Are wheels optimal for locomotion?

Given the large number of wheeled robots coming out of modern factories it’s natural to ask whether wheels are near-optimal for terrestrial locomotion. I believe this question may be partially addressed from a physicists’ perspective although the general problem of finding near-optimal robot models appears to be intractable.

Physics:

A reasonable scientific approach to this problem would involve a laboratory setting of some sort. The scientist may setup an inclined ramp and analyse the rolling motion of different compact shapes with the same mass and volume. In particular it’s natural to ask: Of all the possible shapes having the same material, mass and volume which shape reaches the bottom of the ramp first when released from the top of the ramp?

If we may neglect dissipative forces, we may determine the time for a disk to reach the bottom of the ramp as follows:

1. By the parallel axis theorem, we have:

\begin{aligned} \dot{w} = \frac{\tau}{I}=\frac{mgrsin\theta}{0.5 m r^2} \end{aligned}

where $\theta$ is the ramp’s angle of inclination, $m$ is the mass of the disk and $r$ is the radius of the disk.

2. As a result, the linear acceleration of the disk is constant:

\begin{aligned} \ddot{x}= 2gsin\theta \end{aligned}

3. From this we deduce that given a ramp with length $l$, the total time to reach the bottom of the ramp is given by:

\begin{aligned} T = 2\sqrt{\frac{l}{gsin\theta}} \end{aligned}

This analysis, although useful, is difficult to replicate for other shapes as the moment of inertia will actually depend on the axis chosen. Further, beyond the special case of certain symmetric bodies the total time taken to reach the bottom of the ramp can no longer be determined in an analytic manner. Barring a mathematical method I’m unaware of, we would need to simulate the motion of each body rolling/bouncing down the incline with high numerical precision and then perform an infinite number of comparisons.

Engineering:

At this point it may seem that our task is impossible but perhaps we can make progress by asking a different question…Of all the possible shapes that move down the incline, which have the most predictable and hence controllable motion? The answer to this is of course the spherical bodies(ex. the wheel) and it’s for this reason that engineers build cars with wheels to roll on roads. A different shape that reached the bottom of the incline sooner than the wheel would be useless for locomotion if we couldn’t determine where it would land. For these reasons, our mathematical conundrum may be safely ignored.

In contrast, as natural terrain tends to be highly irregular, animal limbs must be highly versatile. I have yet to come across wheels that simultaneously allow grasping, climbing, swimming and jumping. Furthermore, even if there were a large region of land that was suitable for wheeled locomotion, these animals would inevitably be confined to this region and this situation would be highly detrimental to their ability to explore new terrain.

Given these arguments, it’s clear that wheeled robots aren’t well suited for terrestrial locomotion. Having said this, can we build robots that are near-optimal for terrestrial locomotion? What would we mean by optimal?

Conclusion:

I have thought more broadly about the possibility of using computers to design near-optimal robots for terrestrial locomotion but the problem appears to be quite hard. Given that the cost function isn’t well-defined the search space will be very large. It would make sense to place a prior on our cost function and try to use a method similar to Bayesian Optimisation. However, the task entails a method much more sophisticated than the Bayesian Optimisation methods that are used in industry today.

I’ll probably revisit this challenge in the future but I think it would make sense to first solve an intermediate challenge in Bayesian Optimisation applied to robotics.

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Why don’t mammals have more than 4 limbs?

-a cheetah in action captured by Mark Dumont

Approximately one month ago I wondered why there were no mammals that had more than 4 limbs. I wondered whether they were somehow optimal for the challenges faced by mammals. I was thinking about the Cheetah in particular: would it run faster if it had an additional pair of limbs?

Unsure about how to proceed I posted the question on the biology StackExchange where it generated some interest. The accepted answer by a zoology researcher could be summarized in this way:

“So basically, terrestrial vertebrates have four legs because they evolved from a fish ancestor that had four members that were possibly used as legs (that could “easily” evolved into legs). The explanation is as simple and basic as that. You can have a look to the diversity of terrestrial vertebrates here (click on the branches).”

Unsatisfied with this answer, I asked another zoologist from Cambridge University, Jenny Clack, who also happens to be an expert on tetrapod evolution and she gave me a similar answer:

“If four appendages are optimal, it’s because they were optimal for swimming before tetrapods became terrestrial. Clearly four are not always optimal for tetrapods since many of them lose them, but it would take a major Hox gene duplication – probably deleterious if not fatal – for tetrapods to get more.”

It appears that nobody has tried to do computer simulations or analyses of any kind to determine whether 6 legs may be better or worse for mammals that have particular constraints due to their habitats and functional purposes. After more research I have come to this conclusion. But, after attempting to transform the question into a falsifiable hypothesis(Are 4 legs optimal for certain/all mammals?) I have realised that there are quickly serious difficulties.

Even if we considered a special case of this question: “Are 4 legs optimal for a 100 kg carnivorous savannah mammal?” there are serious challenges. To begin with, there are many important factors that aren’t well understood:
a) type of vascular system used
b) type of respiratory system
c) other variables
d) dynamic relationship between these components

Given the limited understanding of leading researchers of the biological components of land mammals, I realised that doing significant analysis of this problem is basically impossible.

This is the first time I’ve encountered a scientifically interesting problem that can’t lend itself to any kind of reasonable analysis.