Three links

Since visualization is still possible with one more dimension, suppose there are three links, $ {\cal A}_1$, $ {\cal A}_2$, and $ {\cal A}_3$. The C-space can be visualized as a 3D cube with opposite faces identified. Each coordinate $ \theta _i$ ranges from 0 to $ 2 \pi$, for which $ 0 \sim 2 \pi$. Suppose that each link has length $ 1$ to obtain $ a_1 = a_2 =
1$. A point $ (x,y) \in {\cal A}_3$ is transformed as

$\displaystyle \begin{pmatrix}\cos\theta_1 & -\sin\theta_1 & 0  \sin\theta_1 &...
...& 0  0 & 0 & 1  \end{pmatrix} \begin{pmatrix}x  y  1  \end{pmatrix} .$ (4.68)

To obtain polynomials, let $ a_i = \cos\theta_i$ and $ b_i=\sin\theta_i$, which results in

$\displaystyle \begin{pmatrix}a_1 & -b_1 & 0  b_1 & a_1 & 0  0 & 0 & 1  \e...
...& 0  0 & 0 & 1  \end{pmatrix} \begin{pmatrix}x  y  1  \end{pmatrix} ,$ (4.69)

for which the constraints $ a_i^2 + b_i^2 = 1$ for $ i = 1, 2, 3$ must also be satisfied. This preserves the torus topology of $ {\cal C}$, but now it is embedded in $ {\mathbb{R}}^6$. Multiplying the matrices yields the polynomials $ f_1,f_2 \in {\mathbb{R}}[a_1,b_1,a_2,b_2,a_3,b_3]$, defined as

$\displaystyle f_1 = 2 a_1 a_2 a_3 - a_1 b_2 b_3 + a_1 a_2 - 2 b_1 b_2 a_3 - b_1 a_2 b_3 + a_1,$ (4.70)

and

$\displaystyle f_2 = 2 b_1 a_2 a_3 - b_1 b_2 b_3 + b_1 a_2 + 2 a_1 b_2 a_3 + a_1 a_2 b_3,$ (4.71)

for the $ x$ and $ y$ coordinates, respectively.

Again, consider imposing a single constraint,

$\displaystyle 2 a_1 a_2 a_3 - a_1 b_2 b_3 + a_1 a_2 - 2 b_1 b_2 a_3 - b_1 a_2 b_3 + a_1 = 0 ,$ (4.72)

which constrains the point $ (1,0) \in {\cal A}_3$ to traverse the $ y$-axis. The resulting variety is an interesting manifold, depicted in Figure 4.24 (remember that the sides of the cube are identified).

Increasing the required $ f_1$ value for the constraint on the final point causes the variety to shrink. Snapshots for $ f_1 = 7/8$ and $ f_1 = 2$ are shown in Figure 4.25. At $ f_1=1$, the variety is not a manifold, but it then changes to $ {\mathbb{S}}^2$. Eventually, this sphere is reduced to a point at $ f_1 = 3$, and then for $ f_1 > 3$ the variety is empty.

Figure 4.24: The variety for the three-link chain with $ f_1=0$ is a 2D manifold.
\begin{figure}\begin{center}
\centerline{\psfig{file=figs/fourbar4fix.eps,width=4.0in}}
\end{center}
\end{figure}

Figure 4.25: If $ f_1 > 0$, then the variety shrinks. If $ 1 < p < 3$, the variety is a sphere. At $ f_1=0$ it is a point, and for $ f_1 > 3$ it completely vanishes.
\begin{figure}\begin{center}
\begin{tabular}{cc}
\psfig{file=figs/fourbarf78fix....
...h=2.7in}  $f_1 = 7/8$ & $f_1 = 2$ \\
\end{tabular}
\end{center}\end{figure}

Figure 4.26: If two constraints, $ f_1=1$ and $ f_2 = 0$, are imposed, then the varieties are intersected to obtain a 1D set of solutions. The example is equivalent to a well-studied four-bar mechanism.
\begin{figure}\begin{center}
\begin{tabular}{cc}
\psfig{file=figs/onedoffix.eps,...
...file=figs/onedof2fix.eps,width=2.7in} \\
\end{tabular}\end{center}
\end{figure}

Instead of the constraint $ f_1=0$, we could instead constrain the $ y$ coordinate of $ p$ to obtain $ f_2 = 0$. This yields another 2D variety. If both constraints are enforced simultaneously, then the result is the intersection of the two original varieties. For example, suppose $ f_1=1$ and $ f_2 = 0$. This is equivalent to a kind of four-bar mechanism [310], in which the fourth link, $ {\cal A}_4$, is fixed along the $ x$-axis from 0 to $ 1$. The resulting variety,

\begin{displaymath}\begin{split}V(& 2 a_1 a_2 a_3 - a_1 b_2 b_3 + a_1 a_2 - 2 b_...
...1 b_2 b_3 + b_1 a_2 + 2 a_1 b_2 a_3 + a_1 a_2 b_3), \end{split}\end{displaymath} (4.73)

is depicted in Figure 4.26. Using the $ \theta_1,
\theta_2, \theta_3$ coordinates, the solution may be easily parameterized as a collection of line segments. For all $ t \in
[0,\pi]$, there exist solution points at $ (0,2t,\pi)$, $ (t,2\pi-t,\pi+t)$, $ (2\pi-t,t,\pi-t)$, $ (2\pi-t,\pi,\pi+t)$, and $ (t,\pi,\pi-t)$. Note that once again the variety is not a manifold. A family of interesting varieties can be generated for the four-bar mechanism by selecting different lengths for the links. The topologies of these mechanisms have been determined for 2D and a 3D extension that uses spherical joints (see [698]).

Steven M LaValle 2020-08-14