Asteroids orbiting into the highly magnetized and highly relativistic wind of a pulsar offer a favourable configuration for repeating fast radio bursts (FRB). The body in direct contact with the wind develops a trail formed of a stationary Alfv\'en wave, called an \textit{Alfv\'en wing}. When an element of wind crosses the Alfv\'en wing, it sees a rotation of the ambient magnetic field that can cause radio-wave instabilities. In the observer's reference frame, the waves are collimated in a very narrow range of directions, and they have an extremely high intensity. A previous work, published in 2014, showed that planets orbiting a pulsar can cause FRB when they pass in our line of sight. We predicted periodic FRB. Since then random FRB repeaters have been discovered. We present an upgrade of this theory where repeaters can be explained by the interaction of smaller bodies with a pulsar wind. Considering the properties of relativistic Alfv\'en wings attached to a body in the pulsar wind, and taking thermal consideration into account we conduct a parametric study. We find that FRBs, including the Lorimer burst (30 Jy), can be explained by small size pulsar companions (1 to 10 km) between 0.03 and 1 AU from a highly magnetized millisecond pulsar. Some sets of parameters are also compatible with a magnetar. Our model is compatible with the high rotation measure of FRB121102. The bunched timing of the FRBs is the consequence of a moderate wind turbulence. As asteroid belt composed of less than 200 bodies would suffice for the FRB occurrence rate measured with FRB121102. This model, after the present upgrade, is compatible with the properties discovered since its first publication in 2014, when repeating FRB were still unknown. It is based on standard physics, and on common astrophysical objects that can be found in any kind of galaxy. It requires $10^{10}$ times less power than (common) isotropic-emission FRB models.