Configuration Space of a Moving Rigid Object

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Configuration Spaceof a Moving Rigid Object
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Idea Reduce the Robot to a Point? Configuration
Space
Workspace
Configuration space
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Configuration Space of a Disc
Workspace W
Configuration space C
path
y
x
configuration coordinates (x,y) of robots
center configuration space C (x,y) free space
F subset of collision-free configurations
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Configuration Spaceof a Translating Polygon
reference point
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CB B A b-a a?A, b?B
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Linear-Time Computation
(convex polygons)
O(nm)
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Exercise

• Show that, if A and B are both convex set (not
  necessarily polygons), then CB is also a convex
  set

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Configuration Space of a Translating Rotating
Object in 2-D Workspace
Robots configuration is q (x,y,q) ? ?2?0,2p)
What is the placement of the robot in the
workspace at configuration (0,0,0)?
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Configuration Space of a Translating Rotating
Object in 2-D Workspace
workspace
robot
reference direction
q
y
reference point
x
What is the placement of the robot in the
workspace at configuration (0,0,0)?
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Path

• What is a path of a translating and rotating
  object in its 3-dimensional configuration space?

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Configuration Space of a Translating Rotating
Object in 2-D Workspace
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Configuration Space of a Translating Rotating
Object in 2-D Workspace
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Search and Potential Field

• How do they extend to planning a path in the
  3-dimensional configuration space of a
  translating and rotating object?

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Configuration Space of a Translating Rotating
Object in 3-D Workspace
Robots configuration is q (x,y,z,f,q,y)
? 6-dimensional configuration space
z
y
q (0,0,0,0,0,0)
x
q (0,0,0,?,?,?)
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Euler Angles
1 ? 2 ? 3 ? 4
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How many dimensions has the configuration space
of these 3 rings?
Answer 3?5 15
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Metric in Configuration Space
q
configuration space
2p
y
What is the distance between two configurations?
x
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Metric in Configuration Space

• A metric or distance function d in C is a map
  d (q1,q2) ? C?C ? d(q1,q2) gt 0such that
• Non-degeneracy d(q1,q2) 0 iff q1 q2
• Symmetry d(q1,q2) d (q2,q1)
• Triangle inequality d(q1,q2) lt d(q1,q3)
  d(q3,q2)

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d x2-x1 y2-y1 q2-q1
d x2-x1 y2-y1 minq2-q1, 2p-q2-q1
d x2-x1 y2-y1 r?minq2-q1, 2p-q2-q1