Recursive Dynamics - Human Motion Simulation: Predictive Dynamics - page 85

0 1

@τ 1
@θ 2 5

tr @ A 1
@ q 1 @ D 1

g T @ A 1
@ q 1 @ E 1

f 2 @ A 1
@ q 1 @ F 1

@ A 2

q 2 2

@ q 2 @ @ q 2

(4.73)

θ 2 Þθ 1 1 ð 2 θ 2 Þθ 2 2

2 m 2 L 1 l 2 θ 1 θ 2 cos

5 ð 2 2 m 2 L 1 l 2 sin

m 2 L 1 l 2 sin θ 2

2
2 cos

m 2 L 1 l 2 θ

θ 2 2 m 2 gl 2 sin ðθ 1 1 θ 2 Þ 2 fL 2 sin ðθ 1 1 θ 2 Þ

2

@τ 1
@θ 1 5

tr @

A 1
@ q 1 @

D 1
@θ 1

2 m 2 L 1 l 2 θ 2 sin θ 2 (4.74)

52

@τ 1
@θ 2 5

tr @

A 1
@ q 1 @

D 1
@θ 2

2 m 2 L 1 l 2 θ 1 sin 2 m 2 L 1 l 2 θ 2 sin 52 θ 2 2 θ 2 (4.75)

@τ 1
@θ 1 5

tr @

A 1
@ q 1 @

D 1
@θ 1

m 1 l 1 1 m 2 ðL 1 1 l 2 1

I 1 1 I 2 1 2 L 1 l 2 cos θ 2 Þ (4.76)

5

@τ 1
@θ 2 5

tr @

@q 1 @

A 1 D 1

@θ 2

m 2 l 2 1

I 2 1 m 2 L 1 l 2 cos θ 2

5 (4.77)

0
@

1
A 2

2 A 2 2 A 2 2 A 2

@τ 2
@θ 1 5

@

q 1 D 2 1 @

A 2
@ q 2 @

D 2
@

@

f 2 @ g T tr

q 1 E 2 2 q 1 F 2

(4.78)

@ q 2 @ q 1 @ q 2 @ @ q 2 @

m 2 gl 2 sin ðθ 1 1 θ 2 Þ 2 fL 2 sin ðθ 1 1 θ 2 Þ

52

0 1

2 A 2 2 A 2 2 A 2

@τ 2
@θ 2 5

@

q 2 D 2 1 @ A 2 q 2 @ D 2

@

f 2 @

@ A 2

g T

tr

q 2 E 2 2 q 2 F 2 @ q 2 @ @ @ q 2 @ q 2 @ @ q 2 @

θ 2 Þθ 1 1 m 2 L 1 l 2 θ

2
1 cos

5 ð 2

m 2 L 1 l 2 sin θ 2 2 m 2 gl 2 sin ðθ 1 1 θ 2 Þ 2 fL 2 sin ðθ 1 1 θ 2 Þ

(4.79)

@τ 2

@θ 1 5 tr @

q 2 @

A 2
@

D 2
@θ 1

2 m 2 L 1 l 2 θ 1 sin θ 2 (4.80)

5

@τ 2
@θ 2 5

tr @ A 2
@ q 2 @ D 2

0 (4.81)

5

@θ 2

@τ 2
@θ 1 5

tr @

A 2
@ q 2 @

D 2
@θ 1

m 2 l 2 1

I 2 1 m 2 L 1 l 2 cos θ 2 (4.82)

5

@τ 2
@θ 2 5

tr @ A 2
@ q 2 @ D 2

m 2 l 2

I 2 1 (4.83)

5

@θ 2

The foregoing recursive sensitivity equations can be readily verified with the
closed-form solution for this problem as detailed below.