We now introduce some basic differential equations to model motions. The resulting description is often called a *dynamical system*. The first step is to describe rigid body velocities in terms of state. Returning to models that involve one or more rigid bodies, the state corresponds to a finite number of parameters. Let

(8.18) |

denote an -dimensional

(8.19) |

represent the time derivative, or velocity, for each parameter.

To obtain the state at any time , the velocities need to be integrated over time. Following (8.4), the integration of each state variable determines the value at time :

in which is the value of at time .

Two main problems arise with (8.20):

- The integral almost always must be evaluated numerically.
- The velocity must be specified at each time .

Steven M LaValle 2020-01-06