What would a Starship tower catch at Starbase look like in this hour’s weather? Wind and air density are NOAA’s GFS forecast over the tower for the current hour (UTC). Vehicle masses and engines are engineering estimates. You set what a guidance engineer would vary; the rest is under “Fixed for this vehicle and site”.

The lander must reach rest at the target, or as close as it can, on the least fuel. The paper’s Problems 1 and 2, for a point mass on a rotating planet:

The thrust lower bound makes the allowed thrust set a nonconvex annulus; the pointing limit is also nonconvex for , and the mass equation is nonlinear.

Lossless convexification (Problems 3 and 4)

A slack replaces in the thrust bounds and mass equation, with . Theorem 1 proves that, under the paper’s conditions, the relaxed optimum has , so by Lemma 1 it solves the original problem. After the change of variables of Section III-A, each burn time (the paper’s time of flight) is one second-order cone program, and a line search picks . Problem 3 minimises landing error; Problem 4 minimises fuel within that error.

This page adds a cap on the deceleration the structure feels (the sensed acceleration, thrust plus the drag estimate, one second-order cone per node), a thrust-rate limit (per unit reference mass), an approach corridor and thrust bounds that step with engine count. They are convex but outside Theorem 1, so losslessness is not guaranteed; it is checked after every solve. If at every node, Lemma 1 makes the answer optimal for the nonconvex problem as posed here: on the discrete grid, with the cone approximation of the mass-dependent bounds, at the chosen .

Every relation the guidance solves

Problems 3 and 4 as this page poses them, with the limits it adds. Colour marks the kind of symbol; the left rule marks the kind of relation. Click or tap any symbol for its definition, units and value for this vehicle.

Around it

The plan is flown through a truth model the guidance never sees: drag in the forecast air, forecast wind in the unpowered fall, thrust errors and a tracking controller that obeys the same thrust bounds, pointing cone, thrust-rate limit and deceleration cap. A Monte Carlo repeats this over dispersed mass, thrust, wind, gusts and start state, re-solving guidance at each trial’s ignition. If too few trials succeed, the auto-solve moves the burn start and reports why. The cone solver was written for this page; every answer is checked against Problems 1 and 2 before it is shown.

The panel shows each term of the dynamics and how much of each constraint’s allowance is used at this instant. The view opens straight down on the vehicle, eases to a chase view from behind and above that brings the tower in as it nears, then turns smoothly to the side (altitude against north) for the catch; the inset is the plan view. Arrows are the true 3D vectors; a part along the line of sight is shown as ⊙ (toward you) or ⊗ (away). The camera is set by the playback time alone, so scrubbing shows the same frame as playing. Flaps and heading are drawn, not modelled, and the tower is for scale. Colours match the equations under The problem.

Thin traces are Monte Carlo trials: grey met every tolerance, red did not. The paper’s figures follow the simulation’s playback time.

The paper's figures · at the playback time

Guidance and flight

These values are copied into the inputs when the run ends; the log lists what moved and why.

The trials are plotted in the Plots section, behind the nominal answer.