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1
Robot Navigation and
collision avoidance
Copyright © Jacques Saraydaryan
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Robot Navigation and collision avoidance
2
OutLine
Introduction
Mapping
Navigation
Collision avoidance
Copyright © Jacques Saraydaryan
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Introduction: What
navigation means ?
3 Copyright © Jacques Saraydaryan
Robot Navigation and collision avoidance
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What navigation means ?
4
‟ The Process of directing a vehicle so as to reach the intended destination ” IEEE Standard 172-1983
Copyright © Jacques Saraydaryan
‟ Given partial knowledge about its environment and a goal position or a
series of positions, navigation encompasses the ability of the robot to act based on its knowledge and sensors values so as to reach its goal positions as
efficiently and reliably as possible ” Introduction to Autonomous Mobile Robots, MIT Press, Roland SIEGWART, Illah R. NOURBAKHSH 2004
‟ Robot navigation is the problem of guiding a robot towards a goal” Robotics, Vision and Control, Springer, Peter Corke 2011
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5 Copyright © Jacques Saraydaryan
What navigation means ?
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6 Copyright © Jacques Saraydaryan
Data Fusion
Collecting all data from sensors
Transform data into common
languages
Merging data
Convert into same geometric
standard
Clean data
Merge information into a
common representation
What navigation means ?
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7 Copyright © Jacques Saraydaryan
Mapping
Collecting all merged data
Build a cumulative representation
of data
Express the environment obstacle
world into a unique robot
readable data
What navigation means ?
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8 Copyright © Jacques Saraydaryan
Localization
Collect sensors data
Collect encoders data
Process all data regarding a given
map
Express one or many robot
position estimations
What navigation means ?
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9 Copyright © Jacques Saraydaryan
Navigation
Collect one or many robot position
estimation
Use the map as obstacle estimator
Compute path form estimate
position to a targeted position
Re-plan or react in case of new or
dynamic obstacles observation
What navigation means ?
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10 Copyright © Jacques Saraydaryan
Navigation – overview – [ {loc:{x:10,y:5,z:2}, acc:60%}, {loc:{x:5,y:50,z:20}, acc:1%}, {loc:{x:1,y:1,z:14}, acc:5%}, {loc:{x:11,y:6,z:2}, acc:20%}, ... ]
{x:10,y:5,z:2} 80% OR
[ start:{x:1,y:1,z:0} {loc:{x:1,y:10,z:0}, order:1}, {loc:{x:5,y:10,z:0}, order:2}, {loc:{x:7,y:12,z:0}, order:3}, {loc:{x:9,y:14,z:0}, order:4}, goal:{x:1,y:1,z:0} ]
What navigation means ?
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11 Copyright © Jacques Saraydaryan
Navigation – overview – [ {loc:{x:10,y:5,z:2}, acc:60%}, {loc:{x:5,y:50,z:20}, acc:1%}, {loc:{x:1,y:1,z:14}, acc:5%}, {loc:{x:11,y:6,z:2}, acc:20%}, ... ]
{x:10,y:5,z:2} 80% OR
[ start:{x:1,y:1,z:0} {loc:{x:1,y:10,z:0}, order:1}, {loc:{x:5,y:10,z:0}, order:2}, {loc:{x:7,y:12,z:0}, order:3}, {loc:{x:9,y:14,z:0}, order:4}, goal:{x:1,y:1,z:0} ]
What navigation means ?
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12 Copyright © Jacques Saraydaryan
Navigation – strategies–
Behavior-Based
No Localization
External goal
e.g : wall follower
What navigation means ?
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13 Copyright © Jacques Saraydaryan
Navigation – strategies–
Behavior-Based
No Localization
External goal
Reactive-Based
No Localization
Sensor based goal
e.g: Braitenberg Vehicle
What navigation means ?
https://www.youtube.com/watch?v=A-fxij3zM7g
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14 Copyright © Jacques Saraydaryan
Navigation – strategies–
Behavior-Based
No Localization
External goal
Reactive-Based
No Localization
Sensor based goal
Map-Based
Localization
External goal
E.g: Dynamic A*
What navigation means ?
https://www.youtube.com/watch?v=qziUJcUDfBc
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15 Copyright © Jacques Saraydaryan
Behavior Based Architecture
What navigation means ?
Introduction to Autonomous Mobile Robots, MIT Press, Roland SIEGWART, Illah R. NOURBAKHSH 2004
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16 Copyright © Jacques Saraydaryan
Map-Based Architecture
What navigation means ?
Introduction to Autonomous Mobile Robots, MIT Press, Roland SIEGWART, Illah R. NOURBAKHSH 2004
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Mapping
17 Copyright © Jacques Saraydaryan
Robot Navigation and collision avoidance
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Mapping
18 Copyright © Jacques Saraydaryan
Overview
Objectives
Put observed data into a standard view (obstacles, objects, robot)
Use to estimate the robot position
Use to compute a trajectory from a start point to a goal
Summarize the collected data
Map requirement
Map accurancy matches the precision which the robot needs to achieve a goal
Map accurancy matches the precision of the precision robot’s sensor.
Complexity of the map representation has direct impact on the computational
complexity of reasoning about mapping, localization and navigation
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Mapping
19 Copyright © Jacques Saraydaryan
Configuration Space
C = C𝑜𝑏𝑠 ∪C𝑓𝑟𝑒𝑒
C𝑜𝑏𝑠
C𝑓𝑟𝑒𝑒
𝑞 = (𝑥, 𝑦, 𝜃)
𝑞𝐼
𝑞𝐺
𝑞𝐼 ∈ C𝑓𝑟𝑒𝑒 Initial configuration
𝑞𝐺 ∈ C𝑓𝑟𝑒𝑒 Goal configuration
C Set of all possible transformations that may be applied on the robot.
