Cayley-Hamilton for roboticists

Abstract

The Cayley-Hamilton theorem is an important theorem of linear algebra which is well known and used in system theory. Unfortunatelly, this powerful result is practically never used in robotics even though it is of extreme relevance. This article is a review of the use of this result for the calculation of general matrix functions which are very common in robotics. It will be shown how any analytic matrix function like exponential, logarithm and more complicated expressions in robotics, can be easily and analytically calculated in an explicit form. Examples are given for the exponential map, inverse of the exponential map, and the derivative of the exponential map. For the first two examples there exist well known expressions in the literature, but the last one is not as easy to compute without the presented methods.