VFO Path Following Control Strategy for Constrained Motion of Car-Like Robots with Invariant Funnels Computed Using the SOS Optimization

In practical applications of mobile robots it is hard to guarantee perfect tracking of a planned robot path due to uncertainties in measurements and external disturbances. Therefore, one should guarantee that imposed constraints (i.e., collision-free and curvature-constrained robot motion) are satisfied also in a transient stage of the control process when a robot configuration is not aligned with the reference path. This can be achieved using the concept of funnels describing a neighborhood of a reference path, inside of which, all the prescribed constraints are satisfied. In this paper we extend the VFO (Vector Field Orientation) path following control law proposed previously by the authors to car-like robots with strictly limited steering angle and develop a method of computing a safe invariant funnel around a known reference path. Constraints resulting from the steering angle limits and obstacle avoidance are guaranteed to be satisfied in continuous domains of time and configuration space thanks to the application of a control system analysis complemented with the sum-of-squares (SOS) optimization method. Furthermore, a convenient level curve representation is utilized for both reference path and obstacles, rendering explicit computation of a distance to the path and obstacles unnecessary. The proposed method has been verified by simulations of a car-like robot following a reference path starting from non-nominal initial conditions contained in a funnel computed during the motion planning stage.