Differential equations

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

$\displaystyle x = (x_1,x_2,\ldots,x_n)$ (8.18)

denote an $ n$-dimensional state vector. If each $ x_i$ corresponds to a position or orientation parameter for a rigid body, then the state vector puts all bodies in their place. Let

$\displaystyle {\dot x}_i = \frac{dx_i}{dt}$ (8.19)

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

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

$\displaystyle x_i(t) = x_i(0) + \int_0^t {\dot x}_i(s) ds ,$ (8.20)

in which $ x_i(0)$ is the value of $ x_i$ at time $ t = 0$.

Two main problems arise with (8.20):

  1. The integral almost always must be evaluated numerically.
  2. The velocity $ {\dot x}_i(t)$ must be specified at each time $ t$.

Steven M LaValle 2016-12-31