𝑝𝑎𝑡ℎ 𝜏: 0,1 → C𝑓𝑟𝑒𝑒
𝜏 0 = 𝑞𝐼
𝜏 1 = 𝑞𝐺
𝜏 = {𝑞𝐼, … ,𝑞𝑖, … ,
𝑞𝐺}
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Mapping
20 Copyright © Jacques Saraydaryan
Configuration Space
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Mapping
21 Copyright © Jacques Saraydaryan
Configuration Space: Accommodate Robot Size
16-735, Howie Choset with slides from G.D. Hager, Z. Dodds, and Dinesh Mocha
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Mapping
22 Copyright © Jacques Saraydaryan
Configuration Space: Accommodate Robot Size
C𝑜𝑏𝑠
C𝑓𝑟𝑒𝑒
𝑅𝑜𝑏𝑜𝑡
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Mapping
23 Copyright © Jacques Saraydaryan
Configuration Space: Accommodate Robot Size
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Mapping
24 Copyright © Jacques Saraydaryan
Configuration Space: Accommodate Robot Size
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Mapping
25 Copyright © Jacques Saraydaryan
Map representation
Continuous
All objects in the map are represented
Map size depends of the objects density (sparse environment leads to low-memory map)
Decomposition
General decomposition and selection of environment features
Loss of fidelity between map and real environment
Capture useful features and discarding other
Fixed-decomposition and adaptive decomposition
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Mapping
26 Copyright © Jacques Saraydaryan
Continuous representation
Polygone representation
3D polygone map construction
2D polygone map construction
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Mapping
27 Copyright © Jacques Saraydaryan
Continuous representation
Line representation (EPFL)
(a) Real world
(b) Representation with a set of infinite lines
Introduction to Autonomous Mobile Robots, MIT Press, Roland SIEGWART, Illah R. NOURBAKHSH 2004
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Mapping
28 Copyright © Jacques Saraydaryan
Contineous representation
Limitations Avantages + -
High robot location precision
Respect the real world obstacle
position and shape
Low cost memory in case of spare
environnent
High computation and memory
cost in environement with high
objects density
Path planning becomes harder
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Mapping
29 Copyright © Jacques Saraydaryan
Decomposition
Exact cell decomposition
http://cs.stanford.edu/people/eroberts/courses/soco/projects/1998-99/robotics/basicmotion.html
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Mapping
30 Copyright © Jacques Saraydaryan
Decomposition
Fixed decomposition
Introduction to Autonomous Mobile Robots, MIT Press, Roland SIEGWART, Illah R. NOURBAKHSH 2004
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Mapping
31 Copyright © Jacques Saraydaryan
Decomposition
Adaptative decomposition
Introduction to Autonomous Mobile Robots, MIT Press, Roland SIEGWART, Illah R. NOURBAKHSH 2004
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Mapping
32 Copyright © Jacques Saraydaryan
Decomposition
Topological decomposition
http://www.cim.mcgill.ca/~mrl/pubs/saul/iros98.pdf
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Mapping
33 Copyright © Jacques Saraydaryan
Decomposition
Topological decomposition
http://www.cim.mcgill.ca/~mrl/pubs/saul/iros98.pdf
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Mapping
34 Copyright © Jacques Saraydaryan
Decomposition
Limitations Avantages + -
Most of the time the map size is
predictable
Ajustable abstraction is possible
according to the targeted goal
Lot’s of path planning algorihm exist
Could be far from real
environment geometry and
representation
Size of the map could grow with
the size of the environment
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Mapping
35 Copyright © Jacques Saraydaryan
Case of study: Occupancy grid
Fixed cell size decomposition Resulted occupancy grid map
𝑢𝑛𝑘𝑛𝑜𝑤𝑛 𝑎𝑟𝑒𝑎
𝑐𝑒𝑙𝑙 𝑤𝑖𝑡ℎ 𝑜𝑏𝑠𝑡𝑎𝑐𝑙𝑒
𝑓𝑟𝑒𝑒 𝑐𝑒𝑙𝑙
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Mapping
36 Copyright © Jacques Saraydaryan
Case of study: Occupancy grid 𝐶𝑖, 𝑗
𝑜𝑐𝑐(𝑖, 𝑗) 𝐶𝑖, 𝑗 is occupied
𝑝(𝑜𝑐𝑐 𝑖, 𝑗 ) Probability that 𝐶𝑖, 𝑗 is occupied [0,1]
𝑜(𝑜𝑐𝑐 𝑖, 𝑗 ) Odds function has range [0, +∞)
𝑜 𝐴 =𝑝(𝐴)
𝑝(¬𝐴)
