time-to-go approaches zero.
• Nonlinear planar engagement check: fixed-step Euler integration of the point-mass kinematics under the law (dt = 0.005, speed 300 m/s, stationary target, stop at range <= 0.5 m) reaches the target at flight time 33.885000 s with miss range 0.096219 m and a terminal flight path angle of -60.037102 deg against the -60.000000 deg command (terminal-angle error -0.037 deg); the terminal angle stays in the [-61.0, -59.0] deg band from dt = 0.005 down to dt = 0.001. The max sampled command 1436.184250 m/s^2 occurs in the final tgo -> 0 step at range ~1.4 m, the terminal singularity of the time-to-go polynomial command (commands stay below ~55 m/s^2 while the range is >= 50 m); the identity asserted is the terminal flight path angle, not the singular command.

Related leaves

• gnc-autonomy/guidance/proportional-navigation (the unaugmented planar PN law whose collision-course structure this baseline mirrors)
• gnc-autonomy/guidance/augmented-proportional-navigation (maneuvering-target augmentation of the PN baseline)
• gnc-autonomy/guidance/impact-time-control-guidance (the time-constrained member of the same terminal-law family, salvo and simultaneous-impact arrival)
• gnc-autonomy/guidance/midcourse-guidance (waypoint steering, handover, trajectory shaping)
• gnc-autonomy/guidance/impact-point-prediction (open-loop unguided ballistic impact prediction)

Pitfalls

• Do not claim the standalone proportional-navigation law itself: this leaf forms a collision-course nulling baseline from the given engagement geometry, but the PN law, its capture conditions and its variants belong to the proportional-navigation sibling.
• Do not add impact-time or salvo control: impact-time-control-guidance owns the time-constrained member of the terminal-law family; this leaf is angle-only.
• The bias and baseline are deterministic at the given engagement state; this is not a trajectory propagator. The fixed-step Euler checks in the contract test are test-side identity probes, not a leaf capability, and the time-to-go polynomial command has a terminal singularity as t_go approaches zero that a real implementation would saturate at an actuator limit.
• Keep the speed and the time-to-go strictly positive; 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_angle_control_guidance.py (stdlib unittest, offline, deterministic) verifies: the worked-example crossrange velocities, time-to-go, impact-angle error, baseline, bias and total command within 1e-9 relative; the zero-error identity (bias zero, command equal to the baseline); the collision-course identity (baseline zero); the terminal straight-course identity (total command zero); the ValueError rejections (non-positive speed, non-positive closing speed, negative range, non-positive time-to-go); boundary behavior (zero range gives zero time-to-go); the sign physics of the bias for steeper and shallower commanded angles; the linearity of the bias in speed; determinism of repeated calls; the linear-model closed-loop convergence to the terminal crossrange velocity and offset; and the nonlinear planar engagement identity probe showing the terminal flight path angle lands within the anchor's band of the commanded angle. 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 impact-angle-control law itself is paraphrased public guidance-theory literature (Ryoo, Cho and Tahk 2005) and is never reproduced verbatim.