Stable gaits of passive walking mechanisms often have narrow and strangely shaped regions of attraction, which are difficult both to estimate analytically and to enlarge using feedback from just a few actuators. For instance, linearization of the first-return Poincaré map, being decisive for exponential orbital stability of a limit cycle, can be used for estimating neither its region of attraction nor possible deviations from a nominal trajectory in response to small parametric perturbations. In this talk, we reiterate and refine the arguments presented in 2008 and describe a procedure for constructing such estimates based on the notions of virtual holonomic constraints and transverse linearization. The method allows: (a) revealing mechanisms of instability of cycles when dynamics of an underactuated biped is perturbed by imperfections of a walking surface; (b) developing tools for planning gaits insensitive to certain perturbations; (c) designing robustly stabilizing feedback controllers. The results are presented on the examples of a compass-gait biped and a quadruped.