How the controller works

Each demo plans a short section of flight and checks it against nonlinear dynamics. It applies the first actuator command, then plans again from the simulated state.

Both controllers adapt the repository's SCvx approach to tracking a flight reference. They use fixed prediction times and scheduled mission phases. They do not implement the paper's full free-final-time problem or all of its state-triggered constraints, and its convergence guarantees do not carry over to these demos.

State, dynamics and convex subproblem

The state has 14 components: mass m, world position r, world velocity v, a scalar-first unit quaternion q, and body angular velocity ω. Rotation R(q) maps body vectors into the world. The rocket uses position and velocity at its changing center of mass; the car carries gas centrally. 

x = [m, rₓ, rᵧ, r_z, vₓ, vᵧ, v_z, q₀, q₁, q₂, q₃, ωₓ, ωᵧ, ω_z]
ṙ = v
v̇ = g + (R(q) Fᵦ + Fdrag) / m
q̇ = ½ q ⊗ [0, ω]
ω̇ = J(m)⁻¹ [τᵦ − ω × J(m)ω]

Rocket: ṁ = −‖Fᵦ‖ / (Isp g₀)
        τᵦ = (rengine − cᵦ(m)) × Fᵦ + τRCS
Car:    ṁ = −Σ fⱼ / vexit
        Fᵦ = Σ dⱼ fⱼ,   τᵦ = Σ (rⱼ × dⱼ) fⱼ

The predictor integrates these equations with RK4 and normalizes the quaternion after each step. Finite differences around a nominal nonlinear trajectory give the local discrete dynamics. The solver scales the variables before building the sparse conic problem: 

min  Σₖ ‖W (xₖ − xref,ₖ)‖² + 2‖W (xN − xref,N)‖²
     + λu Σₖ ‖uₖ − ūₖ‖² + λv Σₖ ‖νₖ‖₁ + λfuel Σₖ,ⱼ uₖ,ⱼ

s.t. x₀ = xmeasured
     xₖ₊₁ = Aₖ xₖ + Bₖ uₖ + cₖ + νₖ
     ‖xₖ − x̄ₖ‖∞ ≤ ρx,   ‖uₖ − ūₖ‖∞ ≤ ρu
     actuator constraints, dry-mass bound, car gas budget

Aₖ = ∂ΦRK4 / ∂x,   Bₖ = ∂ΦRK4 / ∂u
cₖ = ΦRK4(x̄ₖ, ūₖ) − Aₖ x̄ₖ − Bₖ ūₖ

The car uses the fuel term. Virtual control ν keeps the linearized subproblem feasible, with an L1 penalty of 30,000 in scaled coordinates. Clarabel solves each subproblem as a quadratic cone program. The predictor then simulates the candidate controls with nonlinear dynamics and linearizes around that trajectory for the next iteration.

To accept a plan, Clarabel must report an optimal result. Every control must respect the actuator bounds, all states must be finite, and virtual control must stay below 0.01. The largest scaled nonlinear defect must be below 0.12 for the car or 0.04 for the rocket. A rejected plan stops the simulation and stays visible for inspection. These numerical checks do not establish flight safety.

The blue path shows a prediction available at the displayed time; silver shows the path already flown. The worker measures solve latency. Plan age, nonlinear defect and virtual control show how recent the plan is and how closely it agrees with nonlinear dynamics.

Implementation: SCvx loop, sparse conic builder, RK4 and quaternion math. The original Python formulation is documented in MPC_DESIGN.md and METHODS.md.

Car: six fixed cold-air jets

MuJoCo advances the car throughout ground drive, the 180° C-ramp, flight and landing. Four wheel hinges receive drive or braking torque, and bounded fan suction holds the tires against a ramp made of 160 collision segments. The car leaves when its tires lose ramp contact at the upper lip. SCvx controls the six jets in flight. Each nozzle force acts at its body-fixed mounting point. MuJoCo tire contact cuts the jets and governs the rollout.

The jump uses a flat approach. Choose a landing position and whether to require a 360° barrel roll. The C-ramp always requires recovery from its inverted release. Both maneuvers use the same six jets and 250 kg air limit.

With whole-flight planning enabled, guidance first supplies an initial guess and flight duration. A separate SCvx solve optimizes every airborne control together, using terminal position, velocity, attitude and angular-rate costs plus air use. Intermediate positions are free. Three orientation gates preserve a requested full roll. Local separating planes and body-height constraints keep the trajectory above the ground and clear of the parked vehicles.

The planner uses a 0.12 s grid and up to 35 iterations. A nonlinear merit check backtracks control changes, and a finer rollout verifies clearance before acceptance. Flight duration and the contact sequence remain fixed. The accepted trajectory supplies references to the short-horizon MPC, which replans from measured states. The amber dashed line shows the complete plan; blue shows the current MPC prediction. Failed planning stops the run. Disabling whole-flight planning uses the guidance reference directly.

