Random rotations

Sometimes it is necessary to generate random rotations. For example, if you are testing a stochastic system numerically, you need to perform a Monte Carlo simulation sampling the initial conditions. ReferenceFrameRotations.jl defines a rand function for all rotation representations. For all representations except EulerAngles, it samples a random rotation uniformly in SO(3). rand(EulerAngles) instead samples each angular coordinate uniformly in [0, 2π) and selects an axis sequence uniformly; this coordinate-wise distribution is not uniform over SO(3).

julia> rand(Quaternion)Quaternion{Float64}:
  - 0.171581 - 0.720293⋅i + 0.602687⋅j + 0.297498⋅k
julia> rand(DCM)DCM{Float64}:
  0.329635  -0.365157   0.870632
 -0.864032   0.254992   0.434083
 -0.380513  -0.895343  -0.231453
julia> rand(EulerAngles)EulerAngles{Float64}:
  R(X) :  4.18216 rad  ( 239.62°)
  R(Z) :  5.46777 rad  ( 313.28°)
  R(X) :  1.45175 rad  ( 83.179°)
julia> rand(EulerAngleAxis)EulerAngleAxis{Float64}:
  Euler angle : 2.78874 rad  (159.783°)
  Euler axis  : [-0.087938, -0.724529, -0.683611]
julia> rand(CRP)CRP{Float64}:
  X : + 1.49688
  Y : + 1.51475
  Z : - 0.976916
julia> rand(MRP)MRP{Float64}:
  X : - 0.671787
  Y : + 1.15361
  Z : + 0.581151