Landmark sensors

Many important sensing models can be defined in terms of landmarks. A landmark is a special point or region in the state space that can be detected in some way by the sensor. The measurements of the landmark can be used to make inferences about the current state. An ancient example is using stars to navigate on the ocean. Based on the location of the stars relative to a ship, its orientation can be inferred. You may have found landmarks useful for trying to find your way through an unfamiliar city. For example, mountains around the perimeter of Mexico City or the Eiffel Tower in Paris might be used to infer your heading. Even though the streets of Paris are very complicated, it might be possible to walk to the Eiffel Tower by walking toward it whenever it is visible. Such models are common in the competitive ratio framework for analyzing on-line algorithms [674].

Figure 11.13: The most basic landmark sensor indicates only its direction.

In general, a set of states may serve as landmarks. A common model is to make $ {x_{G}}$ a single landmark. In robotics applications, these landmarks may be instead considered as points in the world, $ {\cal W}$. Generalizations from points to landmark regions are also possible. The ideas, here, however, will be kept simple to illustrate the concept. Following this presentation, you can imagine a wide variety of generalizations. Assume for all examples of landmarks that $ X = {\mathbb{R}}^2$, and let a state be denoted by $ x = (x_1,x_2)$.

For the first examples, suppose there is only one landmark, $ l \in X$, with coordinates $ (l_1,l_2)$. A homing sensor is depicted in Figure 11.13 and yields values in $ Y =
{\mathbb{S}}^1$. The sensor mapping is $ h(x) = \atan2(l_1-x_1,l_2-x_2)$, in which $ \atan2$ gives the angle in the proper quadrant.

Another possibility is a Geiger counter sensor (radiation level), in which $ Y = [0,\infty)$ and $ h(x) = \Vert x-l\Vert$. In this case, only the distance to the landmark is reported, but there is no directional information.

A contact sensor could also be combined with the landmark idea to yield a sensor called a pebble. This sensor reports $ 1$ if the pebble is ``touched''; otherwise, it reports 0. This idea can be generalized nicely to regions. Imagine that there is a landmark region, $ X_l \subset X$. If $ x \in X_l$, then the landmark region detector reports $ 1$; otherwise, it reports 0.

Many useful and interesting sensing models can be formulated by using the ideas explained so far with multiple landmarks. For example, using three homing sensors that are not collinear, it is possible to reconstruct the exact state. Many interesting problems can be made by populating the state space with landmark regions and their associated detectors. In mobile robotics applications, this can be implemented by placing stationary cameras or other sensors in an environment. The sensors can indicate which cameras can currently view the robot. They might also provide the distance from each camera.

Figure 11.14: (a) A mobile robot is dropped into an unknown environment. (b) Four sonars are used to measure distances to the walls.
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\psfig{figure=figs/gaps6.idr,width=2.5in} \\
(a) & (b)

Steven M LaValle 2012-04-20