Stanford OnlineStanford AA274A Principles of Robotic Autonomy | Autumn 2019 | Localization & Filters
Stanford Online
Localization architecture 0:05
The class turns to robot localization and mapping and places them inside the usual perceive, localize, decide, and control stack. Perception filters sensor data. Localization and mapping combine that data with prior information to estimate where the robot is and where objects are. The lesson focus is the map-based approach, not behavior-only methods such as wall following, because maps support broader use even if they are harder to build and maintain.
Belief and probability 7:01
The map-based problem breaks into map representation and belief representation. Here the concern is belief: how to encode position and uncertainty, then update both from sensors such as lidars and cameras. Beliefs may be unimodal or multimodal, and continuous or discretized. To update them, the lecture reviews probability basics: random variables, probability mass and density functions, joint probability, independence, and conditional probability.
Conditional probability 18:31
Use the sample space to read conditional probability. If a point is thrown uniformly into a rectangle, the chance of landing in A is its area, 0.75. If you already know the point is in B, then the chance it is also in A is the overlap divided by B, so 0.25 over 0.5, which is 1/2. If the point is in C, the chance it is in A is 1 because C lies inside A.
Bayes and filtering 25:30
Independence means conditioning gives no new information, so P(X given Y) becomes P(X). The law of total probability lets you sum over all values of Y, or integrate in the continuous case. Bayes’ rule flips the conditioning: P(X given Y) equals P(Y given X)P(X) over P(Y). In robotics, Y is sensor data and X is state, and the normalizer eta does not depend on X.
Expectation and covariance 40:00
Expectation is treated as a linear operator. For a random variable X, the expectation of AX+B equals A times the expectation of X plus B. For a vector of random variables, you take the expectation of each component. Covariance measures joint variability between two random variables. It is zero when they vary independently, and it shows alignment when they move together.
State and belief 42:01
Localization starts by assuming the map is known and the task is to find the robot’s place in it. The robot changes the world through control actions and learns from measurements. A state collects the robot and environment details that affect the future. A belief is the probability of each possible state, and the Bayes filter updates it by prediction and correction.
Two-step update 58:02
Given the belief at time t minus 1, the latest control ut, and the latest measurement zt, the update has two steps. First you predict where xt could be by folding in the control. Then you adjust that prediction with the measurement, one possible state at a time. The result is the belief at time t, normalized so the probabilities sum to one.
Prediction belief 1:10:30
The first step sums over all possible xt minus 1. You weigh each earlier state by how likely it was, then multiply by the chance of moving from that state to xt under ut. This is the total probability law in a continuous form. The second step uses Bayes rule. You multiply by the likelihood of zt given xt, and a normalizer eta keeps the distribution valid.
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