7.4 Planning for Closed Kinematic Chains

This section continues where Section 4.4 left off. The subspace of ${\cal C}$ that results from maintaining kinematic closure was defined and illustrated through some examples. Planning in this context requires that paths remain on a lower dimensional variety for which a parameterization is not available. Many important applications require motion planning while maintaining these constraints. For example, consider a manipulation problem that involves multiple manipulators grasping the same object, which forms a closed loop as shown in Figure 7.19. A loop exists because both manipulators are attached to the ground, which may itself be considered as a link. The development of virtual actors for movies and video games also involves related manipulation problems. Loops also arise in this context when more than one human limb is touching a fixed surface (e.g., two feet on the ground). A class of robots called parallel manipulators are intentionally designed with internal closed loops [693]. For example, consider the Stewart-Gough platform [407,914] illustrated in Figure 7.20. The lengths of each of the six arms, ${\cal A}_1$, $\ldots$, ${\cal A}_6$, can be independently varied while they remain attached via spherical joints to the ground and to the platform, which is ${\cal A}_7$. Each arm can actually be imagined as two links that are connected by a prismatic joint. Due to the total of $6$ degrees of freedom introduced by the variable lengths, the platform actually achieves the full $6$ degrees of freedom (hence, some six-dimensional region in $SE(3)$ is obtained for ${\cal A}_7$). Planning the motion of the Stewart-Gough platform, or robots that are based on the platform (the robot shown in Figure 7.27 uses a stack of several of these mechanisms), requires handling many closure constraints that must be maintained simultaneously. Another application is computational biology, in which the C-space of molecules is searched, many of which are derived from molecules that have closed, flexible chains of bonds [245].

Figure 7.19: Two or more manipulators manipulating the same object causes closed kinematic chains. Each black disc corresponds to a revolute joint.

Figure 7.20: An illustration of the Stewart-Gough platform (adapted from a figure made by Frank Sottile).