1. Probability in Robotics

2. Advantages of Probabilistic Paradigm
Can accommodate inaccurate models
Can accommodate imperfect sensors
Robust in real-world applications
Best known approach to many hard robotics problems
Pays Tribute to Inherent Uncertainty
Know your own ignorance
Scalability
No need for “perfect” world model
Relieves programmers

3. Limitations of Probability
Computationally inefficient
Consider entire probability densities
Approximation
Representing continuous probability distributions.

4. Trends in Robotics Research
Classical Robotics (mid-70’s)
exact models
no sensing necessary
Reactive Paradigm (mid-80’s)
no models
relies heavily on good sensing
Hybrids (since 90’s)
model-based at higher levels
reactive at lower levels
Probabilistic Robotics (since mid-90’s)
seamless integration of models and sensing
inaccurate models, inaccurate sensors

5. Uncertainty Representation

6. Five Sources of Uncertainty
Environment
Dynamics
Approximate
Computation
Random
Action Effects
Inaccurate
Models
Sensor
Limitations

7. Probabilistic Robotics
Key idea: Explicit representation of uncertainty using probability theory
Perception = state estimation
Action = utility optimization

8. Axioms of Probability Theory
Pr(A) denotes probability that proposition A is true. 1) 2) 3)

9. A Closer Look at Axiom 3 B

10. Using the Axioms

11. Discrete Random Variables
X denotes a random variable.
X can take on a finite 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.

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

13. 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)

14. Law of Total Probability
Discrete case
Continuous case

15. Bayes Formula

16. Normalization

17. Conditioning
Total probability:
Bayes rule and background knowledge:

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

19. Causal vs. Diagnostic Reasoning
P(open|z) is diagnostic.
P(z|open) is causal.
Often causal knowledge is easier to obtain.
Bayes rule allows us to use
causal knowledge:
count frequencies!

20. Example z raises the probability that the door is open.
P(z|open) = 0.6 P(z|open) = 0.3
P(open) = P(open) = 0.5
z raises the probability that the door is open.

21. Combining Evidence Suppose our robot obtains another observation z2.
How can we integrate this new information?
More generally, how can we estimate P(x| z1...zn )?

22. Recursive Bayesian Updating
Markov assumption: zn is independent of z1,...,zn-1 if we know x.

23. Example: Second Measurement
P(z2|open) = 0.5 P(z2|open) = 0.6
P(open|z1)=2/3
z2 lowers the probability that the door is open.

24. actions carried out by the robot, actions carried out by other agents,
Often the world is dynamic since
actions carried out by the robot,
actions carried out by other agents,
or just the time passing by
change the world.
How can we incorporate such actions?

25. Typical Actions The robot turns its wheels to move
The robot uses its manipulator to grab an object
Actions are never carried out with absolute certainty.
In contrast to measurements, actions generally increase the uncertainty.

26. Modeling Actions
To incorporate the outcome of an action u into the current “belief”, we use the conditional pdf
P(x|u,x’)
This term specifies the pdf that executing u changes the state from x’ to x.

27. Example: Closing the door

28. State Transitions P(x|u,x’) for u = “close door”:
If the door is open, the action “close door” succeeds in 90% of all cases.

29. Integrating the Outcome of Actions
Continuous case:
Discrete case:

30. Example: The Resulting Belief

31. How all of this relates to Sensors
Sensor fusion

32. Basic statistics – Statistical representation – Stochastic variable
Travel time, X = 5hours ±1hour
X can have many different values
Continous – The variable can have any value within the bounds
Discrete – The variable can have specific (discrete) values

33. Basic statistics – Statistical representation – Stochastic variable
Another way of describing the stochastic variable, i.e. by another form of bounds
Probability distribution
In 68%: µ- < X < µ+
In 95%: µ-2 < X < µ+2
In 99%: µ-3 < X < µ+3
In 100%: - < X < 
The value to expect is the mean value => Expected value
How much X varies from its expected value => Variance

34. Expected value and Variance
The standard deviation X is the square root of the variance

35. Gaussian (Normal) distribution
By far the mostly used probability distribution because of its nice statistical and mathematical properties
What does it means if a specification tells that a sensor measures a distance [mm] and has an error that is normally distributed with zero mean and  = 100mm?
Normal distribution:
~68.3%
~95%
~99%
etc.

36. Estimate of the expected value and the variance from observations

37. Linear combinations (1)
X2 ~ N(m2, σ2)
X1 ~ N(m1, σ1)
Y ~ N(m1 + m2, sqrt(σ1 +σ2))
This property that Y remains Gaussian if the s.v. are combined linearily is one of the great properties of the Gaussian distribution!

