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}:
  - 0.392255 - 0.622145⋅i - 0.602448⋅j - 0.310045⋅k
julia> rand(DCM)DCM{Float64}:
 -0.192215   0.365768  -0.910641
  0.793082  -0.488638  -0.363667
 -0.577991  -0.792116  -0.19616
julia> rand(EulerAngles)EulerAngles{Float64}:
  R(X) :  1.57822 rad  ( 90.4256°)
  R(Z) :  1.94814 rad  ( 111.62°)
  R(X) :  3.23421 rad  ( 185.306°)
julia> rand(EulerAngleAxis)EulerAngleAxis{Float64}:
  Euler angle : 1.91966 rad  (109.988°)
  Euler axis  : [-0.160956, 0.0204801, 0.986749]
julia> rand(CRP)CRP{Float64}:
  X : + 0.0104117
  Y : - 2.39207
  Z : + 1.55929
julia> rand(MRP)MRP{Float64}:
  X : - 0.798497
  Y : + 0.180045
  Z : + 0.118821