occupancy grid mapping · slides adapted from eric westman, cyrill stachniss tas: matthew rockett,...
TRANSCRIPT
![Page 1: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/1.jpg)
1
Sanjiban Choudhury
Occupancy Grid Mapping
Slides adapted from Eric Westman, Cyrill Stachniss
TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle
![Page 2: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/2.jpg)
What is an occupancy map?
2
![Page 3: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/3.jpg)
What is an occupancy map?
2
Probabilistic representation of world
from
noisy, uncertain sensor measurement data,
with the assumption
that the robot pose is known.
![Page 4: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/4.jpg)
What is an occupancy grid map?
316-833, Spring 20185
Example
Courtesy: C. Stachniss
Discretize world into cells
Assign a probability [0,1]to each cell
![Page 5: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/5.jpg)
When do need to map online?
4
![Page 6: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/6.jpg)
When do need to map online?
4
1. Cant rely on floor plans
![Page 7: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/7.jpg)
When do need to map online?
4
1. Cant rely on floor plans 2. Mapping disaster regions
![Page 8: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/8.jpg)
When do need to map online?
4
1. Cant rely on floor plans
3. Mapping unstructured environments
2. Mapping disaster regions
![Page 9: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/9.jpg)
Even floor plans can be wrong…
516-833, Spring 20188
Even when we have prior maps, they can be inaccurate…
CAD map occupancy grid mapTech Museum, San Jose
![Page 10: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/10.jpg)
Why do we need estimation?
616-833, Spring 20189
Occupancy Grids: From scans to maps
False “hits” from people in the environment
Scans are noisy - if you just added them they will contradict each other and create a mess!
![Page 11: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/11.jpg)
Bayes filter is a powerful tool
7Localization Mapping SLAM POMDP
![Page 12: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/12.jpg)
Assembling Bayes filter
8
TasksLocalization
P(pose | data)
Mapping P(map | data)
SLAM P(pose, map | data)
Belief Representations
Probabilistic Models
Bayes Filter
![Page 13: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/13.jpg)
Different tasks as Bayes filtering
9
Tasks Belief Representation Probabilistic Models
Localization P(pose | data) Gaussian / Particles Motion model
Measurement model
Mapping P(map | data) Discrete (binary) Inverse measurement model
SLAM P(pose, map |
data)
Gaussian (pose, landmarks) Motion model, measurement model, correspondence model
![Page 14: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/14.jpg)
Today’s objective
10
1. Understand occupancy grid mapping intuitively
2. Work through Bayes filter derivation
3. Examine when assumptions get violated
![Page 15: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/15.jpg)
Spilling the beans on mapping
11
![Page 16: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/16.jpg)
Spilling the beans on mapping
11
Step 1: Start with an empty map. 0.5 prob of being free.
![Page 17: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/17.jpg)
Spilling the beans on mapping
11
Step 1: Start with an empty map. 0.5 prob of being free.
zt
xt
Step 2: Accept the latest measurement and pose
![Page 18: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/18.jpg)
Spilling the beans on mapping
11
Step 1: Start with an empty map. 0.5 prob of being free.
zt
xt
Step 2: Accept the latest measurement and pose
Step 3: Raycast every beam. Group cells as HIT and MISS.
HIT: Bump down probability. MISS: Bump up.
zt
xt
![Page 19: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/19.jpg)
Spilling the beans on mapping
11
Step 1: Start with an empty map. 0.5 prob of being free.
zt
xt
Step 2: Accept the latest measurement and pose
Step 3: Raycast every beam. Group cells as HIT and MISS.
HIT: Bump down probability. MISS: Bump up.
zt
xt
Step 4: Get new measurement.
Update the map.
![Page 20: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/20.jpg)
12
Mapping as just another Bayes filtering problem
![Page 21: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/21.jpg)
12
Mapping as just another Bayes filtering problem
Task: Mapping P(map | data)
![Page 22: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/22.jpg)
12
Mapping as just another Bayes filtering problem
Task: Mapping P(map | data)
What is the data?
Stream of pose and laser scans x1:t, z1:t
zt
zt�1
xt�1
xt
![Page 23: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/23.jpg)
12
Mapping as just another Bayes filtering problem
Task: Mapping P(map | data)
What is the belief representation?
