Page 228 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

PLANAR REVOLUTE–REVOLUTE (RR) ARM 203

link positions for points S and T , one can find the corresponding values R and
ϕ (or the complement of ϕ to 2π) and then calculate p and q.
One disadvantage of these M-lines is that with them the motion can be non-
monotonic in terms of the joint angles θ 1 and θ 2 (see Figure 5.3). Worse yet,
depending on the arm positions S and T , continuous motion along such an
M-line may be impossible even if the scene contains no obstacles. This happens,
for example, if the M-line is a straight line passing through the arm’s dead zone
(Figure 5.2).
Figuring out beforehand if the nonmonotonic change in joint values or the
M-line passing through the workspace dead zone is a problem in the case at
hand is an extra difficulty, which can be avoided with still another choice for
the M-line, the one that it preferred in this text. We choose the M-line that is a
straight line in the plane of variables θ 1 and θ 2 —that is, a straight line between
points S and T in C-space. When no obstacles are present, this M-line results
in an economical and uniform change of the arm joints from the starting to the
final position. That is, motion along this M-line is monotonic in θ 1 and θ 2 .On
the C-space torus the model of this M-line is a geodesic curve—a tight thread
connecting points S and T . In Figure 5.5 this “shortest route,” M 1 , corresponds
to the positive change of both angles θ 1 and θ 2 .
There are, however, three other ways to obtain a geodesic curve between points
S and T on the torus surface. All four are obtained by using both positive and
negative directions of change for each of the angles θ 1 and θ 2 . These are shown
in Figure 5.5, as well as on the flattened torus in Figure 5.11. We will see that
from time to time the algorithm may use a combination of these routes. One of
the four routes corresponds to the global minimum, and the other three—called
complementary routes—correspond to the local minima of the distance between
S and T in the plane (θ 1 ,θ 2 ), (Figure 5.5). For some special locations of points
S and T , two or more of the four M-lines may become equal. For example, with
points S and T located at the opposite points of the torus’s outer equator, all
four M-lines are of equal length.
Each M-line is characterized by its M k -line segment between points S and T ,
k = 1, 2, 3, 4. (For better visibility, in Figure 5.5 M k also indicates the middle
point of the corresponding segment.) An M-line that represents a straight line in
the plane θ 1 ,θ 2 is given by the equation

θ 2 = p · θ 1 + q (5.1)

To compute coefficients p and q for all M k -lines—we may need this for the
motion planning algorithm—“flatten” the torus by cutting it along two mutually
perpendicular circles passing through point T , one of which is parallel to the
large equator of the torus; this is shown in Figure 5.11. This operation produces
a rectangle, with point S lying somewhere inside of it. Point T is identified;that
is, it produces four points T k ,k = 1, 2, 3, 4, each sitting in the rectangle’s corners
T
S
and corresponding to one of the M k -lines. Denote δ i = sign(θ − θ ), i = 1, 2,
i i