Biomechanical simulation of Horse gaits

The product design of new motorcycle of biomechanical horses.

Art work plan : Biomechanical simulation of natural animals
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Demo reel include all application packages.


About the formula of the voluntary movement of natural animal


A mathematics teacher of Chinon, Mrs Françoise Jacot did describe in her words the construction process for our formulas and made a demos with GeoGebra. These formulas becomes the reference of all voluntary movement.

She explains that this is similar with the Cardoid curve.

The theorem :

The divine formula :


About the next generation industrial revolution on a long-term philosophy

European Horology * Chinese Martial arts * American Dream
= The next generation of Renaissance

1. European Horology (left)

Breguet's theory about the animal force and the principle of voluntary movement
Breguet's tourbillon regulator and escapement

2. Chinese Martial arts (center)

Ng Mui 's Wing Chun (as well as I follow Tai Chi Chuan)
Yin and yang, Wu Xing, Bagua

3. American Dream (right)

William F. Goodwin 's mechanical horse toy
Goodwin 's invention applied to the steam locomotive

1. European Horology


Applying the theory of Breguet to modern time, the movement agent(expansive movement) that he was referring, that is Λ, the cosmological constant to Einstein's equations. It's dark energy that accelerates the expansion of the universe against attraction agent(concentrating movement).

(excerpt from Section IV. Accumulated movement is available.)

"Movement, thus considered, is presented in a different aspect of that physicists gave him until here. I think, in nature, there is the matter and agents, that is to say, bodies and agents, or powers: bodies are the various modifications of matter, called substances, minerals, plants, animals, stars, atmosphere, etc.. : Agents are active principles of the material, or the general laws of nature, namely, attraction and movement. Attraction and movement are perpetually fighting;

they are two opposition movements, which should be called concentrating movement and expansive movement;

the maximum of the latter is the minimum of the other, and the maximum attraction, or concentrating movement is the minimum of expansive movement. It seems that in nature, there is no phenomenon that has not its positive and its negative. There is a principle of movement related or attractive property to each molecule of material, for unite with each other, calling in all directions. ..."

The clock mechanism metaphor to aware of the progress of the universe. It is important to the innovation of humanity.

Our new vehicles and robotics development must be practical and rational.
We can find a natural philosophy for that in the mechanical clock escapements.

Speed of the galaxy that is rotating movement far away from the center of the galaxy cluster is "too large". This is why dark matter's attraction winding up the galaxy clusters.

Abraham-Louis Breguet focusing on the tremendous power of small organisms on the Earth, he said animal movement is brought from the rotational motion of the Earth. Its motion source is the dark energy that is filled in the universe.

I imagine it as a clock mechanism.

Galaxy is a mainspring that is a spiral torsion spring.
Dark matter is our hands winding up the mainspring.
Dark energy is stretched force of the mainspring.
Escape wheel is a metaphor of the earth's land.
Anchor is a metaphor of living matter.
Ankle kicking a escape wheel.
such as,
Animal legs kicking Earth's land.

In other words, These living matters and the earth always are interacting to function as a regulator. (Emergence by the nonlinear interaction.) That movement energy is an open system.

In the following two example escapements, the left side anchor mean the front legs of the animal. and the right side anchor mean the rear legs.

1. The anchor escapement was probably invented by British scientist Robert Hooke around 1657
(Boston dynamics's WildCat demo

2. The grasshopper escapement is an unusual, low-friction escapement for pendulum clocks invented by British clockmaker John Harrison around 1722.

In the theory of relativity, time dilation is a difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from a gravitational mass or masses. so from the local frame of reference (the trotting clock on a Horse), the relatively accelerated galloping clock on a Horse moves slower.

2. Chinese Martial arts

Our formulas construction is attributed to Chinese Internal arts in Tai Chi Chuan. that is Yin and yang, Wu Xing, Bagua.
Also those reference angle of articulated arm is attributed to Wing Chun. (135 degrees is Tan sao)

Basculer aberrant
相背的 突变,蜕变

We can understand well a natural logic from the movement of Tai Chi master.

Dynamic movement of Tai Chi master is ultimate power output of voluntary movement. please take a look at following Master Roberta Lazzeri's demo. After the completely slacked (放鬆), the master emit awesome abrupt power. :

Then, please attention to the formula of voluntary movement (ω1) on the video uploaded.

