Intro/Motivation One of the most common tasks in dynamics and controls is how do we get an object from point A to point B. Sometimes, like in the brachistochrone problem, we get lucky and there is a nice analytical solution. Unfortunately, in most situations, there is no closed form answer, and we need to turn…
Lissajous curves From orbits around Lagrange Points, to double pendulums, we often run into a family of loopy, beautiful, curves. These are called Lissajous curves, and describe complex harmonic motion. In layman terms, Lissajous curves appear when an object’s motion’s have two independent frequencies. Today, we’ll explore another system that produces Lissajous curves, a double…
Recap of Lagrange Points In the circular restricted three body problem, there are a set of 5 points that if we place our spacecraft there, it’ll never move relative to the two bodies. These are called the Lagrange points and are the only equilibrium points for the system. Whenever we find an equilibrium point in…
Good Enough Numerical Methods are all about getting an answer that is good enough. A whole class of numerical methods are iterative methods, which continue running until they are good enough. Newton’s method is the example of an iterative method. It keeps going, making the error smaller and smaller until it finds a zero. A good…
Good Enough Numerical methods are always concerned with the term “good enough”. Whether you are using a finite difference to approximate a derivative, or Newtons method to find the roots of a function, we rarely get an exact answer. Usually, our answer is different from the truth by some amount. We generally don’t know what our error…
Bumpy Terrain Imagine you are halfway up a mountain. Half an hour ago you decided to start bushwhacking and now you’re completely turned around and lost. What’s worse is that you’ve also lost your map and compass. It’s getting late and you need to get back to your home at the foot of the mountain.…
What’s a Finite Difference and Why do we Want it A common situation in real world problems is you have to take a derivative, but you don’t have the underlying function. You might be measuring some natural phenomenon and then only have discrete data points. Another time you wont be able to get a general…
The Klaxon Starts Ringing Imagine it’s the year 2300. Earth and Mars are at war. You’re an Earth warship, blue, based on an asteroid in the asteroid belt. A martian drone, red, lands on the opposite side of the asteroid. It’s out of line of sight so how do you aim your rail-gun to destroy…
What is Newtons Method? Let’s say you had some function f(x), and you wanted to find where that function is equal to zero. If that function is a polynomial like the following we can plug it into the quadratic equation and find that the roots are -2 and -1. What about if the function is…
