Spine Scaling and Contact Modeling

Started: 16 Jul 2004 drawing from notes by MicheleLanzetta? , MarkCutkosky

New Topic: SpineAsperityContactModel? -Main.MarkCutkosky - 09 Dec 2004

---

(more about these ideas in the powerpoint slide set on the CompliantSpinedFoot? page. -- Mark C. %ENDCOMMENT%

Original text from Prof. Cutkosky's email with Main.MicheleLanzetta's comments.

• Private.SpineScale.doc:

I've been thinking about the question: "What is the right spine size for an animal (or robot) of a certain size?" I am interested in the following discussion in spines for rough, non-penetrable surfaces such as rock, brick and concrete. Applications include buildings, caves, cliffs, etc. I hoping that someone will become interested in this and tell me where my thinking is wrong. 1. Let me assume that rough, non-pentrable surfaces are approximately fractal in terms of their surface topology. This is born out by a thesis of a former student, Mike Costa [2000] who found that fractals model concrete and rocks quite well (he was interested in modeling them for haptic rendering). What this means is that the number of asperities per unit area is a function of your length scale. (The smaller you are, the rougher it looks.) For concrete, the fractal surface dimension is something like 2.5 where 2 would be a smooth surface (same as the Euclidean dimension). I have plots of the actual asperity probability density functions and power spectral density (PSD) taken by Mike. 2. Let L be the characteristic length of an animal. Then, as in some biology papers, we assume loading goes as F ~ L^3, using '~' to mean 'proportional to'. (In practice, less because big animals can't afford to be so heavy) 3. Let Ls be the characteristic length of a spine. If the spine is modeled as a cantilever beam, its strength and stiffness can be computed as a function of Ls, assuming constant shape.

Maybe in nature they are taper, but we can make them constant in diameter (for mfctg reason) until the tip, which we want to make very sharp.

The rupture stress goes as Mc/I where M ~ Ls, c ~ Ls, and I ~ Ls^4, where M is the moment, c is the cross section half thickness and I is the inertia. So based on rupture, the spine strength increases as Ls^2 (same if we assume spine is a claw loaded in tension) -- not sure that this is the governing constraint...

I think we want a negligible deflection as a constraint (not rupture) in order not to change the hooking angle. The typical c/Ls ratio is 1/10 for the commercial pins we have. For Ls we must distinguish:

• internal part of spine in compliant material, which will exert active moment/shear and
• the visible part/protrusion.

The length of the internal part is lower limited by the moment and depends on the surrounding material (are we really making a clamp?). The protrusion of a spine must be kept to a minimum in order to minimize the moment and deflection but also long enough for compliance with the wall roughness.

4. So as the animal grows, the loading increases as L^3, but if the spine grows in proportion to the animal, the spine strength increases only as L^2.

Size (protrusion) and strength (diameter) have different non-contrasting constraints:

• Protrusion depends on geometrical aspects (roughness)
• Diameter Strength depends on the load applied to it (animal weight and number of active spines)

In addition, we must consider the tip size, this really dominates for hooking on the very small asperities, which are almost always present. We are free to have the size we want (the smaller is better, but we have more tear and mfctg problems)

5. In practice, it seems like spine or claw dimensions grow more like L^(1/2) if we look at a range of animals from insect to gecko to squirrel? (I'm leaving off large animals that use penetrating claws because that's a different analysis.) I am not able to back out an actual proportionality number across geckos from the Zani papers -- will need Kellar's help in converting PCA data to an exponent... 6. Now, given argument (1.) above, it would seem that more, smaller spines give you an advantage PROVIDED THAT you have appropriate multi-scale conformability (legs, toes, substrates) to bring them all into contact with your surface. The probability of finding N asperities greater than some threshold size, per unit area, increases more than linearly as you reduce the length scale of your spines. 7. But given (4.), spine strength grows faster than L, so larger spines have some advantage as the animal gets bigger. From all this it seems that one should be able to draw some conclusions about the number of spines needed ordesired as a function of spine size and body size. However, it would be good to get some feedback from others before going any further.

The angle of spine with respect to surface is also important. If almost perpendicular, then the tip diameter dominates. But insect legs probably hook almost laterally. Think of a net-like surface pushed to the surface (for protrusions). Demonstration: DistributedSpines?

Already on Twiki

• Feduccia, Alan. "Evidence of Claw Geometry Indicating Arboreal Habits of Archaeopteryx" Science, Vol. 259, Issue 5096, pgs 790-793.

Says that we climber birds have higher claw arc (120-150). For optimal moment the angle with respect to the surface should be 30 – 60 degrees.

• Dai, Z., Gorb S., Schwarz, U.. "Roughness-dependent friction force of the tarsal claw system in the beetle Pachnoda marginata (Coleoptera, Scarabaeidae)" The Journal of Experimental Biology, Vol 205, pgs 2479-2488. Says that we want the spine tip to be as small as the minimum particle size.

• For

Zani, P.A.. "The comparative evolution of lizard claw and toe morphology and clinging performance" The Journal of Evolutionary Biology, Vol 13, no. 2, pgs 316-325.

• Increases in claw curvature for smoother surface.
• Increases in claw thickness for rough surface.

• Fisherman’s claws have 180 degrees or more