Page 230 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

PLANAR REVOLUTE–REVOLUTE (RR) ARM 205

For algorithmic purposes, it is important to (a) detect the fact that a given point in
C-space lies on the currently used M k -line and (b) compute its relative position
on the segment M k . We will use for this the parametric description of the M-
line equation [97]. For example, for M 1 -line, the symmetric presentation using
coordinates of points S and T is,

θ 1 − θ S θ 2 − θ S
1 = 2 = T (5.4)
T
T
θ − θ S θ − θ S
1 1 2 2
from which the parametric equations for θ 1 and θ 2 are found, with t as a param-
eter:
S
T
θ 1 = θ · (1 − t) + θ · t
1 1
(5.5)
S
T
θ 2 = θ · (1 − t) + θ · T
2 2
Apairofangles (θ 1 ,θ 2 ) is recognized as a point on the M-line if both equations
in (5.5) produce the same value of t. The relative position of a point on the
M-line segment is determined as follows:
t = 0 corresponds to point S;
t = 1 corresponds to point T ;
0 <t < 1 corresponds to points inside the segment (S, T );
t< 0 correspond to points outside the segment and closer to S than to T ;
t> 1 corresponds to points outside the segment and closer to T than to S.

We emphasize that in the plane (θ 1 , θ 2 ) where the straight line θ 2 = f(θ 1 ) is
defined, angles θ i and (θ i − 2π) are not equivalent. In determining whether a
given point lies within the segment (S, T ) of a given M k -line, care should be
taken in choosing the right line from (5.2) to represent the angles in (5.5).
We order four M-line segments according to (5.2), with M 1 being the shortest
segment. Here the length of a segment is the Euclidean distance between points S
and T in the plane (θ 1 ,θ 2 ), with coordinates presented as in (5.2). For each angle
S
θ i , i = 1, 2, there is an interval δθ i,k =|θ T − θ | or δθ i,k =|θ T − θ S − 2π|
i,k i,k i,k i,k
related to the corresponding segment M k ; δθ i,k ≤ 2π.
Definition 5.2.7. For two M-line segments M k and M m , k, m = 1, 2, 3, 4, k = m,
their complementarity shows in that, for one or both angles θ i ,intervals δθ i,k and
δθ i,m add to 2π (see Figure 5.5). These two segments are said to be complementary
over the angle θ i .

The following discussion helps clarify how complementary M-lines are used
during the path planning. Assume that M 1 has been chosen as the M-line. Imagine
that the arm, while following M-line, encounters an obstacle (i.e., defines on it
a hit point), tries to pass it around, returns to the hit point without ever meeting