  Ellipsoids are a common representation for reachability analysis, because
they can be transformed efficiently under affine maps, and allow conservative
approximation of Minkowski sums, which let one incorporate uncertainty and
linearization error in a dynamical system by expanding the size of the
reachable set. Zonotopes, a type of symmetric, convex polytope, are similarly
frequently used due to efficient numerical implementation of affine maps and
exact Minkowski sums. Both of these representations also enable efficient,
convex collision detection for fault detection or formal verification tasks,
wherein one checks if the reachable set of a system collides (i.e., intersects)
with an unsafe set. However, both representations often result in conservative
representations for reachable sets of arbitrary systems, and neither is closed
under intersection. Recently, representations such as constrained zonotopes and
constrained polynomial zonotopes have been shown to overcome some of these
conservativeness challenges, and are closed under intersection. However,
constrained zonotopes can not represent shapes with smooth boundaries such as
ellipsoids, and constrained polynomial zonotopes can require solving a
non-convex program for collision checking or fault detection. This paper
introduces ellipsotopes, a set representation that is closed under affine maps,
Minkowski sums, and intersections. Ellipsotopes combine the advantages of
ellipsoids and zonotopes while ensuring convex collision checking. The utility
of this representation is demonstrated on several examples.
