What Are They?

 

Euler angles are a series of rotations about three angles, which are shown by α, β, and γ. (Euler angles can also be referred to as roll, pitch, and yaw.) Euler angles describe the orientation with respect to a fixed coordinate system.

 

 

Intrinsic Rotations: Rotations around a moving body

 

Intrinsic rotations are basic rotations that occur about the axes of a coordinate system XYZ attached to a moving body. Think robotic arm.

 

• x-y-z, or x₀-y₀-z₀ (initial or fixed coordinate system) shown in blue
• x’-y’-z’, or x₁-y₁-z₁ (after first rotation)
• x″-y″-z″, or x₂-y₂-z₂ (after second rotation)
• X-Y-Z, or x₃-y₃-z₃ (final)

 

• α represents a rotation around the z axis,
• β represents a rotation around the x’ axis,
• γ represents a rotation around the z″ axis.

 

Extrinsic Rotations: Rotations around a fixed coordinate system

Extrinsic rotations are basic rotations that occur about the axes of the fixed coordinate system xyz. The XYZ system rotates, while xyz is fixed. This is how the CMM works using a fixed start point. Starting from the fixed xyz coordinate system the XYZ is rotated by a series of three rotations to reach any target orientation for XYZ. The Euler (α, β, γ) are the results of these basic rotations. For instance, in our next example our orientation can be reached as follows:

 

• The XYZ system rotates about the z axis by α. The X axis is now at angle α with respect to the x axis of the fixed system.
• The XYZ system rotates again about the x axis by β. The Z axis is now at angle β with respect to the z axis of the fixed system.
• The XYZ system rotates a third time about the z axis by γ.

 

In summary, the three basic rotations, in both the intrinsic and extrinsic examples, occur about z, x and z. Indeed, this rotation sequence is often known as z-x-z (or 3-1-3).

 

Euler angles in Calypso are reported in Radians. In our examples below we will multiple by the radian conversion 57.295779513 to convert them back to decimal degrees.

How to Report a Euler Angle

 

We will use Result Element to report the results of our Euler angle formula.

 

Let’s break this down rotation by rotation for this first example. Our entire rotation sequence in this example will be using the z-x-z or 3-1-3 example as above. Using the rotations below.

  

About z:        5.000

About x:      40.000

About z:      20.000

  

First, rotate 5.000° about Z for Alignment1

 

 

 

The new X axis becomes our Node (the green line in the above diagram (N)) this gives us the Euler angle α.

 

 

 

 

 

 

 

 

N

 

 

You will see below this is Euler1

 

 

 

 

Now rotate around X 40.000°

 

 

The new Z axis gives us the Euler angle β. In Calypso this will be Euler2.

 

 

 

 

 

Now rotate 20.000° About Z. In Calypso this will be Euler4.

 

 

Using the example above we have rotated Alignment1

 

Using the rotation angles

 

 

  

Euler1 = Y

 

Euler1 (*57.295779519 to convert from Radian to Decimal Degree) is 5.000°

 

Euler2 = X

 

 

 

Euler2 (*57.295779519 to convert from Radian to Decimal Degree) is 40.000°

 

 

 

 

Euler4 = X

 

Euler4 (*57.295779519 to convert from Radian to Decimal Degree) is 20.000°

 

 

 

 

 

This is from the Node to the new X created after the final rotation.

 

 

 

 

 

 

 

 

  

 

In Summary.

 

 

 

α =   5.000°     (Euler1)

 

β = 40.000°     (Euler2)

 

ϒ = 20.000°    (Euler4)

 

 

 

 

 

 

z-y-z or 3-2-3 Rotation

 

 

 

In this example we created an Alignment with the following rotation

 

 

 

 

 

 

 

 

 

 

Rotate 25.000° about Z. This creates the new Y or our Node (N)

 

 

 

The new Y gives us the Euler angle α.

 

 

 

 

 

Rotate 40.000° about Y. This gives us β

 

 

 

 

 

 

Rotate 20.000° about Z. This gives us ϒ

 

In the diagram below the green dashed line is the Node (the new Y location based on the first rotation).

 

 

 

Using Result Element the Euler angles are reported as shown.

 

 

 

Euler 1

 

 

 

115.000° angle (115.00° - 90.00° = 25.000°)

 

 

 

Euler 2

 

 

 

40.000° angle

 

 

 

Euler 4

 

 

 

290.000° angle (290.00° - 270.00° = 20.000°)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Euler angles can be used in formulas, verification of rotational alignments, reporting deviations along a particular axis, etc…. The intent here is to show what they are and how they are reported using Calypso.