Why don't hexapods gallop?

Question

I haven't observed any terrestrial hexapods that gallop like agile tetrapods(cheetahs/horses). Instead, I've observed that all of them scuttle which constrains their maximum velocity. And after further reflection, it occurred to me that there may be a bound on maximum velocity that's strictly due to the hexapod's extra pair of legs. This leads me to the Quadrupedal conjecture:

Four legs are optimal for velocity maximization in a terrestrial robot.

In order to limit the scope of this analysis and make the problem more tractable I would focus on the case where a polyped is moving in a straight line(i.e. linear motion) on a relatively flat surface(ex. gravel road) at maximum velocity.

First, it might be a good idea to define what I mean by gallop:

gallop: I define a gallop to be a gait where the polyped(animal/robot) moves in two phases. One phase where its feet are on the ground and another phase where its feet are off the ground. And I add the following details:

• At maximum velocity the fraction of time spent in either phase is a constant.
• Assuming that the time spent in the phase where the feet are off the ground to be $C_{air}$(ex. 1 sec) and the phase where the feet are on the ground to be $C_{ground}$, then I place the additional restriction that $\frac{C_{ground}}{C_{air}} \ll 0.5$. This means that the polyped doesn't simply 'hop'.
• At high speeds, the kinetic and potential energies of the center of mass(COM) of the galloping polyped are in-phase as in a mass-spring system and in general we have:

$E_{COM}=E_p+E_k$ where $E_k$ and $E_p$ are the kinetic and potential energies of the center of mass respectively[Genin & Williams].

Here's a useful illustration for the case of the cheetah's locomotion:

Secondly, I'd like to establish several things which occurred to me or think are important:

1. A hexapod scuttles(i.e doesn't gallop) only if its acceleration and the angle of the trajectory of its center of mass with respect to the ground is given by: $a < \frac{g}{sin(\theta)}$
2. In the case of a quadruped the front pair of legs and the back pair of legs serve very different functions. The front pair serves the role of traction. And the hind legs are used for propulsion. In fact, if the propulsive force came from the front rather than the rear this would apply a net torque on the quadruped which would cause it to flip over. Moreover, it's interesting to note that even F1 cars are rear-wheel drive rather than front-wheel drive.
3. In order to allow a fair comparison between two polypeds their masses must be equal and the available energy for mechanical work must be constant and equal for both.
4. If we assume that accelerations are mainly due to the torque of the hind limb rather than limb extension, for a quadruped we have: $a_{x} \leq g\frac{L_{caud}}{L_{leg}}$

where $L_{caud}$ is the distance between hips and COM, $L_{leg}$ is the length of the hind limbs and $a_x$ is the horizontal acceleration [Williams and Usherwood].

5. In order for a hexapod to gallop the middle pair of legs must be in phase with the front pair or the back pair. And in order to aid in propulsion they must be in phase with the back pair.

Of all the information presented, I think a careful analysis of the last point may be sufficient to resolve this problem. But, I think there might also be other ways to approach this problem. In my attempt to answer this question below for example I use the legs $\approx$ wheels approximation.

Note: I think this question is important because if the Quadrupedal conjecture can be demonstrated then it would save future robotics experts and engineers the effort of looking for bad solutions to terrestrial locomotion.

References:

1. J.J Genin, P.A. Willems et al. Biomechanics of locomotion in Asian Elephants Journal of Experimental Biology 2010
2. Taylor, Heglund et al. Energetics and Mechanics of Terrestrial Locomotion Journal of Experimental Biology 1982
3. Sarah B. Williams, James Usherwood et al. Pitch then power: limitations to acceleration in quadrupeds 24 June 2009

Answer 1

I really like your points and conjecture, though I disagree! So before getting to the point of the original question, I'd like to offer some comments and thoughts that can stimulate further independent thought.

Definition

Yours is a very good definition, and it is almost the exact definition we (the community) use for differentiating running from walking: any gait which includes phases in which all feet are completely off the ground is classified as a running gait.
This is however a category for several different gaits, including galloping but also trotting (there exist both running and walking trots), bounding, etc.).

As for galloping (and gaits in general) there are several different interpretations using different metrics (from order of foot-strikes to ground-reaction-force profiles), the usefulness of which depends a lot on what interests you. However I think the most widely accepted are the gait diagrams of Hildebrand, well explained in the wiki article. In this definition, the order and length of stance-phases is defining, and thus we have 2 separate types of gallop: a rotary gallop (seen in cheetahs and greyhounds, this involves 2 distinct flight-phases per cycle and is generally faster) and the transverse gallop (with only 1 flight phase per cycle, this is slower but more sustainable). Note that while several animals are able to use both, horses for example use the transverse gallop exclusively.

