Map Building without Localization by Dimensionality Reduction Techniques Takehisa YAIRI RCAST, University of Tokyo Session : VISION, GRAPHICS AND ROBOTICS
Jan 17, 2016
Map Building without Localization by Dimensionality Reduction
Techniques
Takehisa YAIRI
RCAST, University of Tokyo
Session : VISION, GRAPHICS AND ROBOTICS
Outline
• Background
– Motivation, Purpose and Problem to consider
• Related Works
– SLAM, and Mapping with DR methods
• Proposed Framework - LFMDR
– Basic idea, Assumptions, Formalization
• Experiment
– Visibility-Only and Bearing-Only Mappings
• Conclusion
Motivation
• Map building– An essential capability for intelligent agents
• SLAM (Simultaneous Localization and Mapping)– Has been mainstream for many years– Very successful both in theory and practice
• I like SLAM too, but I feel something’s missing..– Are the mapping and localization really inseparable ?– Are the motion and measurement models necessary ?– How about the aspect of map building as an
abstraction of the world ?
• Is there another map building framework ?
Purpose
• Reconsider the robot map building from the viewpoint of dimensionality reduction and propose an alternative framework– Localization-Free Mapping by Dimensionality
Reduction (LFMDR)– No localization, no motion and measurement models– Heuristics : Closely located objects tend to share
similar histories of being observed by a robot
Reduce dimensionality,while preserving localityt=1
t=2t=N
Map Building Problem to Consider
• Feature-based map (i.e. Not topological, not occupancy-grid)
– A map is represented by 2-D coordinates of objects
• There EXIST motion and measurement models – But, they are not necessarily known in advance
1ξ 2ξ Mξ
mPositions of objects
tx1tx1tu
ty ttt mg wxy ,
tttt mf vuxx ,, 11
Observation
State
Move
Motion model(State transition model)
Measurement model(Observation model)
Exist, but may be unknown
Related Works : SLAM [Thrun 02]• Problem :
“Estimate m and x1:t from y1:t , given f and g”• Solutions:
– Kalman Filter with extended state– Incremental maximum likelihood [Thrun, et.al. 98]– Rao-Blackwellized Particle Filter [Montemerlo, et.al. 02]
• Motion and measurement models must be given• Estimations of map and robot position are coupled
Given: Output: Input:
Motion model
Measurement model
Measurement data
ttt mg wxy ,
tttt mf vuxx ,, 11 TT yyyy ,,, 21:1 Map
Robot trajectory TT xxxx ,,, 21:1
Mm ξξξ ,,, 21 jξ̂
Related Works : Dimensionality Reduction and Mapping (1)
• Idea of using DR for robot map building is not new itself ..
• [Brunskill & Roy 05]– PPCA to extract low-dimensional
geometric features (line segments) from range measurements
• [Pierce & Kuipers 97]– PCA to obtain low-level mappings
between robot’s actions and perceptions (sensorimotor mapping)
Point features(High dimensional)
Line segments(Low dimensional)
DR
Related Works : Dimensionality Reduction and Mapping (2)
• Another existing idea is to estimate robot’s states (locations, poses) from a sequence of high dimensional observation data
• Appearance manifolds [Ham, et.al. 05]– LLP + Kalman filter
• Action respecting embedding [Bowling, et.al. 05]– SDE
• Wifi-SLAM [Ferris, et.al. 07]– GP-LVM
DimensionalityReduction
Observation Space
State Space
x1
x2
Related Works : Dimensionality Reduction and Mapping (cont.)
• Treat row vectors as data points• Estimate x1:N and g, from y1:N ,
given f
ty
)()()1(
)()()1(
)(1
)(1
)1(1
MN
jNN
Mt
jtt
Mj
yyy
yyy
yyy
Observation data (from time 1 to N)
Dimension of features
Tim
e
Observation Space
y(1)
y(M)
y(j)
State Space
x1
x2
DR
trajectory
1. All objects are uniquely identifiable
2. Measurement model can be decomposed to homogeneous submodels for individual objects
3. Locations of at least 3 objects are known in advance (Anchor objects)
Proposed Framework : LFMDR (1)Assumptions
An observation about an object is roughly dependent only on its location, given the map and robot’s position
Decomposable
tMmmmM
ttt tttyyy wξξξ xxx ,2,1,
)()2()1( g,,g,g,,,
ttt m wxy ,g
The second assumption may look too restrictive, but, ..
