A large number of problems in mobile robotics can be reduced to a few basic problem formulations. Most problems reduce to some mixture of optimizing something, solving simultaneous linear or nonlinear equations, or integrating differential equations. Well-known numerical methods exist for all of these problems and all are accessible as black boxes in both software applications and general purpose toolboxes. Of course offline toolboxes cannot be used to control a system in real time and almost any solution benefits from exploiting the nature of the problem, so it is still very common to implement numerical algorithms from scratch for many real time systems.

The techniques described in this section will be referenced in many future places in the text. These techniques will be used to compute wheel velocities, invert dynamic models, generate trajectories, track features in an image, construct globally consistent maps, identify dynamic models, calibrate cameras, and so on.

Linearization and Optimization of Functions of Vectors

Perhaps paradoxically, linearization is the fundamental process that enables us to deal with nonlinear functions. The topics of linearization and optimization are closely linked because a local optimum of a function coincides with special properties of its linear approximation.