# 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.

# The Quadrupedal Conjecture

Last summer, I asked myself several related questions concerning the locomotion of quadrupeds:

1) What if the cheetah had more than four legs?
2) Why don’t hexapods gallop?
3) Have we ever had F1 cars with more than 4 wheels?

I found all these related questions interesting but after more reflection I summarised my problem mathematically:

Could four legs be optimal for rapid linear and quasi-linear locomotion on rough planar surfaces?

After searching for papers on this subject and finding none, I decided to name this problem ‘The Quadrupedal Conjecture’ and it may be informally described as saying that four legs allow a creature to travel fastest on a nearly-flat surface. However, I thought it might be interesting to tackle a simpler problem first which we may call the ‘Little Polypedal Conjecture’:

Can we show that given a polyped with $2 n$ legs having pre-defined mass and energy, that there exists $N$ such that for $2 N$ limbs or greater, its maximum linear velocity on a rough planar surface would be reduced?

I believe that there is an elegant solution to the above problem and I think that the quadrupedal conjecture has an elegant solution as well. But it is by no means guaranteed that a solution to such a problem would be simple. Finding the right degree of abstraction would be one of the main challenges as there are many different ways of approaching this problem with varying degrees of realism.

First, I must say that this is not primarily a problem in biology although this question has attracted the attention of biologists on the biology stackexchange. Second, I think this is a question that would heavily involve mathematical physics although this question has been controversial on the physics stackexchange. Further, I think that the solution to this problem would be affected by mass scaling. By this I mean that the optimal number of legs would probably vary with the range of masses available to the polyped.

Finally, I think that this question is highly relevant to roboticists who build legged robots as a thorough investigation of this question would probably lead to better models for polypedal locomotion.

That’s all I can reasonably say for now.

Note: I thought I’d add a touch of melodrama to this blog post as a friend of mine told me that my blog posts can be a bit dry.

# The Poisson Ratio and conservation of volume

This summer, I had the chance to work on modelling robot skin at Hanson Robotics and I’ve learned quite a bit of continuum mechanics in the process. But, one of the most important lessons I’ve learned so far turned out to be quite simple yet counter-intuitive.

Prior to reading any texts on elasticity, I would assume that volume was always conserved as I thought this would be useful for estimating elastic properties. In particular, I used this assumption when estimating the Poisson Ratio and Young’s Modulus of the rectangular sample I was modelling. However, this led to really bad estimates for the Poisson Ratio in particular which is defined to be the negative ratio of transverse to axial strain.

After reflecting on the source of my error, I wondered whether the the volume of an elastic material with a prismatic geometry wasn’t necessarily conserved when subject to small uni-axial strains.

I realised that this fact can be easily demonstrated for Poisson Ratios in the range $-1 \leq \nu < 0.5$:

1. We are given $\nu=-\frac{\Delta{W}/W_0}{\Delta{L}/L_0}=-\frac{\Delta{T}/T_0}{\Delta{L}/L_0}$ where $-1 \leq \nu < 0.5$

2. Let $\frac{\Delta{L}}{L_0}=\alpha > 0$ then $latex \begin{cases} \Delta{W}=-W_0*\nu*\alpha \\ \Delta{T}=-T_0*\nu*\alpha\\ \end{cases}$
3. $V'=L*T*W=(L_0+\alpha L_0)*T_0*(1-\nu \alpha)*W_0*(1-\nu \alpha)=L_0*T_0*W_0*(1+\alpha)*(1-\nu \alpha)^2=V*(1+\alpha)*(1-\nu \alpha)^2$

4. The equation in part 3 actually reduces to:

$\frac{\Delta{V}}{V}=(1+\alpha)*(1-\nu \alpha)^2-1$

Clearly, for $\nu \leq 0$, $\frac{V'}{V}>1$ so the volume has increased. And for $\nu >0$, the volume is constant only when $\nu = \frac{1}{\alpha}*(1-\sqrt{\frac{1}{1+\alpha}})$. Since $\nu$ is supposed to be constant and $\alpha$ is allowed to vary we must conclude that the volume is not constant in this case either.

If I should trust my arguments then for any given non-negative poisson ratio, there is a single non-trivial value for $\alpha=\Delta{L}/L_0$ such that $V'=V$. The graph of the function $f(\alpha)= \frac{1}{\alpha}*(1-\sqrt{\frac{1}{1+\alpha}})$ is given below:

Now, we must deal with the case $\nu = \frac{1}{2}$ separately:

1. Let’s suppose, without loss of generality, that our material has the geometry of a cylinder. Then $V=\pi r^2 L$.
2. If its volume is constant(i.e. it’s incompressible), then: $\quad dV= 2\pi r L dr +\pi r^2 dL = 0$$\nu \equiv \frac{dr/r}{dL/L} = \frac{1}{2}$

So the moral of the story is that for most materials that we interact with in everyday life, their volume isn’t conserved when stretched or compressed unless they happen to have a poisson ratio of $\nu = \frac{1}{2}$.

# Modeling soft actuators

Last summer I worked in the Insect Robotics Lab and got introduced to soft robotics via the maggot robot that is in development there. I worked on C. Elegans movement analysis which led me to the OpenWorm project but the idea of soft robots and soft actuators in particular stuck in my mind. The potential for developing better robot joints was very interesting and given that the field is still in its infancy, quite exciting!

So, just last week the idea occurred to me that it might be interesting to model soft robots. But, where to start? I contacted Adam Stokes, who once worked in the Whiteside lab at Harvard which originated the idea of soft robots, and he referred me to Dylan Ross of the Insect Robotics Lab. After brain-storming with Dylan today we came to the conclusion that it might not be a bad idea to start with modeling soft actuators.

A soft actuator(aka PneuNet) basically has three parts:

• a pressure-pump that injects
• fluid into a
• visco-elastic container(often made of silicon)

Now, this leads to interesting questions that engineers in this field haven’t looked into much detail yet such as:

i) How does the PneuNet behave with variable pressure and/or variable number of cycles?
ii) How does the PneuNet behave with variable material properties(elasticity, viscosity…) assuming fixed pressure?
iii) Assuming a well-defined goal for the PneuNet how can we optimize the energy use and material cost?

These are not easy questions to answer and in fact it turns out that I have a lot of reading to do. Assuming that I’m going to use fine mesh models to describe the theoretical dynamics of this system I would need to read a book on elasticity like Theory of Elasticity by Landau and Lifschitz. And, in order to model the fluid I’ll have to read a book on computational fluid dynamics. Then I’ll need to take into account the fact that the material properties will change over time with the growing number of cycles. This means that I’ll need a tight feedback loop.

The ideal setup would have a webcam which would analyze the variation in the geometry of the visco-elastic container when inflated in real-time, sensors to detect variation in temperature and pressure of the fluid, and of course a supercomputer to solve numerical differential equations. This might be the most challenging project that I have yet to take on.

My goal for next week is to 3D print the visco-elastic container to begin simple experiments and finish reading half of Theory of Elasticity which isn’t a very thick book to be honest.