The Yin and Yang is a joint (γ1) which is coupled to a joint (β1) and a fulcrum (α2). having tilted peacefully to the left, the joint (γ1) is suddenly rocked aggressive to the right.

So when biased to extreme, always recoil occurs powerful.
It is a fundamental natural law of yin and yang. this is because, that the Taiji (Supreme Ultimate) which is a link element coupled together by a joint (β1) and a fulcrum (α1) keep it revolution constantly same speed. the joint (β1) is disposed so as to be capable of revolving in an orbit centered on the fulcrum (α1). the Wuji (Without Ultimate) is a fulcrum (α2).

3. American Dream

William Farr Goodwin of 19th-century mechanician had been tried to give a regularity to the structure of the mechanical element of horse machine legs. (that element form is like a character of "斤") 

Probably, he had expecting a presence of the voluntary movement theorem.

When we look back the history of American industry, we can find pioneers dreams and creative potential that remain even now. For example, William Farr Goodwin had conscious of equivalence between the biomechanical movement of horse legs, and a mechanical linkage for driving the wheels of the locomotive.

In 2005, at the Art and History Museum Neuchâtel, while watching the demonstration of three androids made by Jaquet Droz, I met a engineer of NASA Advanced Supercomputing Division. then I brought him to some craftsmen's ateliers of Switzerland.

He was hoping to my future, gave to me some criticisms about the automaton of past. "Jaquet Droz family works of the 18th century are masterpieces. but french automaton of the 19th century are just toys. there is no formidable idea, however that storyteller's demonstration is fun." etc...

Among them, he so surprised when he saw the idea concept of mechanical horse legs. "Wow, where this great idea come from?" François Junod also did not know well about the meaning of this idea. He said humbly that this is a maybe useless toy, or a representation of mechanical poesy.

I knew that the NAS engineer had been blessed with talent of the long-range outlook. he was a top athlete of rifle shooting that is ability to capture a far away target. from his profond reaction, I recognized that mechanical horse is a most important artifact.

The difference of two masters could be seen also from driving skill. François Junod was a very polite driver like a good teacher. NAS engineer was a perfect racer with a international competition license.

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Comment by Gordon on June 24, 2016 at 5:11pm

Isn't this diy DRONES. Sorry, not seeing much correlation.

Comment by Gary McCray on June 24, 2016 at 6:32pm

Hi Gordon,

Although we are DIY Drones, we are also into "Rovers" robots, boats and even ROVs.

This is perfectly germane to this group and much appreciated as it is basic science related to an interesting means of locomotion.

Even though we call our hardware a flight controller, it is also in constant use as and being actively developed and programmed for terrestrial and aquatic/marine vehicles as well which are even controlled by the same Mission Planner systems.

I think the above represented Goodwin toy mechanical horse would not have worked at all well because lateral discrepancies could not be compensated for so it might go a small way and fall over.

But with modern feedback mechanisms and some means of providing various compensations the basic idea would probably work fine.

In fact, the Boston Dynamics pack mule/dog certainly works well albeit it is considerably mechanically more complicated and has intense feedback systems and capabilities which permit it to remain actively upright while negotiating very difficult terrain.

It is quite possible that eventually various quadrupedal devices could prove very practical for a variety of tasks.

Probably one of the most important functions we serve here is to look at and evaluate emerging and new technologies and come up with interesting ways of implementing and using them and Yutaca's excellent analysis of animal locomotion provides very interesting and novel insights.

Hi Yutaca, Love your work on this and previous Dung beetle.

Best Regards,


Comment by Gordon on June 24, 2016 at 6:43pm

Hi Gary,

I am quite aware of all the various uses for our "flight controller". Just because I post very rarely, does not automatically make me an ignorant newb or someone with nothing worthwhile to contribute.

I wrote the 3 first instructional blogs on how program radios for the 6 "flight" modes and encouraged others to do the same. My blog for the ER9x is essentially now obsolete, because someone smarter came along with a better way to do it, which is fantastic.

