Page 281 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

256 MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS

5.8.4 The General Case of 2-DOF Arm Manipulators
We will now show that in 2D space ULC guarantees that CSO is bounded by
simple closed curves. This result provides an effective tool for reducing the robot
motion problem to the analysis of simple closed curves in CS.

1
1
∼
Lemma 5.8.3. CS of a 2-DOF robot presents one of the following: C = I × I ,
1
1
1
1
∼
∼
the unit square; C = S × I , a cylinder surface; or C = S × S , a torus surface.
1
Proof: The lemma follows from the fact that C i = I for a translational joint,
∼
1
∼
C i = S for a revolute joint, i = 1, 2, and C = C 1 × C 2 . Q.E.D.
Recall that the CS of a 2-DOF manipulator with two revolute joints (RR arm,
Figure 5.37) presents the surface of a common torus. The CS of a PR or RP arm
[arms (c), (d), and (e), Figure 5.1] is a cylinder, which is topologically a piece
of the torus. A cylinder is obtained from the torus (Figure 5.5), for example, by
making two vertical cuts in it. Finally, the CS of a PP arm [arm (b), Figure 5.1]
is a rectangular piece of plane (formally, a unit square). It can be obtained from
the torus by cutting a patch out of it. Figure 5.38 demonstrates how a cylinder
and a patch are obtained from the torus.
The previous discussion and Lemma 5.8.3 suggest an important fact: The RR
arm presents the most general case among all 2-DOF robot arms (see Figure 5.1).
This means that the motion planning algorithm for the RR arm (Section 5.2.2),
can be used to handle any two-link XX arm, X being X = (P or R). This will
result in more calculations than the arm in question really needs, but it will work.
For example, for the PR arm, the general (RR) algorithm may call for the third
and fourth M-line, in which case the control will need to infer that for the PR
arm those M-lines do not exist (see Figure 5.5).

q 1

q 2

Figure 5.37 A 2-DOF robot with two revolute joints.