Page 333 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

308 MOTION PLANNING FOR THREE-DIMENSIONAL ARM MANIPULATORS

As before, assume the robot arm has enough sensing to (a) detect a contact
with an approaching obstacle at any point of the robot body and (b) identify the
location of that point(s) of contact on that body. The act of maneuvering around
an obstacle refers to a motion during which the arm is in constant contact with
the obstacle.
Without loss of generality, assume for simplicity a unit-length limit for joints,
1
1
l i ∈ I and θ i ∈ R , i = 1, 2, 3. Points at infinity are included for convenience.
1
The joint space J is defined as J = J 1 × J 2 × J 3 ,where J i = I if the ith
1
joint is prismatic, and J i = R if the ith joint is revolute. In all combinations of
3
cases, J = I . Thus, by including points at infinity in J i , it is possible to treat
∼
all XXP arms largely within the same analytical framework.
Definition 6.3.1. Let L k be the set of points representing the kth robot link, k =
3 3
1, 2, 3; for any point x ∈ L k ,let x(j) ∈ be the point that x would occupy in
! 3
when the arm joint vector is j ∈ J.Let L k (j) = x(j). Then, L k (j) ⊂
x∈L k
is a set of points the kth robot link occupies when the arm is at j ∈ J. Simi-
3
larly, L(j) = L 1 (j) ∪ L 2 (j) ∪ L 3 (j) ⊂ is a set of points the whole robot arm
occupies at j ∈ J. The workspace (or W-space, denoted W) is defined as
"
W = L(j) (6.1)
j∈J

We assume that L i has a finite volume; thus W is bounded.
Arm links L 1 and L 2 can be of arbitrary shape and dimensions. Link L 3 is
assumed to present a generalized cylinder —that is, a rigid body characterized
by a straight-line axis coinciding with the corresponding joint axis, such that the
link cross section in the plane perpendicular to the axis does not change along
the axis. There are no restrictions on the shape of the cross section itself, except
the physical-world assumption that it presents a simple closed curve—it can be,
for example, a circle (then the link is a common cylinder), a rectangle, an oval,
or any nonconvex curve.
We distinguish between the front end and rear end of link L 3 .The front end
coincides with the arm endpoint p (see Figure 6.13). The opposite end of link L 3
is its rear end. Similarly, the front (rear) part of link L 3 is the part of variable
length between joint J 3 and the front (rear) end of link L 3 .Formally,wehave
the following definition:

+(j)
Definition 6.3.2. At any given position j = (j 1 ,j 2 ,l 3 ) ∈ J, the front part L 3
of link L 3 is defined as the point set
(j) = L 3 (j) − L 3 ((j 1 ,j 2 , 0))
L 3 +
(j) of link L 3 is defined as the point set
Similarly, the rear part L 3 −
(j) = L 3 (j) − L 3 ((j 1 ,j 2 , 1))
L 3 −