Recursive Dynamics - Human Motion Simulation: Predictive Dynamics - page 78

8
<

@ C i 2 1

@ q k

2 @

B i 2 1
@ q k

@ T i

@q i _

q i ðk iÞ

T i 1 ,

@ C i

@ _

2 T i
@

q k 5

2B i 2 1 @

T i
@

2A i 2 1 @

(4.42)

:

q k _

q i ðk iÞ

q k 1

5

0 ðk iÞ

.

8
<
:

@ C i 2 1
@ €

T i ðk iÞ

,

q k

@ C i
@ €

A i 2 1 @

T i (4.43)

q k 5

ðk iÞ

5

@ q k

0 ðk . iÞ

The forward recursive kinematics sensitivity equations are implemented as
follows:
Input state variables q ; _ q ; € q and initial condition A ½ 0 5 I, B ½ 0 5 0, C ½ 0 5 0

Do i n

ð 1 Þ Obtain q i ; _

1 ;

5

q i
ð 2 Þ Calculate and store T i ðq i Þ;@

q i ; €

2 T i ðq i Þ=@

q i ;@

3 T i ðq i Þ=@

q i T i ðq i Þ=@ q i ;@

ð 3 Þ Calculate and store A i
ð 4 Þ Calculate and store B i
ð 5 Þ Calculate and store C i
ð a Þ Calculate and store

(4.44)

@ A i =@ q j

ð b Þ Calculate and store @ B i =@ q j ;@ B i =@ _ q j

ð c Þ Calculate and store @ C i =@ q j ;@ C i =@ _ q j ;@ C i =@ € q j

EndDo

4.5.5 Dynamics sensitivity analysis
Sensitivity of the joint torque with respect to the state variables involves D, E, F,
and G, which correspond to inertia and Coriolis, gravity, external force, and
external moment, respectively.

8
<

C i
@

I i @ T i 1 1 @

D i 1 1
@

ðk iÞ

q k 1

#

q k

@ T i 1 1

@

T i 1 1 @

D i 1 1
@

@ D i

@

D i 1 1 1 ðk i 1 Þ

5 1

(4.45)

q k 5

q k q k

:

T i 1 1 @

D i 1 1
@

ðk . i 1 1 Þ

q k