RollPitchYawMatrix

RollPitchYawMatrix[{α,β,γ}]

gives the 3D rotation matrix formed by rotating by α around the initial axis, then by β around the initial axis, and then by γ around the initial axis.

RollPitchYawMatrix[{α,β,γ},{a,b,c}]

gives the 3D rotation matrix formed by rotating by α around the fixed a axis, then by β around the fixed b axis, and then by γ around the fixed c axis.

Details

  • RollPitchYawMatrix is also known as bank-elevation-heading matrix or Cardan matrix. The angles {α,β,γ} are often referred to as Cardan angles, nautical angles, bank-elevation-heading, or roll-pitch-yaw.
  • RollPitchYawMatrix is typically used to specify a rotation as a sequence of basic rotations around coordinate axes where each rotation is referring to the initial or extrinsic coordinate frame.
  • RollPitchYawMatrix[{α,β,γ}] is equivalent to RollPitchYawMatrix[{α,β,γ},{3,2,1}].
  • RollPitchYawMatrix[{α,β,γ},{a,b,c}] is equivalent to where Rα,a=RotationMatrix[α,UnitVector[3,a]] etc.
  • The default z-y-x rotation RollPitchYawMatrix[{α,β,γ},{3,2,1}]:
  • The rotation axes a, b, and c can be any integer 1, 2, or 3, but there are only twelve combinations that are general enough to be able to specify any 3D rotation.
  • Rotations with the first and last axis repeated:
  • {3,2,3}z-y-z rotation
    {3,1,3}z-x-z rotation
    {2,3,2}y-z-y rotation
    {2,1,2}y-x-y rotation
    {1,3,1}x-z-x rotation
    {1,2,1}x-y-x rotation
  • Rotations with all three axes different:
  • {1,2,3}x-y-z rotation
    {1,3,2}x-z-y rotation
    {2,1,3}y-x-z rotation
    {2,3,1}y-z-x rotation
    {3,1,2}z-x-y rotation
    {3,2,1}z-y-x rotation (default)
  • Rotations with subsequent axes repeated still produce a rotation matrix, but cannot be inverted uniquely using RollPitchYawAngles.

Examples

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Basic Examples  (2)

The standard roll-pitch-yaw matrix:

In[1]:=
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Out[1]//MatrixForm=

Rotate an axes-aligned unit cube:

In[1]:=
Click for copyable input
In[2]:=
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Out[2]=

Scope  (5)

Applications  (6)

Properties & Relations  (11)

Possible Issues  (1)

Neat Examples  (2)

See Also

RollPitchYawAngles  EulerMatrix  EulerAngles  RotationMatrix  RotationTransform

Introduced in 2015
(10.2)