16-833, Spring 20189
Occupancy Grids: From scans to maps
False “hits” from people in the environment
Represent world as a
collection of cells
Each belief is [0,1]0 is occupied, 1 is free,
0.5 unknown
P (m) = P (m1,m2, . . . ,mn)
What is the data?
Stream of pose and laser scans x1:t, z1:t
zt
zt�1
xt�1
xt
![Page 24: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/24.jpg)
Graphical model of mapping
13
xt�1 xt xt+1
zt�1 zt zt+1
m
![Page 25: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/25.jpg)
Problem: Space of maps is huge!!
14
![Page 26: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/26.jpg)
Problem: Space of maps is huge!!
14
The state is a matrix of binary values
The belief assigns a probability to all states
![Page 27: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/27.jpg)
Problem: Space of maps is huge!!
14
Map m
mi
The state is a matrix of binary values
The belief assigns a probability to all states
![Page 28: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/28.jpg)
Problem: Space of maps is huge!!
14
Map m
mi
Example: We are mapping 25m x 25m area at 25 cm resolution.
100 x 100 grid = 10,000 cells
How many possible maps can there be?
2^10000 !!!
The state is a matrix of binary values
The belief assigns a probability to all states
![Page 29: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/29.jpg)
Solution: Approximate by independent cells
15
16-833, Spring 201818
Representation
• The probability distribution of the map is given by the product over the cells
example map(4-dim state)
4 individual cellsCourtesy: C. Stachniss
4 individual cells
Structured map (4-dim state)
Joint probability is approximated by product of individual probabilities
P (m) =Y
i
P (mi)
P (m|x1:t, z1:t) =Y
i
P (mi|x1:t, z1:t)
Binary r.v.
![Page 30: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/30.jpg)
Let’s crank through Bayes filter
16
![Page 31: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/31.jpg)
Let’s crank through Bayes filter
16
P (mi|z1:t, x1:t) =P (mi|z1:t�1, x1:t�1, zt, xt)
[old data] [new data]
![Page 32: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/32.jpg)
Let’s crank through Bayes filter
16
⌘P (zt|mi, z1:t�1, x1:t�1, xt)P (mi|z1:t�1, x1:t�1, xt)=(Bayes)
P (mi|z1:t, x1:t) =P (mi|z1:t�1, x1:t�1, zt, xt)
[old data] [new data]
![Page 33: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/33.jpg)
Let’s crank through Bayes filter
16
⌘P (zt|mi, z1:t�1, x1:t�1, xt)P (mi|z1:t�1, x1:t�1, xt)=(Bayes)
P (mi|z1:t, x1:t) =P (mi|z1:t�1, x1:t�1, zt, xt)
[old data] [new data]
(Cond Ind.) ⌘P (zt|mi, z1:t�1, x1:t�1, xt)P (mi|z1:t�1, x1:t�1)=
[old filter value][old data]
![Page 34: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/34.jpg)
Problem 1: Can we apply conditional indep?
17
P (zt|mi, xt)P (zt|mi, z1:t�1, x1:t�1, xt)=[old data][map value] [sensor pose]
![Page 35: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/35.jpg)
Let’s crank through Bayes filter
18
P (mi|z1:t, x1:t) =P (mi|z1:t�1, x1:t�1, zt, xt)
⌘P (zt|mi, z1:t�1, x1:t�1, xt)P (mi|z1:t�1, x1:t�1, xt)=(Bayes)
(Cond Ind.) ⌘P (zt|mi, z1:t�1, x1:t�1, xt)P (mi|z1:t�1, x1:t�1)=
⌘P (zt|mi, xt)P (mi|z1:t�1, x1:t�1)=(Cond Ind.)
![Page 36: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/36.jpg)
Problem 2: Sensor model is hard to define
19
Why is this hard to specify?P (zt|mi, xt)
ztxt
mi
[ray] [cell]
![Page 37: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/37.jpg)
Problem 2: Sensor model is hard to define
19
Why is this hard to specify?P (zt|mi, xt)
ztxt
mi
[ray] [cell]
P (zt|mi, xt) =P (mi|zt, xt)P (zt|xt)
P (mi|xt)[cell] [ray]
Is this easier to specify?
