1. Mobile Robot Localization (ch. 7)
Mobile robot localization is the problem of determining the pose of a robot relative to a given map of the environment. Because,
Unfortunately, the pose of a robot can not be sensed directly, at least for now. The pose has to be inferred from data.
A single sensor measurement is enough?
The importance of localization in robotics.
Mobile robot localization can be seen as a problem of coordinate transformation. One point of view.

2. Mobile Robot Localization
Localization techniques have been developed for a broad set of map representations.
Feature based maps, location based maps, occupancy grid maps, etc. (what exactly are they?) (See figure 7.2)
(You can probably guess What is the mapping problem?)
Remember, in localization problem, the map is given, known, available.
Is it hard? Not really, because,

3. Mobile Robot Localization
Most localization algorithms are variants of Bayes filter algorithm.
However, different representation of maps, sensor models, motion model, etc lead to different variant.
Here is the agenda.

4. Mobile Robot Localization
We want to know different kinds of maps.
We want to know different kinds of localization problems.
We want to know how to solve localization problems, during which process, we also want to know how to get sensor model, motion model, etc.

5. Mobile Robot Localization (We want to know different kinds of maps. )
At a glance, ….
feature-based, location-based, metric, topological map, occupancy grid map, etc.
see figure 7.2

6. Different kinds of Localization problems. A taxonomy in 4 dimensions
Mobile Robot Localization – A Taxonomy (We want to know different kinds of localization problems.)
Different kinds of Localization problems.
A taxonomy in 4 dimensions
Local versus Global (initial knowledge)
Static versus Dynamic (environment)
Passive versus active (control of robots)
Single robot or multi-robot

7. Mobile Robot Localization
Solved already, the Bayes filter algorithm. How?
The straightforward application of Bayes filters to the localization problem is called Markov localization.
Here is the algorithm (abstract?)

8. Mobile Robot Localization
Algorithm Bayes_filter ( )
for all do
endfor
return

9. Mobile Robot Localization
Algorithm Markov Locatlization ( )
for all do
endfor
return
The Markov Localization algorithm addresses the global localization problem, the position tracking problem, and the kidnapped robot problem in static environment.

10. Mobile Robot Localization
Revisit Figure 7.5 to see how Markov localization algorithm in working.
The algorithm Markov Localization is still very abstract. To put it in work (eg. your project), we need a lot of more background knowledge to realize motion model, sensor model, etc….
We start with Guassian Filter (also called Kalman filter)

11. Bayes Filter Implementations (1)
Kalman Filter
(Gaussian filters)
(back to Ch.3)

12. Bayes Filter Reminder
Prediction
Correction

13. Gaussians -s s m
Univariate m
Multivariate

14. Properties of Gaussians

15. Multivariate Gaussians
We stay in the “Gaussian world” as long as we start with Gaussians and perform only linear transformations.
Review your probability textbook

16. Kalman Filter
Estimates the state x of a discrete-time controlled process that is governed by the linear stochastic difference equation
with a measurement

17. Components of a Kalman Filter
Matrix (nxn) that describes how the state evolves from t to t-1 without controls or noise.
Matrix (nxl) that describes how the control ut changes the state from t to t-1.
Matrix (kxn) that describes how to map the state xt to an observation zt.
Random variables representing the process and measurement noise that are assumed to be independent and normally distributed with covariance Rt and Qt respectively.

18. Kalman Filter Algorithm
Algorithm Kalman_filter( mt-1, St-1, ut, zt):
Prediction:
Correction:
Return mt, St

19. Kalman Filter Updates in 1D

20. Kalman Filter Updates in 1D

21. Kalman Filter Updates in 1D

22. Kalman Filter Updates

23. Linear Gaussian Systems: Initialization
Initial belief is normally distributed:

24. Linear Gaussian Systems: Dynamics
Dynamics are linear function of state and control plus additive noise:

25. Linear Gaussian Systems: Dynamics
This equation can be derived by just looking at the linear function of a Gaussian (ax+bu) and a convolution of two Gaussians

26. Linear Gaussian Systems: Observations
Observations are linear function of state plus additive noise:

27. Linear Gaussian Systems: Observations
Just the multiplication of two Gaussians (ignore c matrix to make it easier
To derive mean, multiply K = sigma / sigma * q into the equation, expand the left mean by sigma + q / sigma + q
To derive variance, same approach: expand left sigma with sigma + q / sigma + q
See page for mathematical derivation.

