Weird results when combined with Quaternions.jl

[Olli Wilkman]

I was testing out Measurements together with some orientation quaternions using Quaternions.jl. Starts out ok, but then something happens with the derivatives.

I have a function to create an orientation quaternion given an axis (a unit vector) and an angle:

using Quaternions, Measurements, LinearAlgebra

orientation(v, angle) = Quaternion(cos(0.5 * angle), sin(0.5 * angle) .* v)

Then I have a function to apply an orientation quaternion to perform a vector rotation (3-vectors are represented as quaternions with zero real part):

vec_quaternion(v) =  Quaternion(0.0, v)
rotate(q, v) = Quaternions.imag(q * vec_quaternion(v) * inv(q))

(Above, Quaternions.imag returns a 3-element Array{Float64,1} with the three imaginary components of the given quaternion.)

Using Measurements, I create an orientation that represents a 45° rotation around the Z axis, with a 1% error in the angle. Looks ok so far:

julia> q = orientation([0.0, 0.0, 1.0], π/4 * (1 ± 0.01))
Quaternion{Measurement{Float64}}(0.9239 ± 0.0015, 0.0 ± 0.0, 0.0 ± 0.0, 0.3827 ± 0.0036, false)

I apply this quaternion to rotate the X-axis unit vector. The printout also looks ok:

julia> u = rotate(q, [1.0, 0.0, 0.0])
3-element Array{Measurement{Float64},1}:
 0.7071 ± 0.0056
 0.7071 ± 0.0056
    0.0 ± 0.0   

But this resulting vector does not behave like expected:

julia> norm(u)
1.0 ± 1.1e-18

julia> u[1]^2 + u[2]^2
1.0 ± 2.2e-18

julia> u[1] + u[2]
1.414213562373095 ± 0.0

[Mosè Giordano]

I'm not familiar with Quaternions, so I don't know what you expect.

I tried to run your code with a plain number 1 instead of 1 \pm 0.01 and I got the following:

julia> q = orientation([0.0, 0.0, 1.0], π/4 * (1))
Quaternion{Float64}(0.9238795325112867, 0.0, 0.0, 0.3826834323650898, false)

julia> u = rotate(q, [1.0, 0.0, 0.0])
3-element Array{Float64,1}:
 0.7071067811865475
 0.7071067811865476
 0.0               

julia> norm(u)
1.0

julia> u[1]^2 + u[2]^2
1.0

julia> u[1] + u[2]
1.414213562373095

so I don't really know what's wrong with what you get when using a Measurement, the nominal values are just the same. Were you expecting different uncertainties?

[Olli Wilkman]

Wait, never mind. Now that you said that, I thought about it more, and looked at it from another angle, and those are reasonable. I was thrown off by the fact that if I do it "manually", the results are quite different, but that's of course due to the history and correlations. I blame a flu I'm currently suffering… Sorry about that.

v = [0.7071 ± 0.0056, 0.7071 ± 0.0056, 0.0 ± 0.0]
3-element Array{Measurement{Float64},1}:
 0.7071 ± 0.0056
 0.7071 ± 0.0056
    0.0 ± 0.0   

julia> norm(v)
1.0 ± 0.0056

julia> v[1]^2 + v[2]^2
1.0 ± 0.011

julia> v[1] + v[2]
1.4142 ± 0.0079

[Mosè Giordano]

No problem, I'm glad you're finding this package useful 🙂 It's always amazing to see how easy it's in Julia to plug different packages together!

[Olli Wilkman]

Yes, now I'm actually rather impressed by how well it works. For what it's worth, what I'm working on is a simple code that lets me integrate orientation and angular velocity using quaternions, similar to a leapfrog integrator for position and velocity. It's technically complete but I have not thoroughly tested it against analytical solutions or other codes, so I haven't really announced it anywhere, https://github.com/dronir/QuaternionIntegrator.jl