Five original simplified meshes represent the parked Model 3, Model Y, Semi, Cybertruck and Cybercab. Dimensions and spacing are approximate. Separate clearance checks use conservative bounding boxes along the flown trajectory. An overlap fails the run. The parked vehicles stay fixed. Tesla's physical barrel-roll demonstration is described in the October 16 reveal report.

The analytical RK4 flight predictor and MuJoCo plant use separate integration and contact code. The predictor includes wheel spin angular momentum but omits tire contact forces. Its COM floor can extend below tire height so MPC commands descent through contact; points beyond contact are flight extrapolations. MuJoCo advances at 0.002 s and handles friction and compliant tire contact. Mass, inertia, nozzle geometry and aerodynamic assumptions remain known, and states are read without sensor noise. Separate integration alone does not establish robustness to model error.

Model Browser implementation
Control Six nonnegative nozzle forces, each capped at 8.5 kN and tapered below 12 kg of remaining gas.
Bottom jets Positions (±1.45, ±0.78, −0.32) m. Force in body +Z.
Side jets Positions (+2.00, ±0.80, −0.08) m. Force inward along body Y, clear of the tires. Both are at the front, so lateral thrust also produces yaw torque.
Mass / exhaust 1,428 kg dry, up to 250 kg gas, 650 m/s effective exhaust velocity. Inertia scales with mass.
Prediction Feedback: 10 steps × 0.12 s; replan every 0.12 s; up to 3 convex iterations. Full-flight planning: a 0.12 s grid over the initialized flight duration, up to 35 iterations.
Plant MuJoCo WASM at 0.002 s: free chassis, four rotating wheels, compliant tire contacts, segmented ramp and parked-vehicle collision boxes.

At the full initial mass of 1,678 kg, weight is 16.46 kN. Four level bottom jets need 4.12 kN each to hover, or 6.21 kN each for the reference takeoff acceleration of 5 m/s². Their 8.5 kN limit provides 34 kN total lift and up to 10.45 m/s² net upward acceleration while level, before drag and attitude-control demands.

Each front side jet has a 2 m yaw moment arm. With yaw inertia 2,900 kg·m², a smooth 180° turn in two seconds needs about 6.58 kN peak side thrust, using the quintic reference's peak angular acceleration. The 8.5 kN limit gives 17 kN·m maximum yaw torque. Opposing jets act through their thrust difference. The complete simulation checks coupled rotation and gas use; the ramp maneuver reaches nearly full side thrust. Bottom thrust differences produce the barrel roll.

A least-squares attitude estimate initializes the first optimization. Only optimized commands reach the physical simulation. With MPC off, the car follows an uncontrolled ballistic path after leaving the ramp. The browser controller has no PD fallback.

The car uses four bottom jets and two front side jets. Stored gas is capped at 250 kg and fan suction at 2 g. Each jet is capped at 8.5 kN; effective exhaust speed is 650 m/s. Gas use follows thrust divided by effective exhaust speed. The total initial mass is 1,678 kg. References: Roadster reveal, engineering interview and thruster and fan explanation.

Source: analytical flight predictor and MPC references; full-flight goal planner; MuJoCo car plant and collision model; Python reference controller.

Rocket: SCvx from liftoff through landing

Every run first builds a feasible initial guess in MuJoCo, then optimizes a full mission using translational SCvx with mass, position, velocity and world force. The objective uses the selected separation altitude, the upright landing location and velocity, and fuel use. Intermediate seed positions have zero tracking weight. The seed supplies the discrete burn schedule and force-direction corridor. Six-DoF MPC tracks the plan, including attitude, gimbal torque and changing inertia. At boostback, entry, landing ignition and the center-engine transition, the mission planner optimizes the remaining flight again from the actual state. Below 300 m, MPC uses a terminal braking reference from measured clearance and the landing goal. Burn order and phase durations are scheduled; the planner does not optimize discrete choices or guarantee a global optimum.

