Sufficient and necessary conditions are established for controllability of affine control systems where the control is constrained to a set whose convex hull contains the origin but is not necessarily, in contrast with previously known results, a neighborhood of the origin. Part of the results, in particular these on global controllability, are specific to systems where all solutions with zero control (drift) are periodic. These conditions are expressed by means of pushforwards along the flow of the drift, rather than in terms of Lie brackets; it turns out that they amount to local controllability of a time-varying linear approximation with constrained controls. Global and local results are given, as well as a few illustrative examples.