Closure constraints

The closure constraints introduced in Section 4.4 can be summarized as follows. There is a set, ${\cal P}$, of polynomials $f_1$, $\ldots$, $f_k$ that belong to ${\mathbb{Q}}[q_1,\ldots,q_n]$ and express the constraints for particular points on the links of the robot. The determination of these is detailed in Section 4.4.3. As mentioned previously, polynomials that force points to lie on a circle or sphere in the case of rotations may also be included in ${\cal P}$.

Let $n$ denote the dimension of ${\cal C}$. The closure space is defined as

$${{\cal C}_{clo}}= \{q \in {\cal C}\; \vert \; \forall f_i \in {\cal P}, f_i(q_1,\ldots,q_n) = 0 \},$$ (7.21)

which is an $m$-dimensional subspace of ${\cal C}$ that corresponds to all configurations that satisfy the closure constants. The obstacle set must also be taken into account. Once again, ${\cal C}_{obs}$ and ${\cal C}_{free}$ are defined using (4.34). The feasible space is defined as ${{\cal C}_{fea}}= {{\cal C}_{clo}}\cap {\cal C}_{free}$, which are the configurations that satisfy closure constraints and avoid collisions.

The motion planning problem is to find a path $\tau: [0,1] \rightarrow {{\cal C}_{fea}}$ such that $\tau(0) = {q_{I}}$ and $\tau(1) = {q_{G}}$. The new challenge is that there is no explicit parameterization of ${{\cal C}_{fea}}$, which is further complicated by the fact that $m < n$ (recall that $m$ is the dimension of ${{\cal C}_{clo}}$).