November 2, 2024
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R3 – Euler and Tait-Bryan Angles

Introduction

This description of Euler and Tait-Bryan angles in R3 is based on the description of Euler Angles in Wikipedia.

According to Euler, any rotation, R, is a combination of 3 Euler angles of rotation. There are two types of Euler Angles. The first are the classic Euler angles, also known as proper angles. The second type are the Tait-Bryan angles, also known as Cardan Angles.

The two types differ in that for each type the unit sphere is in a different orientation and uses a different sequence of rotation around the axes. The classic Euler angles use a unit sphere oriented as the Wave Number Rotation Balls with the z-axis pointing to the north pole, the y-axis to the east and the x-axis to the near pole. The Tait-Bryan angles orient the sphere so that the z-axis is to the north pole, the y-axis to the near pole and the x-axis to the west pole.

The second difference is the types of rotation. Rotation of Classical Euler angles only uses a combination of 2 axes. The 6 combinations possible are:

  • (z-x-z, x-y-x, y-z-y, z-y-z, x-z-x, y-x-y)

Rotation of Tait-Bryan angles uses a combination of 3 axes. The 6 combinations possible are:

  • (x-y-z, y-z-x, z-x-y, x-z-y, z-y-x, y-x-z)

The use of two Wave Number Rotation Balls for visualisation will help understand this post.

Euler Angles

https://upload.wikimedia.org/wikipedia/commons/a/a1/Eulerangles.svg

According to Euler, any rotation, R, is a combination of 3 Euler angles. The frame of a sphere consists of its 3 intersecting axes. When rotating a sphere, the orientation of the axes changes and so the frame changes. The frame is in a standard Euler orientation when it has the point (0, 0,  j^) at the north pole, the point (0,  i^, 0) at the east pole and the point (^, 0, 0) facing. Let this frame be the xyz frame with the standard x,y,z axes and the frames after any rotation be XYZ frame, with the rotated X, Y, Z axes.

When rotating a sphere, the xy (blue circle) and XY plane (red circle) intersect to form a green line of nodes, called N. The 3 Euler angles, α, β, and γ represent angles between the xyz frame and the XYZ frame as follows.

α is the angle between  the x-axis and the N line.

β is the angle between the z-axis and the Z-axis.

γ is the angle between the N line and the X-axis.

Tait-Bryan Angles

The standard Tait-Bryan orientation is the same as for Euler angles with the point (0, 0,  j^) at the north pole, the point (^, 0, 0) at the near pole and the point (0, i^, 0) at the east pole. The diagram above shows the view of the sphere after rotating the x-axis 90ov clockwise around the z-axis so that (0, i^, 0) is facing. This is done in order to show the Tait-Bryant angles.

Let the initial frame be the xyz frame with the standard x,y,z axes and the frames after any rotation be XYZ frame, with the rotated X, Y, Z axes. When rotating a sphere, the xy and YZ planes intersect to form a line of nodes, called N. This is different to the Euler angles where the plane of intersections are the xy and XY planes. The 3 angles φ, θ, and ψ are different to the α, β, and γ Euler angles of the same vector. 

The 3 Tait-Bryan angles represent angles between the xyz frame and the XYZ frame as follows.

φ is the angle between  the N line and the Y-axis.

θ is the angle between the transpose of the N line and the X-axis.

ψ is the angle between the y-axis and the N.

Airplane Flight

https://upload.wikimedia.org/wikipedia/commons/3/30/Plane_with_ENU_embedded_axes.svg

https://upload.wikimedia.org/wikipedia/commons/c/c1/Yaw_Axis_Corrected.svg

Tait-Bryan angles are used for flying aeroplanes as shown in the above diagrams. Here roll corresponds to the ψ angle, is rotation around the x-axis and causes the wings to flap. Pitch corresponds to the θ angle and is rotation  around the y-axis and causes the plane to ascend or descend. Yaw is rotation around the z-axis and causes the plane to turn left or right.

Extrinsic and Intrinsic Rotations

Wikipedia defines extrinsic and intrinsic rotations as follow:

rotations may be extrinsic (rotations about the axes xyz of the original coordinate system, which is assumed to remain motionless), or intrinsic (rotations about the axes of the rotating coordinate system XYZ, solidary with the moving body, which changes its orientation with respect to the extrinsic frame after each elemental rotation).

Extrinsic and intrinsic rotations are easier to understand using two Wave Number Rotation Balls. The first ball stays in its standard orientation with the point (0, 0, j^) at the north pole. It is the fixed frame as the axes are always in this orientation before any rotation. Rotate the second ball. It is the mobile frame as the axes change after each rotation.

This permits 2 ways to rotate the ball that reflect the rotation. The first way to rotate the ball is by an extrinsic rotation where the rotations occur around the axes of the fixed frame. After each rotation of a point the axes are returned to the standard orientation. As a result, the point’s position in relation to the fixed axes changes resulting in a change to the point’s coordinates.

The second way is to rotate the ball by an intrinsic rotation where the rotations occur around the axes of the mobile frame. After each rotation the axes move which has an effect on any subsequent rotation. However, the location of a point is still based on the standard orientation.

Extrinsic and Intrinsic rotations are possible for both Euler and Tait-Bryan angles. The examples that follow will show the same rotation for Euler and Tait-Bryan using Wave Number rotation. They will also look at the equivalent rotation matrices.

Next: Simple Examples

Previous: Using the Wave Number Prop

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