Supplementary Notes: Inverse Kinematics Problems with Exact Hessian Matrices

These supplementary notes are provided in an educational spirit. Section 2 shows how the cross-product formula for computing columns of the Jacobian emerges from direct calculus. Section 5 extends to arbitrary Euler angles. Our work is not limited to homogeneous coordinates or Euler angles. However, these representations are well known and chosen merely to present to a wider audience. In fact our supplementary implementations use quaternions. We refrain from using exponential maps as they in our opinion are less accessible for a general audience, although offer a nice re-parameterization [1]. We outline how to compute the Jacobian in Section 3.1 and in Section 3.2 we work on the Hessian. We also address necessary changes to add a moving root frame in Section 3.3. Our notes finalize by presenting detailed pseudo code algorithms in Section 4. Please see our paper and supplementary code for further information.