Phase Actuators and reference
Ascent Nine engines, up to 9 × 845,000 N. A height-based pitch profile reaches 20° from vertical. The selectable separation altitude is 70–80 km, with 75 km as the default. Cutoff anticipates the three-second coast to that altitude.
Cutoff / separation Booster engines shut down three seconds before separation. The upper engine waits eight more seconds and at least 5 m axial clearance before ignition. Opposite impulses give 3 m/s relative axial speed. MPC turns the booster with RCS while its engines remain off. Boostback begins after at least 6 s, 30 m axial clearance and attitude alignment.
Coast Engines off. MPC commands bounded attitude RCS: 2 MN·m per pitch/yaw axis and 17.5 kN·m in roll. The pitch/yaw authority is an educational assumption.
Boostback / entry / landing Three independent burns follow separation. Boostback uses three engines until the predicted ballistic impact reaches the pad. A twelve-second entry burn starts below 60 km on descent, followed by at least three seconds of coasting. A numerical stopping rollout accounts for propellant depletion and atmospheric drag to schedule landing ignition and shape the braking reference. Landing starts with three engines and reduces to one when single-engine braking can reach the ground. The center engine stays lit until contact. Every lit engine obeys the 57–100% throttle bounds. Horizontal references bring position and velocity toward the launch point over the estimated remaining descent time. Below 80 m, a mass check selects direct landing when minimum thrust prevents hovering. With sufficient mass, the booster instead settles with its feet 2 m above the pad for one second, then lowers at up to 0.7 m/s toward 0.2 m/s at contact. Below 30 m, height tracks the lowest physical foot, including tilt.
Contact Engines cut at actual foot contact. MuJoCo settles the booster for 12 seconds on a 20 m radius pad centered on the launch point.

Body thrust obeys ‖Fxy‖ ≤ tan(6°) Fz, Fz ≥ Tmin and ‖F‖ ≤ Tmax. These are second-order cone constraints. Gimbal force at the engine moment arm produces pitch and yaw torque. A separate bounded RCS input supplies roll torque; powered pitch/yaw RCS is zero.

Three-engine landing uses thirty-two prediction steps of at most 0.06 s. One-engine landing uses sixteen steps of at most 0.25 s during braking and hover, then 0.12 s during final descent. Steps shrink as low as 0.002 s near contact or propellant exhaustion. Coast uses sixteen 0.15 s steps, entry uses ten 0.12 s steps, and ascent and boostback use ten 0.4 s steps. MuJoCo advances at 0.005 s. Three-engine landing replans every 0.1 s; otherwise the interval is 0.4 s, shortening to 0.2 s below 300 m, 0.04 s below 30 m and 0.01 s below 5 m. Mass, tank centers and principal inertia change as propellant burns.

The nominal demonstration separates at 75 km with a full initial first-stage tank of 407,500 kg. Fuel is integrated from commanded thrust and effective specific impulse. Separation speed, remaining fuel and touchdown conditions are outputs of the simulation; residual fuel is not prescribed. The default flight lands directly because its final mass cannot balance minimum engine thrust.

Each engine is bounded to 57–100% throttle with a maximum of 845 kN, held constant throughout flight. Dry mass is 25.6 tonnes, tank capacity is 407,500 kg and effective specific impulse is 282 s. The model includes exponential atmospheric drag and omits altitude-dependent thrust. References: engine data and RTLS burn sequence.

The second stage inherits the separation pose and angular velocity. Both bodies receive opposite axial impulses that conserve momentum. Its engine ignites eight seconds after separation once axial clearance exceeds 5 m, then follows a prescribed axial 981 kN burn outside the booster MPC.

Nozzle and flame directions ease over 0.18 s for display. Physics applies MPC commands immediately. An exponential atmosphere uses sea-level density 1.225 kg/m³, scale height 8.5 km and drag coefficient 0.8 with an attitude-dependent projected area. Drag acts at the center of mass in both the predictor and MuJoCo plant. Wind, lift, aerodynamic moments, actuator lag, Earth curvature, slosh and structural limits are omitted. The native controller has separate guidance and PD modes; browser flights use SCvx.

Calibration references: Fraunhofer separation observation (about 70 km and 6,000 km/h), CRS-7 planned ascent (80 km at cutoff), and the Falcon user guide's sample staging timeline. Missions have different ascent profiles; these references set an approximate range.

The mission planner uses seven translational states and three world-force inputs. It integrates thrust, propellant depletion, gravity and exponential drag. The seed attitude approximates projected area; full rotational dynamics are left to the tracking MPC. The green path shows the current mission plan, blue shows the local MPC prediction, and silver shows the actual path. Replays retain each plan revision.

Source: rocket mission goal planner, mission phases and references, prediction and mass properties, MuJoCo plant, Three.js model.

Live computation on GitHub Pages

GitHub Pages serves the files, and your browser runs both simulations. A module worker loads Clarabel WASM and the official single-thread MuJoCo WASM package for either vehicle. Three.js renders on the main thread. The site needs no Python API, shared memory, server-side solver or special cross-origin headers.

Run live integrates new states and solves new plans. Compute mission runs the same code as fast as the worker permits and records the result. The included recordings let the page open immediately. Their labels distinguish replay from live computation. Stop terminates the worker, including an in-progress solve. Switching vehicle tabs stops the inactive simulation.

Relative URLs let the site run in a repository subdirectory on Pages. Runtime packages ship with the site. If a device cannot keep up with the selected pace, the simulation slows down. The display uses the states it has received and reports the measured solve latency.

Papers and source

Project source on GitHub · MuJoCo · Clarabel · Clarabel WASM provenance and license