Improvements to the models in Section 13.1 can be made by
placing integrators in front of action variables. For example,
consider the unicycle model (13.18). Instead of
directly setting the speed using , suppose that the speed is
obtained by integration of an action that represents
acceleration. The equation
is used instead of , which means that the action sets the *change* in speed. If
is chosen from some bounded interval, then the speed is a
continuous function of time.

How should the transition equation be represented in this case? The set of possible values for imposes a second-order constraint on and because double integration is needed to determine their values. By applying the phase space idea, can be considered as a phase variable. This results in a four-dimensional phase space, in which each state is . The state (or phase) transition equation is

which should be compared to (13.18). The action was replaced by because now speed is a phase variable, and an extra equation was added to reflect the connection between speed and acceleration.

The integrator idea can be applied again to make the unicycle
orientations a continuous function of time. Let denote an
angular acceleration action. Let denote the angular
velocity, which is introduced as a new state variable. This results
in a five-dimensional phase space and a model called the
*second-order unicycle*:

in which is a two-dimensional action vector. In some contexts, may be fixed at a constant value, which implies that is fixed to .

Steven M LaValle 2012-04-20