Kinematics (Description of Aircraft Motion)

Kinematics deals with the description or motion, without any of the physics involved. Physical concepts of rotational motion, such as moments (rotational forces), the inertia tensor with its moments of inertia and products of inertia, and associated equations of motion (the rotational physics counterpart of \(\mathbf{F} = m \mathbf{a}\)) are left on the bench for now. We will study these later.

For now, all we will consider is the rotational motion itself. The simplest question we can ask is, how do we describe aircraft attitude. For this, we need a rotation formalism – a description of attitude using some parameters.

Why do we need a Rotation Formalism for Attitude Description?

Question: But why do we need a rotation formalism? Can’t I just say that my wings aren’t level and that my nose is above the horizon? Answer: Regarding the wings, you can probably see that it is a rotation that puts them into a non-level attitude. As for the nose, it is not really “above” the horizon; it just looks that way looking out of the windshield. In reality the aircraft is “pitched up”, i.e. rotated up. Thus attitude changes really are rotations.

Question: Why do we need to introduce and parameters to describe attitude? Can’t I just read off the aircraft instruments? Answer: What do you think the aircraft instruments are telling you? Values of parameters – heading, roll angle and pitch angle are parameters; they are variables to which you assign a number based on what you read off on the aircraft instruments. Thus, implicitly, whether you are realizing it or not, you are using a rotation formalism with a suitable (intuitive) parametrization, when you think about aircraft attitude.

It is best to think formally about what you think about, rather than just having a diffuse feeling about what you are thinking about. Otherwise you might end up like the German Coast Guard in this video.

Axis Systems (Reference Frames)

Before the rotation formalism can take any foothold, we need to define directions in space: with respect to the Earth, with respect to the aircraft, with respect to the airflow, etc. We saw this before in aerodynamics: we were able to introduce the angle of attack (AOA) \(\alpha\) to describe the orientation of an object (airfoil) in space, with respect to the airflow, but we needed to introduce the direction of the relative wind first as a reference direction, and we also needed to define a reference direction on the airfoil, which we called the chord line. Without defining these two reference directions first, the AOA would have been meaningless. We had to state, the AOA is the angle between what. It is this what, which we now define in three-dimensional space, when we discuss axis systems.

Attitude Parametrization using 3-2-1 Euler Angles

We need a clear, well-defined description of aircraft attitude, which leaves no room for ambiguity. For this, we need a rotation formalism in three dimensions, which manages to describe any aircraft attitude as a 3D rotation from some reference attitude (the latter is usually taken to be an airplane pointing north, wings level, with its nose on the horizon).

Many different rotation formalisms exist, including the rotation matrix (see linear algebra), various Euler angle definitions, the axis-angle formalism (principal rotation vector), quaternions, and several others. We shall introduce one of them here fairly briefly.

We choose the 3-2-1 Euler angle formalism for several reasons:

• It is commonly used in aerospace engineering for attitude descriptions of the kind of atmospheric aircraft we are considering in this course.
• It allows humans to visualize the attitude of the aircraft easily.
• The standard aircraft instruments (attitude indicator and heading indicator) indicate the three parameters of this formalism directly: we can read off their values directly from our instruments in the aircraft.

One of the most attractive features, which make this rotation formalism so popular, is its ease of visualization of aircraft attitude (further below we will encounter a different formalism, which is computationally more convenient, but impossible for humans to visualize). Visualization is tremendously important for pilots, who need to make decisions in real time. The rotation formalism also has some drawbacks, e.g. a coordinate singularity for an airplane flying vertically straight up into the sky, but we find ourselves relatively rarely in such a situation in our applications.

The above preference is clearly application specific. If we were studying the orbital motions of satellites, we may find that 3-1-3 Euler angles are more convenient – a close, but distinctly different relative of the 3-2-1 Euler angles we will be using here. The 3-2-1 Euler angles are called 3-2-1, because – as you can see from the slides below – we first rotate around the third axis, then around the (new) second axis, and then around the (again new) first axis.

Other Rotation Formalisms in 3D

3-2-1 Euler angles are not the only rotation formalism in 3 dimensions. There are many others. Unit quaternion conjugation provides a very elegant and computationally efficient way of computing rotations, while avoiding the coordinate singularity, which Euler angles exhibit. It is not as intuitive though – neither for attitude visualization nor for computations.

Even less visualization intuitive, but very straightforward mathematically (basic linear algebra) is the rotation matrix formalism. In fact, this is the “mother of rotation formalisms,” which also acts as a universal translation tool between all the others: for any rotation formalism out there, there is always a procedure to compute the rotation matrix, and also reversely to infer the parameters of your particular rotation formalism, if the rotation matrix is know.

As an illustration that there are different rotation formalism and that one can translate between them, we therefore explain below how to translate between 3-2-1 Euler angles and the rotation matrix formalism. We show how to compute the rotation matrix (also called Direction Cosine Matrix (DCM)), if the 3-2-1 Euler angles are known. And we also explain the opposite direction, how we can compute the 3-2-1 Euler angles, if the rotation matrix is know. (With the rotation matrix you can rotate any vector or perform a basis transformation of the components of a fixed vector between two different reference frames, as is explained in our Linear Algebra Primer.)

This is certainly an advanced topic for pilots, who are particularly engineering minded. You can safely skip this part, if you like.

A more detailed description of these and other rotation formalisms in three dimensions is given in our article series on 6 popular rotation formalisms in kinematics.

Additionally, if you would like to learn more about the description of attitude and motion, an excellent course on rotation formalisms for this purpose is taught by Prof. Hanspeter Schaub on Coursera, “Kinematics: Describing the Motions of Spacecraft”, which he teaches at the University of Colorado Boulder. You can audit the course on Coursera for free – or even get a completion certificate, if you pay for registration.