### 4 321 kinematic structure

Because all-revolute joint manipulators have good workspace properties, and because a sequence of three intersecting joint axes introduces signiﬁcant simpliﬁcations in the kinematic algorithms, most industrial robot arms now have a kinematic structure as shown in Fig. 8. (Vic Scheinman of Stanford University was, to the best of the authors’ knowledge, the ﬁrst to come up with this design, but he did not write it up in any readily accessible publications...) The design is an example of a 6R wrist-partitioned manipulator: the last three joint axes intersect orthogonally at one point. Moreover, the second and third joints are parallel, and orthogonal to the ﬁrst joint. These facts motivate the name of “321” robot arm: the three wrist joints intersect; the two shoulder and elbow joints are parallel, hence they intersect at inﬁnity; the ﬁrst joint orthogonally intersects the ﬁrst shoulder joint.

The 321 structure can be given a link frame transformation convention that is much simpler [21] than the Denavit-Hartenberg or Hayati-Roberts conventions: its geometry is determined by orthogonal and parallel joint axes, and by only four link lengths and , because the wrist link lengths and are zero. In this simpler convention, the reference frames are chosen to be all parallel when the robot is in its fully upright conﬁguration. This conﬁguration is deﬁned to be the kinematic zero position in the rest of this text, i.e., all joint angles are deﬁned to be zero in this position. The six joints are deﬁned to rotate in positive sense about, respectively, the , and axes, such that positive joint angles make the robot “bend forward” from its kinematic zero position. Many industrial robots have a 321 kinematic structure, but it is possible that the manufacturers deﬁned different zero positions and different positive rotation directions for some joints. These differences are easily compensated by (constant) joint position offsets and joint position sign reversals.

321 kinematic structure with offsets. Many other industrial robots, such as for example the PUMA (Fig 2), have a kinematic structure that deviates a little bit from the 321 structure of Figure 8, [146578]:

1. Shoulder offset: frame in Figure 8 is shifted a bit along the -axis. This brings the elbow off-centre with respect to the line of joint 1.
2. Elbow offset: frame in Figure 8 is shifted a bit along the -axis. This brings the wrist centre point off-centre with respect to the forearm.

The reasons for the offsets will become clear in the Section on singularities (Sect. 15): the offsets move the singular positions of the robot away from places in the workspace where they are likely to cause problems.