Page 325 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

300 MOTION PLANNING FOR THREE-DIMENSIONAL ARM MANIPULATORS

and the wall. Since the Type III + (or III − ) monotonicity property is pre-
served here, the only promising local direction for passing around the Type
III + (III − ) obstacle is downward (upward). This possibility corresponds to
Motion IV below.

Motion III—Along the Intersection Curve Between the V-Plane and a Type
III + or III − Obstacle. This corresponds to moving along the P c (M-line) in C p .
One of the following can happen:

1. The M-line is met. C-point resumes its motion along the M-line as in

Motion I; see path segments bc, cT and b c , c T , Figure 6.9a.

2. A wall is met. This corresponds to the P m (C-point) encountering an obsta-
cle. According to the algorithm A p , P c (C-point) will either reverse its
local direction to move along P c (M-line) or will make a turn to follow the
obstacle. Accordingly, the C-point will either (a) reverse its local direction
to follow the intersection curve between the V-plane and the (Type III)
obstacle or (b) try to go around the union of the wall and the obstacle. For
the latter motion we choose a path along the intersection curve between
the wall and the Type III obstacle.
3. Another Type III + or III − obstacle is met. Since the P m projection of
both Type III obstacles onto C p is not zero, this corresponds to the P c (C-
point) encountering an obstacle, which presents the P m projection of the
intersection curve between both obstacles. According to A p algorithm, the
P c (C-point) will either (a) reverse its local direction to move along P c (M-
line) or (b) make a turn to follow the obstacle. Accordingly, the C-point
will either (a) reverse its local direction to follow the intersection curve
between V-plane and the Type III obstacle or (b) try to go around the
union of two Type III obstacles. For the latter motion, we choose a path
along the intersection curve between the two Type III obstacles in the local
direction left or right decided by A p .

Motion IV—Along the Intersection Curve Between a Type III Obstacle and
a Wall. In C p , this corresponds to the P c (C-point) moving along the bound-

ary of P m ({O}); see segments bcL and b L , Figure 6.11. One of the following

can occur:
1. The C-point returns to the M-plane. Then C-point resumes its motion
along the intersection curve between the M-plane and the wall, similar to
Motion II.
2. The V-plane is encountered (see point L, Figure 6.11). In C p this means that
P c (M-line) is encountered. At this point, algorithm A p will decide whether
the P c (C-point) should start moving along the P c (M-line) or should con-
tinue moving along the obstacle boundary. Accordingly, the C-point will
either (a) continue moving along the intersection curve between the Type
III obstacle and the wall or (b) move along the intersection curve between