Sunday, 27 June 2010

Odd walkers II

Last year I discussed possible gait patterns for animals that walk on an odd number of legs, including various tripod walkers. Around the same time I was contacted by a reader who asked me to help him visualise a gait design. I will not go into details, but the idea was to have an animal, the tiborou, that was to have two paired legs in front and a third unpaired leg in the midline towards the back of the animal.

At that point I thought that this would be similar to the tripod walks discussed in my post, and that all their gaits could be summed up by defining the phase differences between legs. The legs of a walking human have a phase difference of half a cycle or 180 degrees. The hind legs of a hopping kangaroo move together, so the difference is 0 degrees. Back to a tripod walk: a straightforward pattern would be to have legs at 120 degree intervals (there is an animation of that on my old post). Other 'logical' patterns would be to have two legs move together (0 degrees) while the remaining one has an 180 degree difference.

But that was not at all what the reader, Metalraptor, had in mind. He asked for something I at first found very hard to visualise. All the gaits just mentioned have one thing in common, and that is that the walking cycle lasts equally long for all legs. When you think about it, that is how all Earth animals walk, regardless of how many legs they have. The gait of the tiborou had to depart radically from this idea, by having the hind leg go through two cycles in the time it took a front leg to move through one. Now just try to see visualise that...

In the end I thought I should write a small computer programme to illustrate the gait. The following reasoning helped to define its characteristics. Instead of imagining an animal moving over the ground, imagine it staying in place and having its feet slide beneath it (as if it is walking on a treadmill). Now make a film and look at individual frames: when a leg is on the ground its foot will move a certain distance between successive frames. If two legs are on the ground at the same time, that distance between frames must be the same for all legs. If not, one foot will be slipping over the surface. This must be true for any walking gait, and so holds for the tiborou as well. So: the hind leg moves the same distance over the ground between frames as a front leg, and yet moves through two cycles while the front legs move through one. There is only one way out: short steps.

Unfortunately I did not have a handy routine for a lateral view of a leg walking cycle, so I more or less let the matter lie. But later I found out that gaits with different cycle lengths actually occur on Earth! Well, to be precise, I know of just one. I discussed it in my post on brachiation. which showed monkeys that use their tail to grip branches at twice the frequency of the arms.

That knowledge put me back on track and I wrote a rough program to show what such a gait could look like. It is shown above. Please note that the animation makes no attempt at showing feet and it is otherwise also quite unrefined. The white dots are the points where the end of legs end up during a walking cycle.

Here is the animal once more; this time it moves across the surface. That makes it easier to see that the feet of the front and hind limbs do not move relative to one another when on the ground. You can also see that the steps of the hind leg are short, both in time and distance.

Seeing that gaits in which different legs have different cycle frequencies are not just a fictional invention but also something that occurs on Earth, they probably deserves a name. Perhaps it is time for another Furahan neologism, following centaurism and cernuation: how about a 'harmonic gait'? 'Harmonic', when dealing with frequencies, is often used to describe frequencies that differ from one another by multiplying or dividing them by a whole number. That is the case here, as the hind legs moves at twice the frequency of the other ones.

Oh dear... That thought immediately suggests that there could be 'inharmonic' gaits in which different legs have different cycle frequencies, but not related through whole numbers. They probably make little biological sense, but having thought of it I cannot get rid of the concept. I am trying to visualise an animal walking with such a gait right now, and it feels like the insulation of my brain is overheating...


j. w. bjerk said...

If I recall correctly sometimes odd-legged creatures in spore would use a harmonic gait.

My gut reaction is that it looks and feels like a pretty plausible way to make tripods walk.

If you use it for running it may start to look strange, with the odd leg wasting momentum by switching directions too rapidly.

When a creature like this runs, (assuming it can run) it might tilt forward, and fold the 3rd leg away. Or it might use the 3rd leg as an intermittent pusher, touching the ground less frequently than the front legs.

Evan Black said...

Fascinating stuff, though I'm having a hard time understanding why something like this would come about. Mathematically, harmonic gait makes sense, but what practical benefits does it have?

