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On Computing Real Logarithms for Matrices In On computing real logarithms for matrices in the n Lie group of sp ecial Euclidean motions in IR J R Cardoso Instituto Sup erior de Engenharia de Coimbra Quinta da Nora Coimbra Portugal jo carsunisecpt y F Silva Leite Departamento de Matematica Universidade de Coimbra CoimbraPortugal eitematucpt fax Abstract Weshow that the diagonal Pade approximants metho ds b oth for computing the principal logarithm of matrices b elonging to the Lie group SEn IR of sp ecial n Euclidean motions in IR and to compute the matrix exp onential of elements in the corresp onding Lie algebra sen IR are structure preserving Also for the particular cases when n we present an alternative closed form to compute the principal logarithm These low dimensional Lie groups play an imp ortant role in the kinematic motion of manymechanical systems and for that reason the results presented here have immediate applications in rob otics n Keywords Lie group of Euclidean motions in IR matrix logarithms matrix exp onentials Pade approximants metho d Work supp orted in part by Instituto de Sistemas e Robotica Polo de Coimbra y Work supp orted in part by ISR researchnetwork contract ERB FMRXCT and research grant from Gulb enkian Foundation Intro duction It is well known that under some sp ectral conditions anyinvertible real matrix has a real logarithm Culver Lately there has b een an increasing interest in developing computational techniques for real logarithms of real matrices the most signicantwork in this area b eing Kenney and Laub and and Dieci Morini and Papini An imp ortant fo cus of recentworkisondeveloping approximating metho ds for matrix Lie groups that are structure preserving in the sense that they pro duce a matrix in the corresp onding Lie algebra For instance Dieci showed that some of the metho ds already develop ed when applied to an orthogonal matrix resp ectively symplectic matrix always pro duce a real logarithm that is skewsymmetric resp ectively hamiltonian Cardoso and Silva Leite extended the results of Dieci to a muchvaster class of Lie groups This article continues in the same direction and the main ob jective is to show that the Pade approximants metho d is structure preserving for the Lie group n of sp ecial Euclidean motions in IR We also present closed forms for computing the real logarithm for the sp ecial cases when n Our motivation comes from the imp ortance that these Lie groups play in applications to Engineering Wenowintro duce some notation that will b e used throughout the whole pap er Let gln IR denote the real vector space consisting of all n n matrices with real entries gln IR equipp ed with the commutator op eration A B AB BA forms a Lie algebra which is the Lie algebra of the Lie group GLn IR consisting of all invertible matrices in gln IR The n n identity matrix will b e denoted by I Details ab out the theory of matrix Lie groups and corresp onding Lie algebras may b e found n in Sattinger and Weaver Now the rotation group in IR may b e dened as n o T SOn IR X GLn IR X X I detX n and the special group of Euclidean motions in IR by R n R SOn I SEn IR R v IR v Both SOn IR and SEn IR are Lie groups with corresp onding Lie algebras de ned resp ectively by n o T son IR A gln IRA A and S n sen IR S son IR u IR u One imp ortant issue in the control of mechanical systems it that of path planning tra jectories Park and Ravani and Crouch Kun and Silva Leite generalized the classical De Casteljau algorithm for constructing Bezier curves on Lie groups For the classical algorithm see for instance Farin It turns out that the implementation of the De Casteljau algorithm on Lie groups dep ends on sucessiv e computations of matrix logarithms and exp onentials Since displacements of a rigid b o dy form a Lie group these interp olation techniques may b e an ecientway to path planning In n this context the Lie group of sp ecial Euclidean motions in IR whenn plays an imp ortant role For instance the motion at every instant of time of a unicycle which rolls without slipping on a plane is describ ed by a matrix At X t SE IR xt where xt IR describ es the p osition at time t of the center of mass of the unicycle with resp ect to an orthonormal xed frame in the plane and At SO IR describ es its orientation at time t with resp ect to the same inertial frame Similarly the kinematic motion of an autonomous underwater vehicle is describ ed by a matrix At X t SE IR xt where xt IR describ es the p osition at time tofthecenter of mass of the vehicle in space and At SO IR describ es its orientation at time t with resp ect to an