New metrics for rigid body motion interpolation - helix

where S denotes the matrix representation of the twist S. The sum of this series can be computed
explicitly and the resulting expression, when restricted to SO(3), is known as Rodrigues’ formula.
The formula for the sum in SE(3) is derived in [14].
2.4 Riemannian metrics on Lie groups
If a smoothly varying, positive definite, bilinear, symmetric form is defined on the tangent
space at each point on the manifold, such a form is called a Riemannian metric and the manifold is
Riemannian [6]. If the form is non-degenerate but indefinite, the metric is called semi-Riemannian.
On a n dimensional manifold, the metric is locally characterized by a n n matrix of C 1 functions
g ij = where X i are basis vector fields 1 . Note that the entries g ij at each point on the
manifold are functions of the local coordinates at that point. If the basis vector fields can be defined
globally, then the matrix [g ij ] completely defines the metric.
On SE(3) (on any Lie group), an inner product on the Lie algebra can be extended to a Riemannian
metric over the manifold using left (or right) translation. To see this, consider the inner
product of two elements S 1 ;S 2 2 se(3) defined by
j I = s T Ws 1 2; (8)
where s 1 and s 2 are the 6 1 vectors of components of S 1 and S 2 with respect to some basis
and W is a positive definite matrix. If V 1 and V 2 are tangent vectors at an arbitrary group element
A 2 SE(3), the inner product j A
in the tangent space T A SE(3) can be defined by:
j A = : (9)
I
The metric obtained in such a way is said to be left invariant [6] since left translation by any element
A is an isometry.
3 Riemannian metric on SO(n) and SE(n)
In this section we will show that there is a simple way of defining a left (or right) invariant metric
in SO(n) (or SE(n , 1) by introducing an appropriate constant metric in GL(n; IR). Defining a
1 The basis vector fields need not be the coordinate basis; we only require that they are smooth.
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