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
BACKGROUND
Trajectory in a vector space
G
C : R  G
Norbert Weiner (1949) Extrapolation, Interpolation
and Smoothing of Stationary Time Series with
Engineering Applications, Wiley, New York.
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.
BACKGROUND
Trajectory in a Lie group
U : R

G
 G
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
Problem: the latency associated with a system,
that converts 3D mouse position and orientation
measurements to graphic displays, causes loss
of hand-eye coordination
Objective: filter the trajectory, in the rigid
motion group, to predict the mouse’s future
position/orientation.
Approach: lift the trajectory to obtain a
trajectory, in the Lie algebra, that admits
linear predictive filtering.
PREDICTION
lift
U |[ 0,T ]
predict
integrate
obtain
to
I
C |[ 0 ,T ]
C |[ T ,T  T ]

U  CU
U |[ T ,T  T ]
WAVELETS
(1999) “Conjugate quadrature filters” pages 103119 in Advances in Wavelets (Ka-Sing Lau),
Springer, Singapore.
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
WAVELETS
Proof #1. Based on Hardy Spaces, OK
Proof #2. Based on lifting, Incomplete
(a) lift U to C
(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.
INTERPOLATION
with Yongwimon Lenbury (1999) “Interpolatory
solutions of linear ODE’s”
Theorem 2 Let
C
be a dense subspace of
G - value measures on R

with no point masses
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
in
C
ISSUES
Continuous dependence of solutions
U   (C )
Approximation & interpolation of continuous
U : R   G; U(0) I
~
~
~
by solutions U   (C ), C  C
functions
where
M
of
~
C M
is a dense subspace of the space
G-valued measures that vanish on finite sets
Applications and extensions
PRELIMINARIES
Choose a euclidean structure
, : G G R 
with norm
| | : G  R
and let
 : G G R 
be the geodesic distance function defined by the
induced right-invariant riemannian metric
PRELIMINARIES
M
space of
G- valued measures on R  without
point masses whose topology is given by seminorms
|| C ||k  0 | C(t) | dt, k  0.
k
P
topological group of continuous G - valued
functions W on R  that satisfy W(0)  I,
equipped with the topology of uniform convergence
over compact intervals, under pointwise multiplication
functions having bounded variation locally
BP
PRELIMINARIES
Lemma 1
A function
U : R  G
1
 U is in M,
CU
is in
B
then
(1.3) L(U)(t)  0 | C(s) | ds
if and only if
t
gives the distance along the trajectory in
G
and
(1.4) ρ(U(t),I)  L(U)(t), t  R 
PRELIMINARIES
exp : G  G
(1.5)
d
dt
exponential function
exp(tX) X exp(tX), t  R  , X  G
S M subspace of step functions
 0 : S  B map control measures to solutions
 0 (S) contains dense subset of interpolation set
(1.7) B(t 1 ,.., t n , g1 ,.., g n )  B  P(t1 ,.., t n , g1 ,.., g n )
0  t 1    t n  R  , g1 ,..., g n  G
RESULT
0
extends to a continuous  : M  B
that is one-to-one and onto. Furthermore, B
is a subgroup of P and it forms topological
Theorem 1
S M
is dense and
groups under both the topology of uniform
convergence over compact intervals and the
finer topology that makes the function
a homeomorphism.

DERIVATIONS
  ,   : G G G
for matrix groups  X , Y   XY  YX
Lie bracket
Ad : G  Hom ( G,G)
-1
Ad(g)(X)  gXg
Adjoint representation
for matrix groups
d
( 2.1)
dt
-1
Ad (U(t)) ( X )  [U U , Ad (U(t)) ( X ) ]
  0 such that
( 2.2) | [X, Y] |   | X | | Y |
We choose
DERIVATIONS
The proof of Theorem 1 is based on the following
Lemma 2 If
1
satisfy
and
U j  B, C j  U jU j , j  1,2,3,4
(2.3) U 3  U1U 2
1
(2.4) C4  C1  C2 ,
then
(2.5)
and
C 3  C1  Ad (U 3 ) C 2 ,
(2.6) L(U3 )(t)  L(U4 )(t)    | C 2 (s) | L(U4 )(s) K(s, t)ds
t
0
where
K(s, t)  e
 ( L(U2 )(t) - L(U2 )(s))
Proof Apply Gronwall’s inequality to the following
t
L (U )(t)  0 | C (s) | ds
3
t3 3
 0 | C (s) - Ad(U (s))(C (s)) | ds
1
3
2
t
 0 ( | C (s) - C (s) |  | C (s) - Ad(U (s))(C (s)) | ) d
1
2
2
3
2
t

 L(U )(t)  0 | C (s) - Ad(U (s))(C (s)) | ds
4
2
3
2
t
s d
 L(U )(t)  0 | 0 dv Ad(U ( v ))(C ( v ))dv | ds
4
3
2
t s

 L(U )(t)  0 | 0 [ C ( v ), Ad(U ( v ))(C ( v ))]dv | ds
4

 t s
)(t)   0 0 |
L(U
4
3
C
(v)
|  exp(  ( L ( U
3
 L(U
3
)( s )  L ( U
1
4
2
t

)(t)  0 L(U
)( s ))
1
3
)(s)f(s)ds
|
C
(v) |
2
dv ds
RESULT
Theorem 2 Let
C M
be a dense subspace.
,
Then for every positive integer n
and pair of
sequences 0  t 1    t n  R  , g1 ,..., g n  G
 (C) contains a dense subset of
P(t1 ,.., t n , g1 ,.., g n ).
DERIVATIONS
C  B(t 1 ,.., t n , g1 ,.., g n ).
It suffices to approximate C by elements in
( C)  B(t 1 ,.., t n , g1 ,.., g n ).
Lemma 3 Let f : D  M be a homeomorphism
m
of a compact neighborhood of 0  R
into an
N-dimensional manifold M. Then for any mapping
h : D  M that is sufficiently close to f,
f(0)  h(D).
Choose any
DERIVATIONS
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
m
n Define
n by
H:R G
: M G
 (C)  ((C)(t1 ),  , (C)(tn ))
We choose a basis B1 ,  , B d for G and define
X i  ( B1  i ,  , B d  i ), i  1,  , n;  i   ([t i-1 , t i ))
H ( v)   (C  
n
vi  Ad (U) (X i ) ),
1
d n
v  (v1 ,  , v d )  (R )

R
m
DERIVATIONS
We observe that H(0)  (g1 ,  , g n ). To show that H
satisfies the hypothesis of Lemma 3 it suffices, by
d
H ( v) | v0
the implicit function theorem, to prove
dv
m
n by
is nonsingular. We construct
F:R  G
F(v)  (C  (v1  X 1 )   (vn  X n ) ).
where we define the binary operation
C1  C2  C1  Ad ((C1 )) (C2 )
DERIVATIONS
A direct computation shows that
d
d
H ( v) | v0  F ( v) | v0
dv
dv
Furthermore, Lemma 2 and (2.5) imply that
thus
(C1  C2 )  (C1 )(C2 )
M and B are isomorphic topological groups.
Nonsingularity follows since
F(v)  (g1e1,..., g n e1...en ),
ei  exp((t i - t i-1 )X i  vi ), i  1,..., n.