Considering the acceleration-based control model, some constraints can
be placed on the set of allowable vector fields. A
*bounded-velocity model* means that
, for
some positive real value called the *maximum speed*.
This could indicate, for example, that the robot has a maximum speed
for safety reasons. It is also possible to bound individual
components of the velocity vector. For example, there may be separate
bounds for the maximum angular and linear velocities of an aircraft.
Intuitively, velocity bounds imply that the functions , which
define the vector field, cannot take on large values.

A *bounded-acceleration model* means that
, in which is a positive real value called the *maximum acceleration*. Intuitively, acceleration bounds imply that
the velocity cannot change too quickly while traveling along an
integral curve. Using the control model
, this implies
that
. It also imposes the constraint that vector
fields must satisfy
for all
and . The condition
is very
important in practice because higher accelerations are generally more
expensive (bigger motors are required, more fuel is consumed, etc.).
The action may correspond directly to the torques that are
applied to motors. In this case, each motor usually has an upper
limit.

As has already been seen, setting an upper bound on velocity generally does not affect the existence of a solution. Imagine that a robot can always decide to travel more slowly. If there is also an upper bound on acceleration, then the robot can attempt to travel more slowly to satisfy the bound. Imagine slowing down in a car to make a sharp turn. If you would like to go faster, then it may be more difficult to satisfy acceleration constraints. Nevertheless, in most situations, it is preferable to go faster.

A discontinuous vector field fails to satisfy any acceleration bound because it essentially requires infinite acceleration at the discontinuity to cause a discontinuous jump in the velocity vector. If the vector field satisfies the Lipschitz condition (8.16) for some constant , then it satisfies the acceleration bound if .

In Chapter 13, we will precisely specify at every , which is more general than imposing simple velocity and acceleration bounds. This enables virtually any physical system to be modeled.

Steven M LaValle 2020-08-14