Page 214 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

PLANAR REVOLUTE–REVOLUTE (RR) ARM 189

5.2.1 Analysis
Here we will expand to our RR arm manipulator the theory developed in Section
3.3 for mobile robots. One important part of that theory is making use of distinc-
tive topology of obstacles—namely, the fact that any obstacle is a simple closed
curve. Exploiting this fact resulted in elegant motion planning algorithms with
guaranteed convergence. We now intend to establish a similar characteristic of
obstacles faced by our RR arm—namely, that the arm’s complete passing around
an obstacle presents some sort of simple closed curves.
As we will soon observe, this is not so in the arm workspace. Simple examples
will show that paths produced by the arm endpoint when moving around even
simple obstacles are complex and self-intersecting. We will also see, however,
that the said property holds for all virtual obstacles in C-space. It will further be
shown that the number of such closed curves per obstacle is limited—a fact that
is important for the algorithm completeness. These facts will become the basis
of the algorithm design. We will then study the nonuniqueness of choices for the
M-line caused by peculiarities of the arm kinematics, and establish a criterion
for choosing appropriate M-lines. Finally, we will address one side effect of the
developed motion planning procedure, which can sometimes cause the arm to
repeat parts of its path.

Obstacles in W-Space. Consider an example of the arm interaction with
obstacles in the arm workspace (W-space). The formal underpinnings of our
observations will become clearer in the subsequent analysis of C-space.
We begin with a simple circular obstacle A in the arm’s workspace
(Figure 5.3). Starting at position S, the arm moves its endpoint along the M-
line (S, T ) toward the target position T . In this example the M-line happens to
be a straight line. Denote by (a i ,b i ) the ends of link l 2 , where point b i is the arm
endpoint. After traveling for a while in free space, at some moment the arm will
contact obstacle A, at which time the link l 2 position is (a 2 ,b 2 ), and the point

of contact on A is b . Now the arm will attempt to pass around the obstacle in
order to continue its motion along the M-line.
Observe that here the arm has two options for maneuvering around the obstacle
while maintaining a contact with it. With option 1, starting at the link l 2 position
(a 2 ,b 2 ), the arm endpoint moves along the curve b 2 ,b 3 ,. ..,b 6 ,b 7 ,b 8 . Soon
thereafter (between points b 8 and b 9 ), the arm endpoint encounters the M-line
(S, T ) and can continue moving along it toward T .Whenat T , the position of
link l 2 is (a ,T ).

T
With option 2, starting again at point b 2 , the arm endpoint passes through the
curve b 2 ,b 14 ,b 13 ,b 12 . At point b 12 the arm endpoint will encounter the M-line
and then continue along it toward T .Whenat T , the position of link l 2 will be
(a ,T ).

T
In other words, depending on the option taken, the arm endpoint may encounter
the M-line at different points, and the arm may consequently arrive at point T with
different positions of its links. Notice that we can accommodate this discrepancy:
For example, when moving under option 1, after passing point b 8 and reaching