Inverse kinematics (IK) is a central component of systems for motion capture, character animation, robotics motion planning and control. The field of computer graphics has developed fast stationary point methods, such as the Jacobian Transpose method and cyclic coordinate descent. Most of the work that uses Newton’s method and its variants avoids directly computing the Hessian, and instead approximations are sought, such as in the BFGS class of solvers.

In this work, we present a numerical method for computing the exact Hessian of an IK system with prismatic, revolute, and spherical joints. For the latter, formulations are presented for joints parameterized by Euler angles which can be represented for instance by using quaternions. Our method is applicable to human skeletons in computer animation applications and some, but not all, robots. Our results show that using exact Hessians can give performance advantages and higher accuracy compared to standard numerical methods used for solving IK problems. Furthermore, we provide code that allows other researchers to plug-in exact Hessians in their own work with little effort.