Animating Human Motion

"Animating human motion" is the title of a piece behind the cover of
Scientific American (March 1998).

The article explores some computer-based simulations developed by Jessica
Hodgins, an associate professor in the College of Computing at the Georgia
Institute of Technology.

I'm no control theorist or engineer, but my reading of the article suggests
the approach being taken is that of "computing" motions that simulate body
movements; in other words, calculate then execute a program.

Despite all the references to "control systems," there is no mention of
perceptions, reference conditions, error signals, or any other aspect of
closed-loop control.

The article does lead me to ask if PCT could be deployed in making animation
more realistic. If so, the movie studios and the game companies would be a
likely source of funding for research.

Just a thought...

Regards,

Fred Nickols
The Distance Consulting Company
nickols@worldnet.att.net
http://home.att.net/~nickols/distance.htm

[From Richard Kennaway (980225.2045 JST)]

Fred Nickols (980225.1120 GMT):

The article does lead me to ask if PCT could be deployed in making animation
more realistic.

You might be interested in the Build-a-Bug project that a few of us have been
working on. The goal is to simulate a 6-legged insect using a PCT control
hierachy to control the legs. I have a Java applet at
http://www.sys.uea.ac.uk/~jrk/distribution/JRKBug0.7/BugApplet.html which has
a two-level control hierarchy, the bottom-level controlling joint angles and
the top level controlling the position and orientation of the bug's body.

It runs *very slowly* except on very fast machines.

NB. My email address above is temporary for the next 6 weeks, my usual email
address below still works and is preferable.

-- Richard Kennaway, jrk@sys.uea.ac.uk

[From Bill Powers (980225.0806 MST)]

Richard Kennaway (980225.2045 JST)--

Ah, you're safely in Japan. I know your thoughts are going to be mainly on
other subjects for the next six weeks, but perhaps when you need a break
you can think a little about PCT.

I put together a simulation of a pendulum, to serve as a test-bed for an
upside-down pendulum balancing control system (seems that everybody has to
try this once). This brought back an approach to simulation that I tried
with integer arithmetic some years ago, but now can try in floating point
because of a faster computer. I'd like to know what you think of it, and
whether you think it can be generalized to simulating mechanical systems of
other kinds (like legs and arms and bodies).

The pendulum is mounted on a cart at the upper end, and hangs below it on a
rod with a bob on the end. The rod is unbendable but slightly elastic along
its length. The cart has a mass of 0.1 kg, and the bob weighs 1.0 kg; the
rod is massless for now.

It's the elasticity of the rod that makes this different from other
simulations I've seen (you might like to think about how you'd do this, to
compare with the simulation below). The position of the bob is the position
of the mounting point on the cart plus the position of the bob relative to
the mounting point, in x and y (this is two-dimensional for now). The force
acting on the bob is aligned with the direction of the rod, and is equal to

(L - Lo)*ke newtons,

where Lo is the resting length, L is the actual length, and ke is the
constant of elasticity in newtons per meter.

This force is resolved into an x and a y component, and applied to the bob.
Only similar triangles are used -- no trigonometry at all.

The x and y forces produce x and y accelerations of the bob, and the
opposite acceleration of the cart in x (the cart is held on rails in y).
Gravity acts downward on the bob, too. Integrating these forces twice gives
the velocity and position of the bob, and the x position of the cart.

As you will see if you can compile and run the program below, the result
looks very realistic; it might even be physically correct. The actual
dynamical simulation is in the procedure "dynamics."

Note that the rod is very stiff: its coefficient of elasticity is a million
newtons per meter of stretch, or a thousand newtons per millimeter, or one
newton per micron. This approximates the proverbial "stiff rod," but allows
calculating force as a function of rod elongation. It is more physically
correct than the idealized rod, and seems to allow a nice shortcut in the
simulation. Right now I can't think of how I'd do this without the elastic
rod -- I suppose I've have to use centrifugal forces and so on. That's why
I'd like to know how you would do it on the basis of what's in those
classical mechanics books. I think this way is probably a lot simpler.

When the program comes up, the bob is held deflected to one side. Pressing
a key releases the bob and starts the run. Pressing a key again exits the
program. The red trace shows the cart position as it is dragged back and
forth by the pendulum swinging from it. There is a coefficient of friction
that exerts a braking force on the cart proportional to its velocity, so
the oscillations gradually damp out.

There's a very interesting phenomenon that occurs. The bob swings back and
forth, and as the energy of the system is dissipated by the friction in the
cart, the oscillations of the bob die out until it's hanging straight down
from the cart. At the same time, the cart oscillates back and forth,
acquiring a mean velocity which also eventually dies out to zero. What's
interesting is that the cart returns exactly to its initial position before
the bob is released! My intuitive guess would have been that the final
position of the cart would be the mid-point of its oscillations, but it is
not. This doesn't depend on the relative masses of the cart or the bob, or
the coeffient of friction, or the amplitude of the initial displacement of
the bob.

