Page 312 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

THE CASE OF THE PPP (CARTESIAN) ARM 287

l 3
d
l 3max g

e f

c
l 2
l 2max
l′ 1

l 1max
a b
l 1
Figure 6.6 C-space with a Type I obstacle.

Type I Monotonicity. For any obstacle interacting with link l 1 , there are three
directions corresponding to the joint axes l i , i = 1, 2, 3, respectively, along which

the C-obstacle behaves monotonically, as follows: If a position (l ,l ,l ) cannot
1 2 3

be reached by the arm due to an obstacle interference, then no position (l ,l ,l ),
1 2 3

such that l >l in case of the (obstacle’s) contact with the link’s front part,

1 1
or l <l in case of the contact with the link’s rear part, and l ∈ [0,l 2max ],

1 1 2
l ∈ [0,l 3max ], can be reached either.

3
Interaction of Link l 2 with Obstacles—Type II Obstacles
Front Part of Link l 2 —Type II + Obstacles. Consider the case when only the
front part of link l 2 interferes with an obstacle (Figure 6.2). Because link l 2
effectively has two degrees of freedom, the corresponding Type II + obstacle will
look in C-space as shown in Figure 6.7. The monotonicity property in this case
is as follows:
Type II + Monotonicity. For any obstacle interacting with the front part of link
l 2 , there are two axes (directions), namely l 2 and l 3 , along which the C-obstacle
behaves monotonically, as follows: If a position (l ,l ,l ) cannot be reached by

1 2 3
the arm due to an obstacle interference, then no position (l ,l ,l ), such that

1 2 3
l >l and l ∈ [0,l 3max ], can be reached either.

2 2 3
As aresult, aType II + collision, as at point H in Figure 6.7, indicates that any
motion directly upward or downward from H along the obstacle will necessarily
bring the C-point to one of the side walls of the C-space cubicle. This suggests