1. SEIF, EnKF, EKF SLAM, Fast SLAM, Graph SLAM
Pieter Abbeel
UC Berkeley EECS
Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics
TexPoint fonts used in EMF.
Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAA

2. Information Filter From an analytical point of view == Kalman filter
Difference: keep track of the inverse covariance rather than the covariance matrix [matter of some linear algebra manipulations to get into this form]
Why interesting?
Inverse covariance matrix = 0 is easier to work with than covariance matrix = infinity (case of complete uncertainty)
Inverse covariance matrix is often sparser than the covariance matrix for the “insiders”: inverse covariance matrix entry (i,j) = 0 if xi is conditionally independent of xj given some set {xk, xl, …}
Downside: when extended to non-linear setting, need to solve a linear system to find the mean (around which one can then linearize)
See Probabilistic Robotics pp for more in-depth pros/cons and Probabilistic Robotics Chapter 12 for its relevance to SLAM (then often referred to as the “sparse extended information filter (SEIF)”)

3. Ensemble Kalman filter (enKF)
Represent the Gaussian distribution by samples
Empirically: even 40 samples can track the atmospheric state with high accuracy with enKF
<-> UKF: 2 * n sigma-points, n = then still forms covariance matrices for updates
The intellectual innovation:
Transforming the Kalman filter updates into updates which can be computed based upon samples and which produce samples while never explicitly representing the covariance matrix

4. KF enKF ? Keep track of ensemble [x1, …, xN] Keep track of ¹, §
Prediction:
Correction:
Return mt, St
Keep track of ¹, §
Keep track of ensemble [x1, …, xN]
Can update the ensemble by simply propagating through the dynamics model + adding sampled noise ?

5. enKF correction step KF: Current ensemble X = [x1, …, xN]
Build observations matrix Z = [zt+v1 … zt+vN] where vi are sampled according to the observation noise model
Then the columns of
X + Kt(Z – Ct X)
form a set of random samples from the posterior
Note: when computing Kt, leave §t in the format
§t = [x1-¹t … xN-¹t] [x1-¹t … xN-¹t]T
Can also compute C_t from samples

6. How about C? Indeed, would be expensive to build up C.
However: careful inspection shows that C only appears as in:
C X
C § CT = C X XT CT
 can simply compute h(x) for all columns x of X and compute the empirical covariance matrices required
[details left as exercise]

8. References for enKF
Mandel, 2007 “A brief tutorial on the Ensemble Kalman Filter”
Evensen, 2009, “The ensemble Kalman filter for combined state and parameter estimation”

9. KF Summary Kalman filter exact under linear Gaussian assumptions
Extension to non-linear setting:
Extended Kalman filter
Unscented Kalman filter
Extension to extremely large scale settings:
Ensemble Kalman filter
Sparse Information filter
Main limitation: restricted to unimodal / Gaussian looking distributions
Can alleviate by running multiple XKFs + keeping track of the likelihood; but this is still limited in terms of representational power unless we allow a very large number of them

10. Kalman filter property
If system is observable (=dual of controllable!) then Kalman filter will converge to the true state.
System is observable iff
O = [C ; CA ; CA2 ; … ; CAn-1] is full column rank (1)
Intuition: if no noise, we observe y0, y1, … and we have that the unknown initial state x0 satisfies:
y0 = C x0
y1 = CA x0
...
yK = CAK x0
This system of equations has a unique solution x0 iff the matrix [C; CA; … CAK] has full column rank. B/c any power of a matrix higher than n can be written in terms of lower powers of the same matrix, condition (1) is sufficient to check (i.e., the column rank will not grow anymore after having reached K=n-1).

11. Simple self-quiz
The previous slide assumed zero control inputs at all times. Does the same result apply with non-zero control inputs? What changes in the derivation?

12. EKF/UKF SLAM
State: (nR, eR, R, nA, eA, nB, eB, nC, eC, nD, eD, nE, eE, nF, eF, nG, eG, nH, eH)
Now map = location of landmarks (vs. gridmaps)
Transition model:
Robot motion model; Landmarks stay in place R B E H D A G C F

13. Simultaneous Localization and Mapping (SLAM)
In practice: robot is not aware of all landmarks from the beginning
Moreover: no use in keeping track of landmarks the robot has not received any measurements about
 Incrementally grow the state when new landmarks get encountered.

14. Simultaneous Localization and Mapping (SLAM)
Landmark measurement model: robot measures [ xk; yk ], the position of landmark k expressed in coordinate frame attached to the robot:
h(nR, eR, R, nk, ek) = [xk; yk] = R() ( [nk; ek] - [nR; eR] )
Often also some odometry measurements
E.g., wheel encoders
As they measure the control input being applied, they are often incorporated directly as control inputs (why?)

