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):   1.0000 rad (  57.2958 deg)
  R(Y):   1.0000 rad (  57.2958 deg)
  R(X):   1.0000 rad (  57.2958 deg)

julia> EulerAngles(1,1,1.0f0,:XYZ)
EulerAngles{Float32}:
  R(X):   1.0000 rad (  57.2958 deg)
  R(Y):   1.0000 rad (  57.2958 deg)
  R(Z):   1.0000 rad (  57.2958 deg)

julia> EulerAngles(1.,1,1,:XYX)
EulerAngles{Float64}:
  R(X):   1.0000 rad (  57.2958 deg)
  R(Y):   1.0000 rad (  57.2958 deg)
  R(X):   1.0000 rad (  57.2958 deg)

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:

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):   0.0000 rad (   0.0000 deg)
  R(Z):   0.0000 rad (   0.0000 deg)
  R(Y):   0.0000 rad (   0.0000 deg)

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

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

julia> a2*a1
EulerAngles{Float64}:
  R(Y):   1.0000 rad (  57.2958 deg)
  R(Z):   1.0000 rad (  57.2958 deg)
  R(Y):   0.0000 rad (   0.0000 deg)

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):  -0.0000 rad (  -0.0000 deg)
  R(Y):   0.0000 rad (   0.0000 deg)
  R(X):  -0.0000 rad (  -0.0000 deg)

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):  -3.0000 rad (-171.8873 deg)
  R(Y):  -2.0000 rad (-114.5916 deg)
  R(Z):  -1.0000 rad ( -57.2958 deg)

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

julia> a*inv(a)
EulerAngles{Float64}:
  R(X):  -0.0000 rad (  -0.0000 deg)
  R(Y):   0.0000 rad (   0.0000 deg)
  R(X):   0.0000 rad (   0.0000 deg)