Recursive Dynamics - Human Motion Simulation: Predictive Dynamics - page 72

4.3 Dynamic equations of motion
For the purpose of this discussion, consider a kinematic skeleton of a human
body represented as an articulated (jointed) set of rigid bodies as depicted in
Figure 4.1 . A global coordinate system is established at the base of the chain, and
a local coordinate system at each link as mandated by the DH method.
The general form of dynamic equation of motion is derived from
Lagrange's equation. Assuming that only the gravity forces and the driving
joint torques are applied to the human links, the Lagrange's equation is written
as follows:

dt @

d

@ q i 2 @

L L

q i 5 τ i ;

i 5 1 ; ...; n (4.11)

@

where L V is called the Lagrangian. T is the total kinetic energy, V is the

total potential energy, q i is the generalized coordinate of joint i ,

T

5 2

τ i is the general-
ized torque of joint i , n is the number of the total DOF, and t is the time. Here, τ i

is the driving torque actuated by human muscles.
The velocity of the endpoint can be derived as follows:

! i r i

X

0 T j ð q Þ
@

i

@

d
dt ð

d
dt ð

0 r i Þ 5 0 T i i r i Þ 5

v i 5 q j (4.12)

q j

j 5 1

Global coordinate
system

Local coordinate
system

q 1 q 1

x x x x ( q ) ( q ) ( q ) ( q )

Fy, My

Target point

... ...

... ... q 2

q n q n

Fx, Mx
End-effector

Fz,Mz

FIGURE 4.1
Joint-link system with external loads showing the generalized variables—the vector x(q)
and end-effector—with respect to the global coordinate system.