Zeeshan Ali Sayyed What is State Estimation We need to estimate the state of not just the robot itself but also of objects which are moving in the robots environment For instance other cars people ID: 733815
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Slide1
State Estimation and Kalman Filtering
Zeeshan
Ali
SayyedSlide2
What is State Estimation?
We need to estimate the state of not just the robot itself, but also of objects which are moving in the robot’s environment.
For instance, other cars, people,
deers
, etc
.
Localization
TrackingSlide3
Why do we need it?
The world is stochastic and not deterministic
There are errors in the motors or transition mechanism of the robot .
There are errors in the sensors on the robot.
Sometimes, we also need to predict future states so as to plan accordingly. For instance, apply brakes if we are about to collide with another car.Slide4
What is Localization?
Imagine a robot in a simple world.
The robot doesn’t know where it is in the world frame of reference.
Estimating the position and state of the robot in this world making use of the limited information available to the robot is called Localization.Localization is a form of State Estimation where we estimate the state of the robot in the given world.Slide5
Example of LocalizationSlide6
Belief of a Robot
What is belief?
How do we represent it?
How do we start when we have absolutely no information?How do we update belief?Slide7
How do we start?
Uniform Distribution –
This shows we have absolutely no information about the location of the robotSlide8
Quiz
There are 4 possible places where the robot can be. What is the probability that the robot is in the 3
rd
place, given absolutely no other information?Slide9
Incorporating Sensor Measurements
The belief after we incorporate the sensor measurements is called Posterior Belief.Slide10
How do we do that in practice?
There are a variety of techniques for incorporating sensor input into our belief.
The simplest one is a simple product.
For instance, Consider the following worldLet’s say the robot observes Yellow. What do we do?
0.2
0.2
0.2
0.2
0.2
?
?
?
?
?Slide11
Incorporating Transition of Robot
This is technically called Convolution.Slide12
How do we do that in practice?
Assume a cyclic world. What happens, say, if the robot moves 2 steps forward?
0.1
0.2
0.5
0.1
0.1
0.1
0.1
0.1
0.2
0.5Slide13
Final Localization
This technique is referred to as Monte Carlo LocalizationSlide14
Modelling N
oisy Sensors
Sensors are not accurate. To model the error we use probabilistic models to represent.
For eg. If the sensor reports a door, we do not trust it completely. How do we quantify? We do it using our model. For example:
What happens now?
Multiply and normalize!
0.2
0.2
0.2
0.2
0.2Slide15
Modelling Noisy Transition
For example:
0.1
0.2
0.5
0.1
0.1Slide16
Representation of things we have learned
State – X
Measurement – z
Control Actions – uTime – tWhat is
Belief –
Sensor model -
Transition model -
Slide17
Introducing Kalman Filters
Kalman Filters used for both Localization as well as Tracking.
It is very similar to Monte Carlo Localization
It one of the most popular state estimation technique is use, not only in robotics but in many other fields.It deals in Continuous State Spaces (What do they mean?).Slide18
Gaussian
Notation:
Slide19
Comparison of Means and VariancesSlide20
Representing Belief and Measurement
The belief and sensor measurement, both are represented by Gaussians.
Gaussian with high variance implies uncertainty and low variance implies certainty.
Example on boardSlide21
Kalman Filter Algorithm
Incorporate Sensor Measurements
Bayes Rule
Incorporate Transition UpdateTotal ProbabilitySlide22
Incorporating Sensor Measurements
Can you say anything about the posterior?Slide23Slide24
Multiplication of two Gaussians
Addition of two GaussiansSlide25
Incorporating Transition Update
When we move, we tend to lose information. Therefore, the variance of the belief increases.
Simple add the two Gaussians using the previous formula.
That’s the Kalman Filter for a simple one dimensional case!Slide26
Generalized Kalman Filter
We assume we have a linear transition and observation (sensing) models.Slide27
Kalman Filter Algorithm
Algorithm
Kalman_filter
(
m
t-1
,
S
t-1
, u
t
, z
t
):
Prediction:
Correction:
Return
m
t
, St
27Slide28
28
Kalman Filter Summary
Highly efficient
: Polynomial in measurement dimensionality
k
and state dimensionality
n
:
O(k
2.376
+ n
2
)
Optimal for linear Gaussian systems
!
Most robotics systems are
nonlinear
!