Probabilistic Robotics

Probabilistic Robotics Chapter 2 Leon F. Palafox

Probability Distributions

Probability

p(x)

Probability of x

p ( x, y )

Probability of x AND y

Joint Distribution

p ( x, y ) = p ( x ) p ( y )

p( x | y) =

p ( x, y ) p( y)

If they are independent

Total Probability

Total Probability

p( x) = ∑ p( x | y ) p( y )

Discrete Case

y

Bayes Rule

p( x | y ) =

p( y | x) p( x) = p( y)

p( y | x) p( x) ∑ x ' p( y | x' ) p( x' )

p( x | y ) = ηp( y | x) p ( x)

We use a normalizing factor

Bayes Rule

Example • • • • •

P （ D ） = 0.005, since 0.5%. This is the prior probability of D. P(N), This is 1 − P(D), or 0.995. P(+|D) This is 0.99, since the test is 99% accurate. P(+|N), This is 0.01, since the test will produce a false positive for 1% of nonusers. P(+), This is 0.0149 or 1.49%, which is found by adding the probability that a true positive result will appear (= 99% x 0.5% = 0.495%) plus the probability that a false positive will appear (= 1% x 99.5% = 0.995%). This is the prior probability of +.

State

State •Information over the environment and robot •May change over time or remain static •Will be denoted with the variable

xt

•Is called Complete if is the best predictor of the future

State Variables •Robot pose •Actuators •Robot Velocity •Features in its surroundings

Belief

Definition Robot Internal knowledge of the state The state cannot be measured directly

Distributions

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

After measurement

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

Bayes Algorithm

Bayes Algorithm

Step 3 processes the control U. Is the probability that U indices a change in the states. Step 4 Observe the state and thus generate a new set of beliefs

Bayes Algorithm

Example A robot finds a door and is able to open the door using its manipulator.

Initial State

If the robot does NOT know the initial state of the door. (Which is open). Suppose the door is open and the robot uses its manipulator.

Bayes Algorithm

Probabilities bel ( X 0 = open) = 0.5 bel ( X 0 = close) = 0.5 p ( Z t = sense _ open | X t = is _ open) = 0.6 p ( Z t = sense _ closed | X t = is _ open) = 0.4 Very noisy sensors p ( Z t = sense _ open | X t = is _ closed ) = 0.2 p ( Z t = sense _ closed | X t = is _ closed ) = 0.8 p ( X t = is _ open | U t = push, X t −1 = is _ open) = 1 p ( X t = is _ closed | U t = push, X t −1 = is _ open) = 0 p ( X t = is _ open | U t = push, X t −1 = is _ closed ) = 0.8 p ( X t = is _ closed | U t = push, X t −1 = is _ closed ) = 0.2 p ( X t = is _ open | U t = do _ nothing , X t −1 = is _ open) = 1 p ( X t = is _ closed | U t = do _ nothing , X t −1 = is _ open) = 0 We are not p ( X t = is _ open | U t = do _ nothing , X t −1 = is _ closed ) = 0 anything p( X t = is _ closed | U t = do _ nothing , X t −1 = is _ closed ) = 1

doing

Bayes Algorithm

Solution bel ( x1 ) = ∑ p ( x1 | u1 , x 0 )bel ( x0 ) = p ( X 1 | U t = do _ nothing , X 0 = is _ open)bel ( X 0 = is _ open) + X0

p ( X 1 | U t = do _ nothing , X 0 = is _ closed )bel ( X 0 = is _ closed )

bel ( X 1 = open) = 0.5 bel ( X 1 = closed ) = 0.5 bel ( x1 ) = ηp( Z 1 = sense _ open | x1 )bel ( x1 )

bel ( X 1 = open) = 0.75 bel ( X 1 = open) = 0.25 bel ( X 2 = open) = 0.95 bel ( X 2 = closed ) = 0.05 bel ( X 2 = open) = 0.983

bel ( X 2 = open) = 0.017

Final Notes

Notes

• The states are presented in a Markov Chain – Future data are independent if one knows the current state.

• Even though Bayes filter does have future events that may violate Markov Assumption, it proves to be robust. • Overall points to take into account: – Computational efficiency – Accuracy of the approximation – Ease of implementation

Bayes Algorithm

Solution bel ( x1 ) = ∑ p ( x1 | u1 , x 0 )bel ( x0 ) = p ( X 1 | U t = do _ nothing , X 0 = is _ open)bel ( X 0 = is _ open) + X0

p ( X 1 | U t = do _ nothing , X 0 = is _ closed )bel ( X 0 = is _ closed )

bel ( X 1 = open) = 0.5 bel ( X 1 = closed ) = 0.5 bel ( x1 ) = ηp( Z 1 = sense _ open | x1 )bel ( x1 )

bel ( X 1 = open) = 0.75 bel ( X 1 = open) = 0.25 bel ( X 2 = open) = 0.95 bel ( X 2 = closed ) = 0.05 bel ( X 2 = open) = 0.983

bel ( X 2 = open) = 0.017