Rotations

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Rotation matrices (aka Direction Cosine Matrices)

Notation

$R_G^F$ is “frame F expressed in frame G”. The columns of this matrix are the basis vectors ($x,y,z$ axes) of frame F expressed in frame G, and the rows are the G basis vectors re-expressed in F:

$R_G^F = [F_{x_G}, F_{y_G}, F_{z_G}] = [G_{x_F}; G_{y_F}; G_{z_f}]$, where $F_{x_G}=F_x\cdot G_x$

Multiplying by a rotation matrix = projecting onto another set of orthogonal basis vectors.

• See Drake documentation for good review
• columns of rotation matrix are orthonormal vectors (orthogonal vectors (dot product zero between each of them), norm of 1)
  • set of all orthogonal matrices of size n with determinant +1 forms the special orthogonal group $SO(n)$
  • multiplication of a vector by a rotation matrix is finding the projection of the vector onto each of these basis vectors
  • since matrices are orthogonal, $R^{-1} = R^T$
• right-multiply for operations in local frame, left-multiply for operations in global frame
  • Doing Transform 2 in local frame after Transform 1 (in global frame) is the same as doing Transform 2 in global frame first, then Transform 1 in global frame
  • slides reviewing left/right multiplication of transformation matrices

Quaternions

• composed of one real part and three imaginary parts

• Quaternions are related to the axis-angle representation ${v_x,v_y,v_z,\theta}$ with: ${w,x,y,z} = {cos(0.5\theta),v_x sin(0.5\theta),v_y sin(0.5\theta),v_z sin(0.5\theta)}$

• Mirroring quaternions

Tools

• Eigen’s geometry module
  • I wrote a rotation frame visualizer for this with GNU Plot. Largely useless except for sanity checks