Arunkumar Byravan CSE 490R Lecture 3 Interaction loop Sense Receive sensor data and estimate state Plan Generate longterm plans based on state amp goal Act Apply actions to the robot ID: 733817
Download Presentation The PPT/PDF document "State Estimation Probability, Bayes Filt..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
State EstimationProbability, Bayes Filtering
Arunkumar
Byravan
CSE 490R – Lecture 3Slide2
Interaction loop
Sense:
Receive sensor data and estimate “state”
Plan:
Generate long-term plans based on state & goal
Act:
Apply actions to the robotSlide3
State
Byron Boots – Statistical Robotics at
GaTech
Markovian
assumption:
Future
is independent of
past
given presentSlide4
State EstimationEstimate “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 filteringSlide5
Why a probabilistic approach?Explicitly
represent
uncertainty
using
the calculus of probability
theorySlide6
Discrete Random Variables
X
denotes a
random variable
.
X
can take on a countable number of values in {x
1
, x
2
, …,
x
n
}.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.
.Slide7
Continuous Random VariablesX
takes on values in the continuum.
p
(X = x)
,
or
p
(x)
is the
probability density function
.
E.g.Slide8
Gaussian PDFSlide9
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)Joint and Conditional ProbabilitySlide10
Discrete case
Continuous case
Law of Total Probability,
MarginalsSlide11
EventsP(+x, +y) ?
P(+x) ?
P(-y OR +x) ?
Independent?
X
Y
P
+x
+y
0.2
+x
-y
0.3
-x
+y
0.4
-x
-y
0.1Slide12
Marginal Distributions
X
P
+x
-x
Y
P
+y
-y
X
Y
P
+x
+y
0.2
+x
-y
0.3
-x
+y
0.4
-x
-y
0.1Slide13
Conditional ProbabilitiesP(+x | +y) ?
P(-x | +y) ?
P(-y | +x) ?
X
Y
P
+x
+y
0.2
+x
-y
0.3
-x
+y
0.4
-x
-y
0.1Slide14
Bayes Formula
Often
causal
knowledge is easier to obtain than
diagnostic
knowledge.
Bayes rule allows us to use causal knowledge.Slide15
Simple Example of State EstimationSuppose a robot obtains measurement z
What is
P(
open|z
)?Slide16
Example
z
raise
s
the probability that the door is open
.Slide17
Normalization
Algorithm:Slide18
ConditioningBayes rule and
background knowledge
:Slide19
ConditioningBayes rule and background knowledge
:Slide20
Conditional Independence
Equivalent to
andSlide21
Simple Example of State EstimationSuppose our robot obtains another observation
z
2
.
What is
P(open|z
1,
z
2
)?Slide22
Recursive Bayesian Updating
Markov assumption
:
z
n
is
conditionally independent
of
z
1
,...,z
n-1
given
x.Slide23
Example: Second Measurement
z
2
lowers the probability that the door is open
.Slide24
Bayes Filters: FrameworkGiven: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:Slide25
Bayes Filters
Bayes
z
= observation
u
= action
x
= state
Markov
Markov
Total prob.Slide26
Bayes Filter Algorithm
Algorithm
Bayes_filter
(
Bel(x),d
):
n
=0
If
d
is a
perceptual data item z then For all x do For all x do Else if d is an action data item u then For all x do
Return Bel’(x) Slide27
Markov Assumption
Underlying Assumptions
Static world
Independent noise
Perfect model, no approximation errorsSlide28
Bayes Filters for Robot LocalizationSlide29
Bayes Filters are Familiar!Kalman
filters
Particle filters
Hidden Markov models
Dynamic Bayesian networks
Partially Observable Markov Decision Processes (POMDPs)Slide30
SummaryBayes 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.