 
Summary: The De Casteljau algorithm on SE(3)
Claudio Altafini
Optimization and Systems Theory, Royal Institute of Technology
SE10044, Stockholm, Sweden
altafini@math.kth.se
Abstract. Smooth closedform curves on the Lie group of rigid body motions are
constructed via the De Casteljau algorithm. Due to the lack of a biinvariant metric
on SE(3), the resulting curve depends on the choice of the metric tensor. The two
most common cases are analyzed.
1 Introduction
The group of rigid body transformation SE(3) arises naturally as the config
uration space of many robotic systems like aerial [8] and underwater vehicles
[9] or robotic manipulators [10,15]. Motivated by motion planning purposes
for such systems, the search for methods for the generation of smooth tra
jectories on SE(3) has given rise to a rich field of literature. We mention
among others [5,7,11,12,16]. In particular, geometric techniques seem to ap
pear naturally when one wants to construct trajectories in an invariant and
coordinatefree way. For example, one would like to have a notion of distance
that does not change with the way it is measured, but rather that it represents
as much as possible an intrinsic property of a system. It is straightforward
