Quaternions to Rotation Angles

Determine rotation vector from quaternion

• Library:

• Aerospace Blockset / Utilities / Axes Transformations

• 

Description

The Quaternions to Rotation Angles block converts the four-element quaternion vector (q₀, q₁, q₂, q₃), into the rotation described by the three rotation angles (R1, R2, R3). The block generates the conversion by comparing elements in the direction cosine matrix (DCM) as a function of the rotation angles. The rotation used in this block is a passive transformation between two coordinate systems. The elements in the DCM are functions of a unit quaternion vector. Aerospace Blockset™ uses quaternions that are defined using the scalar-first convention. This block normalizes all quaternion inputs. For more information on the direction cosine matrix, see Algorithms.

Limitations

• For the ZYX, ZXY, YXZ, YZX, XYZ, and XZY rotations, the block generates an R2 angle that lies between ±pi/2 radians, and R1 and R3 angles that lie between ±pi radians.

• For the 'ZYZ', 'ZXZ', 'YXY', 'YZY', 'XYX', and 'XZX' rotations, the block generates an R2 angle that lies between 0 and pi radians, and R1 and R3 angles that lie between ±pi radians. However, in the latter case, when R2 is 0, R3 is set to 0 radians.

Ports

Input

q — Quaternion

4-by-1 vector

Quaternion, specified as a 4-by-1 vector.

Data Types: double

Output

[R1,R2,R3] — Rotation angles

3-by-1 vector

Rotation angles, returned 3-by-1 vector, in radians.

Data Types: double

Parameters

Rotation order — Rotation order

ZYX (default) | ZYZ | ZXY | ZXZ | YXZ | YXY | YZX | YZY | XYZ | XYX | XZY | XZX

Output rotation order for the three rotation angles.

Programmatic Use

Block Parameter: rotationOrder

Type: character vector

Values: ZYX | ZYZ |ZXY | ZXZ | YXZ | YXY | YZX | YZY | XYZ | XYX | XZY | XZX

Default: 'ZYX'

Algorithms

The elements in the DCM are functions of a unit quaternion vector. For example, for the rotation order z-y-x, the DCM is defined as:

$$DCM=\left[\begin{array}{lll}\cos \theta \cos \psi \hfill & \cos \theta \sin \psi \hfill & -\sin \theta \hfill \\ (\sin \varphi \sin \theta \cos \psi -\cos \varphi \sin \psi )\hfill & (\sin \varphi \sin \theta \sin \psi +\cos \varphi \cos \psi )\hfill & \sin \varphi \cos \theta \hfill \\ (\cos \varphi \sin \theta \cos \psi +\sin \varphi \sin \psi )\hfill & (\cos \varphi \sin \theta \sin \psi -\sin \varphi \cos \psi )\hfill & \cos \varphi \cos \theta \hfill \end{array}\right]$$

The DCM defined by a unit quaternion vector is:

$$DCM=\left[\begin{array}{lll}(q_0^2+q_1^2-q_2^2-q_3^2)\hfill & 2(q_1{q}_2+q_0{q}_3)\hfill & 2(q_1{q}_3-q_0{q}_2)\hfill \\ 2(q_1{q}_2-q_0{q}_3)\hfill & (q_0^2-q_1^2+q_2^2-q_3^2)\hfill & 2(q_2{q}_3+q_0{q}_1)\hfill \\ 2(q_1{q}_3+q_0{q}_2)\hfill & 2(q_2{q}_3-q_0{q}_1)\hfill & (q_0^2-q_1^2-q_2^2+q_3^2)\hfill \end{array}\right]$$

From the preceding equation, you can derive the following relationships between DCM elements and individual rotation angles for a ZYX rotation order:

$$\begin{array}{c}\varphi =\text{atan}(DCM(2,3),DCM(3,3))\\ =\text{atan}(2(q_2{q}_3+q_0{q}_1),(q_0^2-q_1^2-q_2^2+q_3^2))\\ \theta =\text{asin}(-DCM(1,3))\\ =\text{asin}(-2(q_1{q}_3-q_0{q}_2))\\ \psi =\text{atan}(DCM(1,2),DCM(1,1))\\ =\text{atan}(2(q_1{q}_2+q_0{q}_3),(q_0^2+q_1^2-q_2^2-q_3^2))\end{array}$$

where Ψ is R1, Θ is R2, and Φ is R3.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2007b