log (𝑜 𝑜𝑐𝑐 𝑖, 𝑗 ) Log Odds function has range (−∞, +∞)
Each 𝐶𝑖, 𝑗 holds a value log (𝑜 𝑜𝑐𝑐 𝑖, 𝑗 ) log (𝑜 𝑜𝑐𝑐(𝑖, 𝑗) ) = 𝑙𝑜𝑔𝑝(𝑜𝑐𝑐(𝑖, 𝑗))
𝑝(¬𝑜𝑐𝑐(𝑖, 𝑗))
Definitions
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Mapping
37 Copyright © Jacques Saraydaryan
Case of study: Occupancy grid
Updating grid
On each observation by sensor the following assumption is made
Reminder : Bayes law
𝑝 𝐴|𝐵 =𝑝 𝐵|𝐴 ∗ 𝑝 𝐴
𝑝(𝐵)
𝐴 is occ(i, j)
𝐵 is an observation r giving a value D
𝑝 ¬𝐴|𝐵 =𝑝 𝐵|¬𝐴 ∗ 𝑝 ¬𝐴
𝑝(𝐵)
𝑜 𝐴|𝐵 =𝑝 𝐴|𝐵
𝑝(¬𝐴|𝐵)=
𝑝 𝐵|𝐴 ∗ 𝑝 𝐴
𝑝 𝐵|¬𝐴 ∗ 𝑝 ¬𝐴= 𝐵|𝐴 ∗ 𝑜 𝐴
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Mapping
38 Copyright © Jacques Saraydaryan
Case of study: Occupancy grid
Updating grid
𝐴 is occ(i, j)
𝐵 is an observation r giving a value D 𝑜 𝐴|𝐵 = 𝐵|𝐴 ∗ 𝑜 𝐴
𝑜 𝐴|𝐵 =𝑃 𝐴 𝐵
𝑃(¬𝐴|𝐵)
Probability that 𝐶𝑖, 𝑗 is occupied knowing an observation 𝑟 = 𝐷
Probability that 𝐶𝑖, 𝑗 is not occupied knowing an observation 𝑟 = 𝐷
𝐵|𝐴 =𝑃 𝐵 𝐴
𝑃(𝐵|¬𝐴)
Probability that we made an observation 𝑟 = 𝐷 knowing 𝐶𝑖, 𝑗 is occupied
Probability that we made an observation 𝑟 = 𝐷 knowing 𝐶𝑖, 𝑗 is free
log (𝑜 𝐴 𝐵 = log 𝐵|𝐴 + log (𝑜 𝐴 )
By extension :
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Mapping
39 Copyright © Jacques Saraydaryan
Case of study: Occupancy grid
Update Algorithm
𝐶𝑖, 𝑗 = log (𝑜(𝑜𝑐𝑐 𝑖, 𝑗 ))
𝑟 = 𝐷 1 Sensor (lazer) get information about the
environment r=D on the cell 𝐶𝑖, 𝑗.
2 Information about the map is collected on the
targeted cell 𝐶𝑖, 𝑗 = log(o (𝑜𝑐𝑐 𝑖, 𝑗 ))
3 The new believe on the cell is computed :
log (𝑜 occ(i, j) r = D
= log r = D|𝑜𝑐𝑐 𝑖, 𝑗
+ log(o (𝑜𝑐𝑐 𝑖, 𝑗 ))
p(r = D|𝑜𝑐𝑐 𝑖, 𝑗 )
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Mapping
40 Copyright © Jacques Saraydaryan
Case of study: Occupancy grid
p(r = D|𝑜𝑐𝑐 𝑖, 𝑗 )
p r > D ¬𝑜𝑐𝑐 𝑖, 𝑗 )
r = D|𝑜𝑐𝑐 𝑖, 𝑗 =p r = D 𝑜𝑐𝑐 𝑖, 𝑗 )
p r = D ¬𝑜𝑐𝑐 𝑖, 𝑗 )
Lazer detects at a distance D on the cell 𝐶𝑖, 𝑗.
Lazer passes through the cell 𝐶𝑖, 𝑗
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Mapping
41 Copyright © Jacques Saraydaryan
Case of study: Occupancy grid
Introduction to Mobile Robotics, Mapping with Known Poses, Wolfram Burgard, Cyrill Stachniss, Maren Bennewitz, Kai Arras
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Mapping
42 Copyright © Jacques Saraydaryan
Case of study: Occupancy grid
Introduction to Mobile Robotics, Mapping with Known Poses, Wolfram Burgard, Cyrill Stachniss, Maren Bennewitz, Kai Arras
Using a given grid map occupancy value (e.g 0.5)
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Navigation:
Path Planning
45 Copyright © Jacques Saraydaryan
Robot Navigation and collision avoidance
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Navigation: Path Planning
46 Copyright © Jacques Saraydaryan
Path Planning
Objective:
find continuous path 𝜏 into C𝑓𝑟𝑒𝑒 from start position 𝑞𝐼 to
goal position 𝑞𝐺.
3 main approaches
Road Map path planning
Identify a set of routes within C𝑓𝑟𝑒𝑒
Cell Decomposition path planning
Discrimintate between free and occupied cells
(Exact Cell Decomposition, Adaptative Cell Decomposition)
Environmental based path planning
Environement information drive the algorithm
Potential field, ant colony
𝑝𝑎𝑡ℎ 𝜏: 0,1 → C𝑓𝑟𝑒𝑒
𝜏 0 = 𝑞𝐼
𝜏 1 = 𝑞𝐺
𝜏 = {𝑞𝐼, … ,𝑞𝑖, … ,
𝑞𝐺}
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47 Copyright © Jacques Saraydaryan
RoadMap Planning
Navigation: Path Planning
Methods
Visibility Graph
Voronoi Diagram
Rapid Random Tree
Properties
Produce a graph in C𝑓𝑟𝑒𝑒 such as vertex is in
C𝑓𝑟𝑒𝑒 and edge a collision free path in C𝑓𝑟𝑒𝑒
Mostly based on continuous map (ploygonal
representation of the environment)
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48 Copyright © Jacques Saraydaryan
Visibility Graph
Navigation: Path Planning
Objective
Create a connectivity graph between obstacles
vertices and start/ goal position
Algorithm
Graph computation
vertices : all vertices of obstacles (polygon) +
start point and goal point
edges : edges joining all pair of vertices that can
« see » each other
Path selection
Short path algorithms (Dijkstra , A*)
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49 Copyright © Jacques Saraydaryan
Visibility Graph
Navigation: Path Planning
𝑞𝐼
𝑞𝐺
𝑞𝐼
𝑞𝐺
𝑞𝐼
𝑞𝐺
Initial Situation 1 Vertex and associated edges
