5. roadmaps hyeokjae kwon sungmin kim. 1. roadmap definition
Post on 16-Dec-2015
228 Views
Preview:
TRANSCRIPT
5. Roadmaps
Hyeokjae KwonSungmin Kim
1. RoadMap Definition
1. RoadMap Path Planning
1. Visibility Graph meth-ods
1. The Visibility Graph in Action (1)
1. The Visibility Graph in Action (2)
1. The Visibility Graph in Action (3)
1. The Visibility Graph in Action (4)
1. The Visibility Graph (Done)
Start
Goal
1. Reduced Visibility Graphs
1. The Sweepline Algorithm
1. Sweepline Algorithm Exam-ple
2. Generalized Voronoi Diagram
2. Two-Equidistant
2. Homotopy Classes
Start
Goal
Start
Goal
2. Sensor-Based Construction of the GVD
3. General Voronoi Graph
3. Retract-like Structure Con-nectivity
3. Retract-like Structure Con-nectivity
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.)
Distance
rod
The near-est point
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.
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.
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
Silhouette Meth-ods
Silhouette Methods
The silhouette approaches use extrema of a function defined on a codimension one hyperplane called a slice.
Silhouette Methods
Canny's Roadmap Al-gorithm
Opportunistic Path Planner(OPP)
Canny's Roadmap Algo-rithm
Canny's Roadmap Algorithm is one of the classical motion planning techniques that uses critical points.
critical points
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
How To Find Extrema
In order to find the extrema on a manifold
we will refer to the Lagrange Multiplier Theorem.
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
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.
Connectivity Changes at Criti-cal Points
Connectivity Changes at Critical Points
Silhouette curves on the torus
Connectivity Changes at Critical Points
Connectivity Changes at Critical Points
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
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.
Illustrative Example
This is not a roadmap, it’s not connected.
Illustrative Example
The roadmap is the union of all silhou-ette curves.
Find the critical points .
Opportunistic Path Plan-ner
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
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.
Opportunistic Path Planner
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.
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.
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
Reference
*AnOpportunisticGlobalPathPlan-ner1
JohnF.Canny2andMingC.Lin3
* Robotic Motion Planning:Roadmap Methodshttp://voronoi.sbp.ri.cmu.edu/
~choset
Question &
Answer
top related