5. roadmaps hyeokjae kwon sungmin kim. 1. roadmap definition
TRANSCRIPT
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5. Roadmaps
Hyeokjae KwonSungmin Kim
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1. RoadMap Definition
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1. RoadMap Path Planning
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1. Visibility Graph meth-ods
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1. The Visibility Graph in Action (1)
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1. The Visibility Graph in Action (2)
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1. The Visibility Graph in Action (3)
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1. The Visibility Graph in Action (4)
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1. The Visibility Graph (Done)
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Start
Goal
1. Reduced Visibility Graphs
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1. The Sweepline Algorithm
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1. Sweepline Algorithm Exam-ple
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2. Generalized Voronoi Diagram
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2. Two-Equidistant
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2. Homotopy Classes
Start
Goal
Start
Goal
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2. Sensor-Based Construction of the GVD
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3. General Voronoi Graph
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3. Retract-like Structure Con-nectivity
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3. Retract-like Structure Con-nectivity
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The Rod-Hierarchical Generalized Voronoi Graph
What is different?*a point robotㅡ> a Rod Robot*Non-Euclidean*Sensor Based Approach*Workspace -> Configuration
Space(However, we measure distance in
the workspace, not configuration space.)
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Distance
rod
The near-est point
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Rod-GVG-edges
(a1) Rod-GVG-edges: each of the clus-ters represents a set of configurations equidistant to three obstacles. (a2) The configurations of the rod that are equidistant to three obstacles in the workspace.
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R-edges
(b1) R-edges: the rods are two-way equidistant and tangent to a planar point-GVG edge. (b2) The configura-tions of the rod that are tangent to the planar point-GVG in the workspace.
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rod-HGVG
The rod-HGVG then comprises rod-GVG edges and R-edges(c1) Placements of the rod along the rod-HGVG. (c2) The entire rod-HGVG
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Silhouette Meth-ods
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Silhouette Methods
The silhouette approaches use extrema of a function defined on a codimension one hyperplane called a slice.
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Silhouette Methods
Canny's Roadmap Al-gorithm
Opportunistic Path Planner(OPP)
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Canny's Roadmap Algo-rithm
Canny's Roadmap Algorithm is one of the classical motion planning techniques that uses critical points.
critical points
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The Basic Ideas
• Pick a sweeping surface• As sweeping happens, detect ex-
tremal points and critical points (= places where connectivity changes)
• For each slice where a critical point occurs, repeat this process recursively
• Use this as the roadmap
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How To Find Extrema
In order to find the extrema on a manifold
we will refer to the Lagrange Multiplier Theorem.
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Canny's Roadmap Algo-rithm
Sweep direc-tion
Critical points
The silhouette curves trace the boundary of the environment. Criti-cal points occur when the slice is tangent to the roadmap
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Accessibility and Departa-bility
In order to access and depart the roadmap we treat the slices which con-
tain qstart and qgoal as critical slices and run the algorithm the same way.
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Connectivity Changes at Criti-cal Points
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Connectivity Changes at Critical Points
Silhouette curves on the torus
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Connectivity Changes at Critical Points
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Connectivity Changes at Critical Points
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Building the Roadmap
We can now find the extrema necessary to build the silhouette curves.We can find the critical points where linking is necessaryWe can run the algorithm recur-sively to construct the whole roadmap
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Illustrative Example
Let S be the el-lipsoid with a through hole. Pc is a hyper-plane of codi-mension1 ( x = c ) which will be swept through S in the X direc-tion.
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Illustrative Example
This is not a roadmap, it’s not connected.
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Illustrative Example
The roadmap is the union of all silhou-ette curves.
Find the critical points .
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Opportunistic Path Plan-ner
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Opportunistic Path PlannerThe Opportunistic Path Planner is similar toCanny’s Roadmap but differs in the follow-
ingways• Silhouette curves are now called free-
ways and are constructed slightly differ-ently
• Linking curves are now called bridges• It does not always construct the whole
roadmap• The algorithm is not recursive
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The bridge curves are constructed in the vicinity of interesting critical pointBridge curves are also built when free-ways terminate in the free space at bi-furcation pointsA bridge curve is built leading away from a bifurcation point to another freeway curve.The union of bridge and freeway curves, sometimes termed a skeleton, forms the one-dimensional roadmap.
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Opportunistic Path Planner
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OPP method looks for connectiv-ity changes in the slice in the free configuration space.We are assured that we only need to look for critical points to con-nect disconnected components of the roadmap.If the start and goal freeways are connected, then the algorithm terminates.
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Building the Roadmap
(1) Start tracing a freeway curve from the start configuration, and also from the goal.
(2) If the curves leading from start and goal are not connected enumerate a split point or join point and add a bridge curve near the point. Else stop.
(3) Find all points on the bridge curve that lie on other freeways and trace from these free-ways. Go to step 2.
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Reference*Algorithms for Sensor-Based Robotics:RoadMap MethodsCS 336, G.D. Hager (loosely based on
notes by Nancy Amato and Howie Choset)
*Robot Motion Control and Planninghttp://www.cs.bilkent.edu.tr/~saranli/courses/cs548
*Principles of Robot Motion-Theory, Algorithms, and Implementation
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Reference
*AnOpportunisticGlobalPathPlan-ner1
JohnF.Canny2andMingC.Lin3
* Robotic Motion Planning:Roadmap Methodshttp://voronoi.sbp.ri.cmu.edu/
~choset
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Question &
Answer