Points

2. In the case of a quadruped the front pair of legs and the back pair of legs serve very different functions. [...]

This is true for many but not all morphologies (and for most we simply do not know). As an easy counter-example think of lizards and their sprawling postures (though elephants would also be a good example). Also, it depends not only on morphology but also gait.

In order to allow a fair comparison between two polypeds their masses must be equal and the available energy for mechanical work must be constant and equal for both.

This is a good point, but can (and should!) be circumvented through dimensionless analysis, see [1],[2] and [3] ([1] being imho the most readable). Essentially you need to compare dimensionless speed and time required for different things. Indeed, this sort of analysis is what imho would most strongly contradict your conjecture: you assume that hexapedal "scuttling" is slower than quadrupedal galloping, but I don't think it is! See [4] for a nice study on hexapedal running, and how this inspired robot design [5]. For the more casual reader, take a look at this TED Talk.

In order for a hexapod to gallop the middle pair of legs must be in phase with the front pair or the back pair. And in order to aid in propulsion they must be in phase with the back pair.

This statement seems quite heavily biased. A pseudo-quadrupedal morphology can also be achieved by a hexaped by simply lifting 2 legs! Also, considering the morphology as well as scale is extremely different, I would advise caution in applying assumption 4 (i.e. hind-limbs provide most power): in the quadrupeds we regularly think of (and of which the study you cite treats), mass and therefore gravity-effects are substantial. While hind-limbs are providing most propulsion, forelimbs are very important for providing vertical support! For most insects, gravity-effects can be largely ignored, so this division of roles may (and generally isn't) the same. Indeed think of lizards or other quadrupeds with a sprawling posture: they also do not exhibit the same type of division between hind and fore-limb.
You however point out a very insightful idea! By moving two (or more) legs in phase, you can approximate their effect as a single, virtual leg [6]! This has been a fundamental concept for a lot of early robotic work, as well as in quantifying and matching experimental data of animal locomotion to idealized models.

Why do different animals select different gaits?

This is the root of your original question and still very much an open question. The most substantial answer has been linked to energetics, by the Taylor & Hoyt study[7]: in this famous study, horses were trained to use specific gaits even when they preferred not to, and their oxygen consumption was the measured, in order to estimate the metabolic cost for each specific gait-speed combination. It was then found that horses would have a natural tendency to switch gait (if given the choice) at the speed where one gait became more efficient! You can test this out yourself on a treadmill. At certain speeds (usually around 8 km/h or so), you can still force yourself to walk (i.e. have no flight-phase) but it is actually more tiring and feels less natural than switching to a jog!
While this is quite convincing, there are other hypothesis. One which I myself subscribe to has a lot to do with the dynamic stability of a gait, i.e. the coupling between the mechanics and simple, low-level control. A nice result showcasing the viability of this explanation can be found here [8] (disclaimer, this is from my current lab).

If you'd like a bit more insight, or more math (this essentially starts going into non-linear dynamics from here on out, i.e. limit-cycles, basins of attraction etc.), or more references, please comment (I'm a bit hesitant, for fear of writing an unreadable wall of text...).

References

1: Vaughan, C. L., & O’Malley, M. J. (2005). Froude and the contribution of naval architecture to our understanding of bipedal locomotion. Gait & posture, 21(3), 350-362.
2: Alexander, R., & Jayes, A. S. (1983). A dynamic similarity hypothesis for the gaits of quadrupedal mammals. Journal of zoology, 201(1), 135-152.
3: Miller, B. D., & Clark, J. E. (2015, September). Dynamic similarity and scaling for the design of dynamical legged robots. In Intelligent Robots and Systems (IROS), 2015 IEEE/RSJ International Conference on (pp. 5719-5726). IEEE. Chicago
4: Full, R. J., & Tu, M. S. (1991). Mechanics of a rapid running insect: two-, four-and six-legged locomotion. Journal of Experimental Biology, 156(1), 215-231.
5: Clark, J. E., Cham, J. G., Bailey, S. A., Froehlich, E. M., Nahata, P. K., Full, R. J., & Cutkosky, M. R. (2001). Biomimetic design and fabrication of a hexapedal running robot. In Robotics and Automation, 2001. Proceedings 2001 ICRA. IEEE International Conference on (Vol. 4, pp. 3643-3649). IEEE.
6: Raibert, Marc, Michael Chepponis, and H. B. J. R. Brown. "Running on four legs as though they were one." IEEE Journal on Robotics and Automation 2.2 (1986): 70-82.
7: Hoyt, D. F., & Taylor, C. R. (1981). Gait and the energetics of locomotion in horses. Nature.
8: Owaki, D., Kano, T., Nagasawa, K., Tero, A., & Ishiguro, A. (2013). Simple robot suggests physical interlimb communication is essential for quadruped walking. Journal of The Royal Society Interface, 10(78), 20120669.
9: Smolka, J., Byrne, M. J., Scholtz, C. H., & Dacke, M. (2013). A new galloping gait in an insect. Current Biology, 23(20), R913-R915.

