Asteroids orbiting into the highly magnetized and highly relativistic wind of a pulsar offer a favorable configuration for repeating fast radio bursts (FRB). The body in direct contact with the wind develops a trail formed of a stationary Alfvén wave, called an Alfv\'en wing. When an element of wind crosses the Alfvén wing, it sees a rotation of the ambient magnetic field that can cause radio-wave instabilities. In our 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 to see if they could be explained by the interaction of smaller bodies with a pulsar wind. Considering the properties of relativistic Alfvén wings attached to a body in the pulsar wind, and taking thermal consideration into account (the body must be in solid state) we conduct a parametric study. We find that FRBs can be explained by small size pulsar companions (1 to 10 km) between 0.03 and 1 AU from a pulsar. The intense Lorimer burst (30 Jy) can be explained. Some sets of parameters are also compatible with a magnetar, as suggested for FRB121102. Actually, small bodies orbiting a magnetar could produce (not yet observed) FRBs with $\sim 10^5$ Jy flux density. 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.