Mobile Robotics, 2nd Edition

1Three-dimensional Modeling

This chapter looks at the three-dimensional modeling of a solid (non-articulated) robot. Such modeling is used to represent airplanes, quadcopters, submarines, and so on. Through this modeling we will introduce a number of fundamental concepts in robotics such as state representation, rotation matrices and Euler angles. The robots, whether mobile, manipulator or articulated, can generally be put into a state representation form:

where x is the state vector, u the input vector and y the vector of measurements [JAU 05]. We will call modeling the step which consists of finding a more or less accurate state representation of the robot. In general, constant parameters may appear in the state equations (such as the mass and the moment of inertia of a body, viscosity, etc.). In such cases, an identification step might prove to be necessary. We will assume that all of the parameters are known. Of course, there is no systematic methodology that can be applied for modeling a mobile robot. The aim of this chapter is to present the tools which allow us to reach a state representation of three-dimensional solid robots in order for the reader to acquire a certain experience which will be helpful when modeling his/her own robots. This modeling will also allow us to recall a number of important concepts in Euclidean geometry, which are fundamental in mobile robotics. ...