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Let $f$ be a $C^2$ partially hyperbolic diffeomorphisms of ${\mathbb T}^3$(not necessarily volume preserving or transitive) isotopic to a linear Anosovdiffeomorphism $A$ with eigenvalues $$\lambda_{s}<1<\lambda_{c}<\lambda_{u}.$$Under the assumption that the set $$\{x: \,\mid\log\det(Tf\mid_{E^{cu}(x)})\mid \leq \log \lambda_{u} \}$$ has zero volume insideany unstable leaf of $f$ where $E^{cu} = E^c\oplus E^u$ is the center unstablebundle, we prove that the stable foliation of $f$ is $C^1$ robustly minimal,i.e., the stable foliation of any diffeomorphism $C^1$ sufficiently close to$f$ is minimal. In particular, $f$ is robustly transitive.\par  We build, with this criterion, a new example of a $C^1$ open set of partiallyhyperbolic diffeomorphisms, for which the strong stable foliation and thestrong unstable foliation are both minimal.

Robust minimality of strong foliations for DA diffeomorphisms: $cu$-volume expansion and new examples