13.1.3.2 Flying an airplane

The Dubins car model from Section 13.1.2 can be extended to 3D worlds to provide a simple aircraft flight model that may be reasonable for air traffic analysis. First suppose that the aircraft maintains a fixed altitude and is capable only of yaw rotations. In this case, (13.15) could be used directly by imposing the constraint that $s = 1$ (or some suitable positive speed). This is equivalent to the Dubins car, except that $s = 0$ is prohibited because it would imply that the aircraft can instantaneously stop in the air. This model assumes that the aircraft is small relative to the C-space. A more precise model should take into account pitch and roll rotations, disturbances, and dynamic effects. These would become important, for example, in studying the flight stability of an aircraft design. Such concerns are neglected here.

Now consider an aircraft that can change its altitude, in addition to executing motions like the Dubins car. In this case let ${\cal C}= {\mathbb{R}}^3 \times {\mathbb{S}}^1$, in which the extra ${\mathbb{R}}$ represents the altitude with respect to flying over a flat surface. A configuration is represented as $q = (x,y,z,\theta)$. Let $u_z$ denote an action that directly causes a change in the altitude: ${\dot z}= u_z$. The steering action $u_\phi$ is the same as in the Dubins car model. The configuration transition equation is

$${\dot x} = \cos\theta \qquad {\dot z} = u_z$$

$${\dot y} = \sin\theta \qquad {\dot \theta} = u_\omega .$$ (13.20)

For a fixed value of $u = (u_z,u_\omega)$ such that $u_z \neq 0$ and $u_\omega \neq 0$, a helical path results. The central axis of the helix is parallel to the $z$-axis, and projection of the path down to the $xy$ plane is a circle or circular arc. Maximum absolute values should be set for $u_z$ and $u_\omega$ based on the maximum possible altitude and yaw rate changes of the aircraft.