Page 310 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

THE CASE OF THE PPP (CARTESIAN) ARM 285

6.2.3 Topology of W-Obstacles and C-Obstacles
Monotonicity Property. Obstacles that intersect the W-space volume may
interact with the arm during its motion. As mentioned above, one result of such
interaction is the formation of obstacle shadows. Consider the spherical obsta-
cle O 1 in Figure 6.2. Clearly, no points directly above O 1 can be reached by
any point of the arm body. Similarly, no point of W-space below the obstacle
O 2 or to the left of the cubical obstacle O 3 can be reached. Subsequently, the
corresponding W-obstacles become as shown in Figure 6.4, and their C-space
representation becomes as in Figure 6.5. This effect, studied in detail below, is
caused by the constraints imposed by the arm kinematics on its interaction with
obstacles. Anisotropic characteristics of W-space and C-space present themselves
in a special topology of W-and C-obstacles best described by the notion of the
(W-and C-) obstacle monotonicity:

Obstacle Monotonicity. In all cases of the arm interference with an obstacle,
there is at least one direction corresponding to one of the axes l i , i = 1, 2, 3,

such that if a value l of link l i cannot be reached due to the interference with
i

an obstacle, then no value l >l in case of contact with the link front part, or,
i i
inversely, l <l in case of contact with the link rear part, can be reached either.

i i
In what follows, most of the analysis of obstacle characteristics is done in
terms of C-space, although it applies to W-space as well. Comparing Figures 6.2
and 6.5, note that although physical obstacles occupy a relatively little part of
the arm’s workspace, their interference with the arm motion can reduce, often
dramatically, the volume of points reachable by the arm end effector. The kine-
matic constraints are due to the arm joints, acting differently for different joint
types, and to the fact that arm links are connected in series. As a result, the
arm effectively has only one degree of freedom for control of motion of link l 1 ,
two degrees of freedom for control of link l 2 , and three degrees of freedom for
control of link l 3 . A simple example was mentioned above on how this can affect
path planning: If during the arm motion along M-line the link l 1 hits an obstacle,
then, clearly, the task cannot be accomplished.
The monotonicity property implies that C-obstacles, though not necessarily
convex, have a very simple structure. This special topology of W-and C-
obstacles will be factored into the algorithm; it allows us, based on a given
local information about the arm interaction with the obstacle, to predict impor-
tant properties of the (otherwise unknown) obstacle beyond the contact point.
The monotonicity property can be expressed in terms more amenable to the path
planning problem, as follows:

Corollary 6.2.1. No holes or cavities are possible in a C-obstacle.

W-obstacle monotonicity affects differently different links and even differ-
ent parts—front or rear—of the same link. This brings about more specialized
notions of l i -front and l i -rear monotonicity for every link, i = 1, 2, 3(see more