In many applications, additional constraints may exist on the phase
variables. These are called phase constraints and are
generally of the form
. For example, a car or
hovercraft may have a maximum speed for safety reasons. Therefore,
simple bounds on the velocity variables will exist. For example, it
might be specified that
for some constant
. Such simple bounds are often
incorporated directly into the definition of
by placing limits on
the velocity variables.
In other cases, however, constraints on velocity may be quite
complicated. For example, the problem of computing the re-entry
trajectory of the NASA/Lockheed Martin X-33 reusable
spacecraft14.2 (see Figure 14.2)
requires remaining within a complicated, narrow region in the phase
space. Even though there are no hard obstacles in the traditional
sense, many bad things can happen by entering the wrong part of the
phase space. For example, the craft may overheat or vibrate
uncontrollably [160,201,662]. For a simpler example,
imagine constraints on to ensure that an SUV or a double-decker
tour bus (as often seen in London, for example) will not tumble
sideways while turning.
The additional constraints can be expressed implicitly as
. As part of determining whether some state
lies in
or
, it must be substituted into each constraint to determine
whether it is satisfied. If a state lies in
, it will
generally be called violation-free,
which implies that it is both collision-free and does not violate any
additional phase constraints.
Steven M LaValle 2020-08-14