Attitude control
This page explains the attitude control library for the OMRON NX/NJ series.
It introduces 3D rotation calculation and coordinate transformation using Euler angles, quaternions and rotation matrices.
What Is Attitude Control?
Attitude control is the technology of controlling the orientation of an object.
It is used in robot arms, AGVs, drones, industrial machinery and more.
Euler angles, rotation matrices, quaternions and so on are used to express orientation in 3D space.
Calculating Attitude with Euler Angles
Advantages of Euler angles
- Easy to understand
- Angles are easy for people to handle
- Easy to implement on PLCs
Disadvantages of Euler angles
- Gimbal lock occurs
- Interpolation is difficult
Rotation example
Euler angles X axis = 50 degrees, Y axis = 50 degrees, Z axis = 50 degrees
Example of XYZ-type rotation in the body coordinate system
This video explains the calculation steps; the actual calculation computes the final position (red) directly from the initial position
Euler angles
Consider the nose direction of a paper airplane as the X axis, the wing direction as the Y axis and the vertical fin direction as the Z axis
To change the attitude, each of the X, Y and Z axes is rotated, and the X, Y and Z rotation angles together are called Euler angles
Types of Euler angles
There are 12 types of Euler angles depending on the rotation order, and changing the order changes the result
| X Y Z | X Y X |
| X Z Y | X Z X |
| Y X Z | Y X Y |
| Y Z X | Y Z Y |
| Z X Y | Z X Z |
| Z Y X | Z Y Z |
Body coordinate system and fixed coordinate system
The body coordinate system calculates in the object's own coordinates, so that the nose of the paper airplane is always the X axis direction, while the fixed coordinate system calculates around fixed coordinate axes
In this library, the body coordinate system can be calculated with EulerToMatrix and matRotate3D, and the fixed coordinate system with matRotateXYZ
Points to note
Conversion with Euler angles can cause gimbal lock
Gimbal lock is a phenomenon in which two rotation axes point in the same direction during coordinate transformation, reducing three axes to two so that the lost axis can no longer rotate
It is avoided by changing the rotation order
About the paper airplane
The paper airplane shape is rotated by calculating the rotation of 5 points
By the way
This video is a recording of graphs made in Python for explanation
Calculating Attitude with Quaternions
Advantages of quaternions
- No gimbal lock
- Easy rotation interpolation
- Fast to calculate
- Widely used in robot control
Quaternion
Expresses a rotation with four components made of a rotation axis (3D vector) and a rotation angle
Unlike Euler angles, gimbal lock does not occur
Also, quaternion calculations are simple, reducing computational cost and allowing fast calculation
Concept
To go from the current attitude to the target attitude, you only need to determine the rotation axis and rotation angle to rotate into the target attitude
This works for any target orientation
Considering the rotation of one point gives the following figure
Rotating each of the 5 points of the paper airplane around the rotation axis gives the following
How to find the rotation axis and rotation angle
Quaternion of the initial position
\[
Q1=
\begin{bmatrix}
q_w\\
q_x\\
q_y\\
q_z
\end{bmatrix}
=
\begin{bmatrix}
0\\
point.x\\
point.y\\
point.z
\end{bmatrix}
\]
Convert the Euler angles of the target position to a quaternion
\[
Q2=EulerToQuaternion(th, order)
\]
Rotation angle
\[
\theta=2\arccos(Q2.real)
\]
Rotation axis (unit vector)
\[
T=
\frac{Q2.img}{norm(Q2.img)}
\]
Quaternion representing the rotation
\[
\begin{align}
R.real&=\cos\frac{\theta}{2} \qquad (rotation angle)\\
R.imag&=T\times\sin\frac{\theta}{2} \qquad (rotation axis)
\end{align}
\]
Rotation calculation
\[
out=R\times Q1\times\bar{R}
\]
Relationship between the rotation axis and rotation angle
A quaternion expresses a rotation axis and rotation angle with four values.
FAQ
What is the difference between Euler angles and quaternions?
Euler angles express orientation with angles, while quaternions express it with a rotation axis and rotation amount.
What is gimbal lock?
A loss of a degree of freedom that occurs with Euler angles.
Which is used in industrial robots?
Quaternions and rotation matrices are often used for internal calculations.