Lecture 02:probability review + filtering

Katie DC

Jan. 23, 2019

Notes from Probabilistic Robotics Ch. 2

Admin

• HW1 is on PrairieLearn – due next week• No homework party this week!

• Office hours are now posted and will be starting next week

• Sign up for the CBTF orientation for extra credit

• For all due dates, check out the website:

publish.illinois.edu/ece470-intro-robotics/important-dates-spring-2020/

Logistics: Your ProjectYour goal will be to create a dynamic simulation, in which at least one complex robot interacts with some other object or agent. The purpose of this project is to allow you to explore robotics in an independent, free-form fashion.

• Note that the course staff will help you conceptually design your system and help you with the fundamentals, but they will not provide debugging support

There are 3+1 project updates. You’ll submit (1) a github link to your current codebase for the TAs to run, (2) a well-written readme, (3) a short description of your progress, and (4) a link to video uploaded to youtube demonstrating the deliverable.

Deliverables listed on the course website.

• Project Update 0 (Due Sunday 2/2 at midnight) Form a team (up to 3) and tell us about a task you’d like your robot to perform.

Fun Fact: Who is Bayes?

Bayes was an English statistician, philosopher, and minister who lived from 1701 to 1761, and is known for two works:

1. Divine Benevolence, or an Attempt to Prove That the Principal End of the Divine Providence and Government is the Happiness of His Creatures (1731)

2. An Introduction to the Doctrine of Fluxions, and a Defence of the Mathematicians Against the Objections of the Author of The Analyst (1736), in which he defended the logical foundation of Isaac Newton's calculus ("fluxions") against the criticism of George Berkeley, author of The Analyst

Bayes never published his most famous accomplishment Bayes’ Theorem. These notes were edited and published after his death by Richard Price.

From the HP Autonomy Lab

Probably not Bayes

Robot States and the Environment

Robotic System

Robot States and the Environment

• State represents the environment as well as the robot, for example:• location of walls or objects

• pose of the robot

• Environment interaction comes in the form of• Sensor measurements

• Control actions

• Internal representation (or belief) of the state of the world• In general, the state (or the world) cannot be measured directly

• Perception is the process by which the robot uses its sensors to obtain information about the state of the environment

Axioms of Probability Theory

Random Variable 𝑋Discrete Random Variables

• 𝑋 can take on a countable number of values in 𝑥1, 𝑥2, … , 𝑥𝑛

• 𝑃(𝑋 = 𝑥𝑖), or 𝑃(𝑥𝑖), is the probabilitythat the random variable 𝑋 takes on value 𝑥𝑖

• 𝑃 ∙ is called probability mass function

Continuous Random Variables

• 𝑋 takes on values in the continuum

• 𝑝(𝑋 = 𝑥), or 𝑝(𝑥), is a probability density function

=

b

a

dxxpbax )()),(Pr(

Expectation of a RV

Joint and Conditional Prob. + Total Prob.

Bayes’s Formula

Door example of Bayes Rule

Suppose a robot obtains measurement 𝑧. What is 𝑃 open 𝑧 ?

Robot States and the Environment

Robotic System

Robot’s Belief over statesBelief: Robot’s knowledge about the state of the environment

𝑏𝑒𝑙(𝑥𝑡) = 𝑝(𝑥𝑡|𝑧1:𝑡 , 𝑢1:𝑡)

Posterior distribution over state at time tgiven all past measurements and control

Prediction: 𝑏𝑒𝑙(𝑥𝑡) = 𝑝(𝑥𝑡|𝑧1:𝑡−1, 𝑢1:𝑡)

Calculating 𝑏𝑒𝑙(𝑥𝑡) from 𝑏𝑒𝑙(𝑥𝑡) is called correction or measurement update

Robotic system

tim

e =

1ti

me

= 2

tim

e =

0Discrete Bayes Filter - Illustration

Summary

• Reviewed basic probability theory

• Defined Bayes’s Theorem and applied it to a simple robotics problem

• Defined belief as the robot’s knowledge about the state of the environment and hinted at Bayes Filters (covered in next lecture)