Proposed Framework : LFMDR (2)Interpretation as a DR Problem
• Imagine a mapping between an object position and its history of observation
Tjmjm
TjN
jN
j
Nggyy ξξy xx ,,
)()(1:1
)( ,,,,1
)()()1(
)()()1(
)(1
)(1
)1(1
MN
jNN
Mt
jtt
Mj
yyy
yyy
yyy
ObservationData Matrix
Tim
e
jξ
jm Nξx :1,
Mapping)(
:1j
Ny
Observation History Space XY coordinates (2-dimensional)(N-dimensional)
“If two objects are closely located, their histories of observation are similar ”
Proposed Framework : LFMDR (4)Procedure
1. Explore the environment and obtain observation history data Y1:N
2. Break Y1:N into a set of column
vectors {y(j)1:N}j=1,…,M
3. Apply a DR method to the vectors and obtain a set of 2-D vectors
4. Perform the optimal Affine transformation w.r.t anchor objects, and obtain final estimates
)()()1(
)()()1(
)(1
)(1
)1(1
MN
jNN
Mt
jtt
Mj
yyy
yyy
yyy
)(:1
jNy
Features of LFMDR (1)(Comparison with SLAM)
• Common– Based on state space model
• Different– No assumption that motion and measurement
models are known– Map is directly estimated without robot
localization (localization-free mapping)– Off-line procedure– Larger amount of data required– Assumption of no missing data
ttt mg wxy ,
tttt mf vuxx ,, 11
Adva
ntag
esDi
sadv
anta
ges
Features of LFMDR (2)(Comparison with Other DR-based Approaches)• Comparison with [Brunskill & Roy 05]
– Global vs. Local
• Comparison with [Ham,et.al. 05] [Bowling,et.al. 05] [Ferris,et.al. 07]– Column vectors vs. Row vectors
(i.e., Object positions vs. Robot positions)
v.s.DRDR
1 0 0 1 : 0 : 1 11 1 0 0 : 0 : 1 11 0 0 0 : 1 : 1 10 0 0 0 : 1 : 0 10 0 0 0 : 1 : 0 11 1 1 0 : 1 : 0 1: : : : : : : : :: : : : : : : : :1 1 0 0 : 1 : 0 11 0 0 0 : 1 : 0 1
DRv.s.
DR
)()()1(
)()()1(
)(1
)(1
)1(1
MN
jNN
Mt
jtt
Mj
yyy
yyy
yyy
)()()1(
)()()1(
)(1
)(1
)1(1
MN
jNN
Mt
jtt
Mj
yyy
yyy
yyy
jξ̂
Experiment
• Applied to 2 different situations– [Case 1] Visibility-only mapping– [Case 2] Bearing-only mapping
• Common settings:– 2.5[m]x2.5[m] square region– 50 objects (incl. 4 anchors)– Exploration with random direction
change and obstacle avoidance
• Evaluation– Mean Position Error (MPE)– Mean Orientation Error (MOE)– Averaged over 25 runs
WEBOTS simulator
Triangle Orientation
A-B-C A-C-B
A
BC
A
B C
Difference
DR Methods
1. Linear PCA
2. SMACOF-MDS [DeLeeuw 77]
(a) Equal weights, (b) kNN-based weighting
3. Kernel PCA [Scholkopf,et.al. 98]
(a) Gaussian, (b) Polynomial
4. ISOMAP [Tennenbaum,et.al. 00]
5. LLE [Roweis&Saul 00]
6. Laplacian Eigenmap [Belkin&Niyogi 02]
7. Hessian LLE [Donoho&Grimes 03]
8. SDE [Weinberger, et.al. 05]
Parameters(k, 2, d) were tuned manually
Case 1 : Visibility-Only MappingDescription
• Building a map using only visibility information– i.e., Whether each object is visible (1) or not (0)
• An assumption in this simulation:– An object is visible if its horizontal visual angle of non-occluded
part is larger than 5 deg
Case 1 : Visibility-Only MappingVisibility Measurements
Visibility Observation Data
Object ID
1 2 3 4 … j … M-1 M
Observation history vector of an object
Tim
e
123456::
N-1
N
TjN 1,1,,1,1,1,1,0,0)(
:1 y
1 0 0 1 : 0 : 1 11 1 0 0 : 0 : 1 11 0 0 0 : 1 : 1 10 0 0 0 : 1 : 0 10 0 0 0 : 1 : 0 11 1 1 0 : 1 : 0 1: : : : : : : : :: : : : : : : : :1 1 0 0 : 1 : 0 11 0 0 0 : 1 : 0 1
