In the previous post, we have shown how angular velocities and rotation matrices are linked through the exponential map, or to be specific, the Rodrigues’ rotation formula. In this post I would like to give as an useful application of Rodrigues’ formula, to compute the rotation matrix between two vectors.

Let $a,b\in\mathbb{R}^3$ ($a\neq b$) be two unit vectors expressed in an arbitary coordinate frame. Our goal is to compute the rotation matrix from $a$ to $b$.

The following Python function converts a 3 x nf matrix, axisAngles, whose columns are axis-angles, to a 4 x nf matrix of quaternions under the (x,y,z,w) convention. Note that the quaternions should always have norm 1.