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original post:

I'm currently travelling so I will fill ou and complete my answer later. But as a teaser:

What is your definition for a gait and for galloping specifically? Generally we consider the order and duration (duty factor) each leg spends on the ground (i.e. in stance phase): these are depicted as gait diagrams, see Heglund et al or Alexander.

In this sense, a gallop is an asymmetric gait with flight phase (a phase when all feet are off the ground)... And who says hexapeds can't do it? ; ) indeed, though rare there have been observations of dung beetles galloping! They lift (iirc) their hindmost limbs off the ground completely and gallop with the rest. See [9].

As Orion says, a lot of changes in gait are due to low-level controls (i.e. Central pattern generators and reflexes) as well as mechanical properties od the morphology reacting to mechanical load, but this no reason primitive animals cannot have rich and adaptive gaits. As a counter-example, look up "decerebrate cat" on youtube: this famous experiment showed that indeed very basic low-level control can generate a rich variety of gaits in a cat, including the gallop!

Answer 2

Even tetrapods have many different gaits, each of them has its characteristics (horses are masters of switching between them). Gallop is special because some of the time, all the limbs are in the air. Not a lot of tetrapods gallop.

An important fact here: only mammals developed sufficiently sophisticated motor control over their bodies to be able to move like that. They have special circuits in the brain for coordination of movement and input from vestibular system. Lizards (picture a crocodile) and amphibians (salamander) use the same primitive system as fish: their whole body flexes in an undulating fashion, and the legs just follow that movement. Switching from water to land movement is mostly a response of the spinal chord to increased mechanical load and requires almost no additional circuitry. They made a robot that replicates that (analogous circuit, no digital processing needed). Of course, each limb can still be consciuosly addressed, but the automatic "gait" in these animals is really basic.

Hexapods have even more possible gaits, and the full space of possibile modes of movement is immense. There's little chance that trial and error of evolution would explore all of them. But there's more: hexapods are primitive animals. Insects don't even have proper brains, and more importantly, the segmented nature of their bodies is still very prominent. Think back to segmented worm structure: each segment has its own nerve ganglia, and they are weakly coupled only to the neighboring segments, and these segments all have basically the same circuit. So... no individual addressing of legs, and no out-of-pace movements. In a translationally symmetric control system which only controls locally (and only couples neighbors), the only modes of motion are longituidinal waves. So whatever this leg does, the next leg will do a few moments later. Imagine the longitudinal contract-expand waves in caterpillars and worms. Or imagine the way a shrimp or a centipede moves its many legs. All that happens is that waves of motion propagate front to tail or in reverse. Insects are a bit more complex, but it's still basically 3 pairs of similarly wired legs, and the main mode of motion is still scuttling. Of course, unlike centipedes, they already have high level control over individual limbs, which you can see when flies groom, or when bees do stuff with the pollen, or preying mantises (no comment here).

Concluding summary:

1. Primitive organisms scuttle because that's the only motion their nervous circuitry knows how to do
2. Hexapods gaits would be extremely complex, and probably not even found by evolutionary trial and error
3. The length scale of inects is totally different from larger beasts, and gallop wouldn't make much sense (gravity isn't that big of a deal for insects, having all legs in the air has no advantage, and a lot of them can fly anyway)

Honorable mention: marine cephalopods have extremely well developed control over individual limbs (each has an individual "brain" that's more complex than the main brain itself, especially in octopodes). Some of them do gallop across the sea floor.

Robotic hexapods thus have a huge unexplored space of possible motions - an exciting field of research!

EDIT: I've given a biology answer above. But in terms of robotics... there's no real reason they shouldn't gallop. A general software driver for hexapodes would of course include this as well. However, hexapods are being researched for different purposes. You usually want them for stability, maneuverability and invariance to direction of travel. Galloping is useful when you want to get somewhere fast and reduce the leg contact - but then you might as well get rid of 2 legs and just use 4 (or two and run like an ostrich).

More fundamentally, the answer is, that it's just more complicated. Mostly, robotic locomotion is done quasi-statically. You assume at each time you are stationary, and just correct for deviation from that state. For running and galloping, you need second order equation: you need to be fully aware of your velocity and work with accelerations (you use your present momentum for planning future trajectory). That's a whole different problem. As a physicist, I never considered this much harder than the static version, but in robotics, there is precious little research in this direction compared to the larger base of development for the static approach. So there are just a few groups that do this kind of thing (like the robotic cheetah and similar projects), and they just haven't gotten to hexapods yet.

Answer 3

An insect that gallops was reported in 2013. See http://www.cell.com/current-biology/abstract/S0960-9822(13)01178-0.