An Amplitude-Phase Representation of the North-East-Down Kinematic Differential Equations

Abstract

The overall goal of the paper is to derive a kinematic model for vehicle path-following systems, where the pitch and yaw angles <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(\theta, \psi)$ </tex-math></inline-formula> together with the surge velocity <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$u$ </tex-math></inline-formula> can be treated as control inputs. Then the outer control loop can be designed as a nonlinear path-following guidance law. At the same time, the inner control loops can be stabilized using commercial autopilot systems to control the pitch and yaw angles. The main result of the paper is a novel kinematic amplitude-phase representation of the North-East-Down (NED) positional rates, which can replace the classical Euler angle rotation matrix representation. The proposed kinematic model gives equivalent NED positional rates as the rotation matrix representation for all combinations of the Euler angles and the linear velocities. The main advantage of the kinematic amplitude-phase representation is the design of vehicle guidance laws using nonlinear control theory and Lyapunov stability analysis. Furthermore, it is shown that nonlinear guidance laws can be designed for path following and that the proposed methods guarantee that the origins are uniformly semiglobally exponentially stable (USGES). A case study of an unmanned surface vehicle (USV) demonstrates how a path-following control system can be designed using the amplitude-phase representation.