A returning booster must null three numbers at once by the moment it reaches the catch arms: vertical speed, lateral offset from the tower centerline, and lateral speed. The engines only give you two real controls to do it with — how hard they push, and which way they point.
Thrust: F = throttle × F_max (F_max = 3 sea-level engines)
Mass flow: ṁ = F / v_e v_e = I_sp·g₀ ≈ 3236 m/s
Vertical: m·(dv_y/dt) = F·cos(θ) − m·g
Lateral: m·(dv_x/dt) = F·sin(θ) + wind
Catch test (at h = arm height):
|v_y| ≤ 3 m/s AND |v_x| ≤ 1.5 m/s AND |x| ≤ gap/2
- Throttle sets thrust magnitude, which also sets propellant burn rate via the rocket equation ṁ = F/v_e — mass drops as you burn, so the same throttle gives more deceleration late in the burn than early.
- Gimbal angle tilts the whole thrust vector: part of it still fights gravity (F·cosθ), part of it steers sideways (F·sinθ). Steering hard costs vertical deceleration — you cannot max out both at once.
- Crosswind is a constant lateral force you have to counter continuously with a small opposing gimbal bias.
- Catch-arm gap is the difficulty knob — the shaded capture zone on the flight panel and the ±band on the telemetry chart both resize live with this slider.
Fuel-budget note: the usable landing propellant here is tuned to ~12,000 kg. A numerical check (bang-bang optimal control: free-fall, then full throttle from the best possible ignition height) shows that a smaller 9,000 kg budget — the figure a first pass at this model used — cannot physically null a 95 m/s entry speed within the ±3 m/s catch tolerance even in the theoretically best case (best-case leftover speed ≈ 23 m/s, roughly 8× over tolerance); 12,000 kg brings the theoretical best case to ≈2.4 m/s, leaving a narrow but real margin for gimbal losses and wind once a player is actually flying it.
This is the control problem behind SpaceX's Super Heavy booster catch, first flown successfully in October 2024: instead of legs, the booster's own grid-fin support pins are caught by two arms on the launch tower — but only if the vehicle arrives dead-centered and nearly motionless.