motion and observation models
When a robot tries to figure out where it is, it leans on two little stories it tells itself, written down as equations. The first is the motion model, sometimes called the process model: a rule that says "given where I think I am now, and the command I just sent to my wheels or motors, here is where I should end up next." Drive forward for one second at half a meter per second and the motion model predicts you are roughly half a meter ahead of where you were. The second is the observation model, also called the measurement model: a rule that runs the other direction and says "if I really were standing at this spot, here is what my sensors ought to read." Standing two meters from a wall, the observation model predicts the laser rangefinder should report about two meters. One model predicts how the world changes; the other predicts what the senses will say about it.
These two models are the backbone of nearly every state estimator, because together they let the robot compare a guess against reality. The cycle goes: use the motion model to roll your best guess forward in time (the prediction step), then take a real sensor reading, use the observation model to compute what that reading should have been, and nudge your guess to close the gap between expected and actual (the update step). Crucially, neither model is treated as perfect — each comes bundled with an honest admission of how wrong it tends to be (process noise for the motion model, measurement noise for the observation model), so the robot knows how much to trust each one. Wheels slip, so the motion model is shaky; a good laser is crisp, so the observation model is sharp. The estimator weighs them accordingly.
The art lies in writing models that are accurate enough to be useful but simple enough to compute many times a second. A model can be a clean straight-line equation, or it can be tangled and curved — a camera turning a three-dimensional position into pixel coordinates is a notoriously bendy observation model. When the equations curve, the basic Kalman filter no longer fits, and engineers reach for cousins like the extended or unscented Kalman filter that bend with them. But the two-part skeleton never changes: one model for how I move, one model for how I am seen.
A vacuum robot's motion model says: "I commanded both wheels forward for 0.2 seconds, so I've moved about 6 cm." Its observation model says: "If I'm now 40 cm from the cabinet, my front sonar should ping back ~40 cm." The sonar actually reads 37 cm, so the estimator trusts both models a little and settles its position guess between them.
The motion model predicts where the robot went; the observation model predicts what it should sense there — the estimator reconciles the two.
A handy way to remember the direction of each: the motion model goes state → next state (it predicts the future), while the observation model goes state → expected measurement (it predicts what you'd sense). Both are paired with a noise term, because a model with no admitted uncertainty would make the estimator dangerously overconfident.