All Vertices and associated edges Resulted Graph
𝑞𝐼
𝑞𝐺
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50 Copyright © Jacques Saraydaryan
Visibility Graph
Navigation: Path Planning
Limitations Avantages + -
Very simple
Good candidate if continuous
representation
Fast on sparse environement
The size depends of number of
polygon vertices
Slow on densely populated
environment
Robot tend to be very close to the
obstacles
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51 Copyright © Jacques Saraydaryan
Voronoi Diagram
Navigation: Path Planning
Objective
construct lines from all points that are equidistant
from 2 or more obstacles
Algorithm
Graph Construction
Green et Sibson
Shamos et Hoey
Fortune
Randomized incremental construction
Path selection
Short path algorithms (Dijkstra , A*)
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52 Copyright © Jacques Saraydaryan
(a) random points, k = 25; (b) four points forming a rectangle, k = 4; (c) four walls forming a rectangular environment; (d) rectangular environment with fives polygonal obstacles with pruned parts of the Voronoi diagram outside the freespace of the polygonal environment
Voronoi Diagram
http://www.mdpi.com/1424-8220/15/6/12736/htm
Navigation: Path Planning
http://alexbeutel.com/webgl/voronoi.html Usage sample:
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53 Copyright © Jacques Saraydaryan
Voronoi Diagram
Navigation: Path Planning
Fortune Algorithm
Sweep line : vertical line moving from the left to the
right
Beach line : parabolas compositions dividing the
portion of the plane on the left side of the sweep line
When obstacle is cross by the sweep line a parabol is
added to the beach line such as is point of this
parabol is equidistant from the obstacle to the sweep
line
Vertices of the beach line refers to parabol
intersection points
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54 Copyright © Jacques Saraydaryan
Navigation: Path Planning
Limitations Avantages + -
Allow « safe » navigation
Executability (better for obstacle
avoidance)
Interesting for autonomous mapping
Non optimal navigation path
length
Localization becomes difficult for
short range sensors
Unnatural attraction to
openspace suboptimal path
Voronoi Diagram
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55 Copyright © Jacques Saraydaryan
Probabilistic RoadMap (PRM)
Navigation: Path Planning
Objective
Determining a path between 𝑞𝐼 and 𝑞𝐺 without
obstacle collision by getting successif random point
in C𝑓𝑟𝑒𝑒
Algorithm
Graph Construction
Take random point
Check random point in C𝑓𝑟𝑒𝑒
Try to connect this point current graph through « a local planner »
Path selection
Short path algorithms (Dijkstra , A*)
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56 Copyright © Jacques Saraydaryan
Rapid Random Tree (RRT)
Navigation: Path Planning
Objective
Explore aggressively C by extending possible
locations from initial position 𝑞𝐼
Algorithm
Graph Construction
Incremental algorithm
Path selection
Short path algorithms (Dijkstra , A*)
𝐺. 𝐼𝑛𝑖𝑡(𝑞𝐼) Repeat
𝑞𝑟𝑎𝑛𝑑 → 𝑅𝑎𝑛𝑑𝑜𝑚_𝐶𝑜𝑛𝑓𝑖𝑔(C)
𝑞𝑛𝑒𝑎𝑟 → 𝑁𝑒𝑎𝑟𝑒𝑠𝑡 𝐺, 𝑞𝑟𝑎𝑛𝑑
𝐺.add_edge 𝑞𝑛𝑒𝑎𝑟, 𝑞𝑟𝑎𝑛𝑑
Until 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛
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57 Copyright © Jacques Saraydaryan
Rapid Random Tree (RRT)
Navigation: Path Planning
𝐺. 𝐼𝑛𝑖𝑡(𝑞𝐼) Repeat
𝑞𝑟𝑎𝑛𝑑 → 𝑅𝑎𝑛𝑑𝑜𝑚_𝐶𝑜𝑛𝑓𝑖𝑔(C)
𝑞𝑛𝑒𝑎𝑟 → 𝑁𝑒𝑎𝑟𝑒𝑠𝑡 𝐺, 𝑞𝑟𝑎𝑛𝑑
𝐺.add_edge 𝑞𝑛𝑒𝑎𝑟, 𝑞𝑟𝑎𝑛𝑑
Until 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛
𝑞𝐼
𝑞𝐺
1st it. 𝑞rand
= 𝑞near
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58 Copyright © Jacques Saraydaryan
Rapid Random Tree (RRT)
Navigation: Path Planning
𝐺. 𝐼𝑛𝑖𝑡(𝑞𝐼) Repeat
𝑞𝑟𝑎𝑛𝑑 → 𝑅𝑎𝑛𝑑𝑜𝑚_𝐶𝑜𝑛𝑓𝑖𝑔(C)
𝑞𝑛𝑒𝑎𝑟 → 𝑁𝑒𝑎𝑟𝑒𝑠𝑡 𝐺, 𝑞𝑟𝑎𝑛𝑑
𝐺.add_edge 𝑞𝑛𝑒𝑎𝑟, 𝑞𝑟𝑎𝑛𝑑
Until 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛
𝑞𝐼
𝑞𝐺
𝑞𝐼
𝑞𝐺
2nd it.
2nd it. 𝑞rand
= 𝑞near
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59 Copyright © Jacques Saraydaryan
Rapid Random Tree (RRT)
Navigation: Path Planning
𝐺. 𝐼𝑛𝑖𝑡(𝑞𝐼) Repeat
𝑞𝑟𝑎𝑛𝑑 → 𝑅𝑎𝑛𝑑𝑜𝑚_𝐶𝑜𝑛𝑓𝑖𝑔(C)
𝑞𝑛𝑒𝑎𝑟 → 𝑁𝑒𝑎𝑟𝑒𝑠𝑡 𝐺, 𝑞𝑟𝑎𝑛𝑑
𝐺.add_edge 𝑞𝑛𝑒𝑎𝑟, 𝑞𝑟𝑎𝑛𝑑
Until 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛
𝑞𝐼
𝑞𝐺
𝑞𝐼
𝑞𝐺
3rd it.
