observability (estimation sense)
Observability asks a deceptively simple question: from the measurements I can actually take, can I eventually figure out everything I want to know about the robot's hidden state? The state is the full set of numbers that describe the robot's situation — its position, its heading, maybe its speed, maybe the slow drift of a sensor. Many of those numbers are never measured directly; the robot must infer them from whatever its sensors do report. If, by watching the sensors long enough, there is exactly one possible state that could have produced everything seen, the state is observable. If two genuinely different states would produce identical sensor readings forever, the robot can never tell them apart, and that hidden quantity is unobservable — no amount of clever math will recover it, because the information simply was never captured.
A homely picture helps. Imagine standing in a featureless white room blindfolded, given only a compass. You can always tell which way you are facing, so your heading is observable. But the compass says nothing about whether you are by the north wall or the south wall, so your position is unobservable — every spot in the room looks identical to your single sensor. Now hand yourself a tape measure to the nearest wall and start walking: as the readings change with your movement, the once-hidden position becomes recoverable. This is the deep twist of observability in estimation: it often depends on motion. A quantity invisible while the robot sits still can become visible once it moves the right way, because moving makes the sensors tell a richer, changing story.
Why care? Because if part of the state is unobservable, no filter — not the Kalman filter, not the particle filter, none of them — can pin it down; the best they can do is keep an honest, ever-growing cloud of uncertainty around that piece. Engineers therefore check observability before they trust an estimator, and often design the robot's sensors or its movements specifically to make the things they care about observable. It is the difference between a problem the data can actually answer and one where the robot is doomed to guess, no matter how good its algorithm.
A robot tracking its position with only a single distance sensor to one wall cannot tell left from right along that wall — those positions are unobservable and its uncertainty there grows without bound. Add a second wall sensor at a right angle, or simply let the robot turn and drive, and the two readings together pin the position down: the state becomes observable.
One wall sensor leaves position-along-the-wall unobservable; a second sensor — or motion — makes the full position recoverable.
Observability is about whether the information exists in the measurements at all — it is separate from whether your filter is good enough to extract it. A perfectly observable system can still be estimated badly by a poor algorithm; but an unobservable quantity cannot be estimated by any algorithm, however perfect.