Nonlinear Path Following Method

Abstract

This paper develops a new guidance logic for a vehicle to converge to a specified, desired path of general shape in three dimensions. Conditions are found that ensure exponential convergence to the path for any initial vehicle position and velocity vector, excluding only the velocity vector normal to the desired path. The guidance logic includes a ghost vehicle that follows the desired path. The command accelerations are derived from fictitious forces that act in a frame of reference that moves with the ghost. The forces comprise a springlike force between vehicle and ghost and a drag force created by a fictitious medium that moves with the ghost. The motion of the ghost is determined indirectly by a nonlinear constraint that controls the real vehicle speed and leads to a differential-algebraic system of equations. The path equations are formulated in terms of differential geometry, in which time is replaced by distance along the vehicle path. The geometric equations are transformed into standard form differential equations. These are transformed again into kinetic equations for both a fixed medium and a moving medium. The transformed equations provide more explicit formulas for command accelerations and are used to ensure that commands do not exceed the capabilities of the vehicle. Often, the vehicle limitation is more constraining than the convergence conditions. Simulations of a miniature aircraft, using these equations, illustrate the effectiveness of the method. Other methods are compared.