10th it.
𝑞rand
𝑞near
𝑞rand=
𝑞near
At n th iterations force 𝑞rand= 𝑞G
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Qu’est ce que le Cloud Computing ?
60 Copyright © Jacques Saraydaryan
Rapid Random Tree (RRT)
http://msl.cs.uiuc.edu/rrt/index.html http://msl.cs.uiuc.edu/rrt/gallery_rigid.html
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61 Copyright © Jacques Saraydaryan
Decomposition path planning
Navigation: Path Planning
Methods
Exact cell decomposition
Fixed cell decomposition
Adaptative cell decomposition
Properties
Map (Exact / Fixed / adaptative) gives graph
vertices
Cell connectivities gives graph edges
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62 Copyright © Jacques Saraydaryan
Cell connectivities
Navigation: Path Planning
Exact Cell decomposition
Direct Neighbors cell is not an obstacle
Introduction to Autonomous Mobile Robots, MIT Press, Roland SIEGWART, Illah R. NOURBAKHSH 2004
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Cell connectivities
Navigation: Path Planning
Fixed or adaptative Cell decomposing
1 2 3
4
5 6 7
8 0
All vertices are not
equidistant to
vertex 0
All vertices ARE
equidistant to
vertex 0
0 1
2 3
4 5
6
Origin Cell
Reachable Cell
Cell Connectivity
63
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Cell connectivities
Navigation: Path Planning
64
Real environment Fixed Cell Decompostion
Resulted cell connectivity graph
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65 Copyright © Jacques Saraydaryan
Environmental based path planning
Navigation: Path Planning
Methods
Potential fields
Ant colony
Properties
The environment areas drive the navigation
Robots do not need heavy computation
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66 Copyright © Jacques Saraydaryan
Potential Fields
Navigation: Path Planning
Objective
Generate attractive and repulsive potential fieldon
the environment to drive the robot until it reaches
the goal
Algorithm
Obstacles generate repulsive potential field.
The more the robot is closed to the
obstacle, the higher the repusive potential
field is,
Goal generates attractive potential field
Robotic Motion Planning: Potential Functions,Robotics Institute 16-735, Howie Choset
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67 Copyright © Jacques Saraydaryan
Potential Fields
Navigation: Path Planning
https://www.youtube.com/watch?v=DVnbp9oZZak
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68 Copyright © Jacques Saraydaryan
Potential Fields
Navigation: Path Planning
https://www.youtube.com/watch?v=kpWSDyr7sM0
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70 Copyright © Jacques Saraydaryan
Ant colony
Navigation: Path Planning
Objective
Individues spread on the environment a quantity of
pheromone highlighting their path. Large number of
individues and evaporation process converge to a
solution.
Algorithm
Ants travel on the environment to find food,
Once 1 ant find food, it comes back to the colony spreading pheromone
Other ant are attracted by the pheronome and will reinforce the pheromone if
they find food
If several path are possibles, the evaporation process lead to select the
shortest path
https://www.youtube.com/watch?v=vAnN3nZqMqk
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71 Copyright © Jacques Saraydaryan
Ant colony
Navigation: Path Planning
https://www.youtube.com/watch?v=vAnN3nZqMqk
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Navigation: Short path samples
73 Copyright © Jacques Saraydaryan
Robot Navigation and collision avoidance
