SpaceX landing  —  An Engineer’s Perspective

Here’s a simple experiment. Take a sharpened pencil and try to balance it at its tip using your finger.

Was it easy? Hard? Let’s analyze that in an engineering perspective. Your fingers positioned in the XY plane (assuming pencil was parallel to the Z axis) depending on the angle made by the pencil with the X, Y and Z axis till the angles were 90º, 90 º and 0 º w.r.t to respective axes. What you achieved here is harder said than done, and a very simplified and toned down version of this is popularly known as ‘inverted pendulum’ serves as an introductory problem in a wide branch of engineering called ‘control systems’.

Back in early February 2018, SpaceX pulled off an engineering marvel by landing the ‘Falcon Heavy’ boosters in perfect sync at the Kennedy Space Centre. At its core what SpaceX pulled off was what you tried to achieve with a pencil a couple of seconds back. So, why is this an engineering feat and how did they pull it off?

Let’s go back to the pencil example. Now instead of your finger lets add some vectoring systems (gimbal motors, propellers, etc. anything that can displace the tip at accurate values). A very simple practical example of an inverted pendulum is the popular ‘Segway’. Mathematically, what your finger pulled off is far more complex as compared to what a ‘Segway’ is achieving. Pretty cool huh? ‘Segway’ is achieving a 2 state control of a fixed mass. Now the question is how does ‘Segway’ do what it does? Here comes the next part we call control law and controller design.

Without divulging too much into details and sticking to generic terms, let’s look at the ‘Segway’ more closely. We know its goal is to stand upright like the pencil and there are 2 motors that are used to actuate it to move forward and backward to achieve this goal. But by how much it should move depends on the angle of tilt. The larger the angle of tilt the larger the actuation and vice-versa till a point where the goal is achieved and actuation is no longer needed. That ‘how much’ is determined by control laws. The simplest, yet effective control law is the PID (Proportional Integrative and Derivative) controller. Almost every hobby project involving an unstable system uses a version of PID. It is time tested and needs very less computation. Fun fact, the International Space Station’s attitude (yes, not altitude) control is done using a slightly advanced version of PID.

From academic point-of-view the controllers that are well — established and well — documented are actually quite limited in application. Control laws are broadly divided into classic and modern control. PID and LQR (Linear Quadratic Regulator) come under the first category. Why is there a divide in the control laws? Here’s something very important to understand. To make the math easier for computation, we assume that over a small range the input from the sensors and output from the actuators are linear. (The definition is actually is more detailed, but this will do). But in reality it’s never linear and that’s when classic controllers start failing. The thing is that ‘small range’ mentioned before sometimes is a little too small and doesn’t work in real life because operating ranges are usually pretty large. A little ‘hack’ to deal with this is to tweak the classic controllers and improve the operating range (gain — scheduling, etc.).

Right, now that we have established a pretty comprehensive baseline, let’s address the elephant in the room. Now imagine a pencil 70 meters tall, weighing one million pounds and your hand can generate a million pounds of thrust to slow down it down, considering the non-linearity of varying atmospheric density and changing mass of the pencil to land at a point ensuring atmospheric electromagnetic waves don’t mess with your brain, without crashing on the way and to be just in time without running out of fuel. Yep, that quite mouthful. That’s why this is an engineering marvel. Lars Blackmore, formerly part of NASA JPL ‘Guidance and Control Group’, now one of the senior scientists at SpaceX ‘Reusable Launch Vehicle Program’ co-invented the G-FOLD algorithm widely used for Rocket landing. He developed it initially for the Mars lander and then it was later incorporated into the Falcon. This algorithm autonomously computes the fuel optimal path and reach any target.

There are essentially three control mediums for this landing. They are:
1Compressed air vectoring systems
This system is essentially installed on the top of the falcon lander. It can generate short bursts of air and generate good torque. It is used twice during the duration of landing:

• The first time is when the falcon carrier has to flip 180 degrees to start moving back toward the base
• The second time it’s used to make fine position adjustments just before touch down

2Gimbal Thrusters
They are known as gimbal thruster because it can orient the merlin engine in a specific angle to change trajectories. Gimbal thrusters are also used twice during its descent:

• The first powered descent happens after the 180 degree flip after the boosters detach from the main body. This is a pretty short powered descent till the boosters are brought in to a pre-calculated trajectory and gravity kicks in to do the rest of the work.
• The second powered descent happens couple of thousand meters above the ground to reduce the speed of approach to the ground. This is very crucial to ensure the boosters told get destroyed on impact with the ground

3Grid Panels
Grid panels are located little below the CG of the boosters. They are used right after the first powered descent when the booster are in a calculated free fall. These grids ensure attitude control all the way to the ground.

Using these three actuators and a very solid algorithm SpaceX pulls off the miracle landing that is definitely revolutionizing the Aerospace industry.

– Dibyayan Senagupta