Maths - Euler Angles

We use the term "Euler Angle" for any representation of 3 dimensional rotations where we decompose the rotation into 3 separate angles. When we first start to think about 3D rotations this seems the natural way to proceed but our intuition can be deceptive and there are a lot of problems that arise when we use Euler angles to do calculations.

If you are reading this page in order to write a 3D computer program I suggest you read enough of this page to convince yourself of the problems with Euler angles and to get an intuitive understanding of 3D rotations and then move on to quaternion or matrix algebra representations.

On this page we will also consider theoretical issues such as the number of degrees of freedom in rotations.

Standards Used on this page

As we shall see, there is no single set of conventions and standards in this area, therefore I have chosen to adopt the following to be consistent across this site.

angle applied first	heading
angle applied second	attitude
angle applied last	bank

Right hand coordinate system (for both coordinates and rotations) and angles measured in the frame of reference of the rotating object.

Other websites and books that you come across will use different standards, I cant be compatible with them all, so when I setup the site I chose the conventions which seemed the most popular, that's the best I can do. The standards we choose depends on the properties we want, I have discussed the standards for this website here. See this page for an explanation of the conventions and standards used on this site.

Examples

Aircraft	Telescope
adapted from a diagram from Andy
In this example we would naturally measure the angles in the frame of reference of the aircraft.	In this example we might think that it would make sense to measure the angles with reference to the ground. However the third angle 'tilt' is around the axis of the telescope. It is hard to find an example where all 3 angles are measured relative to the ground.

Types of Euler Angle and Notations

Because, when we are combining rotations, order is important so there are many types of Euler angles. So when we must choose:

• Order of combining angles.
• Relative to rotating object or absolute coordinates.
• Left or right hand rotation.
• Notation for angles.

Unfortunately there is not a widely accepted set of standards so I have tried to combine what standards there are so that we can have consistency across the site.

Order of combining angles.

As a quick shorthand lets use x to represent a pure rotation about the x axis and y to represent a pure rotation about the y axis and so on. so we could represent the order of 3D rotations as: xyz, yzx, zxy or reversing the order zyx, xzy, yxz which gives 6 permutations. There are more because the angles are not independent for instance rotating 90° around x followed by 90° around y then back -90° around x is the same as a single rotation of 90° around z, so we can form any 3D rotation by combining rotations in just 2 planes, but we may have to apply 2 separate rotations in one of the planes. This means that we can also use any of the following sequences: xyx, xzx, yxy, yzy, zxz and zyz. So altogether we have 12 possible angle sequences.

Relative to rotating object or absolute coordinates.

We could measure each of the 3 angles relative to the ground (or some absolute coordinate system) or relative to the object being rotated. So the number of permutations of options goes up from 12 to 24. Our first intuition would probably measure all angles relative to the ground especially if we were writing a computer simulation or game where everything is represented in absolute coordinates. However, as we can see from the examples above, it usually makes sense to measure the angles in the coordinates of the object itself. This has the additional advantage that if we need to calculate the physics (dynamics) of the object then the inertia tensor is in the frame of the object.

Left or right hand rotation.

I don't mean the left or right handedness of the coordinate system I mean the direction along the rotation axis that we consider to be positive. For example, if we are using a right hand coordinate system, then if the thumb of our right hand indicates the direction of the positive rotation axis then the fingers show the direction of rotation.

Notation for angles.

We tend to give the different angles different names depending on the situation where we are using them. Here are some examples:

	airplane	telescope	symbol	angular velocity
applied first	heading	azimuth	θ (theta)	roll
applied second	attitude	elevation	φ (phi)	pitch
applied last	bank	tilt	ψ (psi)	yaw

Successive Rotations

If we have a 3D rotation represented by 3 Euler angles (a1,a2,a3), and then we apply an additional rotation represented by another 3 Euler angles (b1,b2,b3), how do we calculate an equivalent set of 3 Euler angles (c1,c2,c3) which will represent these two successive rotations? This turns out to be very difficult we can't just add the corresponding angles because (c1,c2,c3) is not necessarily (a1+b1,a2+b2,a3+b3) due to the fact that the order of successive rotations is significant, for example

1. Rotate 90°s about x axis
2. Rotate 90°s about y axis
3. Rotate -90°s about x axis

This gives 90° rotation about Z axis,

whereas

1. Rotate 90°s about x axis
2. Rotate -90°s about x axis
3. Rotate 90°s about y axis

This gives 90° rotation about y axis (first 2 lines cancel out).

