Euler Angles

The Euler Angles are defined by the following immutable structure:

struct EulerAngles{T}
    a1::T
    a2::T
    a3::T
    rot_seq::Symbol
end

in which a1, a2, and a3 define the angles and the rot_seq is a symbol that defines the axes. The valid values for rot_seq are:

• :XYX, :XYZ, :XZX, :XZY, :YXY, :YXZ, :YZX, :YZY, :ZXY, :ZXZ, :ZYX, and ZYZ.

The constructor for this structure is:

function EulerAngles(a1::T1, a2::T2, a3::T3, rot_seq::Symbol = :ZYX) where {T1,T2,T3}

in which a EulerAngles with angles a1, a2, and a3 [rad] and rotation sequence rot_seq will be created. Notice that the type of the returned structure will be selected according to the input types T1, T2, and T3. If rot_seq is omitted, then it defaults to :ZYX.

julia> EulerAngles(1, 1, 1)
EulerAngles{Int64}:
  R(Z) :  1 rad  ( 57.2958°)
  R(Y) :  1 rad  ( 57.2958°)
  R(X) :  1 rad  ( 57.2958°)

julia> EulerAngles(1, 1, 1.0f0, :XYZ)
EulerAngles{Float32}:
  R(X) :  1.0 rad  ( 57.2958°)
  R(Y) :  1.0 rad  ( 57.2958°)
  R(Z) :  1.0 rad  ( 57.2958°)

julia> EulerAngles(1., 1, 1, :XYX)
EulerAngles{Float64}:
  R(X) :  1.0 rad  ( 57.2958°)
  R(Y) :  1.0 rad  ( 57.2958°)
  R(X) :  1.0 rad  ( 57.2958°)

Operations

Multiplication

The multiplication of two Euler angles is defined here as the composition of the rotations. Let $\Theta_1$ and $\Theta_2$ be two sequences of Euler angles (instances of the structure EulerAngles). Thus, the operation:

\[\Theta_{2,1} = \Theta_2 \cdot \Theta_1\]

will return a new set of Euler angles $\Theta_{2,1}$ that represents the composed rotation of $\Theta_1$ followed by $\Theta_2$. Notice that $\Theta_{2,1}$ will be represented using the same rotation sequence as $\Theta_2$.

julia> a1 = EulerAngles(1, 0, 0, :ZYX);

julia> a2 = EulerAngles(0, -1, 0, :YZY);

julia> a2 * a1
EulerAngles{Float64}:
  R(Y) :  0.0 rad  ( 0.0°)
  R(Z) :  0.0 rad  ( 0.0°)
  R(Y) :  0.0 rad  ( 0.0°)

julia> a1 = EulerAngles(1, 1, 1, :YZY);

julia> a2 = EulerAngles(0, 0, -1, :YZY);

julia> a2 * a1
EulerAngles{Float64}:
  R(Y) :  1.0         rad  ( 57.2958°)
  R(Z) :  1.0         rad  ( 57.2958°)
  R(Y) :  1.31938e-16 rad  ( 7.55951e-15°)

julia> a1 = EulerAngles(1.3, 2.2, 1.4, :XYZ);

julia> a2 = EulerAngles(-1.4, -2.2, -1.3, :ZYX);

julia> a2 * a1
EulerAngles{Float64}:
  R(Z) : -8.32667e-17 rad  (-4.77083e-15°)
  R(Y) :  3.33067e-16 rad  ( 1.90833e-14°)
  R(X) : -1.11022e-16 rad  (-6.36111e-15°)

Inversion

The inv function applied to Euler angles will return the inverse rotation. If the Euler angles $\Theta$ represent a rotation through the axes $a_1$, $a_2$, and $a_3$ by angles $\alpha_1$, $\alpha_2$, and $\alpha_3$, then $\Theta^{-1}$ is a rotation through the axes $a_3$, $a_2$, and $a_1$ by angles $-\alpha_3$, $-\alpha_2$, and $-\alpha_1$.

julia> a = EulerAngles(1, 2, 3, :ZYX);

julia> inv(a)
EulerAngles{Int64}:
  R(X) : -3 rad  (-171.887°)
  R(Y) : -2 rad  (-114.592°)
  R(Z) : -1 rad  (-57.2958°)

julia> a = EulerAngles(1.2, 3.3, 4.6, :XYX);

julia> a * inv(a)
EulerAngles{Float64}:
  R(X) : -1.92593e-34 rad  (-1.10348e-32°)
  R(Y) :  0.0         rad  ( 0.0°)
  R(X) :  0.0         rad  ( 0.0°)