Page 263 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

238 MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS

partially inside obstacles the corresponding virtual boundaries are simple curves.
Each such curve includes two types of segments: (i) one or two open curve
segments (and these can be followed by the arm) and (ii) one or two segments
of the base circles of the C-space cylinder (and these cannot be accessed by the
arm). Obstacles B and C in Figure 5.27 provide examples of this type.
That the curves are not closed in case (i) produces no algorithmic difficulties
simply because the endpoints of the open curves always correspond to one or
both limit values of link l 1 and hence are easily recognizable. Together with
the simplicity of obstacle virtual boundaries, this fact ensures conditions nec-
essary for convergence of the path planning procedure that we are about to
develop.

The Choice of M-Line. A straight-line M-line in W-space would be a perfectly
legitimate choice for M-line. From the practical standpoint, this choice is, how-
ever, not convenient because it may make it impossible to maintain continuous
motion of both links. Try, for example, to follow a straight line between points
S and T in Figure 5.27a. A discontinuity in the motion of one or both links will
take place somewhere along the path. Similar to the RR arm case, it is preferable
to define the M-line as a straight line in C-space.
A C-space straight M-line is a function θ 2 = p · l 1 + q in the plane of variables
l 1 and θ 2 . Coefficients p and q are found from the coordinates of points S and
T [97]. The image of this M-line, denoted as M 1 -line in Figures 5.27a and
5.27b, is a geodesic line between points S and T on the surface of the C-space
cylinder. If, because of the obstacles, point T cannot be reached from point S
using this M-line, a complementary M-line, denoted as M 2 -line in Figure 5.27,
is used. M 2 -line is defined similar to M 1 -line, with the coordinate of point T
being (θ 2 − 2π) instead of θ 2 .Inother words, M 1 -and M 2 -lines are mutually
complementary over the joint angle θ 2 , adding together to 2π. As one can see, the
overall logic behind the choice of M-line here is very similar (though somewhat
simpler because of two rather than four possible M-lines) to the logic in the case
of RR-arm (Section 5.2).

Special Cases. To complete the study of interaction between the arm and the
obstacles, and before we can formulate the motion planning algorithm proper,
we will consider a few examples that show what kind of special cases can arise
for the PR arm. For the sake of specificity, assume that every time the arm meets
an obstacle and defines on it a hit point, it starts with the local direction “left.”
In the example shown in Figure 5.28, the virtual obstacle A forms a swath
along the C-space cylinder that extends all the way between the cylinder’s two
limit circles—that is, between both limits of l 1 values. That is, in this example
the obstacle virtual boundary includes two separate open curves [passing through
points (4, 1, 2, 3) and (5, 6, 7, 8), Figure 5.28b] and two segments of the cylinder
limit bases [passing through points (4, 5) and (3, 8), accordingly, Figure 5.28b].
After leaving point S, following M-line and meeting obstacle A at point 1, the
arm turns left along the virtual boundary and passes through points 2 and 3. Point