geometry 
    
movements 
  
* 3-D volume model 
* 15 single objects for limbs, torso, head 
forming figure with fixed topology 
* dynamic (kinetic control) 
based on Lagrange equations and 
biomechanical studies 
  
* 13joints with degrees of freedom from 1-3 
(resulting in a total of 22 d.o.f.) 
* uniformly described, flexibly generated partial 
objects with octagonal sectional area, adaptable 
shape and different size (morphology) 
procedural (kinematic description) 
with generator functions of sinusoidal or 
triangular shape 
  
* anthropometric data for 
suitable body properties 
* material definitions and light model 
for realistic rendering 
  
  
key-framing 
using cubic splines for separate inter- 
polation in space and time between chosen 
body poses derived from movement studies 
  
  
Fig. 5: Figure model for animation 
the swinging les as a double pendulum system for the 
swing phase of walking [Alexander 84, Mochon 80 
without external forces and friction, leads to 3 couple 
differential equations yielding typical gait patterns. In 
order to solve these equations, which contain in great 
number sine- and consine-terms of angle differences as 
coefficients, linearizations around the reference state 
(figure standing upright) is done, giving dependent eq. 
1a-c: 
K18—-c,0—-c,U-wo (1a) 
BROP+C1Y-C0=-mo (1b) 
K3 ® + C1, Ÿ — C2, À = —W3 V (1c) 
with ©, ®, W each representing the flexion angle of 
stance leg, thigh and shank of the swing leg relative to 
the vertical (see figure 6), and Ki, Ci, W; as mechanical 
constants and body parameters. 
Further approximations after comparing the sizes of 
remaining coefficients relative to each other, leads to a 
direct solution for three joint angles that still very well 
specify the leg motion during one swing cycle: 
®@ = A11 exp(T1 t) + A12 exp( - TA t) (2a) 
® = A721 cos(T2 f) + A2 sin(T2 f) (2b) 
V = A31 cos(T3 t) + A32 sin(T3 f) (2c) 
with three individual time constants (71, T2 and T3). 
Constants Aij, together with another parameter Ts 
(swing time) are to be calculated from seven initial and 
boundary (final) conditions that are imposed, i.e. 
geometric conditions (a-b), experimental results (c), 
suitable prescriptions (d-g) 
I cos 9g — 4 cos $9 — I2 cos Ug — d sina — 0 (3a) 
! sin ©o + /1 sin Do + 2 sin Uo — d cosa — sp, — d 
(3b) 
l sin D(Ts) — sin ©o = 0.9d (3c) 
W(Ts) = (Ts) (3d) 
D(Ts) = O(Ts) = —arcsin E (3e)