Anonymous said...

a thought - I've seen, among the larger kangaroos, the hind legs move forwards, and then the forelimbs move either once to catch up with the position of hte hind legs - but sometimes the forelimbs move twice to catch up.

does that help any?

Sigmund Nastrazzurro said...

I had not noticed that in Spore (I did not play it much). I checked on it now, and found that harmonic gaits indeed occur on Spore. I found that it depends on step length. I made a creature with short columnar hind legs, that are stretched when it is standing still. As a result, they cannot step forwards or backwards over any appreciable distance. I gave it very long hind legs, attached higher, with considerable bending, so the reach was very large. The result was hind legs moving at twice the frequency of the front ones. Whether there is one hind leg or two does not seem to matter.
As for running, the inventor of the tiborou had similar thoughts, but I will leave their discussion to him.

Suppose you have an animal with paired fins but with just one ventral hind fin. In water this is no problem, but when it moves onto land, this might be the result.
I do not think harmonic gaits make a lot of sense, actually, unless the body shape is for some reason as I described above for the spore creature. If there is a massive difference in reach of limbs, an harmonic gait may be necessary, but perhaps limb legs would then evolve quickly obviating the need for a harmonic gait (the inbetween stage would be intersting).

That would be very interesting, as sofar I only know of brachiating monkeys using a harmonic gait. Kangaroos would be an example of a harmonic gait in a terrestrial animal. Do you know of any videos showing kangaroos moving in this way?

Sigmund Nastrazzurro said...

The following is not from me (S.N.) but from Metalraptor, who came up with the tiborou and its gait:

Thanks for posting the article on the tiborou. Our discussions on these creatures and their strange method of locomotion actually gave me the impetus to further flesh out their biology and evolution. Seeing as you prefer the “show, don’t tell” method of speculative biology though, I won’t go into the details here. However, I would like to answer some of the questions posed in the comments of the article, for the password recognition section is not working when I visit the blog, and I cannot post comments.

J.W. Bjerk’s idea of how the tiborou run is basically spot on. When tiborou pick up speed (basically any gait faster than a trot), their third leg is usually held off the ground, shifting from a tripedal style of locomotion to a bipedal one. The third limb does assist in running, being used as an intermittent pushing leg and a balancing organ to help in maneuvering the running land shark. This is taken to extremes in one species of cheetah-like tiborou, where the hind limb is no longer touches the ground or is used for walking, and instead functions as a balancing organ. In addition, when the tiborou needs to accelerate quickly, the third limb is used to catapult the animal forward.

Of course, not all animals walk the same way, and the same is true for the tiborou. In some extremely large species (i.e., those larger than an elephant), where the limbs have become graviportal, the harmonic gait style of locomotion has been abandoned and a more “normal” style of walking, where each leg moves with a 120 degree difference. Other tiborou run with a bounding gait. Strangely enough, pronking is a common style of locomotion amongst the tiborou, because having all legs bound at the same time doesn’t really require having three or four legs.

I know the fact that the tiborou have three legs doesn’t really make a lot of sense. I first came up with the idea when I was a lot younger, and didn’t know much about speculative evolution and biomechanics at the time. While I’d much rather work with more tetrapod-esque land sharks, I decided to keep the tiborou’s strange design because it distinguished them more from “typical” tetrapods. My Hand Wave for why this occurred is that for tiborou, it was more important to retain the pelvic fins as claspers rather than limbs, leading to the anal fin being modified into the rear limb(s) instead of the pelvic ones. In some species, however, the pelvic fins are modified as well.

I can also confirm that harmonic gaits occur in Spore. Rather than slowing down the movement of the hind legs to match the front ones (for example, if you try to create a Spore version of a Stegosaurus), the computer will create a harmonic gait. In the case of the Spore Stegosaurus, I actually ended up with a gait where the front legs moved three or four times for every step of the hind limbs. As it turns out, rippling rusp-esque gaits can occur in Spore too. You just need lots of legs, and they all need to basically be the same kind of limb pattern (like forward-back-forward or something…is there a proper name for the way that limbs bend?)

