Take $N$ springs, $(s_1, ..., s_N) \in S$ of length $(l_1, ..., l_N)$, and for each spring, label one end "A" and one end "B". Connect the "A" ends of the $N$ springs to a point-like particle on a two-dimensional plane at, $(x_p, y_p) = (0, 0)$, then connect the "B" ends of the springs to anchor points a distance $(d_1, ..., d_N) = (l_1, ..., l_N)$ from $(x_p, y_p)$ at coordinates $((x_1,y_1), (x_2,y_2), ..., (x_N,y_N)) \in C$. For a quick visual, if $N = 8$, the assembly would look like a stretched-out octopus (or spider) with a point-like head at $(x_p, y_p)$.

As all springs are composed of the same material, we can define a spring constant of $k$ for the unit length spring, and assign spring constants for $(s_1, ..., s_N)$ of $(k_1, k_2, ..., k_N) = (\frac{1}{l_1}, \frac{1}{l_2}, ..., \frac{1}{l_N})$. We can write the angle between spring $s_1$ and $s_2$ as $\theta_{(s_1,s_2)}$, the angle between springs $s_2$ and $s_3$ as $\theta_{(s_2,s_3)}$, and so on up to $\theta_{(s_N, s_1)}$.

My question is the following: Say we allow the anchor points, which the "B" ends of the springs are connected to, to take independent two-dimensional random walks, each with $T$ steps and step-size of $D$. Or, if it simplifies things, imagine that we simply add a set of i.i.d. random variables, over the interval $[0, D]$, to the $x$ and $y$ coordinates of each of the anchor point positions in $C$.

After perturbing the anchor points for the springs in this manner, what is the probability distribution for the "body" of the spider $(x_p, y_p)$, i.e. the position of the point-like particle joining the "A"-labeled ends of the $N$ springs in the system? How do we choose the set of anchor positions, $C$, for the $N$ springs to minimize the difference in position for $(x_p, y_p)$ before and after the perturbation of the anchor points, and does this strictly decrease with larger $N$?

Pushing my luck, let's also say that the body of the spider is not a point, but an arbitrarily small circle with a direction vector originally pointing at the position on the contour of the circle half-way between spring $s_N$ and spring $s_1$. Given that the distance between the "A" ends of the springs along the circle's contour are always fixed, what is the angular distribution for the orientation of this vector after anchor point perturbation?

Update - I realize I'm being vague about the distribution of the anchor point positions. The idea was to not discourage simplifications by others that allow for easier analysis (like the anchor points originally being the vertices of a rectangular lattice), but let's imagine that the plane the spider sits on is covered with random points at a density of $P$ points per unit area. The anchor points can thus be a set of points within a range of distances $[B_1, B_2]$ from the body of the spider. Again, I would certainly be amenable to any original lattice configuration of the anchor points.

Update 2 - We can probably also require the further restriction that the "body" of the spider, i.e. the point $(x_p, y_p)$, falls within a convex hull defined by the anchor points in the set $C$. I *think*, though I'm not sure, that this guarantees a unique equilibrium solution for the position of the spider as a function of the lengths of the springs...