38. Linear combinations (2)
We measure a distance by a device that have normally distributed errors,
Do we win something of making a lot of measurements and use the average value instead?
What will the expected value of Y be?
What will the variance (and standard deviation) of Y be?
If you are using a sensor that gives a large error, how would you best use it?

39. Linear combinations (3)
With d and α un-correlated => V[d, α] = 0 (Actually the co-variance, which is defined later)
di is the mean value and d ~ N(0, σd)
αi is the mean value and α ~ N(0, σα)

40. Linear combinations (4)
D = {The total distance} is calculated as before as this is only the sum of all d’s
The expected value and the variance become:

41. Linear combinations (5)
 = {The heading angle} is calculated as before as this is only the sum of all ’s, i.e. as the sum of all changes in heading
The expected value and the variance become:
What if we want to predict X and Y from our measured d’s and ’s?

42. Non-linear combinations (1)
X(N) is the previous value of X plus the latest movement (in the X direction)
The estimate of X(N) becomes:
This equation is non-linear as it contains the term:
and for X(N) to become Gaussian distributed, this equation must be replaced with a linear approximation around To do this we can use the Taylor expansion of the first order. By this approximation we also assume that the error is rather small!
With perfectly known N-1 and N-1 the equation would have been linear!

43. Non-linear combinations (2)
Use a first order Taylor expansion and linearize X(N) around
This equation is linear as all error terms are multiplied by constants and we can calculate the expected value and the variance as we did before.

44. Non-linear combinations (3)
The variance becomes (calculated exactly as before):
Two really important things should be noticed, first the linearization only affects the calculation of the variance and second (which is even more important) is that the above equation is the partial derivatives of:
with respect to our uncertain parameters squared multiplied with their variance!

45. Non-linear combinations (4)
This result is very good => an easy way of calculating the variance => the law of error propagation
The partial derivatives of become:

46. Non-linear combinations (5)
The plot shows the variance of X for the time step 1, …, 20 and as can be noticed the variance (or standard deviation) is constantly increasing.
d = 1/10
 = 5/360

47. The Error Propagation Law

48. The Error Propagation Law

49. The Error Propagation Law

50. Multidimensional Gaussian distributions MGD (1)
The Gaussian distribution can easily be extended for several dimensions by: replacing the variance () by a co-variance matrix () and the scalars (x and mX) by column vectors.
The CVM describes (consists of):
1) the variances of the individual dimensions => diagonal elements
2) the co-variances between the different dimensions => off-diagonal elements
! Symmetric
! Positive definite

51. An n-d Gaussian
distribution is given by:
A 1-d Gaussian
distribution is given by:

52. MGD (2) Eigenvalues => standard deviations
Eigenvectors => rotation of the ellipses

53. MGD (3)
The co-variance between two stochastic variables is calculated as:
Which for a discrete variable becomes:
And for a continuous variable becomes:

54. MGD (4) - Non-linear combinations
The state variables (x, y, ) at time k+1 become:

55. MGD (5) - Non-linear combinations
We know that to calculate the variance (or co-variance) at time step k+1 we must linearize Z(k+1) by e.g. a Taylor expansion - but we also know that this is done by the law of error propagation, which for matrices becomes:
With fX and fU are the Jacobian matrices (w.r.t. our uncertain variables) of the state transition matrix.

56. MGD (6) - Non-linear combinations
The uncertainty ellipses for X and Y (for time step ) is shown in the figure.

57. Probabilistic Action model
p(st|at-1,st-1)
st-1
st-1
at-1
at-1
Continuous probability density Bel(st) after moving 40m (left figure) and 80m (right figure). Darker area has higher probablity.

58. Localization
Localization without knowledge of start location

59. Localization Initial state detects nothing: Moves and
detects landmark:
Moves and
detects nothing:
Moves and
detects landmark:

60. Circular Error Problem
If we have a map:
We can localize!
NOT THAT SIMPLE!
If we can localize:
We can make a map!

61. Expectation-Maximization (EM)
Algorithm
Initialize: Make random guess for lines
Repeat:
Find the line closest to each point and group into two sets. (Expectation Step)
Find the best-fit lines to the two sets (Maximization Step)
Iterate until convergence
The algorithm is guaranteed to converge to some local optima

62. Example:

63. Example:

64. Example:

65. Example:

66. Example:
Converged!

67. Probabilistic Mapping
Maximum Likelihood Estimation
E-Step: Use current best map and data to find belief probabilities
M-step: Compute the most likely map based on the probabilities computed in the E-step.
Alternate steps to get better map and localization estimates
Convergence is guaranteed as before.