![Page 38: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/38.jpg)
Problem 2: Sensor model is hard to define
20
We can’t specify this…
P (zt|mi, xt)
![Page 39: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/39.jpg)
Problem 2: Sensor model is hard to define
20
We can’t specify this…
P (zt|mi, xt)
Solution: Apply Bayes to get an inverse sensor model
P (zt|mi, xt) =P (mi|zt, xt)P (zt|xt)
P (mi|xt)
![Page 40: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/40.jpg)
Let’s crank through Bayes filter
21
P (mi|z1:t, x1:t) =P (mi|z1:t�1, x1:t�1, zt, xt)
⌘P (zt|mi, z1:t�1, x1:t�1, xt)P (mi|z1:t�1, x1:t�1, xt)=(Bayes)
(Cond Ind.) ⌘P (zt|mi, z1:t�1, x1:t�1, xt)P (mi|z1:t�1, x1:t�1)=
⌘P (zt|mi, xt)P (mi|z1:t�1, x1:t�1)=(Cond Ind.)
![Page 41: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/41.jpg)
Let’s crank through Bayes filter
21
P (mi|z1:t, x1:t) =P (mi|z1:t�1, x1:t�1, zt, xt)
⌘P (zt|mi, z1:t�1, x1:t�1, xt)P (mi|z1:t�1, x1:t�1, xt)=(Bayes)
(Cond Ind.) ⌘P (zt|mi, z1:t�1, x1:t�1, xt)P (mi|z1:t�1, x1:t�1)=
⌘P (zt|mi, xt)P (mi|z1:t�1, x1:t�1)=(Cond Ind.)
=⌘
P (mi|zt, xt)P (zt|xt)
P (mi|xt)P (mi|z1:t�1, x1:t�1)(Bayes.)
![Page 42: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/42.jpg)
Let’s crank through Bayes filter
21
P (mi|z1:t, x1:t) =P (mi|z1:t�1, x1:t�1, zt, xt)
⌘P (zt|mi, z1:t�1, x1:t�1, xt)P (mi|z1:t�1, x1:t�1, xt)=(Bayes)
(Cond Ind.) ⌘P (zt|mi, z1:t�1, x1:t�1, xt)P (mi|z1:t�1, x1:t�1)=
⌘P (zt|mi, xt)P (mi|z1:t�1, x1:t�1)=(Cond Ind.)
=⌘
P (mi|zt, xt)P (zt|xt)
P (mi|xt)P (mi|z1:t�1, x1:t�1)(Bayes.)
(Cond Ind.) ⌘
P (mi|zt, xt)P (zt|xt)
P (mi)P (mi|z1:t�1, x1:t�1)=
![Page 43: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/43.jpg)
Let’s look at likelihood ratios
22
P (mi|z1:t, x1:t) ⌘
P (mi|zt, xt)P (zt|xt)
P (mi)P (mi|z1:t�1, x1:t�1)=
There are terms we don’t know and would not like to calculate!
[cell=1]
![Page 44: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/44.jpg)
Let’s look at likelihood ratios
22
P (mi|z1:t, x1:t) ⌘
P (mi|zt, xt)P (zt|xt)
P (mi)P (mi|z1:t�1, x1:t�1)=
There are terms we don’t know and would not like to calculate!
[cell=1]
⌘
P (¬mi|zt, xt)P (zt|xt)
P (¬mi)P (¬mi|z1:t�1, x1:t�1)=
P (¬mi|z1:t, x1:t)
Let’s look at the opposite probability!
[cell=0]
![Page 45: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/45.jpg)
Let’s look at likelihood ratios
22
P (mi|z1:t, x1:t) ⌘
P (mi|zt, xt)P (zt|xt)
P (mi)P (mi|z1:t�1, x1:t�1)=
There are terms we don’t know and would not like to calculate!
Let’s look at the ratio
P (mi|z1:t, x1:t)
P (¬mi|z1:t, x1:t)=
P (mi|zt, xt)
P (¬mi|zt, xt)
P (¬mi)
P (mi)
P (mi|z1:t�1, x1:t�1)
P (¬mi|z1:t�1, x1:t�1)
[cell=1]
⌘
P (¬mi|zt, xt)P (zt|xt)
P (¬mi)P (¬mi|z1:t�1, x1:t�1)=
P (¬mi|z1:t, x1:t)
Let’s look at the opposite probability!
