Following a path of varying curvature as an output regulation problem

AbstractGiven a path of nonconstant curvature, local asymptotic stability can be proven for thegeneral n-trailer whenever the curvature can be considered as the output of an exogenousdynamical system. It turns out that the controllers that provide convergence to zero of thetracking error chosen for the path following problem are composed of a prefeedback thatinput-output linearizes the system plus a linear part that can be chosen in an optimal way. 1 Introduction In the path following problem for nonholonomic wheeled vehicles (see (Canudas de Wit, 1998)for a survey), the longitudinal dynamics, expressing how fast the path is covered, is normallyof secondary importance with respect to the lateral dynamics expressing a notion of distance(i.e. a tracking error) of the vehicle form the path by means of a tracking criterion. This isequivalent to say that the longitudinal speed input can be an a priori given function, for examplea nonnull constant. If the tracking error used is a scalar, the system to analyze is basically aSISO system with drift from the steering input to the tracking error. When the curvature ofa path to follow can be modeled as the output of a neutrally stable dynamical system, thenthe path following problem can be formulated as an output regulation problem in the nonlinearsetting proposed by (Isidori and Byrnes, 1990). The curvature can, in fact, be considered as aknown exogenous disturbance and the output of the system, corresponding to the tracking errorof the path following criterion, can be rendered independent from it by input-output linearizingthe system with a static change of input. With the error independent from the curvature, if therelative degree of the system is well de ned, the output zeroing manifold is the only invariantmanifold that solves the regulation problem. This is equivalent to say that local asymptoticstability to the nonconstant steady state is achieved by and only by the controllers composedof a prefeedback that input-output linearizes the system plus a linear part that can be chosenin an optimal (linear) fashion. If we choose as tracking criterion the one proposed in (Alta niand Gutman, 1998) based on the so-called o -tracking distance, whose peculiarity is that itkeeps the whole vehicle (and not a single guidepoint on the vehicle) at a reduced distance fromthe path, then the relative degree between the steering angle and the corresponding trackingerror is equal to 2 whereas for the criteria normally used it is higher: for example taking asguidepoint the midpoint of the last axle would give a relative degree equal to n+ 1 in the