GEOMETRIC CONTROL OF A BINOCULAR HEAD
Eduardo Bayro-Corrochano and Julio Zamora-Esquivel
Department of Electrical Engineering and Computer Science
CINVESTAV, unidad Guadalajara. Jalisco, Mexico
Keywords:
Conformal Geometry, Kinematics, Tracking.
Abstract:
In this paper the authors use geometric algebra to formulate the differential kinematics of a binocular robotic
head and reformulate the interaction matrix in terms of the lines that represent the principal axes of the camera.
This matrix relates the velocities of 3D objects and the velocities of their images in the stereo images. Our
main objective is the formulation of a kinematic control law in order to close the loop between perception and
action, which allows to perform a smooth visual tracking.
1 INTRODUCTION
In this work we formulate the problem of visual track-
ing and design a control law by velocity feedback
that allows us to close the loop between perception
and action. Geometric algebra allow us to work
with geometric entities like points, lines and planes
and helps in the representation of rigid transforma-
tions. In this mathematical framework we straightfor-
wardly formulate the direct and differential kinemat-
ics of robotic devices like the binocular robot head.
On the other hand we show a reformulation of vi-
sual Jacobean which relates the velocity of a tridimen-
sional object with the velocity of its projection onto
the stereo camera images. Finally we write an ex-
pression that relates the joint velocities in the pan-tilt
unit and velocities of the points in the camera image.
We start this work presenting a brief description of
the geometric entities and conformal transformations
them we show the kinematics of a pan-tilt unit and
formulate its control law.
In contrast to other authors like (C. Canudas de
Wit and Bastin, 1996), (Ruf, 2000) or (Kim Jung-Ha.,
1990) we will use multivectors instead of matrices to
formulate the control law it reduces the computation
and improve the performance of the controller.
2 GEOMETRIC ALGEBRA: AN
OUTLINE
Let
G
n
denote the geometric algebra of n-dimensions,
this is a graded linear space. As well as vector
addition and scalar multiplication we have a non-
commutative product which is associative and dis-
tributive over addition – this is the geometric or Clif-
ford product.
The inner product of two vectors is the standard
scalar or dot product and produces a scalar. The outer
or wedge product of two vectors is a new quantity
which we call a bivector. We think of a bivector as a
oriented area in the plane containing a and b, formed
by sweeping a along b.
Thus, b∧a will have the opposite orientation mak-
ing the wedge product anti-commutative. The outer
product is immediately generalizable to higher di-
mensions – for example, (a ∧ b) ∧ c, a trivector, is
interpreted as the oriented volume formed by sweep-
ing the area a∧ b along vector c. The outer product of
k vectors is a k-vector or k-blade, and such a quantity
is said to have grade k. A multivector (linear combi-
nation of objects of different type) is homogeneous if
it contains terms of only a single grade.
2.1 The Geometric Algebra of n-D
Space
In this paper we will specify a geometric algebra
G
n
of the n dimensional space by G
p,q,r
, where p, q and r
183
Bayro-Corrochano E. and Zamora-Esquivel J. (2007).
GEOMETRIC CONTROL OF A BINOCULAR HEAD.
In Proceedings of the Fourth International Conference on Informatics in Control, Automation and Robotics, pages 183-188
DOI: 10.5220/0001628001830188
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