Speaking of biomechanics and speculative biology, I had a question I wanted to ask you. In the book on Avatar and its creatures, it is stated that the Pandoran direhorse can run at speeds of 60 MPH (90 kilometers per hour). Now, I am assuming that this is hyperbole on the part of Cameron, who wants to establish Pandoran wildlife as “better” than that of Earth’s (For example, he claims a thanator, which is about 5.5 meters long, can successfully kill and eat a Tyrannosaurus, which is about 12 meters long. And don’t get me started on how the two would match up in both Pandora and Earth’s gravity). But that got me wondering, would a six-legged creature be able to run faster than a four-legged one of similar size and anatomy? Would lower gravity have an effect on a creature’s speed?

j. w. bjerk said...

But that got me wondering, would a six-legged creature be able to run faster than a four-legged one of similar size and anatomy?
It all depends on how you define: "similar". For a hexapod to make sense it would generally need quite a few changes. But to the general question: "Does more legs make a creature faster?" I'd say no.

* All the really high speed mammals have long legs.

* The ostrich is one of the fastest land animals with only two legs.

* some of the fastest insects on land, have one pair of over-developed legs, i.e. the grasshopper, or cockroach.

* Some of the faster centipedes (in my experience) have long back legs.

Conclusion: speed is more related to the length and strength of legs than the number of the legs. And since more legs tend to limit how far a leg can step (i.e. each leg has less room), more legs should tend to slow a creature down.

Would lower gravity have an effect on a creature’s speed?
Sure. Energy invested in keeping a creature to falling to the ground can instead be spent pushing it forward. I don't know how much a difference that might make.

Sigmund Nastrazzurro said...

I think J.W. is right as far as the number of legs are concerned: more legs do not necessarily make you faster. While I do agree that a millipede design probably is not the best for high speeds, I do not dare say at which number speed will be compromised: 2, 4, 6, 8 or more?. The threshold may depend on gravity.
The danger of legs clashing into one another can be dealt with in various ways. By coincidence a similar question came up on the message page of my site; I answered it there.

The role of gravity in walking speed is probably complex. The fastest gaits on Earth involve jumps where no leg touches the ground at all, and those would be helped by low gravity. In a very low gravity it might be difficult to move the leg in such a way that it provides a mostly horizontal force on the ground rather than a mostly vertical one, while still obtaining good traction.
This might be a good subject for a post at some point

Anonymous said...

That is generally true, in arms races between species where speed is the primary factor, gaits tend to develop where more and more of the leg is off the ground at any one point.

Having more legs could have the opposite effect though. Think of it this way, more legs mean more force applied to the ground over a period of time, and if the legs are arranged so the movement of one does not interfere with another it could actually help the limbs move faster.

Its kind of odd. The fastest bipeds we know of, the oviraptorosaurs, ornithomimosaurs, and ratite birds all top out in speed at about 45 miles per hour. Some quadrupeds (antelopes, pronghorn, cheetah) on the other hand can reach speeds of up to fifty miles an hour or more. More observation on this phenomenon must occur!

Fuzz Buzz said...

I don't really know anything about biomechanics, so Metalraptor's suggestion that when running the third leg would be used as a tail sounds odd to me. Why would the tiborou run like that instead of bounding, which would have been my first guess? Also, how does it jump?

j. w. bjerk said...

"Having more legs could have the opposite effect though. Think of it this way, more legs mean more force applied to the ground over a period of time, and if the legs are arranged so the movement of one does not interfere with another it could actually help the limbs move faster."

You are assuming that there is no "cost" to adding legs, and that there is unlimited energy available.

Look at it this way. Lets assume our creature will have a mass of 100 kg. With large runner's lungs and everything else that leaves 20kg for legs. The question really comes down to would the creature be faster if it "spent" that 20kg on two heavily muscled legs, or 4 (or more!) lighter legs.

I don't know for sure, but expect if we are merely concerned with producing forward motion, and the creature is of reasonable size so that bones can handle the weight, that 2 legs would be the most efficient. At the very least more legs don't have an obvious advantage. One might be even better if the balance problem could be solved or ignored.

