Stability of Following in Two Dimensions

The mechanism by which objects follow each other on a two-dimensional surface is considered, with the movement of a convoy of ships in mind as a physical example. This is to be compared with single-lane traffic dynamics, which is essentially a one-dimensional following theory applied to cars on a straight highway. At first the assumed laws of following are linear in the position and velocity vectors of the ships, and the form of relation used is one suggested partly by previous work on traffic dynamics and partly from the theory of guided missiles. As a measure of imperfections in speed control and steering, various types of time constants are introduced and the resulting equations tested for stability, yielding criteria for the maximum permissible size of the time constants. Nonlinear control equations are also investigated, and by making linear approximations to the trajectories in these cases, stability criteria are again suggested. Finally a few computed trajectories have been obtained, including an interesting example where the following ship suddenly reverses its direction of motion in course.