Starship Landing Simulator

The problem

A lander must go from its ignition state to rest on the surface, at a target if it can reach it, using the least fuel. The paper's Problems 1 and 2 state this with a point-mass model on a rotating planet:

ẋ = A(ω) x + B ( g + T_c/m ), ṁ = −α ‖T_c‖ ρ₁ ≤ ‖T_c(t)‖ ≤ ρ₂, n̂·T_c(t) ≥ ‖T_c(t)‖ cos θ (thrust bounds and pointing limit) ‖ṙ(t)‖ ≤ V_max, glide slope ≥ γ_gs about the landed point (state constraints)

The lower bound ρ₁ > 0 makes the set of allowed thrust vectors an annulus, which is not convex, and the pointing limit is not convex either when θ > 90°. The mass equation is nonlinear. The paper's convexification solves it anyway, and solves it exactly.

Lossless convexification (Problems 3 and 4)

The paper adds a slack Γ(t) and replaces the thrust constraints by ‖T_c‖ ≤ Γ, ρ₁ ≤ Γ ≤ ρ₂, n̂·T_c ≥ Γ cos θ, with ṁ = −αΓ. Theorem 1 proves the optimum of the relaxed problem has ‖T_c‖ = Γ almost everywhere, so it is also the optimum of the original problem. With the change of variables u = T_c/m, σ = Γ/m, z = ln m (Section III-A) and the second-order cone approximation of the bounds on σ, each fixed time of flight is one second-order cone program. The time of flight is found by a line search, and the two problems are solved in order: first the smallest reachable landing error, then the least fuel for that error.

What sits around it

The guidance model is the paper's point mass. The trajectory it produces is then flown through a truth model that the guidance never sees: an exponential atmosphere with drag, a wind profile the vehicle falls through before ignition, engine thrust errors, and a controller that tracks the plan. A Monte Carlo repeats that flight over dispersed mass, thrust, wind, gusts and start state, re-solving the guidance from each dispersed ignition state. When the success rate falls below the threshold, the auto-solve moves the initial conditions until it clears, and says exactly what it moved and why.

Discretisation is N+1 nodes with thrust linear between them and an exact state transition including planet rotation; the cone solver was written for this page. Every answer is checked against the original nonconvex constraints of Problems 1 and 2 before it is shown.

Inputs · Section IV

Preset The four presets are the two Starship Flight 13 landing burns and a tower catch for each vehicle, built from public reporting and published specifications; the masses and burn-start states are estimates. Every field can be edited.

Vehicle

Initial mass m₀kg

Fuel available m_fkg

Maximum thrust T_maxN

Minimum throttle (ρ₁/T_max)%

Maximum throttle (ρ₂/T_max)%

Fuel rate αs/m

α = 1/(I_sp g_e cos φ) for specific impulse I_sp and engine cant φ; the paper uses 5×10⁻⁴ s/m.

Thrust per engineN

Throttle floor per engine%

Engine schedule (optional), one phase per line: start s, min engines, max engines

With a schedule the bounds become ρ₁(t) = min engines × floor × thrust per engine and ρ₂(t) = max engines × thrust per engine, phase by phase, and the three throttle fields above are ignored.

Planet

Gravity g (x up, y, z)m/s²

Rotation ωrad/s

Include rotation in the dynamics

Frame of Fig. 1: x is altitude along the surface normal, y and z are horizontal. The pointing vector n̂ is the x-axis.

Constraints

Pointing limit θ from verticaldeg

Glide slope γ_gsdeg

Speed limit V_maxm/s

θ = 180° means no pointing limit. γ_gs = 0 or V_max = 0 disables that constraint. Table I of the paper uses θ = 180°, 90° and 45°.

Boundary conditions

Initial position r₀m

Initial velocity ṙ₀m/s

Target q (y, z) at zero altitudem

Time of flight and discretisation

Nodes N

Flight time search ranges

Search grid points

The time of flight is always found by a line search, as in the paper: the grid is evaluated, then the best bracket is refined by golden section, and the range is clipped to what the fuel allows. Raising the shortest allowed time forces longer, gentler burns than the minimum-fuel answer; the auto-solve uses that as its knob.

Auto-solve

Adjust the initial conditions until the Monte Carlo clears the threshold

Success threshold%

Maximum rounds

May move the ignition altitude

May move the time of flight

May move the ignition aim point

It never touches the wind, the atmosphere, gravity, the planet's rotation, the vehicle or the dispersions, and it never loosens the success tolerance. Each round screens the candidates over half the trials and confirms the winner over all of them, so a round costs a few Monte Carlo runs.

Graceful landing and landing site

Peak thrust acceleration capg, 0 = none

Thrust rate limitMN/s, 0 = none

Approach corridor gain k1/s, 0 = none

Touchdown speed v_tdm/s

Landing altitude (tower catch height)m

The corridor demands ‖ṙ‖ ≤ v_td + k (h − h_land) at every node, so speed tapers with height; a burn cannot be shorter than the taper it implies. All three limits are convex and enter both methods.

