Euler angle is a description method of expressing three-dimensional rotation. The calculation of Euler angle requires the aid of a rotation matrix. For the knowledge of rotation matrix, please refer to the first two articles:
Derivation and mnemonic of 3D rotation matrix
Derivation and Mnemonic of 3D Rotation Matrix-Supplement
1 static definition
For a reference system in three-dimensional space, the orientation of any coordinate system can be represented by three Euler angles.
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The reference system, also known as the laboratory reference system, is stationary, and can be simply understood as a geodetic coordinate system, also known as an inertial coordinate system.
- The coordinate system is fixed to the rigid body and rotates with the rotation of the rigid body. For example, the coordinate system of the aircraft itself is also called the carrier coordinate system.
The figure above is a schematic diagram of Euler angles rotated in ZYZ order:
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Let the blue xyz-axis be the reference axis of the inertial system, that is, the 3 axes of the geodetic coordinate system.
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Let the red XYZ axis be the reference axis of the carrier system, that is, the 3 axes of the aircraft coordinate system.
- The intersection of the xy-plane and the XY-plane is called the intersection line, which is represented by the English letter N.
Angle symbol in the picture:
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α is the angle between the x-axis and the line of intersection. The carrier coordinate system is first rotated by α around the Z axis (range 0~2Pi radians).
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β is the angle between the z-axis and the Z-axis, and the carrier coordinate system is rotated by β angle around the current Y axis (range 0~Pi radians).
- γ is the angle between the intersection line and the X-axis, and the carrier coordinate system is rotated by the γ angle around the current Z axis (range 0~2Pi radians).
The sign of the angle here is based on the right-hand rule. For example, the right thumb points to the z-axis, and the rotation direction of the four-finger bend is the positive α direction.
The rotation animation is:
In fact, there is no clear stipulation regarding the order and marking of the included angle, and the designation of the two axes of the included angle. Therefore, when Euler angles are used, it is necessary to clearly indicate the order of the included angles and specify their reference axis. In the legal Euler angle group, the only restriction is that any two consecutive rotations must rotate around different rotation axes. Therefore, there are a total of 12 representations.
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6 kinds of rotation around three axes (Tait-Bryan Angle): XYZ, XZY, YXZ, YZX, ZXY, ZYX
- 6 types of rotation around two axes (Proper Euler Angle): XYX, YXY, XZX, ZXZ, YZY, ZYZ
2 dynamic definition
We can also give two different dynamic definitions of Euler angles. One is the combination of three rotations around the coordinate axis fixed to the carrier; the other is the combination of three rotations around the reference axis of the geodetic coordinate system.
With the dynamic definition, we can better understand the physical meaning and application of Euler angles.
Note that in the following description, the uppercase XYZ coordinate axis is the rotating carrier coordinate axis; the lowercase xyz coordinate axis is the stationary earth reference axis.
Now the two dynamic definitions of Euler angles are described in a way that the rotation sequence is Z, Y, X.
- Definition A: Rotate around the XYZ coordinate axis (carrier coordinate axis):
Initially, the coordinate axes of the two coordinate systems xyz and XYZ overlapped.
Initially, rotate the Z-axis by an angle of α.
Then, rotate the Y-axis by β angle.
Finally, rotate the X-axis by an angle of γ.
- Let the coordinates of any point P1 in the xyz and XYZ coordinate systems be r1 and R1, respectively. Define Z(α) as a rotation around the Z-axis by an angle of α, Y(β) as a rotation around the Y-axis by an angle of β, and X(γ) as a rotation around the X-axis by an angle of γ. Then the definition A can be expressed as follows:
Note that there is another concept of matrix left multiplication and right multiplication. Rotating around the carrier coordinate system is the matrix multiplication to the left in turn, that is, X <- Y <- Z.
- Definition B: Rotate around the xyz coordinate axis (the earth coordinate axis):
Initially, the coordinate axes of the two coordinate systems xyz and XYZ overlapped.
Initially, rotate by an angle of α around the z-axis.
Then, rotate the angle β around the y-axis.
Finally, rotate the x-axis by an angle of γ.
- Let the coordinates of any point P2 in the xyz and XYZ coordinate systems be r2 and R2, respectively. Define z(α) as an angle of rotation around the z-axis, y(β) as an angle of rotation around the y-axis, and x(γ) as an angle of rotation around the x-axis. Then definition B can be expressed as follows:
Note that the rotation around the geodetic coordinate system is the matrix multiplication to the right, that is, z -> y -> x.
Definition A is equal to the static definition . This can be checked directly by geometric drawing methods.
The equality of definition A and definition B can be proved by the rotation matrix: