8.1 Motivation

For many interesting problems, the linkage is arranged in a ``tree'' as shown in Figure 3.21a. Assume here that the links are not attached in ways that form loops (i.e., Figure 3.21b); that case is deferred until Section 4.4, although some comments are also made at the end of this section. The human body, with its joints and limbs attached to the torso, is an example that can be modeled as a tree of rigid links. Joints such as knees and elbows are considered as revolute joints. A shoulder joint is an example of a spherical joint, although it cannot achieve any orientation (without a visit to the emergency room!). As mentioned in Section 1.4, there is widespread interest in animating humans in virtual environments and also in developing humanoid robots. Both of these cases rely on formulations of kinematics that mimic the human body.

Figure 3.21: General linkages: (a) Instead of a chain of rigid bodies, a ``tree'' of rigid bodies can be considered. (b) If there are loops, then parameters must be carefully assigned to ensure that the loops are closed.

Another problem that involves kinematic trees is the conformational analysis of molecules. Example 3.5 involved a single chain; however, most organic molecules are more complicated, as in the familiar drugs shown in Figure 1.14a (Section 1.2). The bonds may twist to give degrees of freedom to the molecule. Moving through the space of conformations requires the formulation of a kinematic tree. Studying these conformations is important because scientists need to determine for some candidate drug whether the molecule can twist the right way so that it docks nicely (i.e., requires low energy) with a protein cavity; this induces a pharmacological effect, which hopefully is the desired one. Another important problem is determining how complicated protein molecules fold into certain configurations. These molecules are orders of magnitude larger (in terms of numbers of atoms and degrees of freedom) than typical drug molecules. For more information, see Section 7.5.

Figure 3.22: Now it is possible for a link to have more than two joints, as in ${\cal A}_7$.