Classical Rodrigues Parameters

The Classical Rodrigues Parameters (CRP) are defined by the following immutable structure:

struct CRP{T}
    q1::T
    q2::T
    q3::T
end

The CRP is a 3-element parameterization of a rotation. In this package, it can be seen as the stereographic parameter obtained from a quaternion by dividing the vector part by the real part. Hence, it is singular for rotations of $180^\circ$.

If a rotation is represented by the Euler angle $\phi$ and the unitary Euler vector $\mathbf{e}$, then the corresponding CRP $\mathbf{c}$ is defined by:

\[\mathbf{c} = \mathbf{e} \tan\left(\frac{\phi}{2}\right)\ .\]

Therefore, if $\phi = \pi$ rad ($180^\circ$), then $\tan(\phi/2)$ is singular and the CRP is not defined.

Initialization

The constructors for this structure are:

CRP(q1::T1, q2::T2, q3::T3) where {T1,T2,T3}
CRP(v::AbstractVector)

in which a CRP with components q1, q2, and q3 will be created. Notice that the type of the returned structure will be selected according to the input types.

julia> CRP(0.1, 0.2, 0.3)CRP{Float64}:
  X : + 0.1
  Y : + 0.2
  Z : + 0.3
julia> CRP([0.1, 0.2, 0.3])CRP{Float64}:
  X : + 0.1
  Y : + 0.2
  Z : + 0.3
julia> CRP(1, 2, 3.0f0)CRP{Float32}:
  X : + 1.0
  Y : + 2.0
  Z : + 3.0

It is also possible to create the identity rotation using I, zero, and one:

julia> CRP(I)CRP{Bool}:
  X : + false
  Y : + false
  Z : + false
julia> zero(CRP)CRP{Float64}:
  X : + 0.0
  Y : + 0.0
  Z : + 0.0
julia> one(CRP)CRP{Float64}:
  X : + 0.0
  Y : + 0.0
  Z : + 0.0

c.q1
c.q2
c.q3

c[1]
c[2]
c[3]

Operations

Sum, subtraction, and scalar multiplication

julia> c1 = CRP(0.1, 0.2, 0.3)CRP{Float64}:
  X : + 0.1
  Y : + 0.2
  Z : + 0.3
julia> c2 = CRP(0.2, -0.1, 0.1)CRP{Float64}:
  X : + 0.2
  Y : - 0.1
  Z : + 0.1
julia> c1 + c2CRP{Float64}:
  X : + 0.3
  Y : + 0.1
  Z : + 0.4
julia> c1 - c2CRP{Float64}:
  X : - 0.1
  Y : + 0.3
  Z : + 0.2
julia> 2c1CRP{Float64}:
  X : + 0.2
  Y : + 0.4
  Z : + 0.6
julia> c1 / 2CRP{Float64}:
  X : + 0.05
  Y : + 0.1
  Z : + 0.15

Multiplication (rotation composition)

The multiplication of two CRPs is defined here as the composition of the rotations. Let $\mathbf{c}_1$ and $\mathbf{c}_2$ be two CRPs (instances of the structure CRP). Thus, the operation:

\[\mathbf{c}_{2,1} = \mathbf{c}_2 \cdot \mathbf{c}_1\]

will return a new CRP $\mathbf{c}_{2,1}$ that represents the composed rotation of $\mathbf{c}_1$ followed by $\mathbf{c}_2$.

julia> c1 = angle_to_crp(0.1, 0.2, 0.3, :ZYX)CRP{Float64}:
  X : + 0.146004
  Y : + 0.107816
  Z : + 0.0348512
julia> c2 = angle_to_crp(-0.2, 0.1, -0.1, :XYZ)CRP{Float64}:
  X : - 0.102865
  Y : + 0.0450321
  Z : - 0.0550765
julia> c2 * c1CRP{Float64}:
  X : + 0.0352059
  Y : + 0.155426
  Z : - 0.00252945

Inversion

The inverse rotation is obtained using inv:

julia> c = angle_to_crp(0.2, -0.1, 0.3, :ZYX)CRP{Float64}:
  X : + 0.156275
  Y : - 0.0349041
  Z : + 0.10798
julia> inv(c)CRP{Float64}:
  X : - 0.156275
  Y : + 0.0349041
  Z : - 0.10798

Functions

Useful helper functions include:

julia> c = CRP(0.1, 0.2, 0.3)CRP{Float64}:
  X : + 0.1
  Y : + 0.2
  Z : + 0.3
julia> norm(c)0.37416573867739417
julia> vect(c)3-element StaticArraysCore.SVector{3, Float64} with indices SOneTo(3):
 0.1
 0.2
 0.3
julia> copy(c)CRP{Float64}:
  X : + 0.1
  Y : + 0.2
  Z : + 0.3
julia> shadow_rotation(c)CRP{Float64}:
  X : + 0.1
  Y : + 0.2
  Z : + 0.3

The shadow rotation of a CRP is the CRP itself.

Kinematics

The time-derivative of a CRP can be computed using:

function dcrp(c::CRP, wba_b::AbstractVector)

julia> c = CRP(0.0, 0.0, 0.0)CRP{Float64}:
  X : + 0.0
  Y : + 0.0
  Z : + 0.0
julia> wba_b = [0.01, 0.02, -0.03]3-element Vector{Float64}:
  0.01
  0.02
 -0.03
julia> dcrp(c, wba_b)CRP{Float64}:
  X : + 0.005
  Y : + 0.01
  Z : - 0.015