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