### Optimization and Systems Theory Seminar

Friday, May 18, 2001, 11.00-12.00, Room 3721, Lindstedtsv. 25

**Claudio Altafini**

Optimization and Systems Theory

KTH

E-mail: claudio.altafini@math.kth.se

####
Modeling and control of redundant robotic chains on Riemannian
manifolds

This talk will overview the first two papers of my doctoral thesis
"Geometric control methods for nonlinear systems and robotic
applications".
For redundant robotic chains composed of simple one-degree of freedom
joints or links, the forward kinematic map from joint space to the
workspace of the end-effector is interpreted geometrically in terms of
Riemannian submersions.
Several properties of redundant robots then admit clear geometric
characterizations, the most remarkable being that the Moore-Penrose
pseudoinverse normally used in Robotics coincides with the horizontal
lift of the Riemannian submersion.
The end-effector of the robot leaves on the Special Euclidean group in
3 dimension. On SE(3), the dynamical equations of the robotic chain
look like a set of controlled Euler-Lagrange equations, or
Euler-Poincare' equations after reduction by group
symmetry. Variational methods are used to generate a geometric spline
for such equations and the extra complications of the corresponding
reduction, due to the semidirect product structure of SE(3), are
analyzed. The loop is closed in workspace using a PD controller which
is then pulled back to joint space by means of the horizontal lift,
all respecting the different geometric structures of the two
underlying model spaces.

Calendar of seminars

*Last update: May 9, 2001 by
Anders Forsgren,
anders.forsgren@math.kth.se.
*