15. Victoria Park Data Set
[courtesy by E. Nebot]

16. Victoria Park Data Set Vehicle
[courtesy by E. Nebot]

17. Data Acquisition
[courtesy by E. Nebot]

18. Estimated Trajectory
[courtesy by E. Nebot]

19. EKF SLAM Application
[courtesy by J. Leonard]

20. EKF SLAM Application odometry estimated trajectory
[courtesy by John Leonard]

21. Landmark-based Localization

22. EKF-SLAM: practical challenges
Defining landmarks
Laser range finder: Distinct geometric features (e.g. use RANSAC to find lines, then use corners as features)
Camera: “interest point detectors”, textures, color, …
Often need to track multiple hypotheses
Data association/Correspondence problem: when seeing features that constitute a landmark --- Which landmark is it?
Closing the loop problem: how to know you are closing a loop?
Can split off multiple EKFs whenever there is ambiguity;
Keep track of the likelihood score of each EKF and discard the ones with low likelihood score
Computational complexity with large numbers of landmarks.

23. KF over very large state spaces
High-dimensional ocean and atmospheric circulation models (106 dimensional state space)
SLAM with 106 landmarks
Becomes computationally very challenging to work with the 106 x 106 covariance matrix (terabytes!)
In SLAM community: information filter which keeps tracks of the inverse covariance matrix, which can often be well approximated by a sparse matrix
In civil engineering community: ensemble Kalman filter, with applications often being in tracking systems described by partial differential equations

24. EKF_localization ( mt-1, St-1, ut, zt, m): Correction:
Predicted measurement mean
Jacobian of h w.r.t location
Pred. measurement covariance
Kalman gain
Updated mean
Updated covariance

25. EKF Prediction Step

26. EKF Observation Prediction Step

27. EKF Correction Step

28. Estimation Sequence (1)

29. Estimation Sequence (2)

30. Comparison to GroundTruth

31. Fast SLAM Rao-Blackwellized particle filter
Robot state = x, y, µ (just like gMapping)
Map = Landmark based (vs. map = gridmap for gMapping)
Key observation (why Rao Blackwellization is so useful):
Location of landmark i is independent of location of landmark j given the entire robot pose sequence
Instead of joint Gaussian over poses of all landmarks, can just keep track of Gaussian for each landmark separately
Linear scaling with number of landmarks (rather than quadratic)

32. SLAM thus far Landmark based vs. occupancy grid
Probability distribution representation:
EKF vs. particle filter vs. Rao-Blackwellized particle filter
EKF, SEIF, FastSLAM are all “online”
Currently popular 4th alternative: GraphSLAM

33. Graph-based Formulation
Use a graph to represent the problem
Every node in the graph corresponds to a pose of the robot during mapping
Every edge between two nodes corresponds to the spatial constraints between them
Goal: Find a configuration of the nodes that minimize the error introduced by the constraints

34. Problem Formulation The problem can be described by a graph
Goal:
Find the assignment of poses to the nodes of the graph which minimizes the negative log likelihood of the observations:
Observation of from
error
nodes

35. Approaches 2D approaches: Lu and Milios, ‘97 Montemerlo et al., ‘03
Howard et al., ‘03
Dellaert et al., ‘03
Frese and Duckett, ‘05
Olson et al., ‘06
Grisetti et al., ’07
Tipaldi et al.,’ 07
3D approaches:
Nuechter et al., ‘05
Dellaert et al., ‘05
Triebel et al., ’06
Grisetti et al., ’08/’09
Say more on 3D approaches

36. Graph-Based SLAM in a Nutshell
Problem described as a graph
Every node corresponds to a robot position and to a laser measurement
An edge between two nodes represents a data-dependent spatial constraint between the nodes
[KUKA Hall 22, courtesy P. Pfaff & G. Grisetti]

37. Graph-Based SLAM in a Nutshell
Problem described as a graph
Every node corresponds to a robot position and to a laser measurement
An edge between two nodes represents a data-dependent spatial constraint between the nodes
[KUKA Hall 22, courtesy P. Pfaff & G. Grisetti]

38. Graph-Based SLAM in a Nutshell
Once we have the graph, we determine the most likely map by “moving” the nodes
[KUKA Hall 22, courtesy P. Pfaff & G. Grisetti]

39. Graph-Based SLAM in a Nutshell
Once we have the graph, we determine the most likely map by “moving” the nodes
… like this.
[KUKA Hall 22, courtesy P. Pfaff & G. Grisetti]

40. Graph-Based SLAM in a Nutshell
Once we have the graph, we determine the most likely map by “moving” the nodes
… like this.
Then we render a map based on the known poses
[KUKA Hall 22, courtesy P. Pfaff & G. Grisetti]

41. Graph-based Visual SLAM
Visual odometry
Loop Closing
Real world example: flying robot in corridor. Map inconsistent. Construct graph, optimize.
[ courtesy B. Steder]

42. The KUKA Production Site

43. The KUKA Production Site

44. The KUKA Production Site
scans
total acquisition time , seconds
traveled distance , meters
total rotations radians
size x 110 meters
processing time < 30 minutes