Page 253 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

228 MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS

that this information is known before the motion takes place, from coordinates
of points S and T .
If during its motion along the M-line from S to T the arm encounters an
obstacle, such a local direction is chosen for passing around the obstacle for
which the corresponding change in joint value l 1 coincides with the established
M-line direction of change of l 1 . In the special case of l 1 being constant in the
vicinity of the hit point, the local direction should be chosen as corresponding
to decreasing values of l 2 ; otherwise the arm will not be able to pass around the
obstacle.
If, while passing around an obstacle, the current value l 1 moves outside of
S
T
the interval (l ,l ), then, clearly, point T lies in the shadow of the obstacle and
1 1
cannot be reached. If the M-line direction of change of l 1 is “0”—which will be
so if the M-line happen to be parallel to link l 2 —and an obstacle is met along
the way, then, again, point T cannot be reached because it is in the shadow
of the obstacle. In other words, Lemma 5.4.1 holds, and this helps simplify the
planning procedure.
While the PP arm planning procedure will work for the arm links of any
shapes, it can be further simplified and made more efficient if link l 2 is assumed
to present an elongated rectangle whose sides are parallel to the joint axes l 1 and
l 2 , respectively. If link l 2 happens to be of a more complex shape, the algorithm
can replace it with a minimum rectangle that contains the link. Hence link l 2 in
the example of Figure 5.23 would be treated as a rectangle of width zero.
Now the whole path planning procedure for the PP-Arm Algorithm can be
formulated; L o = S.

1. Establish the M-line direction of change of link l 1 (see above). Set j = 1.
Go to Step 2.
2. From point L j−1 , the arm moves along M-line until one of the following
occurs:
(a) Target T is reached. The procedure stops.
(b) An obstacle is encountered and a hit point, H j , is defined. Choose the
local direction using the M-line direction of change of l 1 .GotoStep 3.
3. The arm follows the obstacle virtual boundary until one of the following
occurs:
(a) The target T is reached. The procedure stops.
(b) The M-line is met at a distance d from T such that d< d(H j ,T ); point
L j is defined. Increment j.GotoStep2.
T
S
(c) The current value l 1 is outside of the interval (l ,l ). Target T cannot
1 1
be reached. The procedure stops.
It can be shown that the algorithm will work correctly if the arm links and
obstacles in the scene are of arbitrary chapes.