On computing real logarithms for matrices in the Lie group of special Euclidean motions in Rn

We show that the diagonal Pade approximants methods, both for computing the principal logarithm of matrices belonging to the Lie groupSE (n, IR) of special Euclidean motions in IRn and to compute the matrix exponential of elements in the corresponding Lie algebra se(n, IR), are structure preserving. Also, for the particular cases when n == 2,3 we present an alternative closed form to compute the principal logarithm. These low dimensional Lie groups play an important role in the kinematic motion of many mechanical systems and, for that reason, the results presented here have immediate applications in robotic

Padé and Gregory error estimates for the logarithm of block triangular matrices

In this paper we give bounds for the error arising in the approximation of the logarithm of a block triangular matrix T by Padé approximants of the function f(x)=log[(1+x)/(1-x)] and partial sums of Gregory's series. These bounds show that if the norm of all diagonal blocks of the Cayley-transform B=(T-I)(T+I)-1 is sufficiently close to zero, then both approximation methods are accurate. This will contribute for reducing the number of successive square roots of T needed in the inverse scaling and squaring procedure for the matrix logarithm.http://www.sciencedirect.com/science/article/B6TYD-4G5BJ9P-2/1/398a212a906943d2474a2cd6166c1d3