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74 Copyright © Jacques Saraydaryan
Wavefront: a Breadth-first search
Navigation: Path Planning
Principle
Explore the frontier by launching a wavefront
that marks each hit cells with a distance to the
original point http://www.redblobgames.com/pathfinding/a-star/introduction.html
𝑈𝑛𝑣𝑖𝑠𝑖𝑡𝑒𝑑= 𝑞𝐼 𝑑𝑖𝑠𝑡 𝑞𝐼 = 0 𝑝𝑟𝑒𝑣 𝑞𝐼 = 𝑁𝑜𝑛𝑒
For each 𝑢
∈ 𝑈𝑛𝑣𝑖𝑠𝑖𝑡𝑒𝑑
𝑟𝑒𝑚𝑜𝑣𝑒 𝑢 𝑓𝑟𝑜𝑚 𝑈𝑛𝑣𝑖𝑠𝑖𝑡𝑒𝑑 For each 𝑣 ∈ 𝑁𝑒𝑖𝑔ℎ𝑏𝑜𝑟 𝑢
If dist v ∄ 𝑎𝑑𝑑 𝑣 𝑡𝑜 𝑈𝑛𝑣𝑖𝑠𝑖𝑡𝑒𝑑 𝑑𝑖𝑠𝑡 𝑣 = 𝑑𝑖𝑠𝑡 𝑢 + 1 𝑝𝑟𝑒𝑣 𝑢 = 𝑣
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75 Copyright © Jacques Saraydaryan
Wavefront: a Breadth-first search
Navigation: Path Planning
http://www.redblobgames.com/pathfinding/a-star/introduction.html
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76 Copyright © Jacques Saraydaryan
Wavefront: a Breadth-first search
Navigation: Path Planning
https://www.youtube.com/watch?v=ylnH9GctlTA
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77 Copyright © Jacques Saraydaryan
Wavefront: a Breadth-first search
Navigation: Path Planning
http://www.redblobgames.com/pathfinding/a-star/introduction.html
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78 Copyright © Jacques Saraydaryan
Dijkstra’s
Navigation: Path Planning
Principle
Explore the frontier by selecting candidate points
according to their distance to the origine. Cell
weight is taken into account in the distance
measure
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79 Copyright © Jacques Saraydaryan
Dijkstra’s
Navigation: Path Planning
Algorithm
For each 𝐶 ∈ Cfree
𝑎𝑑𝑑 𝐶 𝑡𝑜 𝑈𝑛𝑣𝑖𝑠𝑖𝑡𝑒𝑑 f𝑠𝑐𝑜𝑟𝑒[𝐶] = +∞ 𝑝𝑟𝑒𝑣[𝐶] = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
f𝑠𝑐𝑜𝑟𝑒[𝑞𝐼] = 0 Repeat
𝑢 ← 𝑀𝑖𝑛𝐹𝑠𝑐𝑜𝑟𝑒(𝑈𝑛𝑣𝑖𝑠𝑖𝑡𝑒𝑑)
𝑟𝑒𝑚𝑜𝑣𝑒 𝑢 𝑓𝑟𝑜𝑚 𝑈𝑛𝑣𝑖𝑠𝑖𝑡𝑒𝑑 For each 𝑣 ∈ 𝑁𝑒𝑖𝑔ℎ𝑏𝑜𝑟 𝑢
𝑐𝑢𝑟𝑟𝑒𝑛𝑡_𝑠𝑐𝑜𝑟𝑒 = f𝑠𝑐𝑜𝑟𝑒 𝑢 + 𝑙𝑒𝑛𝑔𝑡ℎ(𝑢, 𝑣) If 𝑐𝑢𝑟𝑟𝑒𝑛𝑡_𝑠𝑐𝑜𝑟𝑒 < f𝑠𝑐𝑜𝑟𝑒[v]
f𝑠𝑐𝑜𝑟𝑒[𝑣] = 𝑐𝑢𝑟𝑟𝑒𝑛𝑡_𝑠𝑐𝑜𝑟𝑒 𝑝𝑟𝑒𝑣[𝑢] = 𝑣
Until 𝑈𝑛𝑣𝑖𝑠𝑖𝑡𝑒𝑑 = ∅
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80 Copyright © Jacques Saraydaryan
Dijkstra’s
Navigation: Path Planning
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81 Copyright © Jacques Saraydaryan
Dijkstra’s
Navigation: Path Planning
http://www.redblobgames.com/pathfinding/a-star/introduction.html
![Page 78: Robot Navigation and collision avoidance · What navigation means ? 4 ‟ The Process of directing a vehicle so as to reach the intended destination ” IEEE Standard 172-1983 Copyright](https://reader036.vdocuments.site/reader036/viewer/2022062607/6044bcc04fe4c42e0a1409a6/html5/thumbnails/78.jpg)
82 Copyright © Jacques Saraydaryan
Greedy Best First Search
Navigation: Path Planning
Principle
Explore the frontier by selecting candidate points
according to their distance estimate to the goal.
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83 Copyright © Jacques Saraydaryan
Greedy Best First Search
Navigation: Path Planning
Algorithm
For each 𝐶 ∈ Cfree
𝑎𝑑𝑑 𝐶 𝑡𝑜 𝑈𝑛𝑣𝑖𝑠𝑖𝑡𝑒𝑑 f𝑠𝑐𝑜𝑟𝑒[𝐶] = +∞ 𝑝𝑟𝑒𝑣[𝐶] = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
f𝑠𝑐𝑜𝑟𝑒[𝑞𝐼] = 0 Repeat
𝑢 ← 𝑀𝑖𝑛𝐹𝑠𝑐𝑜𝑟𝑒(𝑈𝑛𝑣𝑖𝑠𝑖𝑡𝑒𝑑)
𝑟𝑒𝑚𝑜𝑣𝑒 𝑢 𝑓𝑟𝑜𝑚 𝑈𝑛𝑣𝑖𝑠𝑖𝑡𝑒𝑑 For each 𝑣 ∈ 𝑁𝑒𝑖𝑔ℎ𝑏𝑜𝑟 𝑢
f𝑠𝑐𝑜𝑟𝑒[𝑣] = ℎ𝑒𝑢𝑟𝑖𝑠𝑡𝑖𝑐𝐶𝑜𝑠𝑡𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒 𝑣, 𝑞𝐺 𝑝𝑟𝑒𝑣[𝑢] = 𝑣
Until 𝑈𝑛𝑣𝑖𝑠𝑖𝑡𝑒𝑑 = ∅
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84 Copyright © Jacques Saraydaryan
Greedy Best First Search
Navigation: Path Planning
http://www.redblobgames.com/pathfinding/a-star/introduction.html
