For purposes of illustration, suppose that
. A
configuration is expressed as
, and a velocity is
expressed as
. Each
is an element of
the tangent space
, which is a twodimensional vector
space at every . Think about the kinds of constraints that
could be imposed. At each
, restricting the set of
velocities yields some set
. The parametric
and implicit representations will be alternative ways to express
for all
.
Here are some interesting, simple constraints. Each yields a
definition of as the subset of
that satisfies the
constraints.

: In this case, imagine that you are
paddling a boat on a swift river that flows in the positive
direction. You can obtain any velocity you like in the direction,
but you can never flow against the current. This means that all
integral curves increase monotonically in over time.

: This constraint allows you to
stop moving in the direction. A velocity perpendicular to the
current can be obtained (for example, causes motion with unit
speed in the positive direction).

,
: Under this constraint, integral
curves must monotonically increase in both and .

: In the previous three examples,
the set of allowable velocities remained twodimensional. Under the
constraint
, the set of allowable velocities is only
onedimensional. All vectors of the form
for any
are allowed. This means that no motion in the direction
is allowed. Starting at any , the integral curves will be
of the form
for all
, which confines
each one to a vertical line.

: This constraint is qualitatively the
same as the previous one. The difference is that now the motions can
be restricted along any collection of parallel lines by choosing
and . For example, if , then only diagonal motions are
allowed.

: This constraint is
similar to the previous one, however the behavior is quite different
because the integral curves do not coincide. An entire half plane
is reached. It also impossible to stop becasue
violates the constraint.

: This constraint was
used in Chapter 8. It has no effect on the existence
of solutions to the feasible motion planning problem because motion in
any direction is still allowed. The constraint only enforces a
maximum speed.

: This constraint
allows motions in any direction and at any speed greater than . It
is impossible to stop or slow down below unit speed.
Many other constraints can be imagined, including some that define
very complicated regions in
for each . Ignoring the
fact that and represent derivatives, the geometric
modeling concepts from Section 3.1 can even be used to define
complicated constraints at each . In fact, the constraints
expressed above in terms of and are simple examples of
the semialgebraic model, which was introduced in Section
3.1.2. Just replace and from that section
by and here.
If at every there exists some open set such that
and
, then there is no effect on the existence of
solutions to the feasible motion planning problem. Velocities in all
directions are still allowed. This holds true for velocity
constraints on any smooth manifold [924].
So far, the velocities have been constrained in the same way at every
, which means that is the same for all
. Constraints of this kind are of the form
, in which could be , , , , or
, and is a function from
to
. Typically, the
relation drops the dimension of by one, and the others
usually leave it unchanged.
Now consider the constraint
. This results in a different
onedimensional set, , of allowable velocities at each
. At each , the set of allowable velocities must be
of the form
for any
. This means that as
increases, the velocity in the direction must increase
proportionally. Starting at any positive value, there is no way
to travel to the axis. However, starting on the axis, the
integral curves will always remain on it! Constraints of this kind
can generally be expressed as
, which
allows the dependency on or .
Steven M LaValle
20120420