I made a suggestion to incorporate dead reckoning to help maintain positional accuracy years ago when we moved from the oilpan to APM2. I knew it was possible, because the F/A-18A's ( yes, the vintage A model) I maintained did it on a daily basis. I couldn't provide the code because it is not my forte, so I was shot down and told it was too hard and beyond the capabilities of the hardware. Guess what, about a year later, the likes of Tridge, Leonard, Randy & co (people with brains the size of a planet) came along and made it happen, on the hardware that it couldn't be done on.

I have also known about the Boston Dynamics mules and am in awe of them, ever since their existence was first made public, which if i recall correctly was way back in popular mechanics or science, when they were only available in print form.

You hit the nail on the head when you mentioned "intense feedback".

It is simply my opinion that whatever form the vehicle takes, doe not become a "drone" until it has Electronic Control & Feedback.

Like many others, I am tired of seeing this site being perverted by people posting the following:

1 Pure click bait.

2 Non "drone" stuff

3 Non DIY

Rant over. Back to where I guess I must belong because I'm not in the fold.



Comment by Gary McCray on June 24, 2016 at 7:06pm

Hi Gordon, not trying to be rude, but basic stuff like Yutaca is presenting is often very insightful and capabnle of providing interesting possibilities.

the presence of "Drones" in our title is sometimes unfortunate, because to me at least it infers a limitation not actually present.

The studies of basic mechanical motion are very necessary for an understanding of ways to produce locomotion and manipulation and Yutaca is one of the most gifted of this group.

I actually linked your 6 way setup in the wiki and on my sites months ago.

You are definitely a bona fide member and in the fold, quite possibly more so than I am because I definitely tend to rub the wrong way periodically.

In any case, just saying there is plenty of room here for Yutaca and others with novel approaches and ideas.

Best regards,


Comment by Yutaca Sawai on June 24, 2016 at 7:50pm

Hi there,

I'm sorry for the confusion. and appreciated for the nice follow.

This is a presentation of the universal mechanical element that can represent any voluntary movement of natural animal, can replaces conventional propeller, wheel or screw. that is physical and metaphysical law of nature. This mechanical horse design is a 110th generation in the manual design. It can be optimized and efficiently by the artificial intelligence or genetic algorithm.

While combine to emerging technologies following
- Tesla motors's electric car technology
- nVidia and Google's self-driving technology with artificial intelligence
- Boston Dynamics's self-balancing robotics technology
It might become a machine animal like an element which make up the universe.

so true,
mechanical fins for underwater exploration might be more easy to realize.

Best regards,


Comment by Gary McCray on June 24, 2016 at 8:26pm

Hi Yutaca,

There are some interesting University underwater designs based on fish and fins where efficiencies actually exceed conventional prop by a considerable margin.

Usually patterned after high efficiency fish like Tuna, as I recall, - both flexible body and fins.

Rapidly these days I think artificial intelligence and genetic algorithm are becoming indistinguishable with massively parallel computing and deep learning.

Nvidia making serious strides in those fields.

Animals,and even bacteria evolved to become many times highly optimized in their environment, so your concentration on the mechahincal capabilities of these systems provides many possible insights.

I am in awe of your work.

Thank You,


Comment by Yutaca Sawai on June 24, 2016 at 9:36pm

Hi Gary,

Thanks for a good understanding, despite the lack of my description.

Would it be better to present the complete formula of the wing of bats and insects in our forum?

Due to the fact that voluntary movement can be calculated with a simple mathematical formula,

I think UAV machines will become like a natural animal.

Then what will we be in finally?

so I will deliver the new creating opportunity to you.

Thank You,


Comment by Mark Harrison on June 25, 2016 at 2:45pm

Hey Gordon, not officially speaking for all moderators, but the general rule of thumb is that postings should be "generally interesting to diy-drone types" and this qualifies (based on sampling myself - I thought it was interesting, and made me think of what it would take to control these kind of bio-mimicking mechanical systems).

BTW I remember your er9x posts... they were great, hope you can put some more like that!

Comment by Mark Harrison on June 25, 2016 at 2:47pm

Yutaca, this is great stuff.  I learned a lot, and the videos are beautiful.  

Hope you post more of you work here!

Comment by Yutaca Sawai on June 26, 2016 at 12:59am

I agreed.

Please don't worry. I learned that all conversation and reaction is worth can take advantage to the next improvements. and I'm like an actor which can change the personality flexibly. now I'm roleplaying as Mr. Spock.


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