Direction Cosine Matrices

Direction cosine matrices, or DCMs, are $3 \times 3$ matrices that represent a coordinate transformation between two orthonormal reference frames. Let those frames be right-handed, then it can be shown that this transformation is always a rotation. Thus, a DCM that rotates the reference frame $a$ into alignment with the reference frame $b$ is:

\[\mathbf{D}_{ba} = \left[\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix}\right]\]

In ReferenceFrameRotations.jl, a DCM is a $3 \times 3$ static matrix, i.e. it is immutable.

Initialization

Usually, a DCM is initialized by converting a more "visual" rotation representation, such as the Euler angles (see Conversions). However, it can be initialized by the following methods:

• Identity DCM.

julia> DCM(I)  # Create a Boolean DCM, this can be used to save space.
DCM{Bool}:
 1  0  0
 0  1  0
 0  0  1

julia> DCM(Int64(1)I)  # Create an Integer DCM.
DCM{Int64}:
 1  0  0
 0  1  0
 0  0  1

julia> DCM(1.f0I) # Create a Float32 DCM.
DCM{Float32}:
 1.0  0.0  0.0
 0.0  1.0  0.0
 0.0  0.0  1.0

julia> DCM(1.0I)  # Create a Float64 DCM.
DCM{Float64}:
 1.0  0.0  0.0
 0.0  1.0  0.0
 0.0  0.0  1.0

• User-defined DCM.

julia> DCM([-1 0 0; 0 -1 0; 0 0 1])
DCM{Int64}:
 -1   0  0
  0  -1  0
  0   0  1

julia> DCM([-1.f0 0.f0 0.f0; 0.f0 -1.f0 0.f0; 0.f0 0.f0 1.f0])
DCM{Float32}:
 -1.0   0.0  0.0
  0.0  -1.0  0.0
  0.0   0.0  1.0

julia> DCM([-1.0 0.0 0.0; 0.0 -1.0 0.0; 0.0 0.0 1.0])
DCM{Float64}:
 -1.0   0.0  0.0
  0.0  -1.0  0.0
  0.0   0.0  1.0

Operations

Since a DCM is a static matrix (<: StaticMatrix), then all the operations available for general matrices in Julia are also available for DCMs.

Orthonomalization

A DCM can be orthonormalized using the Gram-Schmidt algorithm by the function:

function orthonormalize(dcm::DCM)

julia> D = DCM([2 0 0; 0 2 0; 0 0 2]);

julia> orthonormalize(D)
DCM{Float64}:
 1.0  0.0  0.0
 0.0  1.0  0.0
 0.0  0.0  1.0

julia> D = DCM(3.0f0I);

julia> orthonormalize(D)
DCM{Float32}:
 1.0  0.0  0.0
 0.0  1.0  0.0
 0.0  0.0  1.0

julia> D = DCM(1, 1, 2, 2, 3, 3, 4, 4, 5);

julia> Dn = orthonormalize(D)
DCM{Float64}:
 0.408248   0.123091   0.904534
 0.408248   0.86164   -0.301511
 0.816497  -0.492366  -0.301511

julia> Dn * Dn'
DCM{Float64}:
  1.0          -2.87528e-16  -8.47673e-16
 -2.87528e-16   1.0           3.4516e-16
 -9.03184e-16   3.31283e-16   1.0