Introduction
This
work
develops an Ellipsoidal
Calculus based on two operations: propagation
and fusion.
Propagation refers to
obtaining an ellipsoid that must satisfy an affine relation
with another ellipsoid, and fusion to that of computing the ellipsoid
that
tightly bounds the intersection of two other ellipsoids. These two
operations
supersede the Minkowski sum and difference, affine
transformation and intersection
tight bounding of ellipsoids on which
other Ellipsoidal Calculi are
based.
Actually, a Minkowski operation can be seen as a fusion followed by a
propagation,
and an affine transformation as a particular case of propagation.
Moreover,
the presented formulation is robust in the sense that it is inmune to
degeneracies
of the involved ellipsoids (they may extend infinetely in any
direction) and/or of the affine relations.
Altogether, these tools permit a set-membership approach to dealing
with uncertainty in many contexts. We here illustrate how they can be
used in two specific domains: the spatial
interpretation of line
drawings and the computation of force/manipulability ellipsoids
for parallel manipulators. Other possible applications are
to estimation theory, and to robot self-localization and
map building (SLAM).
The
calculus has been implemented
in MAPLE, and
is distributed
here under the name ECT - Ellipsoidal
Calculus Toolbox. Most of the routines are general, in the
sense that they work for n-dimensional ellipsoids. ECT is accompanied
with
several MAPLE worksheets examplifying its use, and with several
routines to plot the results.
Snapshots
|
This sequence shows
one application of the propagation and fusion operators to line drawing
interpretation. The figure in red is a projection of a truncated
tetrahedron.
Note that it is only correct when the three edges joining the
two
red triangles intersect at a common point, the imaginary apex
of
such a tetrahedron. Since the apex must lie on any pair of two such
edges,
we can use the propagation operator to derive the yellow
elliptical
bounds for the apex, given the green
elliptic uncertainties for the vertices.
Next,
the three yellow ellipses
are intersected by repeatedly using the fusion operator. The resulting
elliptical envelope of such intersection is shown in yellow, in
the second figure.
|
Related papers
L. Ros, A. Sabater, and F.
Thomas. "An
Ellipsoidal Calculus based on Propagation and Fusion." IEEE
Transactions on Systems Man and Cybernetics, Part B. Vol. 32, n. 4, pp.
430-442. August 2002. IEEE Press. ISSN 1083-4419.
A. Sabater
and F. Thomas, "Set Membership
Approach to the
Propagation of Uncertain Geometric Information," Proceedings of the
1991 IEEE International Conference on Robotics and Automation, Vol.
III, pp. 2718-2723, abril 9-11, 1991, Sacramento, USA.
Maple code
(For MAPLE 8.0 or higher)
This code is freely
distributed only for academic purposes.
Please contact the authors for permission to use it commercially.
The
Ellipsoidal Calculus Toolbox is distributed as a single file, ect-1.0.txt,
included below. There is no written manual for ECT 1.0 yet. However, all functions
of
the library are extensively documented inside ect-1.0.txt. These
comments, together with the papers above, and the example
worksheets below, should suffice to
develop your own application of ECT. If you have troubles using ECT,
you can also e-mail us,
we'll be glad to help you.
So where do you start? We recommend that you first read the first paper
above (which explains the the calculus in the form implemented in ECT),
then download all files below, and try the execution of the example
worksheets in MAPLE 8.0 or higher. HTML versions of these files are
also
available, to browse their contents without executing them. If
you prefer, you can download all files below collected in a zip file (2.6
Mb).
- ECT's primary
library file:
- Simple examples
of propagation:
- Simple examples
of fusion:
- Simple examples
of Minkowski sums:
Two applications
of ECT are included, related to dealing with uncertainty in line
drawing interpretation and isotropy analysis of parallel manipulators:
- Application to
line drawing interpretation:
- Application to
isotropy analysis:
