"Riemannian" collar theorem

Question

Let $(M,g)$ be a compact manifold of dimension $n$ with boundary. If $\partial M$ is smooth then one has a control on the determinant of the Jacobian of the diffeomorphism in the collar theorem, i.e. for $\epsilon$ small enough the diffeomorphism $F:\partial M\times (0,\epsilon)\to M_\epsilon$ can be given by $F(x,t)=\exp_x(t\,{\bf n}(x))$. Moreover there exist a universal constant $C>0$ such that $C^{-1}\le|\det DF|\le C$ (one can take C=2 here). Let say that a family of manifolds satisfies a "Riemannian collar theorem" if for each manifold in the family there exists a collar diffeomorphism $F$ with a universal bounded on $|\det DF|$ as above. Does the family of manifolds with piecewise smooth boundary satisfies a "Riemannian collar theorem"? For example, assume that $\partial M=\cup_{i=1}^m N_i$, where $N_i$ are smooth $(n-1)$-manifolds, and each point of singularity lies on a transversal intersection of exactly two elements of $\{N_i\}$. Do we have a "Riemannian collar theorem" for such family of $(M,g)$? Probably not in general but can we say anything about when it does?