  In this paper we investigate the relation between robustness of periodic
orbits exhibited by systems with impulse effects and robustness of their
corresponding Poincar\'e maps. In particular, we prove that input-to-state
stability (ISS) of a periodic orbit under external excitation in both
continuous and discrete time is equivalent to ISS of the corresponding 0-input
fixed point of the associated \emph{forced} Poincar\'e map. This result extends
the classical Poincar\'e analysis for asymptotic stability of periodic
solutions to establish orbital input-to-state stability of such solutions under
external excitation. In our proof, we define the forced Poincar\'e map, and use
it to construct ISS estimates for the periodic orbit in terms of ISS estimates
of this map under mild assumptions on the input signals. As a consequence of
the availability of these estimates, the equivalence between exponential
stability (ES) of the fixed point of the 0-input (unforced) Poincar\'e map and
ES of the corresponding orbit is recovered. The results can be applied
naturally to study the robustness of periodic orbits of continuous-time systems
as well. Although our motivation for extending classical Poincar\'e analysis to
address ISS stems from the need to design robust controllers for limit-cycle
walking and running robots, the results are applicable to a much broader class
of systems that exhibit periodic solutions.