28. The Prediction-Correction-Cycle

29. The Prediction-Correction-Cycle

30. The Prediction-Correction-Cycle

31. Kalman Filter Summary
Highly efficient: Polynomial in measurement dimensionality k and state dimensionality n: O(k n2)
Optimal for linear Gaussian systems!
However, most robotics systems are nonlinear, unfortunately!

32. Nonlinear Dynamic Systems
Most realistic robotic problems involve nonlinear functions

33. Linearity Assumption Revisited

34. Non-linear Function

35. EKF Linearization (1)

36. EKF Linearization (2)

37. EKF Linearization (3)

38. EKF Linearization: First Order Taylor Series Expansion
Prediction:
Correction:

39. EKF Algorithm Extended_Kalman_filter( mt-1, St-1, ut, zt): Prediction:
Correction:
Return mt, St

40. Bayes Filter Implementations (2)
Particle filters

41. Sample-based Localization (sonar)

42. Particle Filters Represent belief by random samples
Estimation of non-Gaussian, nonlinear processes
Monte Carlo filter, Survival of the fittest, Condensation, Bootstrap filter, Particle filter
Filtering: [Rubin, 88], [Gordon et al., 93], [Kitagawa 96]
Computer vision: [Isard and Blake 96, 98]
Dynamic Bayesian Networks: [Kanazawa et al., 95]

43. Particle Filters

44. Sensor Information: Importance Sampling

45. Robot Motion

46. Sensor Information: Importance Sampling

47. Robot Motion

48. Particle Filter Algorithm
Algorithm particle_filter( St-1, ut-1 zt):
For Generate new samples
Sample index j(i) from the discrete distribution given by wt-1
Sample from using and
Compute importance weight
Update normalization factor
Insert
For
Normalize weights

49. Particle Filter Algorithm
Importance factor for xit:
draw xit from p(xt | xit-1,ut-1)
draw xit-1 from Bel(xt-1)

50. Importance Sampling Weight samples: w = f / g

51. Importance Sampling with Resampling

52. Importance Sampling with Resampling
Weighted samples
After resampling

53. Resampling Given: Set S of weighted samples.
Wanted : Random sample, where the probability of drawing xi is given by wi.
Typically done n times with replacement to generate new sample set S’.

54. Resampling Stochastic universal sampling Systematic resamplingw2 w3 w1 wn
Wn-1 w2 w3 w1 wn
Wn-1
Stochastic universal sampling
Systematic resampling
Linear time complexity
Easy to implement, low variance
Roulette wheel
Binary search, n log n

55. Also called stochastic universal sampling
Resampling Algorithm
Algorithm systematic_resampling(S,n):
For Generate cdf
Initialize threshold
For Draw samples …
While ( ) Skip until next threshold reached
Insert
Increment threshold
Return S’
Also called stochastic universal sampling

75. Sample-based Localization (sonar)

76. Initial Distribution

77. After Incorporating Ten Ultrasound Scans

78. After Incorporating 65 Ultrasound Scans

79. Estimated Path

80. Using Ceiling Maps for Localization

81. Vision-based Localization
h(x) z
P(z|x)

82. Under a Light
Measurement z:
P(z|x):

83. Next to a Light
Measurement z:
P(z|x):

84. Elsewhere
Measurement z:
P(z|x):

85. Global Localization Using Vision

86. Robots in Action: Albert

87. Limitations The approach described so far is able to
track the pose of a mobile robot and to
globally localize the robot.
How can we deal with localization errors (i.e., the kidnapped robot problem)?

88. Approaches
Randomly insert samples (the robot can be teleported at any point in time).
Insert random samples proportional to the average likelihood of the particles (the robot has been teleported with higher probability when the likelihood of its observations drops).