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108 Application Examples in the Robotics Domain

(2 DOF) offers sidewards gyring of and full adduction with two addi-
tional coupled joints (one further DOF). Fig. 8.1 illustrates the workspace
with a stroboscopic image.

(b) (c)

(d)

(a)

Figure 8.1: a–d: (a) stroboscopic image of one finger in a sequence of extreme
joint positions.
(b–d) Several perspectives of the workspace envelope r, tracing out a cubical

10 10 10 grid in the joint space . The arrow marks the fully adducted posi-
tion, where one edge contracts to a tiny line.

For the kinematics in the case of our finger, there are several coordi-
nate systems of interest, e.g. the joint angles, the cylinder piston positions,
one or more finger tip coordinates, as well as further configuration depen-
dent quantities, such as the Jacobian matrices for force / moment trans-
formations. All of these quantities can be simultaneously treated in one
single common PSOM; here we demonstrate only the most difficult part,
the classical inverse kinematics. When moving the three joints on a cubical
10 10 10 grid within their maximal configuration space, the fingertip (or
more precisely the mount point) will trace out the “banana” shaped grid
displayed in Fig. 8.1 (confirm the workspace with your finger!) Obviously,