 
Summary: Motion on submanifolds of noninvariant holonomic constraints for a
kinematic control system evolving on a matrix Lie group
Claudio Altafini
SISSAISAS, International School for Advanced Studies
via Beirut 24, 34014 Trieste, Italy; altafini@sissa.it
Ruggero Frezza
Dipartimento di Elettronica e Informatica, Universit`a di Padova
Via Gradenigo 6/A 35100 Padova, Italy; frezza@dei.unipd.it
April 11, 2003
Abstract
For a control system on a matrix Lie group with one or more configuration constraints that
are not left/right invariant, finding the combinations of (kinematic) control inputs satisfying
the motion constraints is not a trivial problem. Two methods, one coordinatedependent and
the other coordinatefree are suggested. The first is based on the WeiNorman formula; the
second on the calculation of the annihilator of the coadjoint action of the constraint oneform
at each point of the group manifold. The results are applied to a control system on SE(3) with
a holonomic inertial constraint involving the noncommutative part in a nontrivial way. The
difference in terms of compactness of the result between the two methods is considerable.
Keywords: matrix Lie groups, constrained motion, WeiNorman formula, noninvariant one
forms, coadjoint action.
