
Equations for Organic Motion - trueduke
http://codepen.io/soulwire/full/kqHxB
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tlb
None of these look organic to me. I spent some time trying to get robot
movements to look organic, and motion-capturing human movement. I believe no
simple trig formula can generate plausible animal movements.

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dljsjr
Simple is the operative word. I work for a robotics lab specializing in
bipedal locomotion and we're developing a robot that exploits the periodic,
superpositioned waveform nature of passive dynamics to produce a stable gait
at varying speeds while only actuating at the hip. The mechanics behind the
project involve tuning the sinusoid that governs the hip joint pitch in
conjunction with the up/down motion at the knee. This tuning is done using a
fairly neat tendon network since we only introduce mechanical power at the hip
joint. The resultant behavior is clearly a non-linear controls problem, but at
it's heart it's still trig. Just not simple trig.

The robot is modeled after an ostrich. You can read more about it on our
organization's website if you want: <http://www.ihmc.us/groups/fastrunner/>

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lutusp
Yes, if periodic = organic. Otherwise these are generic trig equations with no
particular relation to "organic" motion.

Are the orbital paths of planets "organic" or just interesting physics?

To me, a truly "organic motion" would be a differential equation that
describes an organic process as a function of time, like the logistic
function:

<http://en.wikipedia.org/wiki/Logistic_function>

This has the advantage of possessing an actual relation to organic processes,
rather than being a member of a large class of functions that have no specific
connection to life.

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jessedhillon
Curious, this may be more of a philosophical question: what is the reason why
these organic-seeming movements can be modeled mathematically, specifically
with trigonometric functions? Why do biological processes seem to obey trig
like so?

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dietrichepp
Biological processes are composed of chemical processes, and chemical
processes are often modeled with differential equations. The trig functions
(sine and cosine) and the exponential function are all solutions to the
simplest second and first order linear differential equations, respectively.
You might equally ask why the planets follow the trig functions so closely
when they orbit the sun, or why the shape of the Earth is close to a sphere,
or why the first digit of the amount of money in your pocket has a roughly 30%
chance of being 1, but only an 18% chance of being 2.

~~~
rodh
Would second order differentials imply linear acceleration?

Would we then always get better approximations the higher the order?

EDIT: I ask this because linear acceleration doesn't seem like something which
might readily occur in nature.

~~~
dietrichepp
You can get _constant_ (not linear) acceleration with a second order
differential equation. Here is an example:

    
    
        d^2 x
        ----- = -g
        dt^2
    

Solving this gives us the parabola we are all familiar with, when an object
falls under constant acceleration:

    
    
        x = -1/2 g t^2 + v_0 t x + x_0
    

This is a second order differential equation, but it is too trivial an
example. Here is a more interesting differential equation:

    
    
        d^2 x
        ----- = -x
        dt^2
    

All solutions to this equation take the form:

    
    
        x = a sin t + b cos t
    

There are a lot of physical materials in nature which produce this kind of
relationship between force and position. Basically, anything that is kind of
like a spring. Reach over and flick a glass on your desk with your fingernail.
Hear it ring? The sine wave that it's ringing with is predicted by the
relationship between force and position that we understand. Every tuned
musical instrument in the world, with one exception, generates notes that can
be modeled by some differential equation. Pianos, violins, and horns are
relatively easy; drums are much harder but still tractable.

The chemical and physical systems in living tissue are much more complicated.
Our best models for them often incorporate nonlinear elements (the sine wave
example is linear) or a large number of variables. But periodic behavior is
still quite common, and the sine wave is in some senses the simplest function
that is smooth and periodic.

Note: The one musical instrument for which differential equations are not the
best model is the digital synthesizer.

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scotty79
I'm curious what equation would approximate best the human limb movement such
as when I'm moving my hand in front of me from left to right.

I'm not sure why but for me even motioncaptured movements seem unnatural and
give no illusion of life.

~~~
delinka
"...even motioncaptured movements seem unnatural and give no illusion of
life."

I often agree with this. When digging deeper into the details of why this is
the case, it's often because the motion captured data is played back on a
model of different proportions than the actor. I think I first noticed this
when a magazine article did a writeup on The Polar Express movie. There's a
photo of Tom Hanks in a mocap suit beside a rendering of the train conductor
in the same pose. The conductor's proportions were completely different from
the real Mr. Hanks. I've also noticed this problem when ballet is mocapped and
applied to computer models.

My opinion, since I'm no CGI expert nor hobbyist, is that actual natural
movement has no place in a cartoon. You'll notice that each of Pixar's worlds
maintains its own naturalness - Mr. Incredible moves the right way with
respect to the 3D model in his own universe as do the other characters.
Although they modeled Syndrome's swagger on a Pixar employee (or at least a
real human if not a fellow Pixarite), that doesn't mean they applied an actual
mocap session's data to the character model. Every time I've seen anything
"behind the scenes" of a Pixar film, they're animating movement 'manually' and
maintaining a 'natural' feeling within the context if the current project.

~~~
scotty79
I agree about cartoons. The creation of movement is essential part of the
uniqueness of given art piece. But I'm talking about endeavors that aim to
create virtual version of reality, some games, movie fx etc.

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wtracy
I get a mostly-blank page on Firefox 17. What browsers does this support?

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nshepperd
Firefox 17 here. Mostly blank page, even after allowing whatever wanted to run
in Noscript. But the page started working (although, without colours for some
reason) when I disabled Noscript and restarted firefox.

~~~
wtracy
Huh. I'm a NoScript user, so that might be it. Thanks.

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zyb09
nice, you can use these to get some really nice polishing in iOS apps. Thanks
for sharing.

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herdrick
Those aren't right. There should be an abs() around each trigonometric
function.

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huhtenberg
This is very hypnotoad-ish.

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twiceaday
Wouldn't it be both more robust and more efficient to use B-splines?

~~~
caster_cp
It would be certainly more efficient to actually _simplify_ many of the
obviously overly complicated "formulas" there. Just for the sake of argument,
you achieve the same "effect" of sin(t) _cos(t) with a simple sin(2_ t)
(ignoring the 0.5 multiplying the sin that made the amplitude of the movement
unnecessarily small).