Wind, air and the fall before ignition

Apply wind and drag when flying the plan

Wind profile, one line per level: altitude m, speed m/s, from deg

Drag area C_dA during the burnm²

Drag area C_dA broadside, fallingm²

Lift to drag, broadside0 = no lift

Flip to engines first, durations, 0 = none

The bellyflop: with a flip time the vehicle falls broadside at the larger drag area, banking its lift at the ignition point to steer, then rotates to engines-first over that many seconds and arrives at the ignition altitude with the smaller area. The flip is unpowered, so it costs altitude and gains speed. Set the flip time to 0 for a vehicle that comes back engines first, as the booster does.

Unpowered fall starts atm, 0 = none

Feedback tracking when flying the plan

The guidance model is the paper's: no aerodynamics. The truth model that flies the answer has drag with wind (exponential atmosphere), engine errors and, when enabled, a controller that tracks the plan. With a fall altitude the vehicle drops through the wind first and the burn starts from wherever that leaves it.

Monte Carlo

Trials0 = off

Mass error% 1σ

Engine thrust error% 1σ

Wind speed error% 1σ

Wind direction errordeg 1σ

Steady gustm/s 1σ per axis

Start position errorm 1σ per axis

Start velocity errorm/s 1σ per axis

Re-solve guidance at ignition

Success tolerance: position, speedm, m/s

Random seed

Each trial draws the errors, drops the vehicle through its own wind, re-solves the convex guidance from the ignition state it arrives at, then flies the plan with tracking through the truth. A trial succeeds when it lands within the tolerances with fuel to spare.

Saved cases

A case is every input on this page under a name you choose. Save one that works, load it later as a template, change the Monte Carlo or anything else, and run it again. Cases live in this browser; the download and load buttons move them to and from files on disk.

Load from file

Run

Idle. Everything runs in your browser, in a background thread.

Problems 3 and 4

The whole set of relations this run solved, and the answer they produced. Colour says what a symbol is; the rule down the left says what kind of relation the line is. Hover any symbol for what it means.

What each of these relations is doing to the vehicle at any instant is drawn on the trajectory plot below, not tabulated here.

Simulation

Drag to rotate · wheel to zoom. The overlay draws what every relation of Problems 3 and 4 is doing to the vehicle at this instant: each term of the dynamics as an arrow from the lander, and every constraint as a gauge of how much of its allowance is spent. Colours are those of the equation set above.

t = 0.0 s speed scope view flown path equation overlay

Plots

Every plot in one place. Thin traces behind each answer are the Monte Carlo trials, grey where the trial met every tolerance and red where it did not, so each plot carries its own spread of outcomes. The paper's figures follow the playback cursor of the simulation above.

The paper's figures · at the playback time

Fig. 2 of the paper. Planar thrust bounds ρ₁ ≤ ‖T_c‖ ≤ ρ₂ (annulus) cut by the pointing limit θ from n̂ (wedge). The dark region is the set of thrust vectors allowed in Problems 1 and 2. Ring: the thrust vector now; grey dots: the same instant in each trial.

Fig. 5 of the paper. U(Γ) = {‖T_c‖ ≤ Γ, n̂·T_c ≥ Γ cos θ} for Γ(t) now. Its extreme points (thick arc) satisfy ‖T_c‖ = Γ; Theorem 1 says the optimal thrust sits there.

Fig. 3 of the paper. Top view. The circle of radius d* = ‖E r*(t_f) − q‖ from the minimum landing error step; Problem 4 may land anywhere inside it, but the closest point is on the circle. Ground track, current position, and where every trial touched down.

Guidance and flight

Throttle against time with the bounds ρ₁ and ρ₂. The paper's Fig. 6 and 7 show the same profile: a bang-bang switch between the two bounds. Thin traces: what each trial actually burned.

Thrust angle from vertical against the pointing limit θ. Thin traces: each trial, where the tracking controller has to lean the vehicle further than the plan.

Thrust vectors on the plane (horizontal magnitude, vertical component). Dark: the original nonconvex set. Light: its convex relaxation projected on the thrust plane. Lossless means every convex answer lands on the dark set. Thin traces: the path each trial's thrust took through the same plane.

Slack Γ against ‖T_c‖ at every node of the convex answer. Theorem 1 says the points sit on the diagonal. The trials are not drawn here: Γ lives inside the convex problem, and the flown vehicle has no such variable.

Altitude against horizontal distance from the landed point, with the glide slope line. Thin traces: each trial from its own dispersed ignition point, which is why they start wide.

Speed against time with V_max. Thin traces: the speed each trial actually flew.

Mass against time. Thin traces: each trial, dispersed in starting mass and in what it burned.

Line search on the time of flight: landing error (step 1) and fuel (step 2) at every time tried. Gaps are infeasible times. Dots: the flight time and propellant each trial actually used, after the guidance re-solved and retimed at its own ignition state.

Auto-solve

These values are written into the input form as soon as the run ends; the run log above lists what moved and why.

Monte Carlo

The plots of these trials are in the plots section above, drawn behind the answer they disperse.

Lossless convexification (Problems 3 and 4)

What sits around it

The paper's figures · at the playback time

Guidance and flight

Monte Carlo outcomes