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PLANAR REVOLUTE–REVOLUTE (RR) ARM 187

5.2 PLANAR REVOLUTE–REVOLUTE (RR) ARM

Let us reiterate, with a bit more specifics of the RR-arm, the arm’s model given
in Section 5.1.1. The arm consists of two links, l 1 and l 2 , and two revolute joints,
J 0 and J 1 (Figure 5.2). Joint J 0 is fixed. Strictly for better visualization, links
will be drawn as line segments. (As mentioned above, the shape of the arm links,
or the fact of their being smooth or convex or concave, will be of no importance
to the planning algorithm.) Link l i ,i = 1, 2, is hence a straight-line segment of
length l i . It can rotate indefinitely about the corresponding joint producing an
angle θ i , called the joint value.If W-space (workspace) is free of obstacles, the
arm endpoint b can reach any point within the W-space boundaries.
The arm’s W-space is formed by a circle of radius (l 1 + l 2 ) (the outer circle,
Figure 5.2) and by a circular “dead zone” (the inner circle, Figure 5.2) of radius
|l 1 − l 2 |. The middle circle in Figure 5.2 is a locus of points reachable by joint
J 1 . For a given position P of the arm endpoint in W-space, the corresponding
p p
pair of values (θ ,θ ), or the set of Cartesian coordinates of the link endpoints
1 2
a p and b p , represent an arm solution (arm position) for P . It is easy to see that,
in general, any position of the arm endpoint in W-space, except for points along
the W-space boundaries, corresponds to two arm solutions.
An obstacle in W-space is a closed curve of finite length homeomorphic to a
circle; that is, it cannot have self-intersections or double points. This also means

q = 0
2
b
l 2

q 2 a
J 1
l 1 q 1

q = 0
1
J o
O

Figure 5.2 Revolute–revolute (RR) arm. l 0 and l 1 are joints; θ 1 and θ 2 are joint values;
b is the arm endpoint.