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

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

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

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

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

and

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

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

Again, consider imposing a single constraint,

$$2 a_1 a_2 a_3 - a_1 b_2 b_3 + a_1 a_2 - 2 b_1 b_2 a_3 - b_1 a_2 b_3 + a_1 = 0 ,$$ (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.

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.

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.

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,

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