Multi-input control-affine systems static feedback equivalent to a triangular form and their flatness

In this paper, we give a complete geometric characterisation of systems locally static feedback equivalent to a triangular form compatible with the chained form, for m = 1, respectively with the m-chained form, for m ≥ 2, where the number of controls is m + 1. They are x-flat systems. We provide a system of first-order PDE's to be solved in order to find all x-flat outputs, for m = 1, respectively all minimal x-flat outputs, for m ≥ 2. We illustrate our results by examples, in particular, by an application to a mechanical system: the coin rolling without slipping on a moving table.