![Page 81: Robot Navigation and collision avoidance · What navigation means ? 4 ‟ The Process of directing a vehicle so as to reach the intended destination ” IEEE Standard 172-1983 Copyright](https://reader036.vdocuments.site/reader036/viewer/2022062607/6044bcc04fe4c42e0a1409a6/html5/thumbnails/81.jpg)
85 Copyright © Jacques Saraydaryan
Greedy Best First Search
Navigation: Path Planning
http://www.redblobgames.com/pathfinding/a-star/introduction.html
![Page 82: Robot Navigation and collision avoidance · What navigation means ? 4 ‟ The Process of directing a vehicle so as to reach the intended destination ” IEEE Standard 172-1983 Copyright](https://reader036.vdocuments.site/reader036/viewer/2022062607/6044bcc04fe4c42e0a1409a6/html5/thumbnails/82.jpg)
86 Copyright © Jacques Saraydaryan
Alorithm frontier selection
Navigation: Path Planning
Breadth-first search Greedy Best First Search Dijkstra’s
Unvisited min jump j(u)= number of jump to reach u f𝒔𝒄𝒐𝒓𝒆(u)= j(u)
Unvisited min distance to origin g(u)= cost so far to reach u f𝒔𝒄𝒐𝒓𝒆(u)=g(u)
Unvisited min estimate distance distance to goal h(u)= heuristic estimate distance to the goal f𝒔𝒄𝒐𝒓𝒆(u)=h(u)
𝑞𝐼
𝑞𝐺 𝑞𝐺 𝑞𝐺
𝑞𝐼 𝑞𝐼
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87 Copyright © Jacques Saraydaryan
Alorithm frontier selection
Navigation: Path Planning
Breadth-first search Greedy Best First Search Dijkstra’s
Unvisited min jump j(u)= number of jump to reach u f𝒔𝒄𝒐𝒓𝒆(u)= j(u)
Unvisited min distance to origin g(u)= cost so far to reach u f𝒔𝒄𝒐𝒓𝒆(u)=g(u)
Unvisited min estimate distance distance to goal h(u)= heuristic estimate distance to the goal f𝒔𝒄𝒐𝒓𝒆(u)=h(u)
𝑞𝐼
𝑞𝐺 𝑞𝐺 𝑞𝐺
𝑞𝐼 𝑞𝐼
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88 Copyright © Jacques Saraydaryan
A*
Navigation: Path Planning
Principle
Combine Dijkstra’s (𝑔(𝑢))and Greedy Best First
Search (ℎ(𝑢)) frontier selection
𝑓𝑠𝑐𝑜𝑟𝑒(𝑢) = 𝑔 𝑢 + ℎ(𝑢)
ℎ(𝑢) 𝑔 𝑢
𝑞𝐼
𝑞𝐺
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89 Copyright © Jacques Saraydaryan
A*
Navigation: Path Planning
Algorithm
𝑐𝑙𝑜𝑠𝑒𝑑𝐿𝑖𝑠𝑡 = ∅ 𝑜𝑝𝑒𝑛𝐿𝑖𝑠𝑡 = 𝑞𝐼 For each 𝐶 ∈ Cfree
g𝑠𝑐𝑜𝑟𝑒[𝐶] = +∞ f𝑠𝑐𝑜𝑟𝑒 𝐶 = +∞ prevNode [𝐶] = ∅
While 𝑜𝑝𝑒𝑛𝐿𝑖𝑠𝑡 ≠ ∅ u = min f𝑠𝑐𝑜𝑟𝑒 If u == 𝑞𝐺
reconstructPath(u) 𝑟𝑒𝑚𝑜𝑣𝑒 𝑢 𝑓𝑟𝑜𝑚 𝑜𝑝𝑒𝑛𝐿𝑖𝑠𝑡 𝑎𝑑𝑑 𝑢 𝑡𝑜 𝑐𝑙𝑜𝑠𝑒𝑑𝐿𝑖𝑠𝑡 For each 𝑣 ∈ 𝑁𝑒𝑖𝑔ℎ𝑏𝑜𝑟 𝑢 If 𝑣 ∈ 𝑐𝑙𝑜𝑠𝑒𝑑𝐿𝑖𝑠𝑡
𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑒 𝑣𝑠𝑐𝑜𝑟𝑒 = 𝑔𝑠𝑐𝑜𝑟𝑒 𝑢 + 𝑙𝑒𝑛𝑔𝑡ℎ 𝑢, 𝑣
If 𝑣 ∉ 𝑜𝑝𝑒𝑛𝐿𝑖𝑠𝑡 𝑎𝑑𝑑 𝑣 𝑡𝑜 𝑜𝑝𝑒𝑛𝐿𝑖𝑠𝑡
ElseIf 𝑣𝑠𝑐𝑜𝑟𝑒 ≥ 𝑔𝑠𝑐𝑜𝑟𝑒 𝑣 continue
prevNode 𝑣 = u 𝑔𝑠𝑐𝑜𝑟𝑒 𝑣 = 𝑣𝑠𝑐𝑜𝑟𝑒 f𝑠𝑐𝑜𝑟𝑒 𝑣 = 𝑔𝑠𝑐𝑜𝑟𝑒 𝑣 + ℎ𝑒𝑢𝑟𝑖𝑠𝑡𝑖𝑐𝐶𝑜𝑠𝑡𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒 𝑣, 𝑞𝐺
Return Failure
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90 Copyright © Jacques Saraydaryan
A*
Navigation: Path Planning
𝑐𝑙𝑜𝑠𝑒𝑑𝐿𝑖𝑠𝑡 = ∅ 𝑜𝑝𝑒𝑛𝐿𝑖𝑠𝑡 = 𝑞𝐼 For each 𝐶 ∈ Cfree
g𝑠𝑐𝑜𝑟𝑒[𝐶] = +∞ f𝑠𝑐𝑜𝑟𝑒 𝐶 = +∞ prevNode [𝐶] = ∅
While 𝑜𝑝𝑒𝑛𝐿𝑖𝑠𝑡 ≠ ∅ u = min f𝑠𝑐𝑜𝑟𝑒 If u == 𝑞𝐺
reconstructPath(u) 𝑟𝑒𝑚𝑜𝑣𝑒 𝑢 𝑓𝑟𝑜𝑚 𝑜𝑝𝑒𝑛𝐿𝑖𝑠𝑡 𝑎𝑑𝑑 𝑢 𝑡𝑜 𝑐𝑙𝑜𝑠𝑒𝑑𝐿𝑖𝑠𝑡 For each 𝑣 ∈ 𝑁𝑒𝑖𝑔ℎ𝑏𝑜𝑟 𝑢 If 𝑣 ∈ 𝑐𝑙𝑜𝑠𝑒𝑑𝐿𝑖𝑠𝑡
𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑒 𝑣𝑠𝑐𝑜𝑟𝑒 = 𝑔𝑠𝑐𝑜𝑟𝑒 𝑢 + 𝑙𝑒𝑛𝑔𝑡ℎ 𝑢, 𝑣
If 𝑣 ∉ 𝑜𝑝𝑒𝑛𝐿𝑖𝑠𝑡 𝑎𝑑𝑑 𝑣 𝑡𝑜 𝑜𝑝𝑒𝑛𝐿𝑖𝑠𝑡
ElseIf 𝑣𝑠𝑐𝑜𝑟𝑒 ≥ 𝑔𝑠𝑐𝑜𝑟𝑒 𝑣 continue
prevNode 𝑣 = u 𝑔𝑠𝑐𝑜𝑟𝑒 𝑣 = 𝑣𝑠𝑐𝑜𝑟𝑒 f𝑠𝑐𝑜𝑟𝑒 𝑣 = 𝑔𝑠𝑐𝑜𝑟𝑒 𝑣 + ℎ𝑒𝑢𝑟𝑖𝑠𝑡𝑖𝑐𝐶𝑜𝑠𝑡𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒 𝑣, 𝑞𝐺
Return Failure
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91 Copyright © Jacques Saraydaryan
Dijkstra’s A* Greedy Best
First Search
Navigation: Path Planning
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93 Copyright © Jacques Saraydaryan
Exercices
Navigation: Path Planning
𝑞𝐼 𝑞𝐼
𝑞𝐺 𝑞𝐺
Wavefront Dijkstra
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94 Copyright © Jacques Saraydaryan
