Mobile Robotics, 2nd Edition

4Guidance

In previous chapters, we have studied how to build a control law for a robot described by nonlinear state equations (see Chapter 2) or when the robot’s behavior is known (see Chapter 3). Guidance is performed on a higher level and focuses on the setpoint given to the controller in order for the robot to be able to accomplish its assigned mission. It will, therefore, have to take into account the knowledge of its surroundings, the presence of obstacles, the roundness of the Earth, and so forth. Conventionally, guidance is applied in four different environments: terrestrial, marine, aerial and spatial. Given the fields of application covered in this book, we will not study the spatial environment.

4.1. Guidance on a sphere

For longer paths over the surface of the Earth, the Cartesian coordinate system – which assumes a flat Earth – can no longer be considered. We then have to rethink our control laws by navigating relative to a spherical coordinate system (also referred to as geographical coordinates), which rotates together with the Earth. Let us denote by ℓ_x the longitude and by ℓ_y the latitude of the point being considered. The transformation in the geographical coordinate system is written as:

[4.1]

When ρ = 6, 370 km, we are on the surface of the Earth, which we will assume to be spherical (see Figure 4.1(a)).

Let us consider two points a and m on the surface of ...