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REVOLUTE–PRISMATIC (RP) ARM WITH PARALLEL LINKS 229

5.5 REVOLUTE–PRISMATIC (RP) ARM WITH PARALLEL LINKS

The revolute–prismatic parallel kinematic configuration (Figure 5.1c) is a com-
mon major linkage in commercial robot arm manipulators. The so-called Stanford
manipulator robot (see, e.g., Ref. 7) is a typical example of this type. Joint values
of this arm are the angle θ 1 of the first (revolute) joint and the variable length
l 2 of the second (prismatic) joint; 0 ≤ l 2 ≤ l 2max (see Figure 5.24a). The outer
boundary of the arm’s workspace (W-space) is a circle of radius (l 1 + l 2max ). Its
inner boundary is a circle of radius l 1 , which defines the dead zone inaccessible
to the arm endpoint. Unlike in Figure 5.1c, in order to not overcrowd the picture,
no dead zone appears in the arm in Figure 5.24a; that is, here l 1 = 0. With the
arm endpoint b in some position of W-space—say, S —the position of the link’s
rear end, a S , can be found by passing a line segment of length l 2max from b S
through the origin O.
For specificity, let us define the M-line as a straight-line segment connecting
points S and T ; denote it M 1 -line. In the example in Figure 5.24a, one path
from S to T would be as shown, the curve (S, 1,2,3, 4,5,6,7, 8,9,10, T ).
Observe that if due to obstacles the arm cannot reach point T by following the
M 1 -line, it might be able to reach T “from the other side,” by changing angle
θ 1 in the direction opposite to that of the M 1 -line. Hence, similar to how we
did it with the RR arm (Section 5.2), a complementary M-line, the M 2 -line, is

introduced, defined as consisting of three parts: two straight-line segments, (S, S )

and (T, T ), continuing the M 1 -line segment outwards until its intersection with
the W-space outer boundary, and a segment of the W-space outer boundary that
corresponds to the interval of θ 1 complementing that of the M 1 -line to 2π.This
choice for the complementary M-line is largely arbitrary; any M 2 -line will do, as
long as it is uniquely defined, is computationally simple, and complements the
θ 1 interval of M 1 -line to 2π.
As the arm moves along the M-line, obstacles will interfere with its motion
in two different ways. In our example the arm endpoint will be forced to leave
M-line two times: The first time is when obstacle A interferes with the rear part
of link l 2 , creating the curve (1, 2, 3, 4, 5), (Figure 5.24a), and the second time
is when obstacle B interferes with the front part of link l 2 , creating the curve
(6, 7, 8, 9, 10) (Figure 5.24a). Note two shadows formed during this process
(shaded in Figure 5.24a): Under no circumstances the arm endpoint will be able
to reach any point within the figures with boundaries (0, 1, 2, 3, 4, 5, 0) and
(6, 7, 8, 9, 10, 11, 12).
Define a front contact of link l 2 as a situation where a part of the link between
its front end (which is the part containing the arm endpoint) and the origin is in
contact with the obstacle. A rear contact of the link refers to a situation where a
part of the link between its rear end and the origin is in contact with the obstacle.
Correspondingly, a front contact forms the front shadow of the obstacle, and a
rear contact forms the rear shadow . In Figure 5.24a, the rear shadow of obstacle
A is limited by the line (1, 0, 5, 4, 3, 2, 1), and the front shadow of obstacle B is
limited by the line (6, 7, 13, 9, 10, 11, 12, 6).