inertial frame The organization of the pap er is as follows We start with some basics ab out logarithms of matrices and show that the principal logarithm of elements in SEn IR b elong to the corresp onding Lie algebra Section is devoted to show that the Pade approximants metho ds for computing the principal logarithm is structure preserving for the sp ecial Euclidean group Wealsoprove a dual result for the exp onential of elements in the Lie algebra sen IR An algorithm to compute the principal logarithm of a matrix in SEn IR is included Finally we present closed forms for the principal logarithm of elements in SE IRandSE IR The principal logarithm in SEn IR X Consider the matrix equation e T where T is a given matrix b elonging to the general linear Lie group GLn IR All solutions X of this equation not necessarily real are called logarithms of T It turns out however see for instance Culver or Horn and Johnson that if the sp ectrum of T denoted by T do es not intersect IR thenT has a unique real logarithm whose sp ectrum lies in the strip C Imz g This logarithm is called the principal logarithm of T and fz will b e denoted by Log T It also happ ens that if kI T k for any matrix norm k P I T kk then the p ower series converges to the principal logarithm of T So k k it makes sense to write k X I T Log T kI T k k k We also recall a result ab out matrix square ro ots namely that if a real matrix then there exists a unique real square ro ot of T having T satises T IR eigenvalues with p ositive real part see for instance De Prima and Johnson This square ro ot of T will b e the only one used along the pap er and will b e denoted by T without any further reference In the particular situation when R T SEn IR v then R T v R I SOn IRwhenever R SOn IR we conclude that and using the fact that R T SEn IR The next result stresses the imp ortance of the principal logarithm of a matrix b elonging to the Lie group SEn IR Theorem If T SEn IR and T IR then Log T sen IR Pro of We rst note that as proved in Dieci the analogue of this theorem for the case when the Lie group SEn IR is replaced by the rotation group SOn IR is also true For a generalization to other Lie groups see also Cardoso and Silva Leite If kI T k the theorem follows by applying this result after having used the series expansion of Log T On the other hand if kI T k there exists a p ositive k k k where T is obtained from T after k sucessive integer k such that k I T k k k square ro ots So Log T sen IR Now since Log T Log T see Kenney and Laub it follows that Log T sen IR Pade approximants metho d We refer to Baker and Baker and GravesMorris for more details concerning to general theory of Pade approximants Here we recall how to obtain the diagonal Pade approximants of a scalar function P i c x is the MacLaurin series of f and that R x Assume that f x i mm i P x m with Q is the m mdiagonalPade approximantof f Then one may m Q x m write m P x a a x a x m m R x mm m Q x b x b x m m where the constants b i m are uniquely determined by solving the system i of linear algebraic equations A X B with c c c c b c m m m c c c c b c m m m c c c c b c m m m A X B b c c c c c m m m m m and the a s i m are then obtained through the following formulas i a c P i a c b c i m i i k ik k The following result will b e essencial to prove the main result in this section Lemma If f is an odd function then the coecients of the diagonal Pade approx imants of f satisfy a b i i i Pro of If f is o dd then the even co ecients c in its MacLaurin series vanish In i this situation a simple manipulation with prop erties of determinants will show that any matrix obtained from A ab ove by replacing each of its even columns by column B will have zero determinant It then follo ws from applying Cramers rule to the system A X B that b i Replacing this in wealsoget a i i i It is well known that square Pade approximants R A of the matrix function mm f ALog I Amay b e used to approximate the principal logarithm of any matrix T I A with kI T k It turns out that some imp ortant simplications take place if instead of using the m mdiagonalPade approximantof f ALog I A one uses the m m diagonal Pade approximantof g B Log I B I B where B AA I We denote this approximantby S B These twoPade mm approximants are related through the identity R A S AA I mm mm The following result shows the advantage of working with S AA I instead mm of R Aasanapproximation for Log I T mm X of the matrix function g X Log I Lemma The Pade approximant S mm X I X is of the form g X X X where is a polynomial function of the odd powers of X and is a polynomial function of the
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