  In the standard orbit model on shape analysis, a group of diffeomorphism on
the ambient space equipped with a right invariant sub-riemannian metric acts on
a space of shapes and induces a sub-riemannian structure on various spaces. An
important example is given by the Large Deformation Diffeomorphic Metric
Mapping (LDDMM) theory that has been developed initially in the context of
medical imaging and image registration. However, the standard theory does not
cover many interesting settings emerging in applications. We provide here an
extended setting, the graded group action (GGA) framework, specifying
regularity conditions to get most of the well known results on the orbit model
for general groups and shape spaces equipped with a smooth structure of Banach
manifold with application to multi-scale shape spaces. A specific study of the
Euler-Poincar{\'e} equations inside the GCA framework leads to a uniqueness
result for the momentum map trajectory lifted from shape spaces with different
complexities deciphering possible benefits of over-parametrization in shooting
algorithms.
