Quaternion

This page explains the quaternion functions of the calculation library for the OMRON NX/NJ series.
For the basic theory of quaternions, see Attitude Control (Euler Angles and Quaternions) .

What Is a Quaternion?

A quaternion is a mathematical model for representing rotations in 3D space.
It is widely used in robot control, drone control, attitude control, 3D graphics and more. Euler angles can suffer from gimbal lock, but quaternions avoid that problem.

Examples of quaternion use

• Calculating orientation in real time from the gyro in a phone
• Rotations in games calculated by engines such as Unity and Unreal
• Calculating the attitude control of a SpaceX rocket in real time
• Robot control
• Autonomous driving
• AR (augmented reality)

Related pages

• Matrices
• Rotation conversion

Quaternion operations

Quaternion creation (Quaternion)

Quaternion

Input

w

Scalar part (real part)

x

Vector part x (imaginary part)

y

Vector part y (imaginary part)

z

Vector part z (imaginary part)

Output

out

Quaternion

Example

Input

w

1.0

x

2.0

y

3.0

z

4.0

Output

out

\[ \begin{bmatrix} q_w \\ q_x \\ q_y \\ q_z \\ \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \\ \end{bmatrix} \]

Formulas

\begin{align} q=q_w+q_xi+q_yj+q_zk=& \begin{bmatrix} q_w \\ q_x \\ q_y \\ q_z \\ \end{bmatrix} \\ i^2+j^2+k^2=&-1 \\ ij=&k \\ jk=&i \\ ki=&j \end{align}

Comparison (qtEQ)

qtEQ

Input

Q1

Quaternion 1

Q2

Quaternion 2

Example

Addition (qtADD)

qtADD

Input

Q1

Quaternion 1

Q2

Quaternion 2

Output

out

Quaternion

Example

Input

Q1

[1.0, 1.0, 1.0, 1.0]

Q2

[0.5, 2.0, 1.0, 0.0]

Output

out

[1.5, 3.0, 2.0, 1.0]

Formulas

\[ Q1+Q2 = \begin{bmatrix} a_{w} \\ a_{x} \\ a_{y} \\ a_{z} \\ \end{bmatrix} + \begin{bmatrix} b_{w} \\ b_{x} \\ b_{y} \\ b_{z} \\ \end{bmatrix} = out \]

Subtraction (qtSUB)

qtSUB

Input

Q1

Quaternion 1

Q2

Quaternion 2

Output

out

Quaternion

Example

Input

Q1

[1.0, 1.0, 1.0, 1.0]

Q2

[0.5, 2.0, 1.0, 0.0]

Output

out

[0.5, -1.0, 0.0, 1.0]

Formulas

\[ Q1-Q2 = \begin{bmatrix} a_{w} \\ a_{x} \\ a_{y} \\ a_{z} \\ \end{bmatrix} - \begin{bmatrix} b_{w} \\ b_{x} \\ b_{y} \\ b_{z} \\ \end{bmatrix} = out \]

Cross product (qtCross)

qtCross

Input

Q1

Quaternion 1

Q2

Quaternion 2

Output

out

Quaternion

Example

Input

Q1

[1.0, 1.0, 1.0, 1.0]

Q2

[0.5, 2.0, 1.0, 0.0]

Output

out

[-2.5, 1.5, 3.5, -0.5]

Formulas

\[ Q1\times Q2 = \begin{bmatrix} a_{w} \\ a_{x} \\ a_{y} \\ a_{z} \\ \end{bmatrix} \times \begin{bmatrix} b_{w} \\ b_{x} \\ b_{y} \\ b_{z} \\ \end{bmatrix} = out \]

Conjugate (qtCQ)

qtCQ

Input

qt

Quaternion

Output

out

Quaternion

Example

Input

qt

[0.5, 2.0, 1.0, 0.0]

Output

out

[0.5, -2.0, -1.0, -0.0]

Formulas

\[ \bar{qt} = \begin{bmatrix} a_{w} \\ a_{x} \\ a_{y} \\ a_{z} \\ \end{bmatrix} = out \]

Set real part (qtSetReal)

qtSetReal

Input

w

Scalar part (real part)

In-out

qt

Quaternion

Example

Input

w

0.5

In-out

qt

[0.5, 0.0, 0.0, 0.0]

Set imaginary part (qtSetImag)

qtSetImag

Input

vec

Vector part (imaginary part)

In-out

qt

Quaternion

Example

Input

vec

[1.0, 2.0, 3.0]

In-out

qt

[0.0, 1.0, 2.0, 3.0]

FAQ

What is a quaternion?

A mathematical model that expresses rotation with four components.

How does it differ from Euler angles?

No gimbal lock and smooth rotation interpolation are possible.

Are they used in robot control?

Widely used in industrial robots, drones and more.