# An Ellipsoidal Calculus Based on Propagation and Fusion - MAPLE code freely available for academic purposes -  Joint work by
Assumpta Sabater (1),
Federico Thomas (2), and Lluís Ros (2)

(1) Applied Mathematics Department - UPC
(2) Institut de Robòtica i Informàtica Industrial - CSIC / UPC

assumpta dot sabater-pruna followed by at upc dot es
fthomas followed by at iri dot upc dot es
llros followed by at ir dot upc dot es

Introduction  |  SnapshotsRelated papersMaple code

## 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.