Exercices
Navigation: Path Planning
𝑞𝐼 𝑞𝐼
𝑞𝐺 𝑞𝐺
Greedy Best First Search A*
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95 Copyright © Jacques Saraydaryan
Exercices
Navigation: Path Planning
𝑞𝐼 𝑞𝐼
𝑞𝐺 𝑞𝐺
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96 Copyright © Jacques Saraydaryan
Exercices
Navigation: Path Planning
𝑞𝐼 𝑞𝐼
𝑞𝐺 𝑞𝐺
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References
110
References
Copyright © Jacques Saraydaryan
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111
References (1/2)
IEEE Standard 172-1983 Introduction to Autonomous Mobile Robots, MIT Press, Roland SIEGWART, Illah R. NOURBAKHSH 2004 Robotics, Vision and Control, Springer, Peter Corke 2011 PLANNING ALGORITHMS, Steven M. LaValle,University of Illinois, 2006 http://planning.cs.uiuc.edu/ Introduction to Mobile Robotics, Mapping with Known Poses, Wolfram Burgard, Cyrill Stachniss, Maren Bennewitz, Kai
Arras, Uni Freiburg http://ais.informatik.uni-freiburg.de/teaching/ss14/robotics/slides/08-occupancy-mapping-mapping-with-known-poses.pdf
Introduction to Mobile Robotics, Robot Motion Planning, Wolfram Burgard, cyrill stachniss, Maren Bennewitz, Kai Arras, Uni Freiburg, 2011
http://ais.informatik.uni-freiburg.de/teaching/ss11/robotics/slides/18-robot-motion-planning.pdf Occupancy Grids, Robotics, Benjamin Kuipers https://www.cs.utexas.edu/~kuipers/slides/L13-occupancy-grids.pdf http://cs.stanford.edu/people/eroberts/courses/soco/projects/1998-99/robotics/basicmotion.html http://www.cs.cmu.edu/afs/cs/project/jair/pub/volume11/fox99a-html/node10.html http://www.geometrylab.de/applet-30-en http://cs.smith.edu/~streinu/Teaching/Courses/274/Spring98/Projects/Philip/fp/visibility.htm http://theory.stanford.edu/~amitp/GameProgramming/MapRepresentations.html https://fr.wikipedia.org/wiki/Diagramme_de_Vorono%C3%AF https://en.wikipedia.org/wiki/Voronoi_diagram http://www.barankahyaoglu.com/robotics/voronoirobot/ https://en.wikipedia.org/wiki/Fortune%27s_algorithm http://msl.cs.uiuc.edu/rrt/index.html https://en.wikipedia.org/wiki/Probabilistic_roadmap http://msl.cs.uiuc.edu/rrt/gallery_rigid.html http://www.redblobgames.com/pathfinding/a-star/introduction.html https://en.wikipedia.org/wiki/A*_search_algorithm http://theory.stanford.edu/~amitp/GameProgramming/AStarComparison.html http://buildnewgames.com/astar/
Copyright © Jacques Saraydaryan
References
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https://en.wikipedia.org/wiki/D* http://idm-lab.org/bib/abstracts/papers/aaai02b.pdf images http://www.ros.org/news/2013/03/ http://library.isr.ist.utl.pt/docs/roswiki/attachments/pr2_simulator(2f)Tutorials(2f)BasicPR2Controls/rviz_move_base_div
erge.png http://www.youbot-store.com/developers/software/ros/youbot-ros-navigation-stack https://www.fsb.unizg.hr/acg/yaw_rate_estim_fig1.jpg http://library.isr.ist.utl.pt/docs/roswiki/costmap_2d.html http://joydeepb.com/Publications/biswas-rgbd11-plane-filtering.pdf http://www.cim.mcgill.ca/~mrl/pubs/saul/iros98.pdf http://www.sfbtr8.spatial-cognition.de/project/r3/HGVG/graphics/3DMzh.jpg http://www.mdpi.com/1424-8220/15/6/12736/htm http://a4academics.com/images/ProjSeminarImages/Ant-behavior-real-world.png
videos https://www.youtube.com/watch?v=A-fxij3zM7g https://www.youtube.com/watch?v=qziUJcUDfBc https://www.youtube.com/watch?v=zZLQ8Yh2iEE https://www.youtube.com/watch?v=DVnbp9oZZak https://www.youtube.com/watch?v=vAnN3nZqMqk https://www.youtube.com/watch?v=ylnH9GctlTA
Copyright © Jacques Saraydaryan
References