The nonlinear path following guidance law is a relatively simple way to guide a vehicle on a desired path. How it works is briefly explained here.
For a given desired path, the guidance law designates a reference point on the desired path with a certain distance L, in front of the aircraft . Then, it creates acceleration command using the velocity vector and the L vector based on this simple vector formula.
According to our analysis, there are a number of good features in this method. It contains proportional and derivative controls on the crosstrack error as well as an anticipation element. The anticipation comes from the fact that it uses the reference point forward of the vehicle. The bandwidth of this guidance law depends on the ratio of the speed and the distance L. So we can use this information to choose the distance L and to check stability when we add an inner-loop.
It can be shown that the acceleration vector lies on the plane defined by the vector V and L, and also it is perpendicular to the velocity vector. In other words this acceleration is rather for changing the direction as opposed to changing the speed. The corresponding magnitude of the acceleration command is
So it depends on the angle defined between vector V and L. Now, let me briefly explain why this simple formula may be very useful by investigating the angle h.
In the diagram above there is an airplane which is currently deviated from the desired path. Here it is assumed that, at least locally, the desired path is approximately a circular arc. The investigation of this diagram reveals that
Here, h1 comes from the fact that there is a cross-track error. So it corresponds to a proportional control over the cross-track error. h2 comes from the fact that the velocity direction is such that the vehicle is getting away from the desired path . So it corresponds to a derivative control over the cross-track error. h3 can be shown to be closely related with the curvature of the local circular arc. So this components tells the vehicle how much and in what direction the upcoming path command is curved.
For more details, please refer to "Performance and Lyapunov Stability of a Nonlinear Path-Following Guidance Method", S. Park, J. Deyst , J. How, Journal of Guidance, Control, and Dynamics 2007.