Modeling Asteroid Rotation Using Quaternions: Seeking Guidance on Init

[Codezero]

TL;DR Summary

I am writing a paper on modeling asteroid rotation using quaternions and establishing a differential equation to track its changes over time. I'm struggling with the initial modeling and comparative analysis against other mathematical approaches like Euler angles. I seek guidance on starting points for modeling, formulating differential equations, structuring comparative analysis, and any additional concepts to enrich my analysis

Hello everyone,

I am an International Baccalaureate (IB) student working on my extended essay, which is a mandated 4,000-word research paper. My chosen topic is Quaternions, a mathematical concept I find highly intriguing. The primary aim of my paper is to model the rotation of an asteroid using quaternions and establish a differential equation to track its change over time. Additionally, I aim to compare the practicality and efficiency of using quaternions with other mathematical approaches such as the Euler angles to highlight their unique advantages.

Background and Preliminary Work
I have equipped myself with math knowledge beyond the typical high school curriculum, delving into resources to understand quaternion rotations and some preliminary calculations concerning them. I have also begun exploring quaternion differential equations. Here’s a summary of what I've learned and attempted so far:
Understanding Quaternions: Grasped the basics of quaternions and their application in rotational dynamics through various resource i have also learnt about euler angles and the common problem that arises from their use, gimbal lock
Exploration of Differential Equations: Began understanding the formulation of differential equations to describe quaternion rotations over time.

Challenges and Queries
While I have made some progress, I am struggling with how to model an asteroid’s rotation using quaternions from scratch. Here are my specific questions:
Modeling Start Point: What could be a logical starting point for modeling an asteroid's rotation using quaternions? Are there any established methodologies or examples I could follow?
Differential Equation Formulation: How might I go about establishing a differential equation to track the asteroid's rotational change over time?
Comparative Analysis: What might be some other mathematical approaches for modeling rotations, and how can I structure a comparative analysis to highlight the advantages of quaternions?
Further Ideation: Are there any additional concepts or applications of quaternions that could enrich my paper's analysis and discussion?

This question arises from my self-driven exploration and ambition to contribute a well-researched paper on a challenging and engaging topic. The motivation stems from my fascination with both space phenomena and advanced mathematics, and the desire to explore an intriguing intersection of the two.

Any guidance, references, or suggestions on how to proceed would be immensely appreciated. I am particularly interested in practical advice on modeling and comparative analysis, and any resources that could further deepen my understanding of quaternions and their applications in rotational dynamics.

Thank you in advance for your time and insights!

 

[jedishrfu]

Welcome to PF!

This is a very interesting topic. Qaternions were quite popular in Physics due to Hamillton's work. H envisioned all of Physics being transformed by them. They had the built-in property of rotation when using them to describe the motion of a body in #D space. However, Gibb's felt they were unnecesarily complicated and so he borrowed some of their notation namely i,j,k and created Vector Analysis from them.

More recently there's been a resurgence of interest in computer graphics systems.

For starters, I found this reference on Wikipedia:

https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation

and these references on Youtube that may be some help:



3Brown1Blue is an excellent presenter of mathematical knowledge.

https://wirewhiz.com/quaternion-tips/

For alternative methods, you might look to these systems of math:
-vector analysis and its use in Classical Mechanics (Goldstein's book on CM)
- geometric algebra and its use in CM (in particular David Hestenes work)

 

[FactChecker]

To start with, you might look at the short "Algorithms" paragraph in the Mathworks documentation of their quaternion algorithm. An important step is to find a good error correction gain, $K$, that works for your application. That might be a part of your report. Your choice of asteroid rotation might not be the best for displaying the advantage of quaternions if you can find a fixed rotation axis and avoid the problem of gimbal-lock. If your asteroid is jetting gasses and the rotations change, then there would be a real advantage for quaternions.

 

[fresh_42]

You can find some of the basic bijections explicitly noted here:
https://www.physicsforums.com/insig...c-part/#3-The-3-Sphere-or-Glome-or-SU2mathbbC