When I get to the control system, it will push on the cart, also through a
stiff spring. I wish we could create force feedback to the mouse, because
then you'd feel exactly how this dynamic system would feel to a hand
pushing on it. But it should still be possible to balance the pendulum
upside down by operating the mouse, and then compare that performance
(after learning is complete) with a simulated control system doing the same
thing.

This is another answer to the skeptics who think that PCT applies only to
tracking and linear systems. Of course they won't remember it.

Best,

Bill P.

···

-------------------------------------------------------------------------

program pendcart;

uses crt,dos,graph,grutils;

var bob: record x, y, dx, dy, fx, fy, vx, vy, mass: real end;
     cart: record x, y, dx, dy, fx, fy, vx, vy, mass: real end;
     dt: real;
     l,l0,f,g,ke: real;
     iters: longint;
     ch: char;

procedure initpendulum;
begin
dt := 2e-4; { sec }
ke := 1e6; { newton/meter}
l0 := 1.0; { meter}
g := 9.8; { meter/sec^2}
iters := 0;

with bob do
begin
  vx := 0.0; {m/sec}
  vy := 0.0;
  x := -l0*sin(0.8); {l0 = L-nought, not 10.0}
  y := -l0*cos(0.8); {meter}
  mass := 1.0 {kg}
end;

with cart do
begin
  vx := 0.0;
  vy := 0.0;
  x := 0.0;
  y := 0.0;
  mass := 0.1;
end;
end;

procedure showpendulum;
const oldx: integer = 0;
      oldy: integer = 0;
      oldcx: integer = 0;
      oldcy: integer = 0;
begin
if iters > 0 then
begin
  setcolor(black);
  rectangle(hcenter - 30 + oldcx,
            vcenter,hcenter + 30 + oldcx,vcenter- 15);
  circle(hcenter + oldx, vcenter - oldy,10);
  circle(hcenter + oldcx - 20, vcenter - oldcy,7);
  circle(hcenter + oldcx + 20, vcenter - oldcy,7);
  line(hcenter + oldcx,vcenter - oldcy,
       hcenter + oldx,vcenter - oldy);
end;

oldx := round(200*bob.x);
oldy := round(200*bob.y);
oldcx := round(200*cart.x);
oldcy := round(200*cart.y);

setcolor(white);
rectangle(hcenter - 30 + oldcx,
            vcenter,hcenter + 30 + oldcx,vcenter- 15);
circle(hcenter + oldx, vcenter - oldy,10);
circle(hcenter + oldcx - 20, vcenter - oldcy,7);
circle(hcenter + oldcx + 20, vcenter - oldcy,7);
line(hcenter + oldcx - 20,vcenter + 7,hcenter + oldcx + 20,vcenter+ 7);
line(hcenter + oldcx,vcenter - oldcy,
      hcenter + oldx,vcenter - oldy);
line(0, vcenter + 7,hsize,vcenter + 7);

end;

procedure dynamics;
begin
  with bob do
  begin
   dx := bob.x - cart.x;
   dy := bob.y - cart.y;
   l := sqrt(sqr(dx) + sqr(dy));
   f := (l0 - l)*ke; {l0 = L-nought, not 10.0 -- sorry}
   fx := f*dx/l;
   fy := f*dy/l - mass*g;
   vx := vx + fx/mass* dt;
   x := x + (vx + 0.5*fx*dt/mass)*dt;
   vy := vy + fy* dt;
   y := y + (vy + 0.5*fy*dt/mass)*dt;
  end;
  with cart do
  begin
   fx := -bob.fx - 0.1*vx;
   vx := vx + fx/mass*dt;
   x := x + (vx + 0.5*fx/mass*dt)*dt;
  end;
end;

begin
initgraphics;
initpendulum;
l := l0; {l0 = L-nought, not 10.0 -- sorry}
outtextxy(0,0,'PRESS ANY KEY TO START OR QUIT');
setcolor(lightred);
outtextxy(0,50,'CART POSITION');
setcolor(white);

while not keypressed do
begin
  dynamics;
  if (iters mod 50) = 0 then
  begin
   showpendulum;
   putpixel(iters div 500, 70 - round(100*cart.x),lightred);
   putpixel(iters div 500, 70,white);
  end;
  if iters = 0 then
  begin
   line(hcenter,vcenter- 20,hcenter,vcenter - 35);
   ch := readkey;
  end;
  inc(iters);
end;
closegraph;
end.