Last lecture Configuration Space Free-Space and C-Space Obstacles - - PowerPoint PPT Presentation
Last lecture Configuration Space Free-Space and C-Space Obstacles Minkowski Sums 1 Free-Space and C-Space Obstacle How do we know whether a configuration is in the free space? Computing an explicit representation of the free-
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Configuration Space Free-Space and C-Space Obstacles Minkowski Sums
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How do we know whether a configuration is in
Computing an explicit representation of the free-
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How do we know whether a configuration is in the free
Computing an explicit representation of the free-space is
Solution: Compute the position of the robot at that
If colliding, the configuration is within C-space obstacle Otherwise, it is in the free space
Performing collision checks is relative simple
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CLEAR(q)
Is configuration q collision free
LINK(q, q’)
Is the straight-line path between q and q’ collision-
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Running time increases exponentially with the
For a d-dimension grid with 10 grid points on each
Several variants of the path planning problem
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Complete algorithm Slow
A complete algorithm finds a path if one exists and
Example: Canny’s roadmap method
Heuristic algorithm Unreliable
Example: potential field
Probabilistic completeness
Intuition: If there is a solution path, the algorithm will
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[Kavraki, Svetska, Latombe,Overmars, 96]
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Probabilistic Roadmaps for Path Planning in High-
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Static obstacles Many queries to be processed in the same
Examples
Navigation in static virtual environments Robot manipulator arm in a workcell
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Uniform sampling Resampling (expansion)
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Input: geometry of the moving object & obstacles Output: roadmap G = (V, E) 1: V ← ∅ and E ← ∅.
2: repeat
3: q ← a configuration sampled uniformly at random from C.
4: if CLEAR(q)then
5: Add q to V. 6: Nq ← a set of nodes in V that are close to q. 6: for each q’∈ Nq, in order of increasing d(q,q’) 7: if LINK(q’,q)then 8: Add an edge between q and q’ to E.
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The graph G is called a probabilistic roadmap. The nodes in G are called milestones.
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Many small connected components
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Failure rate Weight Resampling probability
p
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Connect qinit and qgoal to the roadmap Start at qinit and qgoal, perform a random walk, and
Try multiple times
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If a path is returned, the answer is always
If no path is found, the answer may or may not
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A small number of milestones almost “cover”
Rigorous definitions and proofs in the next
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What probability distribution should be used for
How should milestones be connected? A path generated by a randomized algorithm is
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Path Planning Using Lazy PRM, R. Bohlin & L. Kavraki,
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Nodes
Randomly chosen configurations, which may or may
No call to CLEAR
Edges
an edge between two nodes if the corresponding
no call to LINK
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1.
2.
3.
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Find the shortest path in the roadmap
A* algorithm Dijkstra’s algorithm
Check whether nodes and edges are collisions
CLEAR(q) LINK(q0, q1)
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Select nodes that close the boundary of F
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Unit interval
Unit square Unit cube
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What shall we do?
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1.
Pick x uniform at random from [-1,1]
2.
Set
2
1 x y − =
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1.
Pick θ uniformly at random from [0, 2π]
2.
Set x = cosθ and y = sinθ
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Both are uniform in some sense. For sampling orientations in 2-D, the second
The definition of uniform sampling depends on
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Unit quaternion
2 + ny 2+ nz 2 = 1.
Sample n and θ separately Sample ξ from [0, 2π] uniformly at random
n = (nx, ny, nz)
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Longitude and latitude
= = = cos sin sin cos sin θ ϕ θ ϕ θ
z y x
n n n x y z ϕ θ
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Choose θ and ϕ uniformly at random from [0, 2π]
and [0, π], respectively.
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Spherical patches of
Suppose U1 and U2 are
2 2 1
U R n U R n U n
y x z
π π x y z ϕ θ 1 where
2 1
U R − =
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Use medial axis based sampling
Medial axis: similar to internal Voronoi diagram; set of
Compute approximate Voronoi boundaries using
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Sample the workspace by taking points on the
Medial axis of the workspace (works well for
How can we handle robots with rotational degrees of