1. State Estimation Probability, Bayes Filtering
Arunkumar Byravan
CSE 490R – Lecture 3

2. Plan Act Sense Interaction loop
Sense: Receive sensor data and estimate “state”
Plan: Generate long-term plans based on state & goal
Act: Apply actions to the robot
Plan
Act
Sense

3. State
Byron Boots – Statistical Robotics at GaTech
Markovian assumption: Future is independent of past given present

4. State Estimation Estimate “state” using sensor data & actions:
Robot position and orientation
Environment map
Location of people, other robots etc.
Indoor localization for mobile robots (given map)
Fuse information from different sensors (LIDAR, Wheel encoders)
Probabilistic filtering
Show slide with sense -> plan -> act loop
What we’d like to do through sensing -> state estimation
What is state?
Examples -> for self-driving car it could be it’s own position/orientation in map, detecting people, lane markers, other cars, buildings etc
Different types of sensors that can be used: camera, LIDAR, IMU etc
One main problem in state estimation -> localization
Estimating robot’s own state (position/orientation) in an environment
Outdoors: GPS is a good proxy that provides good estimates
Indoors: We need to do this explicitly

5. Why a probabilistic approach?
Explicitly represent uncertainty using the calculus of probability theory

6. Discrete Random Variables
X denotes a random variable.
X can take on a countable number of values in {x1, x2, …, xn}.
p(X = xi), or p(xi), is the probability that the random variable X takes on value xi.
p( ) is called probability mass function.
E.g. .

7. Continuous Random Variables
X takes on values in the continuum.
p(X = x), or p(x) is the probability density function.
E.g.

8. Gaussian PDF

9. Joint and Conditional Probability
P(X = x and Y = y) = P(x, y)
If X and Y are independent then P(x, y) = P(x) P(y)
P(x | y) is the probability of x given y P(x | y) = P(x , y) / P(y) P(x , y) = P(x | y) P(y)
If X and Y are independent then P(x | y) = P(x)
Two neighboring laser beams
Venn diagram for P(x|y)
Independent:
P(x|y) = P(x,y)/P(y) = P(x)P(y)/P(y)= P(x)

10. Law of Total Probability, Marginals
Discrete case
Continuous case
x is laser measurement, want to know P(x).
Don’t have a good model for P(x), but a pretty good one
for P(x) given the distance to a wall. Now we just have
to compute weighted sum over all possible wall positions to get P(x).

11. Events P(+x, +y) ? P(+x) ? P(-y OR +x) ? Independent? X Y P +x +y 0.2
0.3 -x
0.4
0.1
0.2
0.5
0.6
Independent?
P(+x): 0.5
P(+y): 0.6
P(+x,+y)=0.3

12. Marginal DistributionsX P +x -x X Y P +x +y
0.2 -y
0.3 -x
0.4
0.1 Y P +y -y
0.5, 0.5
0.6, 0.4

13. Conditional Probabilities
P(+x | +y) ?
P(-x | +y) ?
P(-y | +x) ? X Y P +x +y
0.2 -y
0.3 -x
0.4
0.1
1/3
2/3
0.6

14. Bayes Formula
Often causal knowledge is easier to obtain than diagnostic knowledge.
Bayes rule allows us to use causal knowledge.

15. Simple Example of State Estimation
Suppose a robot obtains measurement z
What is P(open|z)?

16. Example z raises the probability that the door is open.
Do the same for P( not open | z) /
z raises the probability that the door is open.

17. Normalization
Algorithm:

18. Conditioning Bayes rule and background knowledge:
P(x|y,z)) = P(x,y,z) / P(y,z) = P(y|x,z)P(x,z)/P(y,z)
= P(y|x,z)P(x|z)P(z) / P(y|z)P(z)
= P(y|x,z)P(x|z) / P(y|z)
P(x|y) = P(x,y)/P(y) = S_z P(xyz) / P(y)
= S_z P(x|y,z)P(y,z) / P(y)
= S_z P(x|y,z)Pz|y)P(y)/P(y)
= S_z P*x|y,z)P(z}y)

19. Conditioning Bayes rule and background knowledge:
Just replace all conditionals p(x|y) by their p(x,y)/p(y) form.

20. Conditional Independence
Equivalent to
and
Two laser beams again
Proof as homework

21. Simple Example of State Estimation
Suppose our robot obtains another observation z2.
What is P(open|z1, z2)?

22. Recursive Bayesian Updating
Markov assumption: zn is conditionally independent of z1,...,zn-1 given x.

23. Example: Second Measurement
z2 lowers the probability that the door is open.

24. Bayes Filters: Framework
Given:
Stream of observations z and action data u:
Sensor model P(z|x).
Action model P(x|u,x’).
Prior probability of the system state P(x).
Wanted:
Estimate of the state X of a dynamical system.
The posterior of the state is also called Belief:

25. Bayes Filters z = observation u = action x = state Bayes Markov
Total prob.
Markov

26. Bayes Filter Algorithm
Algorithm Bayes_filter ( Bel(x),d ):
n=0
If d is a perceptual data item z then
For all x do
Else if d is an action data item u then
Return Bel’(x)

27. Markov Assumption Underlying Assumptions Static world
Independent noise
Perfect model, no approximation errors

28. Bayes Filters for Robot Localization

29. Bayes Filters are Familiar!
Kalman filters
Particle filters
Hidden Markov models
Dynamic Bayesian networks
Partially Observable Markov Decision Processes (POMDPs)

30. Summary
Bayes rule allows us to compute probabilities that are hard to assess otherwise.
Under the Markov assumption, recursive Bayesian updating can be used to efficiently combine evidence.
Bayes filters are a probabilistic tool for estimating the state of dynamic systems.