This section continues the discussion from Section 3.4. Suppose that a collection of links is arranged in a way that forms loops. In this case, the C-space becomes much more complicated because the joint angles must be chosen to ensure that the loops remain closed. This leads to constraints such as that shown in (3.80) and Figure 3.26, in which some links must maintain specified positions relative to each other. Consider the set of all configurations that satisfy such constraints. Is this a manifold? It turns out, unfortunately, that the answer is generally no. However, the C-space belongs to a nice family of spaces from algebraic geometry called varieties. Algebraic geometry deals with characterizing the solution sets of polynomials. As seen so far in this chapter, all of the kinematics can be expressed as polynomials. Therefore, it may not be surprising that the resulting constraints are a system of polynomials whose solution set represents the C-space for closed kinematic linkages. Although the algebraic varieties considered here need not be manifolds, they can be decomposed into a finite collection of manifolds that fit together nicely.4.11
Unfortunately, a parameterization of the variety that arises from closed chains is available in only a few simple cases. Even the topology of the variety is extremely difficult to characterize. To make matters worse, it was proved in  that for every closed, bounded real algebraic variety that can be embedded in , there exists a linkage whose C-space is homeomorphic to it. These troubles imply that most of the time, motion planning algorithms need to work directly with implicit polynomials. For the algebraic methods of Section 6.4.2, this does not pose any conceptual difficulty because the methods already work directly with polynomials. Sampling-based methods usually rely on the ability to efficiently sample configurations, which cannot be easily adapted to a variety without a parameterization. Section 7.4 covers recent methods that extend sampling-based planning algorithms to work for varieties that arise from closed chains.