[cell=0]
![Page 46: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/46.jpg)
Log likelihood ratios
23
P (mi|z1:t, x1:t)
P (¬mi|z1:t, x1:t)=
P (mi|zt, xt)
P (¬mi|zt, xt)
P (¬mi)
P (mi)
P (mi|z1:t�1, x1:t�1)
P (¬mi|z1:t�1, x1:t�1)
(inverse sensor model) (new ratio)(prior)
![Page 47: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/47.jpg)
Log likelihood ratios
23
P (mi|z1:t, x1:t)
P (¬mi|z1:t, x1:t)=
P (mi|zt, xt)
P (¬mi|zt, xt)
P (¬mi)
P (mi)
P (mi|z1:t�1, x1:t�1)
P (¬mi|z1:t�1, x1:t�1)
(inverse sensor model) (new ratio)(prior)
Taking logs of all terms
log
✓P (mi|z1:t, z1:t)P (¬mi|z1:t, z1:t)
◆log
✓P (mi|zt, xt)
P (¬mi|zt, xt)
◆log
✓P (mi)
P (¬mi)
◆log
✓P (mi|z1:t�1, x1:t�1)
P (¬mi|z1:t�1, x1:t�1)
◆= � +
![Page 48: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/48.jpg)
Log likelihood ratios
23
P (mi|z1:t, x1:t)
P (¬mi|z1:t, x1:t)=
P (mi|zt, xt)
P (¬mi|zt, xt)
P (¬mi)
P (mi)
P (mi|z1:t�1, x1:t�1)
P (¬mi|z1:t�1, x1:t�1)
(inverse sensor model) (new ratio)(prior)
Taking logs of all terms
log
✓P (mi|z1:t, z1:t)P (¬mi|z1:t, z1:t)
◆log
✓P (mi|zt, xt)
P (¬mi|zt, xt)
◆log
✓P (mi)
P (¬mi)
◆log
✓P (mi|z1:t�1, x1:t�1)
P (¬mi|z1:t�1, x1:t�1)
◆= � +
lt l(mi|zt, xt) lt�1l0= +�(updated belief) (old belief)(inverse model) (prior)
![Page 49: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/49.jpg)
Pseudo code of occupancy mapping
24
zt
xt
for every ray r in (xt, zt)
for every cell mi in r
if mi is MISS
li = li + l(HIT)� l0
li = li + l(MISS)� l0else
![Page 50: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/50.jpg)
What is the inverse sensor model P(m|z,x)?
25
P (zt|mi, xt) =P (mi|zt, xt)P (zt|xt)
P (mi|xt)
P (zt|mi, xt) =P (mi|zt, xt)P (zt|xt)
P (mi|xt)
P (mi)� �
P (mi) + �
P (mi)=
if ray passes through cell
if ray does not intersect
cell
if ray stops in cell
[cell] [ray]
![Page 51: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/51.jpg)
Problem: Dynamic obstacles
26
t=0 t=2 t=4 t=6 t=8 t=10 t=12
Laser array
What will the occupancy map look like?
![Page 52: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/52.jpg)
Problem: Dynamic obstacles
27
t=2 t=4 t=6
t=8 t=10 t=12
But is this what we want??
![Page 53: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/53.jpg)
Solution: Don’t use independence
28
t=2 t=4 t=6
t=8 t=10 t=12
If we directly guessed the best explanation, we can come up with this!
![Page 54: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/54.jpg)
Solution: Don’t use independence
28
t=2 t=4 t=6
t=8 t=10 t=12
If we directly guessed the best explanation, we can come up with this!
What is wrong with independence? If I know there is ONE obstacle and if a cell gets a MISS
and neighbor gets a HIT, then cell must be FREE (P=1).
![Page 55: Occupancy Grid Mapping · Slides adapted from Eric Westman, Cyrill Stachniss TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle. What is an occupancy map? 2. What is an occupancy map?](https://reader030.vdocuments.mx/reader030/viewer/2022040720/5e2b3817b9e70a331869c653/html5/thumbnails/55.jpg)
Dynamic obstacle mapping in general
29
“Map building with mobile robots in dynamic environments” D. Hähnel, R. Triebel, W. Burgard, and S. Thrun. 2003
“Occupancy Grid Models for Robot Mapping in Changing Environments” D. Meyer-Delius, M. Beinhofer and W. Burgard 2003