Weyl tube formula for manifolds with boundary

A manifold with boundary is a set with positive reach. The tube formula and kinematic formulas for such sets were described by Federer 60 years ago in this paper.

I think Alfred Gray's book "Tubes" https://www.springer.com/gp/book/9783764369071 is relevant. See also https://www.sciencedirect.com/science/article/pii/0040938382900052 (Comparison theorems for the volumes of tubes as generalizations of the Weyl tube formula, by the same author).