  Positive spanning sets span a given vector space by nonnegative linear
combinations of their elements. These have attracted significant attention in
recent years, owing to their extensive use in derivative-free optimization. In
this setting, the quality of a positive spanning set is assessed through its
cosine measure, a geometric quantity that expresses how well such a set covers
the space of interest. In this paper, we investigate the construction of
positive $k$-spanning sets with geometrical guarantees.Our results build on
recently identified positive spanning sets, called orthogonally structured
positive bases. We first describe how to identify such sets and compute their
cosine measures efficiently. We then focus our study on positive $k$-spanning
sets, for which we provide a complete description, as well as a new notion of
cosine measure that accounts for the resilient nature of such sets. By
combining our results, we are able to use orthogonally structured positive
bases to create positive $k$-spanning sets with guarantees on the value of
their cosine measures.
