9.1: Motivating Example

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Imagine a floor with three doors, two of which are closer together, and the third farther down the corridor (Figure 9.1). Imagine know that your robot is able to detect doors, i.e., is able to tell whether it is in front of a wall or in front of a door. Such features can serve the robot as a landmark. Given a map of this simple environment and no information whatsoever where our robot is located, we can use landmarks to drastically reduce the space of possible locations once the robot has passed one of the doors. One way of representing this belief is to describe the robot’s position with three Gaussian distributions, each centered in front of a door and its variance a function of the uncertainty with which the robot can detect a door’s center. (This approach is known as a multi-hypothesis belief.) What happens if the robot continues to move? From the error propagation law we know:

The Gaussians describing the robot’s 3 possible locations will move with the robot.
The variance of each Gaussian will keep increasing with the distance the robot moves.

What happens if the robot arrives at another door? Given a map of the environment, we can now map the three Gaussian distributions to the location of the three doors. As all three Gaussians will have moved, but the doors are not equally spaced, only some of the peaks will coincide with the location of a door. Assuming we trust our door detector much more than our odometry estimate, we can now remove all beliefs that do not coincide with a door. Again assuming our door detector can detect the center of a door with some accuracy, our location estimate’s uncertainty is now only limited by that of the door detector.

Things are just slightly more complicated if our door detector is also subject to uncertainty: there is a chance that we are in front of a door, but haven’t noticed it. Then, it would be a mistake to remove this belief. Instead, we just weight all beliefs with the probability that there could be a door. Say our door detector detects false-positives with a 10% chance. Then, there is a 10% chance to be at any location that is not in front of door, even if our detector tells us we are in front of a door. Similarly, our detector might detect false-negatives with 20% chance, i.e., tell us there is no door even though the robot is just in front of it. Thus, we would need to weigh all locations in front of a door with 20% chance and all locations not in front of a door with 80% likelihood if our robot tells us there is no door, even if we are indeed in front of one.