Balance systems
This page documents the various "balance systems" that are used as a running examplse throughout the text. A detailed description of the dynamics of this class of systems is presented in Chapter 2 - System Modeling. This page contains a description of the system, including the models and commands used to generate many of the plots in the text.
Introduction
A balance system is a mechanical system in which the center of mass is balanced above a pivot point. Some common examples of balance systems are shown below.
The Segway human transportation system (left) uses a motorized platform to stabilize a person standing on top of it. When the rider leans forward, the vehicle propels itself along the ground, but maintains its upright position. Another example is a rocket (\figref{fig:modeling:balance}b), in which a gimbaled nozzle at the bottom of the rocket is used to stabilize the body of the rocket above it. Other examples of balance systems include humans or other animals standing upright or a person balancing a stick on their hand.
The figure to the right show a schematic diagram of a simple balance system. To model this system, we choose state variables that represent the position and velocity of the base of the system, 
and 
, and the angle and angular rate of the structure above the base, 
and 
. We let 
represent the force applied at the base of the
system, assumed to be in the horizontal direction (aligned with ), and choose the position and angle of the system as outputs. With this set of definitions, the dynamics of the system can be computed using Newtonian
mechanics and has the form:
