# The equilibrium position for the body of a spider-like spring system after randomly perturbing the anchor positions of its legs

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...

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I suspect it might be easier to solve the continuous version of this, where Brownian motion is considered instead of random walks. Off the top of my head I'd guess you'd have something like a coupled Langevin system. –  Steve Huntsman Aug 23 '12 at 13:52
@Steve Huntsman If you have a good idea for solving the system by approximating the anchor point perturbations as Brownian motion, I'd be very excited to hear about it. –  CKura Aug 23 '12 at 13:58

Not an answer, just an illustration.

Perhaps this is one version of what you have in mind? Here I have five feet wandering 10 steps, while the equilibrium position (red) wanders at a reduced scale, essentially $\frac{1}{5}$ the feet displacements:

Here are some responses to the comments questions of CKura. For the spider's body, I chose to use the point $c$ whose sum of vectors to the feet $p_i$ balances: $\sum_i (p_i - c) = 0$. Thus, $c$ is the centroid of the feet, $c=\frac{1}{n} \sum_i p_i$. Therefore, the statistics of the random walk for $c$ are just $\frac{1}{n}$-th of those for the feet. Symmetric arrangement of the legs is irrelevant in this formulation.

Above, each of the five feet moved uniformly randomly within $\pm \frac{1}{2}$ at each step. Below the three feet move within $\pm 1$. (The start position of each foot is marked in yellow.)

I emphasize that I am not necessarily addressing your exact questions, but rather riffing on your theme.

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@Joseph O'Rourke Absolutely beautiful. =) Intuitively I'd think that, if one has a roughly symmetric arrangement of $N$ feet, the body of the spider will wander at scale of roughly $\frac{1}{N}$ the displacement of the anchor positions... and as you mention, that's what seems to be going on here. If you don't mind me asking, could you share some details of your simulation? What step size did you use for the random walk, and (just guessing) is the position of the spider's body a weighted average where the weights are based on the original distances from $(x_p, y_p)$ to the anchor points? –  CKura Aug 24 '12 at 1:28
Actually, I'm wondering just how much the symmetric arrangement of the legs matters for minimizing the drift in the position of the spiders body... –  CKura Aug 24 '12 at 4:41
@CKura: Where you see a spider, I Rorschach a joyous jumping jack. :-) –  Joseph O'Rourke Aug 24 '12 at 10:13
@Joseph O'Rourke I see the jumping jack too. =) –  CKura Aug 24 '12 at 10:28