Sub-optimal motion planning of one-chained, two-input nonholonomic systems

In this paper three algorithms of motion planning for two-input, one-chained nonholonomic systems are presented. The classical Murray-Sastry algorithm is compared with two original algorithms aimed at optimizing energy of controls. Based on the generalized Campbell- Baker-Hausdorff-Dynkin formula applied to the systems, some observations are made concerning the optimal relationship between amplitudes and phases of harmonic controls. The observations help to optimize a selection of controls and to design new algorithms for planning a sub- optimal trajectory between given boundary configurations. It was also shown that for those particular systems the generalized C-B-H-D formula is valid not only locally (as in a typical case) but also globally. Simulations performed on the five-dimensional chain system facilitate distinguishing the proposed algorithms from the Murray-Sastry algorithm and to illustrate their features. Systems in a chained form are important from a practical point of view as they are canonical for a class of systems transformable into this form. The most prominent among them are mobile robots with or without trailers.