Equations for Organic Motion

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/

Any motion can be modeled mathematically. There are probably infinite equations that provide "organic looking motion". I could make one that made really good looking motion but was really long and complex and you could have a parallel person asking "why is nature so messy and not giving us simple motion?". The insight here is to not attach any special significance just because "it looks organic". In reality what we are probably saying is that it "looks smooth". (In fact you could look at Bezier curves which are often used for animations to see non-trigonemtric examples, which while easy to define are relatively complex to actually run through http://en.wikipedia.org/wiki/Bézier_curve . All of iOS's built-in famous smooth animations use these.)

So let's break these down. All of these examples are periodic. So for starters we need a good periodic function. We could have started with y = |(x%2) - 1| ( http://tolmasky.com/letmeshowyou/notsmooth.png ). That would give us the back and forth motion we want, but would have not looked smooth, since we linearly increase, then linearly decrease, and in the middle abruptly change. Your eye would catch that abruptness and that's what would give it the "inorganic feel" (despite the fact that there are plenty of things in biology that are super abrupt).

So what we want instead is the same kind of back and forth, but rounding off the edges. You could imagine forming a function by taking a normal parabola (y=x^2) and an inverted one (y=-x^2), and putting them together to get the rounded bottom and top corners ( http://tolmasky.com/letmeshowyou/twoparabolas.png ).

Luckily for us, we've already discovered a class of functions that give us something similar in the form of sine and cosine. Now, the reason why these are smooth and periodic is simply because the values of these functions are derived from tracing along a circle ( http://www.galaxygoo.org/math/sineCurve.html ). Now that you know this, you have a "tool" for your mathematical bag of tricks: every time you want to do something smooth and periodic, you are probably going to reach for sine or cosine. Similarly, if you want to now tweak this and have things ramp up in speed or down, you could try using exponents, etc.

Hopefully this sheds a little light on what's going on here.

I'm not sure I would say that biological processes obey anything - merely that we can find it useful to describe real world behaviour in terms of these abstract mathematical functions. Of course, it's interesting to speculate if there is a link between the level of maths required to usefully model the real world and the level of maths we are happy working with - i.e. if the real world was much more complex would we also have developed a much more richer natural ability for mathematics.

If you look at your arm joint from the elbow up, keep your elbow in a fixed point you move your arm in a circular motion on a plane. The same applies for your hand on your wrist joint.

Extending that your arm moves in 3 dimensions so fixed point elbow means you can move your arm in a sphere.

But you cant reach every point on the sphere, because of the way the muscles work, which is why you need to limit it. It's really hard to get those limits right, which is why the artificial characters dont look natural .. if their hand moves to a point where your subconscious mind see's that position would be painful or impossible, then it looks unnatural.

My old calculus teacher used to always tell us to flip the paper around when you're drawing a curve, so the curve you're trying to draw moves naturally with the arc of your arm .. you get a much sexier curve that way.

The movement is almost linear, except for the rapid acceleration when you hand begins to move and the rapid deceleration as it slows to focus on a point.

[1]: Example, with onion skinning: http://gizma.com/easing/ [requires Flash] (Check x and y under the "Qnt" column on the bottom row, then click anywhere in the box to move the ball.

Movement time = Start/stop time + Speed * log (1 + Distance / Width of target)

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.

http://www.shadmehrlab.org/book/minimum_jerk/minimumjerk.htm