TRAJECTORIES IN LIE GROUPS Wayne M. Lawton Dept. of Mathematics, National University of Singapore 2 Science Drive 2, Singapore

Norbert Weiner (1949) Extrapolation, Interpolation and Smoothing of Stationary Time Series with Engineering Applications, Wiley, New York. BACKGROUND Trajectory in a vector space Rudolph E. Kalman (1960) “A new approach to linear filtering and prediction problems”, Trans. American Society of Mechanical Engineers, J. Basic Engineering, vol. 83, pp

with Timothy Poston and Luis Serra (1995) “Time-lag reduction in a medical virtual workbench”, pages in Virtual Reality and its Applications (R. Earnshaw, H. Jones, J. Vince) Academic Press, London. BACKGROUND Trajectory in a Lie group

Objective: filter the trajectory, in the rigid motion group, to predict the mouse’s future position/orientation. BACKGROUND Problem: the latency associated with a system, that converts 3D mouse position and orientation measurements to graphic displays, causes loss of hand-eye coordination Approach: lift the trajectory to obtain a trajectory, in the Lie algebra, that admits linear predictive filtering.

lift PREDICTION predict integrate to obtain

(1999) “Conjugate quadrature filters” pages in Advances in Wavelets (Ka-Sing Lau), Springer, Singapore. WAVELETS orthonormal wavelet bases are determined by CQF’s (sequences satisfying certain properties) CQF’s are parametrized by loops in SU(2) every loop in SU(2) can be approximated by (trigonometric) polynomial loops

Proof #1. Based on Hardy Spaces, OK WAVELETS (a) lift U to C Proof #2. Based on lifting, Incomplete (b) approximate C by polynomial D, that is the lift of a loop V (c) approximate loop V by a polynomial W Proof #3. A. Pressley and G. Segal, Loop Groups, Oxford University Press, New York 1986.

with Yongwimon Lenbury (1999) “Interpolatory solutions of linear ODE’s” INTERPOLATION be a dense subspace ofTheorem 2 Let Then any continuous trajectory in G can be uniformly approximated (over any finite interval) & interpolated (at any finite set of points) by a trajectory having lift is with no point masses- value measures on in

ISSUES  Continuous dependence of solutions  Approximation & interpolation of continuous  Applications and extensions by solutions whereis a dense subspace of the space of-valued measures that vanish on finite sets functions

PRELIMINARIES Choose a euclidean structure with norm be the geodesic distance function defined by the induced right-invariant riemannian metric and let

PRELIMINARIES space of- valued measures on point masses whose topology is given by seminorms topological group of continuous functionsonthat satisfy equipped with the topology of uniform convergence over compact intervals, under pointwise multiplication - valued without functions having bounded variation locally

PRELIMINARIES Lemma 1 is inif and only if then gives the distance along the trajectory in is in A function and

PRELIMINARIES subspace of step functions exponential function map control measures to solutions contains dense subset of interpolation set

RESULT is dense andTheorem 1 extends to a continuous that is one-to-one and onto. Furthermore, is a subgroup ofand it forms topological groups under both the topology of uniform a homeomorphism. convergence over compact intervals and the finer topology that makes the function

DERIVATIONS Lie bracket Adjoint representation for matrix groups We choose such that

Lemma 2 If satisfy and then where DERIVATIONS The proof of Theorem 1 is based on the following

Proof Apply Gronwall’s inequality to the following

RESULT be a dense subspace.Theorem 2 Let Then for every positive integer contains a dense subset of sequences and pair of

DERIVATIONS It suffices to approximateby elements in Choose any Lemma 3 Letbe a homeomorphism of a compact neighborhood ofinto an N-dimensional manifoldThen for any mapping that is sufficiently close to

DERIVATIONS We choose a basisfor Lemma 3 follows from classical results about the degree of mappings on spheres. To prove Theorem 2 we will first construct then apply Lemma 3 to a map and define Defineby

DERIVATIONS To show that H where we define the binary operation We observe that is nonsingular. We construct by satisfies the hypothesis of Lemma 3 it suffices, by the implicit function theorem, to prove

DERIVATIONS thus A direct computation shows that Furthermore, Lemma 2 and (2.5) imply that andare isomorphic topological groups. Nonsingularity follows since