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Generalising a seminal result of Epstein and Penner for cusped hyperbolicmanifolds, Cooper and Long showed that each decorated strictly convexprojective cusped manifold has a canonical cell decomposition. Penner used theformer result to describe a natural cell decomposition of decoratedTeichm\"uller space of punctured surfaces. We extend this cell decomposition tothe moduli space of decorated strictly convex projective structures of finitevolume on punctured surfaces.  The proof uses Fock and Goncharov's $\mathcal{A}$-coordinates for doublydecorated structures. In addition, we describe a simple, intrinsicedge-flipping algorithm to determine the canonical cell decompositionassociated to a point in moduli space, and show that Penner's centres ofTeichm\"uller cells are also natural centres of the cells in moduli space. Weshow that in many cases, the associated holonomy groups are semi-arithmetic.

On Moduli Spaces of Convex Projective Structures on Surfaces: Outitude and Cell-Decomposition in Fock-Goncharov Coordinates