This section continues where Section 4.4 left off. The subspace of 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 . For example, consider the Stewart-Gough platform [407,914] illustrated in Figure 7.20. The lengths of each of the six arms, , , , can be independently varied while they remain attached via spherical joints to the ground and to the platform, which is . Each arm can actually be imagined as two links that are connected by a prismatic joint. Due to the total of degrees of freedom introduced by the variable lengths, the platform actually achieves the full degrees of freedom (hence, some six-dimensional region in is obtained for ). 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 .