Stanford OnlineStanford AA274A Principles of Robotic Autonomy | Autumn 2019 | Parametric Filtering
Stanford Online
Localization review 0:05
Localization is framed as estimating your position with respect to a known map. The Bayes filter does that by repeating two steps. First it predicts where you might be after a control input. Then it corrects that guess with the newest sensor reading. The prediction step sums over every prior state that could lead to the current one. The correction step checks whether the belief fits the measurement and lowers or raises the probability accordingly.
Making filtering practical 9:03
The full Bayes filter is simple in idea but hard to compute directly, because it ranges over continuous states and uses high-dimensional integrals. Two main ways make it manageable. Parametric filters assume a shaped belief, often Gaussian, so you track only its mean and variance. Nonparametric filters keep no fixed shape and instead use approximations such as samples or particles. The Gaussian choice is useful because it leads to fast computation when the belief stays roughly unimodal.
Gaussian assumptions 25:30
Adding two Gaussian random variables gives another Gaussian. Their expected values add, and so do their covariance matrices. Multiplying two Gaussian density functions also gives a Gaussian density, after completing the square. These three facts are taken as given and will be used repeatedly. The lecture points you to Probabilistic Robotics for more reading and says the same ideas will keep appearing later.
Linear Bayes form 28:03
The goal is to make the Bayes filter computable by assuming linear dynamics and linear measurements. The state update is written as a linear map of the previous state and current control, plus Gaussian process noise with zero mean and covariance Rt. The measurement model is also linear, with Gaussian noise of zero mean and covariance Qt. If the initial belief is Gaussian too, then the belief stays Gaussian at every time.
Kalman update 51:30
The state prediction is handled by adding two independent Gaussian terms, so the result stays Gaussian. Its mean becomes the linear prediction, and its covariance comes from the state term plus the process noise. The same logic gives the first Kalman filter equations without doing the long square-completing derivation.
Extended filter idea 59:01
When the motion model or the measurement model is nonlinear, the belief is no longer exactly Gaussian. The fix is to linearize the nonlinear function around the current mean, using the first-order Taylor expansion. That gives a local line, lets you push a Gaussian through it, and leads to the extended Kalman filter with Jacobians replacing the original matrices.
UKF versus EKF 1:16:30
The UKF is presented as more robust, though harder to implement. EKF linearizes the transformation. UKF instead approximates the input random variable, passes it through the nonlinear function, and reconstructs the output density. That gives a better Gaussian fit when uncertainty is wide, so it often works better under high uncertainty.
AI-generated summary. It can be wrong or incomplete - check anything that matters against the original.