68. but computed backward in time
The E-Step
P(st|d,m) = P(st |o1, a1… ot,m) . P(st |at…oT,m)
t  Bel(st)
Markov Localization t
Analogous to 
but computed backward in time

69. The M-Step # of times l was observed at <x,y>
Updates occupancy grid
P(mxy=l | d) =
# of times l was observed at <x,y>
# of times something was obs. at <x,y>

70. Probabilistic Mapping
Addresses the Simultaneous Mapping and Localization problem (SLAM)
Robust
Hacks for easing computational and processing burden
Caching
Selective computation
Selective memorization

74. Markov Assumption Future is Independent of Past Given Current State
“Assume Static World”

75. Probabilistic Model
Action Data
Observation Data

76. Derivation : Markov Localization
Bayes
Markov
Total Probability
Markov

77. Nature of Sensor Data: Uncertainty
Range Data
Odometry Data

78. Mobile Robot Localization
Proprioceptive Sensors: (Encoders, IMU) - Odometry, Dead reckoning
Exteroceptive Sensors: (Laser, Camera) - Global, Local Correlation
Scan-Matching
Scan 1
Scan 2
Iterate
Displacement Estimate
Initial Guess
Point Correspondence
Scan-Matching
Correlate range measurements to estimate displacement
Can improve (or even replace) odometry – Roumeliotis TAI-14
Previous Work - Vision community and Lu & Milios [97]

79. Explicit models of uncertainty & noise sources for each scan point:
Weighted Approach
Explicit models of uncertainty & noise sources for each scan point:
Sensor noise & errors
Range noise
Angular uncertainty
Bias
Point correspondence uncertainty
Correspondence Errors
Combined
Uncertanties
Improvement vs. unweighted method:
More accurate displacement estimate
More realistic covariance estimate
Increased robustness to initial conditions
Improved convergence
1 m
x500

80. Goal: Estimate displacement (pij ,fij )
Weighted Formulation
Goal: Estimate displacement (pij ,fij )
Measured range data from poses i and j
sensor noise
bias
true range
{uk} = measured data
{rk} = true range measurements
{duk} = zero mean gaussian noise process
{bk} = noise process
Error between kth scan point pair
= rotation of fij
Noise Error
Bias Error
Correspondence Error

81. Covariance of Error Estimate
Correspondence
Sensor Noise
Bias
Covariance of error between kth scan point pair =
Lik sq sl
Sensor Noise
Pose i
{uk} = measured data
{rk} = true range measurements
{duk} = zero mean gaussian noise process
{bk} = noise process
Assume correspondence erors noise and bias errors are mutually independent
Sensor Bias
neglect for now

82. Correspondence Error = cijk
Estimate bounds of cijk from the geometry
of the boundary and robot poses
Max error
Assume uniform distribution
where

83. Finding incidence angles aik and ajk
Hough Transform
-Fits lines to range data
-Local incidence angle estimated from line tangent and scan angle
-Common technique in vision community (Duda & Hart [72])
-Can be extended to fit simple curves
aik
Scan Points
Fit Lines

84. Maximum Likelihood Estimation
Likelihood of obtaining errors {eijk} given displacement
Non-linear Optimization Problem
Position displacement estimate obtained in closed form
Orientation estimate found using 1-D numerical
optimization, or series expansion approximation methods

85. Weighted vs. Unweighted matching of two poses
Experimental Results
Weighted vs. Unweighted matching of two poses
512 trials with different initial displacements within :
+/- 15 degrees of actual angular displacement
+/- 150 mm of actual spatial displacement
Initial Displacements
Unweighted Estimates
Weighted Estimates
Increased robustness to inaccurate initial displacement guesses
Fewer iterations for convergence

86. Unweighted Weighted

87. Displacement estimate errors at end of path
Eight-step, 22 meter path
Displacement estimate errors at end of path
Odometry = 950mm
Unweighted = 490mm
Weighted = 120mm
More accurate covariance estimate
Improved knowledge of
measurement uncertainty
- Better fusion with other sensors

88. Conclusions Possible Future Work:
Developed general approach to incorporate uncertainty into scan-match displacement estimates.
range sensor error models
novel correspondence error modeling
Method can likely be extended to other range sensors (stereo cameras, radar, ultrasound, etc.)
requires some specific sensor error models
Showed that accurate error modelling can significantly improve displacement and covariance estimates as well as robustness
Possible Future Work:
Weighted correspondence for 3D feature matching

89. Uncertainty From Sensor Noise and Correspondence Error
x500