## Motivation:

Considering that the fastest observed land animals are quadrupeds that employ a rectilinear gallop, we are led to consider three natural questions. First, in the presence of important gravitational effects(i.e. greater than 1 Newton) are four limbs necessary and sufficient for the rectilinear gallop? Second, what mathematical argument could explain the relative lack of galloping hexapods? Finally, if we think of gait transitions what other gaits are improbable for hexapods if galloping gaits are excluded?

In order to clarify the difficulty of these questions, the first two sections involve back-of-the-envelope calculations which provide a partial understanding of the problem. While it’s acknowledged that these mathematical arguments are insufficent, the point of these calculations is to motivate an altogether different approach to these questions. In particular, the author conjectures that all three questions may be suitably addressed by combining developments in soft robotics and unsupervised reinforcement learning.

## Four limbs are necessary and sufficient for galloping:

Due to the absence of consistent axiomatic foundations for biomechanics, which precludes the possibility of a mathematical proof, what follows is a back-of-the-envelope calculation.

Given that physical space is approximately Euclidean(i.e. $\mathbb{R}^3$), rapid rectilinear locomotion is a matter of survival for any organism that must traverse the shortest distance between two points on approximately flat terrain. This is the case for many terrestrial mammals such as the rabbit rushing to its burrow and-by symmetry-its terrestrial predators. Mammalian quadrupeds such as the cheetah, greyhound and rabbit provide an existence proof that four legs are sufficient for the rectilinear gallop.

To address the issue of necessity, we shall define the gallop as an asymmetric gait on rough surfaces with three properties at terminal velocity:

1. It has two phases: one phase where its feet are on the ground and another phase where its feet are off the ground.
2. The forelimbs $\mathcal{T}$ are specialised for traction whereas the hind limbs $\mathcal{P}$ are specialised for propulsion.
3. In order to nullify the effect of moments around the organism’s center of mass, a rectilinear gallop must satisfy:

and from this it follows four legs are necessary for galloping gaits(i.e. rotary or transverse gallop).

## Are hexapods unsuitable for galloping?:

Before we continue it’s important to note that there is a serious problem with the previous argument. Due to the existence of galloping quadrupeds the argument of sufficiency can’t be contested but what about the argument of necessity? Have we actually disproven that an organism with three limbs can’t gallop? No, we haven’t.

Now, let’s consider the challenge of laying out an argument that hexapods aren’t suitable for galloping. This may be true but we would need to have a general model of any physically-realisable hexapod. Even if we had irrefutable axioms of biomechanics it’s not clear that such a conclusion may be deduced from pure analysis alone. For such a complex system we may not be able to gain more than partial insights and computer simulations of some kind would probably be necessary.

## Limitations of reductionist models:

The challenge for any mathematically convincing explanation is that it implicitly requires that we use dimensional analysis which depends on the identification of a finite set of parameters governing the dynamics of locomotion. As pointed out by the authors of [4], this is problematic for a complex system such as an animal as an infinite number of parameters may be necessary to describe the system. There are all kinds of biological constraints such as the maximal heart rate, VO2 max, the organisation of the animal’s nervous system, as well as biological or environmental constraints that we ignore.

In practice, a reduction to a small-parameter model is possible by specifying a particular set of dynamic observables to model(ex. animal speed, specific locomotory behaviours). However, it must be noted that the original organism has now been projected onto a potentially arbitrary set of dynamic observables that are particularly easy for humans to measure in a lab. Such a simplified model might provide us with useful insights but how do we arbitrate between a number of such models. In fact, given enough time we may discover an infinite number of mathematical laws associated with the quadrupedal gallop but none which explain why the quadrupedal gallop emerges in an ecologically realistic setting.

An even more serious problem in our case is that there are practically no galloping hexapods. In this case, how should a model of galloping hexapods be developed? Should we attempt to construct hexapods that gallop and analyse the determinants of their locomotory behaviour? How much time and money should we spend searching through the space of plausible hexapod morphologies? And what is a good method for conducting such a search?

## The method of the artificial:

To simplify my task, I should probably start by addressing the core assumption in this article. Is galloping somehow optimal for rectilinear motion in quadrupeds? This question requires the construction of a quadrupedal model embedded in an ecologically realistic setting(i.e. with quadrupedal predators/prey) which isn’t hardwired to gallop and from which galloping gaits spontaneously emerge. Clarifying what I mean by ecologically realistic will be an important challenge. In any case, it would probably require a departure from reductive approaches to addressing the nonexistence of galloping hexapods.

Two reasonable approaches to analysing complex systems are described by Pierre Oudeyer in [8]:

The first, used by mathematicians and some theoretical biologists, consists in abstracting from the phenomenon of language a certain number of variables along with the rules of their evolution in the form of mathematical equations. Most often this resembles systems of coupled differential equations, and benefits from the framework of dynamic systems theory. The second type, which allows for modelling of more complex phenomena than the first, is that used by researchers in artificial intelligence: it consists in the construction of artificial systems implemented in computers or in robots. These artificial systems are made of programs which most often take the form of artificial software or robotic agents, endowed with artificial brains and bodies. These are then put into interaction with an artificial environment (or a real environment in the case of robots), and their dynamics can be studied. This is what one calls the “method of the artificial” (Steels, 2001)

I think the method of the artificial would probably lead to more important insights into quadruped locomotion than exclusive use of reductive methods. That said, I am also not advocating the rejection of reductive methods. On the contrary, I find that they are a very useful starting point for analysing complex systems and generally provide economical insights into the behaviour of a complex system.

## Discussion:

This question of whether quadrupeds are optimal for galloping locomotion first occurred to me approximately four years ago. In fact, one of my most popular questions on the biology stackexchange queried the nonexistence of six-limbed mammals. Two years later in 2016, I wondered whether the nonexistence of galloping hexapods could be explained by Newtonian mechanics. I actually had a positive email exchange with Steve Heim, a roboticist that answered my question. We agreed that my mathematical arguments couldn’t settle the problem in a definitive manner and at the end of our exchange I considered that it might be impossible to resolve this problem in a definitive manner regardless of my mathematical ability.

Two years after, it’s 2018 and I think I now have a reasonable strategy for approaching this problem. I am no longer going to aim for a resolution of this problem in a definitive manner using reductive approaches and my earlier suppositions may be considered working hypotheses in a weak sense. Instead I think that I can gain insight into this problem using a constructive approach whereby the morphology of the hexapod is an experimental variable and locomotory behaviours are discovered by the organism using unsupervised reinforcement learning.

# References:

1. The co-ordination of insect movements. G.M. Hughes. 1951.
2. A new galloping gait in an insect. Smolka. 2013.
3. Froude and the contribution of naval architecture to our understanding of bipedal locomotion. Vaughan et al. 2005.
4. Gait and speed selection in slender inertial swimmers. Gazzola, Argentina & Mahadevan. 2015.
5. Criteria for dynamic similarity in bouncing gaits. Bullimore & J.M. Donelan. 2007.
6. Comparing Smooth Arm Movements with the Two-Thirds Power Law and the Related Segmented-Control Hypothesis. Richardson & Flash. 2002.
7. Why Change Gaits? Dynamics of the Walk-Run Transition. F. Diedrich and W. Warren. 1995.
8. Self-Organization: Complex Dynamical Systems in the Evolution of Speech. P. Oudeyer. 2011.