Or to look at it another way, do you think an ostrich, could be made quicker by adding more legs? Besides the additional weight, i doubt it has any excess oxygen or metabolic energy to power an additional pair of legs without taking away from it's original pair.

Anonymous said...

Found your blog recently and have been addicted... very awesome stuff, as a spec bio fan and also writer and amateur artist.

Anyhow, I see someone already mentioned Spore, but yes... it's not exactly science, but it is interesting to play with leg patterns and see how they work.

I made made several tripods that move quite nicely. It may seem obvious to some, but it seems the most convenient arrangement is two limbs up front, and one in back, as you did in your example video.

One thing Spore doesn't seem to do is change the gaits much between walking or running. Your gait concept makes much more sense for a normal pace, but I think that at higher running speeds, it makes more sense to have the front legs sort of "pull", where they move in unison (except for turning and such), and the rear leg to be the stabilizer and thrusting force.

Interestingly, I also created a monopedal creature which moves absolutely wonderfully. In a way, it basically moves like a kangaroo with fused hind limbs. However, I realistically speaking, I doubt my creation would have any biological advantage, and certain movements would be quite awkward.

Anyhow, thanks for this blog... sometimes I just like seeing creature speculative biology, but other times I really like the science of it. You do a great job really analyzing these things.

Sigmund Nastrazzurro said...

Hello Anonymous,

Thank you. You make me curious about your monopodal creature; can we see it somewhere?

Andrew Broeker said...

It seems to me that the problem in animation which makes the gait look unnatural is that there is frequently only one foot touching the ground.

Imagine a large animal with two legs capable of hauling it around and a third more akin to a ski pole, where the third leg simply assists balance for the forward leg that's making ground contact. Only one foot would be off the ground at a time, but the third leg would still reposition between primary steps to assist balance.

David Adderman said...

Searched up "odd walkers" on this blog and found this post, and really got me thinking if this harmonic gait would be feasible for a tribbethere. Here's a picture of said tribbethere to see how it might compare to the "tiborou" in the picture:

The tribbethere, as described by creator Sheather888 in his "Serina" project, is actually a mammal-like fish! Descended from a mudskipper-like ancestor, it eventually became terrestrial and mammalian, with its pectoral fins and tail developing into three walking limbs.

The question is, would the hind leg realistically move twice as often as either front leg in a loose approximate of a quadruped "pacing"? Or would the tribbethere be restricted to a bound as its only gait, akin to a rabbit?

Also, it's probably worth pondering if the tribbethere's limb design is feasible at all. Would the single rear limb need to be significantly thicker than either forelimb? Would its gait have an awkward "hop" with each step of the hind leg, given that the single back limb carries the entire weight of its torso?

Sigmund Nastrazzurro said...

David Adderman: it;s a good thing I get an email every time someone writes a comment anywhere, or else I would never have noticed your comment to a 2010 post.
That aside, your question is interesting. But when I reviewed other's people's work, I always tried contacting them beforehand. Would it be right to ask Sheather888 what he thinks of the matter? Are you in contact with him/her?

UnassumingEchidna said...

This kind of brings to mind Tumblr user Alphynix's concept on land dolphins...

Their idea was that dolphins would beach themselves to hunt prey ashore, sort of like orcas hunting seals. Eventually, some of the dolphins with stronger, more flexible tails began using their tail as a sort of pseudo-leg, curling it under their body and bounding along with a motion akin to a sea lion.

Over time, the dolphins's flippers become shorter and stronger, developing hoof-like keratinous pads on each flipper and on the tail, using them as three walking limbs. Now fully terrestrial, the land dolphins start taking up niches of omnivores and carnivores, with their example species being a unicorn-like land predator resembling an entelodont.

It's an interesting premise and worth a look. It certainly challenges the idea that some animals are "too specialized to evolve any further".

Sigmund Nastrazzurro said...

