A feedback plan, , is called a *solution* to the problem
in Formulation 8.2 if from all
, the
integral curves of (considered as a vector field) arrive in
, at which point the termination action is applied. Some
words of caution must be given about what it means to ``arrive'' in
. Notions of stability from control theory [523,846]
are useful for distinguishing different cases; see Section
15.1. If is a small ball centered on ,
then the ball will be reached after finite time using the inward
vector field shown in Figure 8.5b. Now suppose that
is a single point, . The inward vector field
produces velocities that bring the state closer and closer to the
origin, but when is it actually reached? It turns out that
convergence to the origin in this case is only *asymptotic*; the origin is
reached in the limit as the time approaches infinity. Such stability
often arises in control theory from smooth vector fields. We may
allow such asymptotic convergence to the goal (if the vector field is
smooth and the goal is a point, then this is unavoidable). If any
integral curves result in only asymptotic convergence to the goal,
then a solution plan is called an *asymptotic solution
plan*. Note that in general it may be
impossible to require that is a smooth (or even continuous)
nonzero vector field. For example, due to the *hairy ball
theorem* [834], it is known that no such vector field exists
for
for any even integer . Therefore, the strongest
possible requirement is that is smooth except on a set of
measure zero; see Section 8.4.4. We may also allow
solutions for which *almost all* integral curves arrive
in .

However, it will be assumed by default in this chapter that a solution
plan converges to in finite time. For example, if the inward
field is normalized to produce unit speed everywhere except the
origin, then the origin will be reached in finite time. A constraint
can be placed on the set of allowable vector fields without affecting
the existence of a solution plan. As in the basic motion
planning problem, the speed along the path is not important. Let a
*normalized vector field* be any
vector field for which either
or , for all . This means that all velocity vectors are either unit vectors
or the zero vector, and the speed is no longer a factor. A normalized
vector field provides either a direction of motion or no motion.
Note that any vector field can be converted into a normalized
vector field by dividing the velocity vector by its magnitude
(unless the magnitude is zero), for each .

In many cases, unit speed does not necessarily imply a constant speed
in some true physical sense. For example, if the robot is a floating
rigid body, there are many ways to parameterize position and
orientation. The speed of the body is sensitive to this
parameterization. Therefore, other constraints may be preferable
instead of
; however, it is important to keep in mind
that the constraint is imposed so that provides a *direction* at . The particular magnitude is assumed unimportant.

So far, consideration has been given only to a *feasible feedback
motion planning problem*. An *optimal feedback motion planning
problem* can be defined by introducing a cost functional. Let
denote the function
, which
is called the *state trajectory* (or *state history*). This
is a continuous-time version of the state history, which was defined
previously for problems that have discrete stages. Similarly, let
denote the *action trajectory* (or *action
history*),
. Let denote a cost
functional, which may be applied from any to yield

in which is the time at which the termination action is applied. The term can alternatively be expressed as by using the state transition equation (8.38). A normalized vector field that optimizes (8.39) from all initial states that can reach the goal is considered as an

Note that the state trajectory can be determined from an action history and initial state. In fact, we could have used action trajectories to define a solution path to the motion planning problem of Chapter 4. Instead, a solution was defined there as a path to avoid having to introduce velocity fields on smooth manifolds. That was the only place in the book in which the action space seemed to disappear, and now you can see that it was only hiding to avoid inessential notation.

Steven M LaValle 2020-08-14