We Will Control the Vertical

[Title from The Outer Limits intro.]

Recently, my research has involved working with the control systems of the LISA spacecraft, which allow them to remain properly oriented. One of the main tools we use for this is a State Space Model. This type of structure models a physical system as a set of inputs, outputs, and states, which change in time. The system is defined by 4 matrices:

where u is a vector of inputs, y is a vector of outputs, and x is a vector of states. The dot indicates a time derivative, which tells us how the states change in time. Because this is a linear model, it will only work if we stay close to some equilibrium, but that's exactly what we hope to achieve. If we choose our inputs in relation to the outputs, we can try to stabilize the system.

I decided to play around with this model a bit by using the classic example of an inverted pendulum. This type of system may be familiar to you if you've ever tried to balance a pole on the palm of your hand. We want to keep the pole straight by moving its base. Gravity makes it tip in one direction or the other, and we react by moving our hand in the same direction. The trouble is how to avoid overshooting. For my model, I used a sliding cart in place of a hand. If we don't apply any external force to the cart, the rod will quickly fall over:

We want to apply a force to keep the cart underneath the pole, but I found that if I applied a force exactly opposite to the one making the pole tip, the pole would stay at a fixed angle while the cart zoomed off the screen. I played around a bit with different forces based on the pole's angle and speed, and found a configuration that looks pretty good:

It been interesting to learn some of the techniques that are more engineering-focused. I'm just glad I have more experienced people to help me out – I don't want to be responsible for another Mars Climate Orbiter!

Searching in Vein

Earlier this week, I had one of my periodic MRIs as a cancer survivor – All clear! During the scan, I get a chemical injected partway through called gadolinium, which allows the scanner to pick up blood flow. Since chemotherapy, my veins have been more constricted than most people's, making it difficult to get the IV in. After a couple failed attempts to get a vein by feel, the tech got out a type of vein-finding device I had never seen before:

AccuVein

This is a little different from the one that was used on me, but the tech was already flustered enough from the two failed pokes without me pulling a camera out. The device projects a square of light with shadows anywhere there's a vein. As was the case when I got an ultrasound, I started asking questions about how it works, only to be met with shrugs.

It turns out the device uses near-infrared light to scan over the area. Blood vessels contain lots of water, which absorbs in the near-infrared region:

Wikipedia

This plot shows the amount of light absorbed by liquid water at different wavelengths. It dips down for the visible spectrum, which is why water appears transparent, but rises quickly on either side. That means the blood vessels will absorb the light put out by the device, instead of reflecting it. It can detect where these gaps in reflection occur, and project an image of the veins on skin.

I've been a cancer patient and survivor for 10 years now, and I'm delighted to keep finding interesting physics in my medical experiences!

Out of My Element

Recently, a friend shared a meme about how COVID has made the idea of bowling particularly unappealing: Sticking your fingers into a ball handled by many people, which has also been rolling on the floor, then eating greasy food with those same fingers. After getting that image out of my head, I started thinking about the physics of bowling. In particular, I was interested in the curve that skilled bowlers are able to give to the ball:

Typically in physics, we make the assumption of "rolling without slipping," meaning that an object rotates at the correct speed for the contact between the two surfaces not to slide, i.e.

where v is the velocity along the floor and ω and r are the angular velocity and radius of the rotating object. If you're mathematically minded, you may notice this is the derivative of the relation between angle and arc length for circles. Bowling balls though have significant slip, as illustrated in this animation:

Wikipedia

When the player throws the ball, they give it some initial linear velocity and angular velocity, or spin. When the ball makes contact with the floor, friction begins pulling on the ball, which applies a torque. Since the ball is slipping, we need to know the relative velocity between the surface of the ball and the floor:

Note that the angular velocity is given according to the right-hand rule: If you point your thumb in the direction (ωx, ωy), your fingers curl in the direction of rotation. Using this, we can get the force and torque on the ball:

where μ is the coefficient of friction, m is the mass of the ball, g is the acceleration from gravity, and X is the cross product.