Because this is so difficult, it is usual to convert to matrix notation or quaternions and calculate the product and then convert back to euler angles. But if this is done many times the rounding errors of all these conversions will build up, leading to distortions. So if we need to do a lot of such calculations it may be best to work entirely in matrices or quaternions even though they are less intuitive than Euler angles.

I can think of different ways to combine rotations so I have put a discussion of this here.

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Rotation about single axis

First we will consider rotation purely about each of the three axes individually, then we will consider the different ways of combining them.

Attitude (About Z axis)

First assume a rotation purely around the z axis, measuring from the x axis, as shown here:

In order to combine rotations using matrices we need to be clear about what conventions that we are using. The coordinate directions are represented by a right hand coordinate system and the rotation directions are represented by a the right hand rule . The identification of cells of matrix and ordering of rows and columns (as explained here) is as follows:

v_out0 v_out1 v_out2

=

m00 m01 m02 m10 m11 m12 m20 m21 m22

v_in0 v_in1 v_in2

So point x=1, y=0, z=0

is transformed to x=cos(attitude), y =sin(attitude) , z=0

and point x=0, y=1, z=0

is transformed to x=-sin(attitude), y =cos(attitude) , z=0

so the complete matrix for rotation about the z axis is:

[R1] =

cos(attitude) = c φ -sin(attitude) = -sφ 0 sin(attitude) = sφ cos(attitude) = c φ 0 0 0 1

About z axis using axis angle

Axis = (0,0,1) Angle =φ

About z axis using quaternions

The axis-angle above can be converted to quaternion as described here.

cos(φ/2) + (0 i + 0 j + 1k) * sin(φ/2)

cos(φ/2) + k * sin(φ/2)

Heading (About Y axis)

Similarly rotation about the y axis, measuring from the z axis, gives:

So point x=1, y=0, z=0

is transformed to x=cos(heading), y =0, z=- sin(heading)

and point x=0, y=0, z=1

is transformed to x=sin(heading), y=0, z =cos(heading)

so the complete matrix for rotation about the z axis is:

[R2] =

cos(heading) = c θ 0 sin(heading) = sθ 0 1 0 -sin(heading) = -sθ 0 cos(heading) = c θ

About y axis using axis angle

Axis = (0,1,0) Angle =θ

About y axis using quaternions

The axis-angle above can be converted to quaternion as described here.

cos(θ/2) + (0 i + 1 j + 0k) * sin(θ/2)

cos(θ/2) + j * sin(θ/2)

Bank (About X axis)

and rotation about the x axis, measuring from y, gives:

So point x=0, y=1, z=0

is transformed to x=0, y=cos(bank), z =sin(bank)

and point x=0, y=0, z=1

is transformed to x=0, y=-sin(bank), z =cos(bank)

so the complete matrix for rotation about the x axis is:

[R3] =

1 0 0 0 cos(bank) = c ψ -sin(bank) = -sψ 0 sin(bank) = sψ cos(bank) = c ψ

About x axis using axis angle

Axis = (1,0,0) Angle =ψ

About x axis using quaternions

The axis-angle above can be converted to quaternion as described here.

cos(ψ/2) + (1 i + 0 j + 0 k) * sin(ψ/2)

cos(ψ/2) + i * sin(ψ/2)

Example

Imaging you are in a cuboid room, you on the floor of one corner looking along along the base of one wall, you want to turn to look at the top of the opposite corner. What angle do you have to turn through?

