Minkowski sum-difference
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Transcript Minkowski sum-difference
Configuration Spaces for
Translating Robots
Minkowsi Sum/Difference
David Johnson
C-Obstacles
• Convert
– robot and obstacles
– point and configuration space obstacles
Workspace robot and obstacle
C-space robot and obstacle
Translating Robots
• Most C-obstacles have mysterious form
• Special case for translating robots
• Look at the 1D case
-7 -6 -5 -4 -3 -2 -1 0 1
obstacle
2 3
4
robot
5 6
7
Translating Robots
• What translations of the robot result in a
collision?
-7 -6 -5 -4 -3 -2 -1 0 1
obstacle
2 3
4
robot
5 6
7
Minkowski Difference
• The red C-obs is the Minkowski difference of
the robot and the obstacle
-7 -6 -5 -4 -3 -2 -1 0 1
obstacle
2 3
4
robot
5 6
7
Minkowski Sum
• First, let us define the Minkowski Sum
Minkowski Sum
A B {a b | a A, b B}
A
B
Minkowski Sum
A B {a b | a A, b B}
A B {a B | a A}
Minkowski Sum
Minkowski Sum
A B {a b | a A, b B}
Minkowski Sum Example
• Applet
• The Minkowski sum is like a convolution
• A related operation produces the C-obs
– Minkowski difference
A B {a b | a A, b B}
Back to the 1D Example
• What translations of the robot result in a
collision?
O R {a b | a O, b R}
-7 -6 -5 -4 -3 -2 -1 0 1
obstacle
2 3
4
robot
5 6
7
Tracing Out Collision Possibilities
Minkowski Difference
-B
From sets to polygons
• Set definitions are not very
practical/implementable
• For polygons, only need to consider vertices
– Computationally tractable
Properties of Minkowski Difference
• For obstacle O and robot R
– if O - R contains the origin
Opt Rpt 0,0
Ox Rx 0
Oy Ry 0
Ox Rx
Oy Ry
Opt Rpt
Collision!
Another property
• The closest point on the Minkowski difference
to the origin is the distance between polygons
• Distance between polygons
d ( A, B) min a b , a A, b B
d ( A, B) min z , z A B
Example
• Applet
Discussion
• Given a polygonal, translating robot
• Polygonal obstacles
• Compute exact configuration space obstacle
• Next class – how will we use this to make
paths?