6 Popular Rotation Formalisms in Kinematics

6 Popular Rotation Formalisms in Kinematics

Overview

Rotation formalisms are essential in kinematics for aircraft attitude description. This article covers several frequently used rotation formalisms in three dimensions. It is composed of several individual pages linked below.

Aircraft attitude can be described as an imagined rotation from an agreed upon reference attitude. This reference attitude is in flat earth approximation typically chosen to be straight-and-level flight to the north.

In addition to the above imagined rotation describing attitude, we also wish to describe the actual rotation (change in attitude) of the aircraft over time. This will lead us to the differential kinematic equation for the rotation parameters of each rotation formalism. This differential kinematic equation is one of the four equations we need to describe aircraft motion (see our article on equations of motion).

Together, these components are the key ingredients of kinematics, the description of motion of a rigid body in three-dimensional space as a function of time (without any of the physics involved, like mass and forces, which are part of dynamics/kinetics, and to be distinguished from kinematics, which deals only with the geometric aspects of the problem).

Several rotation formalisms can be used, each with its own advantages and disadvantages. This article describes some of the most commonly used ones. The reader may find it helpful to read our less technical, introductory article on aircraft attitude and Euler angles first, which gives a concise overview over one of the oldest and most commonly used rotation formalisms. This article here goes much deeper and explores the topic of attitude and rotational motion description much more systematically.

Description of Individual Rotation Formalisms

This article covers the following rotation formalisms on separate individual pages. It is suggested you read them in the order listed, but you can jump around between them, as long as you are familiar with the notation used.

If one of the above formalisms does not have a link to a page yet, it means we are still in the process of writing that page; it will be posted soon.

References and Further Reading

Firstly, would like to refer the reader to Prof. Hanspeter Schaub’s course on Coursera, “Kinematics: Describing the Motions of Spacecraft”, which he teaches at the University of Colorado Boulder. You can either get the paid version and receive a completion certificate, or audit it for free (in which case you will be able to do the associated exercises, but not be able to submit them for grading – an option you can always enable later). Schaub’s kinematics course is the first course in his series of four courses  comprising the “Spacecraft Dynamics and Control Specialization” on Coursera.

Our article here was inspired in part by Schaub’s kinematics course and follows parts of it to some extent, including using similar notation and terminology in some sections. Schaub’s course goes into much more detail in certain aspects than our article and covers additional topics as well, which is why we highly recommend you take it, in addition to reading our article. Doing the quizzes and programming exercises associated with Schaub’s course will do you good. On the other hand, Schaub’s course takes a very application oriented approach in some regards and some formal buildup had to be omitted in order to keep the course series at manageable length, which ultimately targets spacecraft and not atmospheric aircraft. For instance, the quaternions are defined solely based on the principal rotation vector formalism and are never treated algebraically as an extension of the complex numbers. As a result, their application to rotations in the course is restricted to translating them ultimately to the rotation matrix (just as one would do with Euler angles); conjugation with a unit quaternion as a three-dimensional rotation formalism in its own right – the way it was originally conceived by Hamilton and with some of its computational benefits – is not covered. So in our article we chose a somewhat different, complementary emphasis.

We would further also like to refer the reader to the Wikipedia article on rotation formalisms in three dimensions and links and references therein, which provide some complementary information.  In addition, each page above in this article contains links to further references for each of the rotation formalisms.