UnassumingEchidna: I am not familiar with Alphynix' work, but it sounds similar to the Tiborou that Metalraptro came up with 10 years ago. Funny: at one time I was looking very hard for anything on speculative biology, and if I look at everything now I won't have the time to do anything myself...
As for being too specialised to evolve, I tend to agree that its is not a 'law of nature', even though we are talking about fiction.

tripodcat said...

I love the concept of the harmonic gait! It certainly reminds me of my family's amputee rescue cat, who was born with a severely deformed right hind leg that was amputated by the vet when she was about a couple months old. I've always noticed that her one remaining hind leg tends to have a "hopping" quality, and when she runs at a moderate-speed trotting pace her back leg visibly takes faster, shorter steps compared to both front legs. That said, she DOES have a tail, so I'm not sure how well she compares to the gait of a naturally-tripedal species that modified their tail into a third leg...

(Also: I've seen that most speculative tripods, like the tiborou or the aforementioned tribbetheres and land dolphins, tend to have two front legs and one hind one. Would it make any anatomical sense to have one front leg and two hind ones, and how would that evolve?)

tribbetherium said...

You mentioned that in my thorngrazer animation the creature has to "throw up its hind body" before taking a step, and suggested that the hind leg take twice as many half-steps.

So with that in mind, I made this animation of an Armox, a very large ceratopsian-like tribbethere:

Not sure if it looks more stable? It's mentioned to be a "slow runner" so I suppose it could maintain some moderate speed without becoming unbalanced?

Sigmund Nastrazzurro said...

tripodcat: I wonder which gaits 'tripod cats use. I just looked on YouTube, and there are quite a few videos of cats with various missing legs. There are even cats with just one and just one hind leg, and they go about their business...
As for one front leg and two hind ones; I can't come up with a moderately believable scheme for that either.

tribbetherium: I think the period in which the hind part of the torso ifs off the ground is shorter, and that must be an advantage in such animals. By the way, the cat videos I talked about above might help to provide inspiration for these very nice animations of yours.
Of course, what works well for small animals might not work for large ones, but I just found out that there was a three-legged male lion ('Clarence'):

You can see it walking in the video, but compared to the house cats, its walk is ungainly. It is either too heave to get away with such a gait, or hasn't adapted yet.

Chunderbird said...

I imagine the "disharmonic gait" could work for a tripod with very long frontlimbs and a single short hindlimb. Think something akin to a chalicothere but tripedal, with large powerful front limbs and a single hind leg that mostly just stabilizes the weight of the rear torso.

Sigmund Nastrazzurro said...

Chunderbird: it might work; I still find it difficult to visualise, though.

Rotary Pork said...

this may be the most bizarre example of a harmonic gait i've ever seen. or perhaps it indeed is an example of a disharmonic gait? it is difficult to tell:

Hoofed Mammal said...

Would it make sense for tripod centaurism to involve the unpaired limb? I had a concept of a clade of alien animals that had two front legs and one hind leg much like the tiborou example, but since one-legged hoppers always seemed iffy to me I had the idea that one vaguely terror bird-like predator would evolve to walk on its front limbs while using its single hind limb as a grasping claw for catching prey. My issue is, however, the fact that the weight of the belly and hip part would make it tip backwards if it walks on just its front legs, so how plausible would hand-standers be (I find the Nightstalker of After Man questionable for that same reason.)

Sigmund Nastrazzurro said...

Hoofed mammal: 'Would it make sense...' hard to say, but just reason it through. Suppose you now have a former tripod animal that has now become a biped, so the posterior limb is now a sort of tail. To prevent the mass of the tail to topple the animal, that mass has to be minimal or brought forward. If the tail isn't doing anything useful, shrinking it seems a good option.But perhaps, before it is gone, it evolves a new purpose, as a weapon. You can either use swing it over the back and head, like a scorpion's tale, or through the front legs. (I once suggested to Dougal that the Nightstalker might have looked better if the hind legs had swung between the front legs, because that seemed easier and would have given it much more freedom; he liked it, but it was too late in the Nightstalker's life history to change it.)

But does all this sound probable? not really... But who would have thought that running dinosaurs would give rise to flying animals that then became fast-swimming fish hunters? The trick is that every stage should work on its own.