Learning Nice Robust Trajectories for a Car-Like Robot Enrico Ferretti ∗ , Giuseppe Oriolo ∗ , Stefano Panzieri † , Giovanni Ulivi † * Dipartimento di Informatica e Sistemistica, Universit`a di Roma “La Sapienza” Via Eudossiana 18, 00184 Roma, Italy † Dipartimento di Discipline Scientifiche, Terza Universit`a di Roma Via della Vasca Navale, 84, 00146 Roma, Italy {ferretti,oriolo,panzieri,ulivi}@labrob.ing.uniroma1.it Abstract A finite-dimensional iterative learning controller is presented that can steer a car-like robot in finite time between given configurations while optimizing a given performance criterion. The robustness property of the controller guarantees that the obtained trajectory achieves exact steering even in the presence of perturba- tions with respect to the nominal model. By properly choosing the performance criterion, it is possible to minimize the trajectory length, to avoid saturations of the steering angle as well as to generate collision-free trajectories among workspace obstacles. Simulation results are reported to show the satisfactory performance of the method. 1 Introduction The problem of moving a a car-like robot between given configurations with trajectories that are optimal in some sense but at the same time can be executed in perturbed conditions is quite entangled. On the one hand, a number of planning algorithms have been proposed for generating trajectories that give rise to natural and smooth motions (e.g., see [1]). On the other hand, these trajectories are computed in open-loop, and a feedback action is mandatory to obtain real-time accurate tracking. However, the peculiar features of the car-like robot, and namely its being a nonholonomic system, make the synthesis of trajectory tracking controllers difficult. In particular, to prove asymptotic stability one is forced to make the somewhat unrealistic assumption that the trajectory never stops [2]. In this paper, we propose a solution approach based on the use of a learning controller. Learning is a methodology for improving the performance of a control system through iterative training [3]. A general framework for solving the finite-time steering problem through learning has been proposed in [4]; specific applications include nonholonomic mobile robots [5] and flexible structures [6, 7]. Our learning algorithm makes use of a parameterized control chosen in a finite-dimensional class. In particular, the input vector is the linear combination of a suitable set of generating functions. By augmenting the dimension of the coefficient vector with respect to the number of the state variables n, we obtain an overparameterized controller that can meet other performance goals, typically expressed as cost criteria to be minimized. The main advantages of this approach are (i) the fact that it is not necessary to provide a trajectory connecting the initial and final configurations, (ii) the flexibility in the choice of the trajectory cost criterion, and (iii) the robustness properties of the obtained trajectories against disturbances and model perturbations. In particular, we shall apply the above approach in order to plan robust trajectories that minimize the total length, the maximum steering angle or the distance from workspace obstacles inside the chosen control class. This is obtained by expressing these cost criteria as functions of the control parameters and performing a constrained optimization process in the parameter space. Simulations results for a laboratory prototype are reported in order to show the effectiveness of the approach.