Several common sensor models can be defined by observing particular
coordinates of while leaving others invisible. This is the
continuous version of the selective sensor from Example
11.4. Imagine, for example, a mobile robot that rolls
in a 2D world,
, and is capable of rotation. The state
space (or configuration space) is
. For
visualization purposes, it may be helpful to imagine that the robot is
very tiny, so that it can be interpreted as a point, to avoid the
complicated configuration space constructions of Section
4.3.^{11.7} Let
denote the coordinates of the point, and let
denote its orientation. Thus, a state in
is
specified as
(rather than
, which may
cause confusion with important spaces such as , , and ).

Suppose that the robot can estimate its position but does not know
its orientation. This leads to a *position sensor* defined as
, with and (also denoted as
). The third state variable, , of the state remains unknown.
Of course, any of the previously considered nature sensing action
models can be added. For example, nature might cause errors that are
modeled with Gaussian probability densities.

A *compass* or *orientation sensor* can likewise be made by
observing only the final state variable, . In this case,
and . Nature sensing actions can be included. For
example, the sensed orientation may be , but it is only known that
for some constant , which is the
maximum sensor error. A Gaussian model cannot exactly be applied
because its domain is unbounded and
is bounded. This can be
fixed by truncating the Gaussian or by using a more appropriate
distribution.

The position and orientation sensors generalize nicely to a 3D world,
. Recall from Section 4.2 that in this
case the state space is , which can be represented as
. A position sensor measures the first three
coordinates, whereas an orientation sensor measures the last three
coordinates. A physical sensor that measures orientation in
is often called a *gyroscope*. These are usually based on the
principle of precession, which means that they contain a spinning disc
that is reluctant to change its orientation due to angular momentum.
For the case of a linkage of bodies that are connected by revolute
joints, a point in the state space gives the angles between each pair
of attached links. A *joint encoder* is a sensor that yields one
of these angles.

Dynamics of mechanical systems will not be considered until Part
IV; however, it is worth pointing out several sensors. In
these problems, the state space will be expanded to include velocity
parameters and possibly even acceleration parameters. In this case,
a *speedometer* can sense a velocity vector or a scalar speed.
Sensors even exist to measure *angular velocity*, which indicates
the speed with which rotation occurs. Finally, an *accelerometer*
can be used to sense acceleration parameters. With any of these
models, nature sensing actions can be used to account for
measurement errors.

Steven M LaValle 2020-08-14