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Redundant robotic chains on Riemannian manifolds Claudio Alta ni

Summary: Redundant robotic chains on Riemannian manifolds
Claudio Alta ni 
February 27, 2002
For redundant robotic chains composed of simple one-degree of freedom joints or links, a geometric
interpretation of the forward kinematic map in terms of Riemannian submersions is proposed. 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. Furthermore, this enables us to use all the techniques for motion control of rigid
bodies on Riemannian manifolds (and Lie groups in particular) to design workspace control algorithms
for the end-e ector of the robotic chain and then to pull them back to joint space, all respecting the
di erent geometric structures of the two underlying model spaces. The application to the control of a
holonomic mobile manipulator is described.
Keywords: redundant robotic chains, Riemannian submersions, product of exponentials formula, pseu-
doinverse, motion control, holonomic mobile robot.
1 Introduction
From a mathematical viewpoint, a robotic chain can be seen as a mechanical control system having as
con guration space the manifold in which the joint variables i.e. the parameters describing the angles of
rotation (respectively, the lengths) of each of the joints (resp. links) are living, and control inputs that are the
torques (resp. the forces) applied at the same joint/link. See any of the many books on modeling and control


Source: Altafini, Claudio - Functional Analysis Sector, Scuola Internazionale Superiore di Studi Avanzati (SISSA)


Collections: Engineering; Mathematics