sensor based exploration : incremental construction of the hierarchical generalized voronoi graph
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Sensor Based Exploration : Incremental Construction of the Hierarchical Generalized Voronoi Graph. Why Sensor Based. Classical work is based on the assumption that a robot has a full knowledge of the world - PowerPoint PPT PresentationTRANSCRIPT
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Sensor Based Exploration:
Incremental Construction of the Hierarchical
Generalized Voronoi Graph
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Why Sensor Based
1. Classical work is based on the assumption that a robot has a full knowledge of the world
2. The problem: realistic deployment of robots into unknown environments and into environments that are too difficult to model
3. Sensor based planning is important because:
1. the robot often has no priori knowledge of the world or may have only a coarse knowledge of the world
2. the world model is bound to contain inaccuracies or unexpected changes
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1. One of the first motion planning techniques that
1. relies only on line-of-sight sensor information.
2. functions in higher dimensions
3. offers completeness guarantees
2. A numerically well posed and complete algorithm for sensor based robot mapping of unknown environments.
3. The robot generates a small portion of the a roadmap edge and then follow this portion to generate the next segment.
4. The robot traces an edge until it reaches a node, at which it branches to explorer all edges emanating from that node.
1. When all nodes have no explored directions, the algorithm finishes.
This Algorithm
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Generalized Generalized Voronoi Graphs Voronoi Graphs
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Generalized Voronoi Graph vs.
Generalized Voronoi Diagram
• Generalized Voronoi Diagram (GVD):for planar environment only.
• Generalized Voronoi Graph (GVG): a generalization of the GVD into higher dimensions. One dimensional. A more concise representation of the workspace or configuration space.
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• Distance function: the distance between a point x and a convex set
• Multi-object distance function:
• All the above distance related function can be computed from sensor data directly.
Distance Function
||||)(||||min)(
0
00
0 cx
cxxdandcxxd i
Cci
i
)(min)( xdxD ii
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• The building block of the GVG is two-equidistant face:
Equidistant Face
})()(,)()()(0:{ xdxdandjihxdxdxdRxf jihjim
ij
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• Three-equidistant face:
• By continuing intersection of the two-equidistant faces, a m-equidistant face is formed, which is a one dimensional set of points.
• A m+1-equidistant face can be formed also, which is a meet point.
• The GVG is the collection of m-equidistant faces(edges) and m+1-equidistant faces(meet point).
Equidistant Face
jkikijijk ffff
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Hierarchical GVG• GVG is not necessarily
connected in dimensions greater than two, and thus is not a roadmap and insufficient for path planning.
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• Higher order GVG is defined to connect the GVG: recursively defined on lower dimensional equidistant
faces. • HGVG: the collection of all GVG and all higher order
GVG.
We will focus on R3 only in the rest of this paper.
Hierarchical GVG
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Second-Order GVG• Second order two-equidistant face:
})()(,,,)()()()()(:{| xdxdandlkjihxdxdxdxdxdfxf kljilkhijijfkl
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• The cycle of the second order GVG implies the existence of GVG inside of it.
• Linking from outer second order GVG to GVG is achieved via gradient descent of the distance to the second closest obstacle.
Second-Order GVG
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GVG Tracing Function
• Edge tracing:
trace the roots of the expression
as is varied.
x: point on the GVG.
z1: in the tangent direction of x.
At x, let the hyperplane
spanned by local coordinates
z2-zm be termed the normal
plane. The tracing function
This function assumes a zero
value only on the GVG.
0),(1 yG
),)((
),)((
),)((
),(
1
31
21
1
ydd
ydd
ydd
yG
m
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GVG Edge Construction
• edge construction:
• predictor step: moves the robot for a small distance along the tangent direction of the GVG
• corrector step: find the intersection of the GVG and the correcting plane. This is achieved through the Newton method:
It can be proved that the Jacobian matrix is always nonsingular.
),()( 11
11 kk
ykk yGGyy
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Some Details
• The tangent to the graph is defined by the vector orthogonal to the hyper plane, which contains the m closest points of the m closest obstacles
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• Meet point detection: by watching for an abrupt change in the direction of the gradients to the m closest obstacles.
Some Details
Accessibility: using gradient ascent on the multi-object distance function, moving in a direction to which the sensor with the smallest value is facing.
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Construction of the Second- Order GVG
• This section applies the same tracing method for GVG to trace the edges of second-order GVG.
• The tracing function is:
• Tangent direction: is the null space of the Jacobian of G2:
),)((
),)((
),)((
),(
3
43
21
2
ydd
ydd
ydd
yG
m
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Simulations• Planar Simulations
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• Planar Simulations
Simulations
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• Three-Dimensional: Three-Dimensional: GVG only, not connected; HGVG, GVG+GVG2, connected
Simulations
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Simulations
• Three-DimensionalThree-Dimensional: GVG only, not connected; HGVG, GVG+GVG2, connected
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• Circular mobile robot base in the planar case
• Jagged due to crudely approximated tangent
• However, CVG is connected and maxim clearance from the workspace boundary.
Experiments
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Conclusions on GVG
• An incremental procedure to construct the GVG and the HGVG is introduced
• Requires only local sensor distance data
• Future work: exploit geometries of the HGVG to locate itself on the partially explored map or conclude the robot has entered new territory.
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The Basic The Basic Motion Motion
ProblemProblem
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Let’s Assume…
• We have an a priori map of the environment OR
• We have sufficient sensor information toreconstruct the environment
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Supporting References
• “Motion Planning Using Potential Fields,” R.
Beard & T. McClain, BYU, 2003
• You should download this from the course page
and read it
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Lecture Objectives• Examine alternate approaches to motion planning
Roadmap Approach:– Visibility Graph Methods
Cell Decomposition:– Exact Decomposition
– Approximate: Uniform discretization & quadtree approaches
Potential Fields
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1. Sensor Based
2. Algorithm
3. Voronoi Diagrams
4. Hierarchical Voronoi Diagrams
5. Second Order Voronoi Diagrams
6. Simulations
7. The Basic Motion Problem
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Sensor Based Exploration: Incremental Construction of the Hierarchical Generalized Voronoi
Graph
Howie Choset, Sean Walker, Kunnayut Eiamsa-Ard, Joel BurdickFebruary 2000
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