Minimum divergence methods are popular tools in a variety of statistical applications. We consider tubular model adequacy tests, and demonstrate that the new divergences that are generated in the process are very useful in robust statistical inference. In particular we show that the family of $S$-divergences can be alternatively developed using the tubular model adequacy tests; a further application of the paradigm generates a larger superfamily of divergences. We describe the properties of this larger class and its potential applications in robust inference. Along the way, the failure of the first order influence function analysis in capturing the robustness of these procedures is also established.