Page 270 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

TOPOLOGY OF ARM’S FREE CONFIGURATION SPACE 245

• Distances are Euclidean distances along M-line in the plane (l 1 ,θ 2 ).
• For specificity only, the local direction for passing around an obstacle is
taken as “left” when the first hit point is defined.

The PR-Arm Algorithm consists of the following steps:
1. M 1 -line is designated as the M-line. Set the flag down. Set j = 1. Go to
Step 2.
2. Set counter C 2 to zero. Set F = 0. From point L j−1 , the arm moves along
the 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 “left”. Turn on counter C 2 .GotoStep3.
3. The arm follows the virtual boundary until one of the following occurs:
(a) Target is reached. The procedure stops.
(b) M-line is met at a distance d from T such that d< d(H j ,T ); point
L j is defined. Set the flag down. Increment j. Go to Step 2.
(c) The arm reaches one of the limits of link l 1 (this corresponds to an
endpoint of one open curve of a virtual boundary) without ever meeting
the M-line. Go to Step 4.
(d) The arm returns to H j (i.e., a closed curve along the virtual boundary
has been completed) without ever encountering the M-line. The target
cannot be reached. The procedure stops.
4. Depending on the value F, the flag condition, and the current arm position,
one of the following occurs:
(a) F = 0. Set F to +1or −1 according to the rule above. Return to the
last hit point. Choose the local direction “right.” Go to Step 3.
(b) Value F corresponds to the current curve endpoint (i.e., +1for the
upper limit and −1 for the lower limit of l 1 ); this means that the
whole obstacle has been explored. The target cannot be reached. The
procedure stops.
(c) Value F does not correspond to the current curve endpoint, and the flag
is down; this means the first open curve of a Type II obstacle has been
explored. Set j = 1. Set the flag up. Designate M 2 -line as the M-line.
Return to S.GotoStep 2.
(d) Value F does not correspond to the current curve endpoint, and the flag
is up; this means that the second open curve of a Type II obstacle has
been investigated. The target cannot be reached. The procedure stops.

5.8 TOPOLOGY OF ARM’S FREE CONFIGURATION SPACE

In the previous sections of this chapter we have considered an exhaustive list
of five kinematic configurations of two-link arm manipulators (see Figure 5.1).