Rotations

This example was generated on 2024-11-18T22:29:37.060.

Before diving into the creation of the axes graph, it is worth highlighting that transformations that express the relative orientation or its time-derivatives between two generic set of axes are represented by a Rotation object, which stores a Direction Cosine Matrix (DCM) and its derivatives. This package leverages the already available ReferenceFrameRotations.jl to define the DCM objects.

A time-fixed rotation between two axes and its derivative can then be expressed as follows:

using StaticArrays
using FrameTransformations
using ReferenceFrameRotations

dcm = angle_to_dcm(π / 3, :Z)
δdcm = DCM(0I)

R = Rotation(dcm, δdcm)

2-element Rotation{2, Float64}:
 [0.5000000000000001 0.8660254037844386 0.0; -0.8660254037844386 0.5000000000000001 0.0; 0.0 0.0 1.0]
 [0.0 0.0 0.0; 0.0 0.0 0.0; 0.0 0.0 0.0]

R[1]

DCM{Float64}:
  0.5       0.866025  0.0
 -0.866025  0.5       0.0
  0.0       0.0       1.0

R[2]

DCM{Float64}:
 0.0  0.0  0.0
 0.0  0.0  0.0
 0.0  0.0  0.0

A rotation object is returned by all the rotation functions that are applied to the FrameSystem. It provide overloads to the basic algebraic operations so that multiplication and inversions can be efficiently computed leveraging the properties of rotation matrixes.

For example, to rotate a generic vector v, we can simply do:

v = [1.0, -6.0, 3.0, 0.0, 5.0, 0]
R * v

6-element StaticArraysCore.SVector{6, Float64} with indices SOneTo(6):
 -4.696152422706632
 -3.8660254037844393
  3.0
  4.330127018922193
  2.5000000000000004
  0.0

For a static vector vector sv:

sv = SA[1.0, -6.0, 3.0, 0.0, 5.0, 0]
R * sv

6-element StaticArraysCore.SVector{6, Float64} with indices SOneTo(6):
 -4.696152422706631
 -3.8660254037844393
  3.0
  4.330127018922193
  2.5000000000000004
  0.0

And for a Translation

t = Translation(1.0, -6.0, 3.0, 0.0, 5.0, 0)
R * t

Translation{2, Float64}:
 [-4.696152422706631, -3.8660254037844393, 3.0]
 [4.330127018922193, 2.5000000000000004, 0.0]

The inverse can instead be taken as:

inv(R)

2-element Rotation{2, Float64}:
 [0.5000000000000001 -0.8660254037844386 0.0; 0.8660254037844386 0.5000000000000001 0.0; 0.0 0.0 1.0]
 [0.0 0.0 0.0; 0.0 0.0 0.0; 0.0 0.0 0.0]

See the Rotation API for more information on this object.

This page was generated using Literate.jl.