  Planning trajectories for nonholonomic systems is difficult and
computationally expensive. When facing unexpected events, it may therefore be
preferable to deform in some way the initially planned trajectory rather than
to re-plan entirely a new one. We suggest here a method based on affine
transformations to make such deformations. This method is exact and fast: the
deformations and the resulting trajectories can be computed algebraically, in
one step, and without any trajectory re-integration. To demonstrate the
possibilities offered by this new method, we use it to derive position and
orientation correction algorithms for the general class of planar wheeled
robots and for a tridimensional underwater vehicle. These algorithms allow in
turn achieving more complex applications, including obstacle avoidance,
feedback control or gap filling for sampling-based kinodynamic planners.
