Page 308 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

THE CASE OF THE PPP (CARTESIAN) ARM 283

3. In a special case of motion along the M-line, the directions are ST =

forward and TS = backward.
Consider the motion of a C-point in the M-plane. When, while moving along
the M-line, the C-point encounters an obstacle, it may define on it a hit point, H.
Here it has two choices for following the intersection curve between the M-plane
and the obstacle surface: Looking from S toward T , the direction of turn at H
is either left or right. We will see that sometimes the algorithm may replace the
current local direction by its opposite. When, while moving along the intersection
curve in the M-plane, the C-point encounters the M-line again at a certain point,
it defines here the leave point, L. Similarly, when the C-point moves along a
V-plane, the local directions are defined as “upward” and “downward,” where
“upward” is associated with the positive and “downward”—with the negative
direction of l 3 axis.

6.2.2 The Approach

Similar to other cases of sensor-based motion planning considered so far, con-
ceptually we will treat the problem at hand as one of moving a point automaton
in the corresponding C-space. (This does not mean at all, as we will see, that C-
space needs to be computed explicitly.) Essential in this process will be sensing
information about interaction between the arm and obstacles, if any. This infor-
mation—namely, what link and what part (front or rear) of the link is currently
in contact with an obstacle—is obviously available only in the workspace.
Our motion planning algorithm exploits some special topological character-
istics of obstacles in C-space that are a function of the arm kinematics. Note
that because links l 1 , l 2 ,and l 3 are connected sequentially, the actual number of
degrees of freedom available to them vary from link to link. For example, link l 1
has only one degree of freedom: If it encounters an obstacle at some value l ,it

1
simply cannot proceed any further. This means that the corresponding C-obstacle

occupies all the volume of C-space that lies between the value l and one of the
1
joint limits of joint J 1 .This C-obstacle thus has a simple structure: It allows
the algorithm to make motion planning decisions based on the simple fact of a
local contact and without resorting to any global information about the obstacle
in question.
A similar analysis will show that C-obstacles formed by interaction between
link l 2 and obstacles always extend in C-space in the direction of one semi-axis
of link l 2 and both semi-axes of link l 3 ; it will also show that C-obstacles formed
by interaction between link l 3 and obstacles present generalized cylindrical holes
in C-space whose axes are parallel to the axis l 3 . No such holes can appear, for
example, along the axes l 1 or l 2 .Inother words, C-space exhibits an anisotropy
property; some of its characteristics vary from one direction to the other. Further-
more, C-space possesses a certain property of monotonicity (see below), whose
effect is that, no matter what the geometry of physical obstacles in W-space, no
holes or cavities can appear in a C-obstacle.