Recursive Dynamics - Human Motion Simulation: Predictive Dynamics - page 74

Therefore, the total potential energy P due to gravity is the sum of each link's
potential energy:

X

n

X

n
i 5 1 2

m i g T

0 T i i r i Þ

P P i 5 ð (4.19)

5

i 5 1

The kinetic energy Equation (4.15) and the potential energy Equation (4.19)
are substituted into the Lagrange's equation ( Equation 4.11 ) to derive the equa-
tions of motion, based on the procedure given in Fu et al. (1987) .
As a result, the equation of motion is obtained as follows:

X

J i T m i g M ð q Þ q ; q Þ 1 τ 5 1 V ð q (4.20)

i

where, V ð q

; q Þ is the Coriolis and Centrifugal torque vector

" # T !

X X X

2 0 T j ð q Þ
@

n n n 0 T j ð q Þ

@

Tr @ I j @

; _ V i ð q q Þ 5 q k _ q m ; _ i ; k ; m 1 ; 2 ; ...; n

5

q k @ q m q i

k 5 1 m 5 1

j 5 max ði;k;mÞ

(4.21)

and P i J i T m i g is the joint torque vector due to gravity force.
Again, from the principle of superposition, the general equation of motion
including several external loads can be obtained by adding the restoring torque
term ( Equation 4.20 ) and static torque term ( Equation 4.10 ) to the above equation
( Equation 4.20 ). As a result, the final version of general equation of motion with
external loads, in vector-matrix form is

X X

_ J i T m i g

| {z }

J k F k
| {z }
external 2 forces

q N τ 5 M ð q Þ

| {z }

q € 1 V ð q Þ

| {z }

; 1 1 1 K ð q Þ

| {z }

2 (4.22)

i k

massinertia matrix CoriolisCentrifugal muscle 2 elasticity

gravity forces

2

Equation 4.22 is the most common form of the equations of motion as each
terms is readily identifiable. Note that these equations govern the motion and are
highly nonlinear.

4.4 Formulation of regular Lagrangian equation
The regular form of the Lagrangian equation can be written in vector-matrix form
(Fu et al., 1987):

X

J i m i g J s f s M ð q Þ q ; q Þ 1

τ 5 1 V ð q 1 (4.23)

i