TRAJECTORIES IN LIE GROUPS

Wayne M. Lawton

Dept. of Mathematics, National University of Singapore

2 Science Drive 2, Singapore 117543

wlawton@math.nus.edu.sg

Norbert Weiner (1949) Extrapolation, Interpolation and Smoothing of Stationary Time Series with Engineering Applications, Wiley, New York.

BACKGROUND

GR:C

Trajectory in a vector space G

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. 35-45.

with Timothy Poston and Luis Serra (1995) “Time-lag reduction in a medical virtual workbench”, pages 123-148 in Virtual Reality and its Applications (R. Earnshaw, H. Jones, J. Vince) Academic Press, London.

BACKGROUND

GR: U

Trajectory in a Lie group G

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.

liftlift

PREDICTION

CUU

I ]T,0[|U

predictpredict

]T,0[|C

]TT,T[|C integrateintegrate

toto

obtainobtain ]TT,T[|U

(1999) “Conjugate quadrature filters” pages 103-119 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 C

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

G with no point masses - value measures on R

Cin

ISSUES

Continuous dependence of solutions

Approximation & interpolation of continuous

Applications and extensions

MC~

by solutions

I U(0)G;R : U

)(U C

C CC~

),~

(U~

where is a dense subspace of the space

M of G -valued measures that vanish on finite sets

functions

PRELIMINARIES

Choose a euclidean structure

with norm

R: GG,

R:|| G

be the geodesic distance function defined by the induced right-invariant riemannian metric

and let

R: GG

PRELIMINARIES

space of RM - valued measures on

point masses whose topology is given by seminorms

k

0k0.k dt,|C(t)|||C||

topological group of continuousP Gfunctions Ron that satisfyequipped with the topology of uniform convergence

W I,W(0)

over compact intervals, under pointwise multiplication

- valued

G without

PB functions having bounded variation locally

PRELIMINARIES

Lemma 1

M,is in

t

0ds|C(s)|L(U)(t)(1.3)

if and only if

then

gives the distance along the trajectory in

is in

A function

RtL(U)(t),I)ρ(U(t),(1.4)

and

G R:U B UUC

1

G

PRELIMINARIES

subspace of step functions

exponential function

G X,Rt exp(tX),X exp(tX)dt

d(1.5)

map control measures to solutions

MS

G:exp G

B: S0

)g,..,g,t,..,P(tB)g,..,g,t,..,B(t(1.7)n1n1n1n1

)(S0 contains dense subset of interpolation set

Gg,...,g,Rtt0n1n1

RESULT

is dense andTheorem 1

extends to a continuous

MSB: M

0

that is one-to-one and onto. Furthermore, BPis a subgroup of and 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

GGG :,

)(HomG:Ad GG,for matrix groups YXXYY,X

for matrix groups-1gXgAd(g)(X)

])X((U(t))Ad,1-

UU[)X((U(t))Addtd

)1.2(

We choose 0 such that

|Y||X||]Y[X,|)2.2(

Lemma 2 If1,2,3,4j,UUCB,U 1

jjjj

satisfy

and

1

213 UUU)3.2(

,214 CCC)4.2(

and

then,2313 C)(UAdCC(2.5)

t0 4243

t)dsK(s,)(sL(U|(s)C|)(t)L(U)(t)L(U(2.6) )

where))(s)L(U-)(t)L(U(

et)K(s, 22

DERIVATIONSThe proof of Theorem 1 is based on the following

t0 ds| (s))

2(s))(C

3Ad(U-(s)

1C|

t0 ds|(s)

3C|)(t)

3(U

3L

t0 ds)| (s))

2(s))(C

3Ad(U-(s)

2C||(s)

2C-(s)

1C|(

t0 ds| (s))

2(s))(C

3Ad(U-(s)

2C|)(t)

4L(U

t0

s0 ds| v))dv(

2))(Cv(

3Ad(Udv

d|)(t)

4L(U

t0

s0 ds| v))]dv())(Cv(Ad(U),v(C[|)(t)L(U

2334

t0

s0 ds vd|)v(

2C|))s)(

1U(L)s)(

1U(L(exp(|)v(

3C|)(t)

4L(U

Proof Apply Gronwall’s inequality to the following

t0 )(s)f(s)ds

3L(U)(t)

4L(U

RESULT

be a dense subspace.Theorem 2 Let MCThen for every positive integer ,n

)(C).g,..,g,t,..,P(t n1n1

Gg,...,g,Rtt0n1n1

contains a dense subset of

sequences

and pair of

DERIVATIONS

It suffices to approximate

).g,..,g,t,..,B(tC n1n1by elements in

Choose any

C).g,..,g,t,..,B(t)( n1n1 C

Lemma 3 Let

h(D).f(0)

MD:f be a homeomorphism

of a compact neighborhood of mR0 into an

N-dimensional manifold M. Then for any mappingMD:h that is sufficiently close to f,

DERIVATIONS

nm GR:H

d,,

1BB We choose a basis for G

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

))t,([tn;,,1i),(i i1-iiid,,

i1BBX

,))(X(U)Adv(Cv)(H n1 ii

Define nG: M by

))(C)(t,),(C)(t((C)n1

mnd

d1R)(R)v,,(vv

DERIVATIONS

nm GR:F

To show that H

where we define the binary operation

We observe that

is nonsingular. We construct

).)Xv()Xv((Cv)(F nn11

by

).g,,g(0)(Hn1

satisfies the hypothesis of Lemma 3 it suffices, by

the implicit function theorem, to prove 0v|v)(H

dv

d

)C())C((AdCCC 21121

DERIVATIONS

thus

A direct computation shows that

Furthermore, Lemma 2 and (2.5) imply that

0v0v|v)(|v)(H F

dv

d

dv

d

)C()C()CC( 2121 M and B are isomorphic topological groups.

Nonsingularity follows since

),...eeg,...,e(g)v(F n1n11n.1,...,i),v)Xt-exp((te ii1-iii