Kinematics

Currently, only the kinematics of Direction Cosine Matrices and Quaternions are implemented.

Direction Cosine Matrices

Let A and B be two reference frames in which the angular velocity of B with respect to A, and represented in B, is given by

\[\boldsymbol{\omega}_{ba,b} = \left[\begin{array}{c} \omega_{ba,b,x} \\ \omega_{ba,b,y} \\ \omega_{ba,b,z} \end{array}\right]~.\]

If $\mathbf{D}_{b}^{a}$ is the DCM that rotates the reference frame A into alignment with the reference frame B, then its time-derivative is

\[\dot{\mathbf{D}}_{b}^{a} = -\left[\begin{array}{ccc} 0 & -\omega_{ba,b,z} & +\omega_{ba,b,y} \\ +\omega_{ba,b,z} & 0 & -\omega_{ba,b,x} \\ -\omega_{ba,b,y} & +\omega_{ba,b,x} & 0 \end{array}\right] \cdot \mathbf{D}_{b}^{a}~.\]

In this package, the time-derivative of this DCM can be computed using the function:

function ddcm(Dba, wba_b)

julia> wba_b = [0.01, 0, 0];

julia> Dba = angle_to_dcm(0.5, 0, 0, :XYZ)
DCM{Float64}:
  1.0   0.0       0.0
 -0.0   0.877583  0.479426
  0.0  -0.479426  0.877583

julia> ddcm(Dba, wba_b)
3×3 StaticArrays.SMatrix{3, 3, Float64, 9} with indices SOneTo(3)×SOneTo(3):
 -0.0   0.0          0.0
  0.0  -0.00479426   0.00877583
  0.0  -0.00877583  -0.00479426

Quaternions

Let A and B be two reference frames in which the angular velocity of B with respect to A, and represented in B, is given by

\[\boldsymbol{\omega}_{ba,b} = \left[\begin{array}{c} \omega_{ba,b,x} \\ \omega_{ba,b,y} \\ \omega_{ba,b,z} \end{array}\right]~.\]

If $\mathbf{q}_{ba}$ is the quaternion that rotates the reference frame A into alignment with the reference frame B, then its time-derivative is

\[\dot{\mathbf{q}}_{ba} = \frac{1}{2} \cdot \left[\begin{array}{cccc} 0 & -\omega_{ba,b,x} & -\omega_{ba,b,y} & -\omega_{ba,b,z} \\ +\omega_{ba,b,x} & 0 & +\omega_{ba,b,z} & -\omega_{ba,b,y} \\ +\omega_{ba,b,y} & -\omega_{ba,b,z} & 0 & +\omega_{ba,b,x} \\ +\omega_{ba,b,z} & +\omega_{ba,b,y} & -\omega_{ba,b,x} & 0 \end{array}\right] \cdot \mathbf{q}_{ba}~.\]

In this package, the time-derivative of this quaternion can be computed using the function:

function dquat(qba, wba_b)

julia> wba_b = [0.01, 0, 0];

julia> qba = angle_to_quat(0.5, 0, 0, :XYZ);

julia> dquat(qba, wba_b)
Quaternion{Float64}:
  - 0.00123702 + 0.00484456⋅i + 0.0⋅j + 0.0⋅k