Normalization)(
:1)(
:1)(
:1~ j
Nj
Nj
N yyy
)(:1
~ jNy
Observation HistorySpace
(Binary matrix)
Column
Compensate variety of the frequencies the objects are observed
Euc. norm
Case 1 : Visibility-Only MappingMaps After 2000 Time Steps
LPCA KPCA(Gaussian, 2=0.5)
Isomap(k=6)SMACOF (k=5)
LLE (k=8) LEM (k=6) SDE (k=7)HLLE (k=8)
Case 1 : Visibility-Only MappingMean Position Errors
0 400 800 1200 1600 20000
0.5
1
1.5
2
Number of Steps
Mea
n Po
sitio
n E
rror
(m
)
LPCASMACOF (WGT,K=5)
KPCA (GAUS,2=0.5)ISOMAP (K=6)LLE (K=8)LEM (K=6)HLLE (K=8)SDE (K=7)
Case 1 : Visibility-Only MappingFinal Map Errors
Gold
Silver
Bronze
LPCA(CMDS) - 1.055 10 18.19 8SMA(UNWGT) - 0.421 7 5.86 6SMA(WGT) K=5 0.206 4 4.83 4KPCA(GAUS) 2 = 0.5 0.926 8 23.29 9KPCA(POLY) d=8 0.953 9 27.03 10ISOMAP K=6 0.177 2 4.11 2LLE K=8 0.241 5 5.4 5LEM K=6 0.352 6 8.17 7HLLE K=8 0.192 3 4.24 3SDE K=7 0.138 1 3.65 1
DR methodsOpt. param.MPE [m] Rnk MOE[%] Rnk
Case 2 : Bearing-Only MappingDescription
• Building a map only with bearing measurements
– Motivated by recent popularity of Bearing-Only SLAM– Assuming all objects are always visible (No missing
observation)
(Relative direction angles to objects)
Bearingangles
Case 2 : Bearing-Only MappingBearing Measurements
1,1 2,1 : j,1 : M,1
1,2 2,2 : j,2 : M,2
1,N 2,N : j,N : M,N
Original Bearing DataObject ID
Tim
e
1 2 j M
1
2
N
cos1,1 cos2,1 : cosj,1 : cosM,1
sin1,1 sin2,1 : sinj,1 : sinM,1
cos1,2 cos2,2 : cosj,2 : cosM,2
sin1,2 sin2,2 : sinj,2 : sinM,2
cos1,N cos2,N : cosj,N : cosM,N
sin1,N sin2,N : sinj,N : sinM,N
1 2 j M
Unit directional vectors
Tim
e
1
2
N
)(:1
~ jNy
Observation History Space
2N-dimensional
Use a unit directional vector
instead of bearing angle
Discontinuity
tj , T
tjtj ,, sin,cos
Case 2 : Bearing-Only Mapping Maps After 2000 Time Steps
LPCA SMACOF (k=8) Isomap (k=9)
LLE (k=8) LEM (k=7) SDE (k=7)
Case 2 : Bearing-Only Mapping Mean Position Errors
500 1000 1500 20000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Number of Steps
Mea
n Po
sitio
n E
rror
(m
)
LPCASMA-WGT (K=8)ISOMAP (K=9)LLE (K=8)LEM (K=7)SDE (K=7)
Case 2 : Bearing-Only Mapping Final Map Errors
LPCA(CMDS) - 0.168 5 2.33 5SMA(UNWGT) - 0.101 4 1.38 3SMA(WGT) K=8 0.0609 1 1 1KPCA(GAUS) 2=1.0 3.47 9 49.2 9KPCA(POLY) d=2 0.605 8 9.15 8ISOMAP K=9 0.0979 3 1.83 4LLE K=8 0.173 6 3.03 6LEM K=7 0.367 7 8.46 7HLLE - NA - NA - SDE K=7 0.0741 2 1.36 2
DR methodsOpt. param.MPE [m] Rnk MOE[%] Rnk
Gold
Silver
Bronze
(*) It might imply the distribution approaches to linear
Conclusion
• Reconsidered robot map building from the viewpoint of dimensionality reduction
• Proposed a new framework named LFMDR – Motion and measurement models are not required– Not need to estimate robot’s poses (localization-free)– However, larger amount of data is needed
• Tested on two types of sensor measurements– Visibility information, and Bearing angles
• Compared a variety of DR methods
Future Works
• Relaxation of restrictions– Missing measurements
– Data association problem
• Scalability– Mapping of a larger number of objects
• On-line algorithm– Tracking of moving objects
• Multi-sensor fusion– e.g. mapping with bearing and range measurements