Most people would say that you would need to turn through a heading of 45° and then rotate up at an attitude of 45°. This belief is so strong that programmers will spend days rewriting and debugging their code if they don't get this answer.

In fact we need to turn through a heading of 45° and then rotate up at an attitude of 35°. You turn through 45° and you are now looking at the base of the diagonal corner. You now want to look up to the top of the diagonal side. This forms a triangle where the adjacent side is √2 and the opposite side is 1. So the angle is arctan(0.7071)=35°.

This is a warning not to trust intuition when working with 3D angles and avoid using euler angles whenever possible. (see lookat method for other methods)

Linearising Rotations

In order to investigate the non-linear nature of Euler rotations we can try showing the angles on a linear 3D grid and mapping the rotations onto it.

The first issue is that we must apply the angles in a given order, in the example below we rotate heading 90° (2a) followed by attitude 90° (3a) and the aircraft is going up with the rudder pointing toward us, however if we rotate attitude 90° (2b) followed by heading 90° (3b) and the aircraft is going up with the rudder pointing away from us.

I think we can still use this diagram but we have to be careful to interpret it as implying an order and we cannot change the order.

Another example below shows that just by rotating the heading and attitude we can produce a bank.

But again this is not valid on this diagram because we have altered the order of the Euler angles. All this really shows is that we cannot add the angles if we are combining them in different orders.

Even if we do apply the angles in our chosen order it does not necessarily mean that each point on the diagram represents a unique rotation, it may be that the same rotation is represented at several places on the diagram, this happens at the singularities attitude= ±π/2.

So can we distort this model so that it becomes linear? I don't know the answer to this, I suspect we need to investigate the mathematics of topology and manifolds which I have started on this page, I would welcome any help with this. I think that we can't linearise in 3D which is why we need to embed in a higher dimensional space (such as quaternions) but I don't know how to prove this.

NASA Standard Aeroplane

In order to try to explain things I thought it might help to work out a simple case where rotations are only allowed in multiples of 90°s. This should make it easier to illustrate the orientation with a simple aeroplane figure, we can rotate this either about the x,y or z axis as shown here. There is also a program to show the orientations for any Euler angles.

adapted from a diagram from Andy

angles: φ: heading = rotation about y (applied first) = ay θ: attitude = angle about z (applied second) = az ψ: bank= angle about x (applied last) = ax Coordinate System: right hand Positive angle: right hand screw rule. Order: heading,attitude then bank (about y,z then x) [R3][R2][R1] = ([Ry][Rz])[Rx] Method for combining rotations: local coordinate system [resulting transform] = [first rotation] * [second transform] Singularity at θ attitude = +- PI/2 normal range: heading: -pi and pi attitude: -pi/2 and pi/2 bank: -pi and pi

Note: I cant find a definition of this standard, but I think it uses a slightly different coordinate definition from VRML (z and y axis swapped). Also when working with aeroplanes we often work in terms of the position of external objects relative to the aircraft (i.e. the inverse of its position transform as explained here). therefore to get the expression normally used with NASA aeroplane we invert all inputs (change sign of every term with odd number of sine terms) invert output (transpose matrix) swap z and y for inputs (swap z and y columns) swap z and y for outputs (swap z and y rows). This gives the matrix which is quoted for NASA aeroplane, which is: ch*ca sh*ca -sa ch*sa*sb - sh*cb sh*sa*sb + ch*cb ca*sb ch*sa*cb + sh*sb sh*sa*cb - ch*sb ca*cb The derivation of this is shown here. However, for this site I think it is better to use the transform derived below as that is in the form which compatible with the rest of this site.

This gives a combined transformational matrix of,

[R] = [R3][R2][R1] = ([Ry][Rz])[Rx]

This is expanded out here. To save space cos(bank) is written as cb and so on:

Prerequisites

Euler Angles are one possible way to represent the orientation, or other rotational quantity, associated with a solid 3D object.

If you are not familiar with this subject you may like to look at the following pages first:

• Rotations

Standards

Rules of Mathematical Operations

Successive Rotations

Example Dish Example

Matrix vs. Quaternion