Page 215 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

190 MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS

b 1
S b 5
b 4
b 6
b 3
P 1
b 2
A b 9
b'
b 10
b 7 A
b''
b 14
a 3 b 8
a 13 a 4 b 11
a 2 a 12 b 13
a 5
a 1 a 11a T '' b 12
a 6
a 10 b T
a 14 T
O a 9
a 7 a' T
a 8
B
B

P 2

Figure 5.3 Obstacles A and B form “shadows”; the arm endpoint cannot reach points
inside a shadow. For example, point P 1 is in the shadow of the circular obstacle A and
thus cannot be reached. The shadow of the circular obstacle B forms two disconnected
“subshadows.”

the M-line, the arm endpoint can continue through points b 9 ,b 10 ,b 11 , meeting
the M-line again at point b 12 as under option 2.
In fact, when starting at point b 2 under any of the two options, if one con-
tinues “rotating” the arm around obstacle A while keeping in contact with it,
the arm endpoint will make a complete closed curve, passing through the points
b 2 ,b 3 ,b 4 ...,b 8 ,b 9 ...,b 13 ,b 14 and eventually arriving at the same point b 2 .
This indicates that the paths produced under both options are complementary to
each other: When added together, they form a closed curve.
Regarding this curve, consider the area whose curvilinear boundary passes
through points b ,b 2 ,b 3 ,b 4 , then the segment b 4 ,b 10 of the workspace boundary,

then points b 10 ,b 9 ,b of our curve, and finally the smaller part of the obstacle

A boundary between points b and b . This area is called the shadow of obstacle

A: Though this is a part of free space, no point (such as P 1 ) inside this area can
be reached by the arm endpoint.
This suggests that an obstacle shadow will be perceived by the arm as an
obstacle, as real as an actual physical obstacle. The arm cannot penetrate either