00The shape of the problem

A booster returning to a pad is a rigid body with one force it can point and throttle, an atmosphere that pushes back, and a mass that shrinks as it works. It has to arrive at a specific point on the ground with almost no velocity, almost no tilt, and almost no rotation, using less propellant than it has, and it gets exactly one attempt.

Four things make it genuinely hard, and they are worth naming before any equations appear, because every design decision later is a response to one of them.

The vehicle cannot hover. At the end of the flight it is nearly empty, and the engine has a minimum throttle. If minimum thrust divided by mass exceeds local gravity, then any lit engine accelerates the vehicle upward. There is no steady state to settle into. You get one continuous braking maneuver, and it has to end at the ground.

Propellant is the binding constraint. The fuel-optimal solution is to fall as long as possible and brake as hard as possible at the last moment. That solution is also the one with the least margin, because braking late means every error has to be corrected with the authority left over from an already saturated engine.

Translation and rotation are coupled through the same actuator.The only way to accelerate sideways is to tilt the whole vehicle, and the only way to tilt is to point the engine off-axis, which steals thrust from the vertical channel. Wanting to move left and wanting to slow down are the same actuator fighting itself.

Control authority is not constant. It depends on thrust, air density and speed, all of which change by orders of magnitude during the descent. A controller tuned for the conditions at three kilometres is tuned for a different vehicle than the one that exists at twenty metres.

01Kinematics: choosing what to track

Kinematics is the bookkeeping of motion without asking what causes it. It sounds like the easy part. It is where most of the subtle mistakes live, because kinematics is where you choose what the numbers mean, and a wrong choice there is invisible until it produces a landing that is off by fifteen metres for no apparent reason.

The state vector

Pick the smallest set of numbers that determines the future. For a booster with a single centred engine:

Eleven numbers. Everything the vehicle knows about itself.

Three position, three velocity, two attitude angles, two angular rates, one propellant mass. Notice what is missing: there is no roll angle and no roll rate.

Why two attitude angles and not three

A single engine on the centreline produces a force through the centreline. Rotate the vehicle about its own long axis and nothing about that force changes. The engine has no lever arm to produce roll torque, so roll is uncontrollable with this actuator, and worse, it is unobservable in the dynamics that matter. Real boosters solve roll separately with cold gas thrusters or differential fin deflection, which is a different subsystem with a different bandwidth.

Dropping it is not laziness. Carrying an unactuated third angle would let the vehicle drift into orientations where the two remaining angles stop being independent, which is gimbal lock: a coordinate singularity where two of your three rotation parameters describe the same physical rotation and the equations lose a degree of freedom. Two angles about two fixed horizontal axes cannot lock, and for a vehicle that never intentionally tilts past about thirty degrees they are exact enough.

If you did need full three-axis attitude, the standard answer is quaternions, which have no singularities at the cost of one redundant parameter and a normalisation constraint you must enforce every step.

The body axis as a unit vector

Both angles collapse into one useful object, the direction the vehicle is pointing:

Thrust acts along +û. Everything else is bookkeeping around this vector.

02Translational dynamics

Now the causes. Three forces act on the centre of mass: thrust along the deflected body axis, gravity down, drag opposing motion through the air.

Drag is written with |v|·v rather than v² so the sign follows the direction automatically in all three components.

Two details in that line carry real weight.

First, thrust points along the deflected axis,û(φ + δ_φ, ψ + δ_ψ), not the body axis. The gimbal angle is small, at most fifteen degrees, but it is the entire steering mechanism, so it belongs in the force equation and not just the torque equation.

Second, the relative velocity v_rel = v − v_wind is what generates drag, not ground-relative velocity. A vehicle descending vertically in a crosswind is flying at an angle of attack and feels a sideways force, which is how wind actually disturbs a booster: not by pushing on it directly, but by changing the direction of the air it is moving through.

Air density is not a constant

Scale height H is the altitude over which density falls by a factor of e.

An exponential atmosphere is the simplest model that gets the physics right, and the consequence is stark. At eight kilometres the density is about a third of sea level. Every aerodynamic quantity in this article, drag, fin authority, damping, weathervane stiffness, is proportional to ρ, so all of them vary by a factor of three over a stretch of descent that takes under a minute.

Mass is a state, not a parameter

Integrated over a burn, this is the Tsiolkovsky rocket equation.

Specific impulse I_sp is exhaust velocity expressed in seconds, a convention that survives because it is unit-system independent. The vehicle in the header carries 5 200 kg of propellant on a 22 200 kg dry mass, so it burns about nineteen percent of its total mass during a landing.

This matters more than it first appears. Available acceleration isT/m, and as mass falls, the same throttle setting produces more acceleration. A controller that is well behaved at the start of the burn is more aggressive at the end of it, and a pure feedforward plan computed at ignition will overshoot. The mass state is what lets the guidance law recompute against the vehicle that actually exists right now.