Page 247 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

222 MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS

The arm’s first joint angle, θ 1 , is responsible for motion in one plane; for speci-
ficity, assume this is a horizontal plane. The second joint angle, θ 2 , is responsible
for the arm motion in the vertical plane. Together they allow the arm endpoint
(end effector) to reach any point on the surface of a sphere of radius l 2 .The
workspace (W-space) of this arm is hence a sphere. Any point P on the sphere
P
P
P
P
corresponds to two joint solutions, (θ ,θ ) and (π + θ ,π − θ ).
1 2 1 2
Since the body of link l 2 moves in the three-dimensional (3D) space inside
the W-space sphere, it can interact with 3D obstacles that may appear inside
the workspace sphere, thus presenting a potential collision avoidance issue. The
fact that one end of link l 2 is fixed (at the base J 1 ) and the motion of its other
end is limited to the sphere surface constrains the link interaction with obstacles
significantly. In terms of motion planning, this is equivalent to motion along a
curve rather than around a “real” 3D object. This means that the 2D motion
planning algorithms of Section 5.2 fully apply here.
Proceeding in this direction, we want to chose an M-line, the line that the arm
endpoint would go through if no obstacle interfered with the arm motion. Since
the C-space (configuration space) of this arm is a torus, the choice is among
four M-lines (Section 5.2). These are shown in Figure 5.20. Denote the positive
direction of change of angle θ i ,i = 1, 2, by “+” and denote the negative one by
“−”. Then the four M-lines are four geodesic curves, as shown in Figure 5.20b,
with the corresponding joint angles changing as follows:
θ 1 θ 2
M 1 : + +
M 2 : − +
M 3 : + −
M 4 : − −
The choice of the M-line and the motion planning procedure will proceed accord-
ing to the RR-Arm Algorithm (Section 5.2.2).

RR Arm of Figure 5.19c. A detailed picture of this arm configuration is shown
in Figure 5.21a. The only difference between this configuration and the one in
Figure 5.20a is that here the arm’s two joints are at a distance from each other,
equal to the length of the first link, l 1 . Links l 1 and l 2 lie in the same plane—in
general, depending on link l 1 shape, in parallel planes. The arm’s endpoint moves
along the surface of a torus, and so its W-space is a torus. This torus may or may
∗
not have a hole depending on the relation between the lengths of links l 1 and l 2 :
l 2 >l 1 produces a W-space torus with no hole;
l 2 <l 1 produces a W-space torus with a hole.

Projections of W-space onto the horizontal (xy) and vertical (xz or yz) planes for
both cases, l 2 >l 1 and l 2 <l 1 , are shown in Figures 5.21b and 5.21c, respec-
tively.

∗ To emphasize, it is not the configuration space that forms a torus here, as we had it before, but the
workspace.