Recursive Dynamics - Human Motion Simulation: Predictive Dynamics - page 86

4.6.5 Closed-form equations of motion
In order to compare the Lagrangian formulation with the closed form, we will
derive below the closed-form equations. We shall impose gravity g and a force f
at the end-effector.
The closed-form Lagrangian equation of a two-link rigid arm is well studied
and can be written as follows:

θ 2 ÞÞθ 1 1 ðI 2 1 θ 2 Þθ 2 m 1 l 1 1 m 2 ðL 1 1 l 2 1 m 2 l 2 1 τ 1 5 ðI 1 1 I 2 1 2 L 1 l 2 cos m 2 L 1 l 2 cos

2 m 2 L 1 l 2 θ 1 θ 2 sin m 2 L 1 l 2 θ

2
2 sin

θ 2 2 θ 2 1 m 2 gl 2 cos ðθ 1 1 θ 2 Þ 1 m 1 gl 1 cos θ 1

2

m 2 gL 1 cos θ 1 1 fL 2 cos ðθ 1 1 θ 2 Þ 1 fL 1 cos θ 1

1

(4.84)

2
1 sin

m 2 l 2 Þθ 2 1 ðI 2 1 θ 2 Þθ 1 1 m 2 L 1 l 2 θ m 2 l 2 1

τ 2 5 ðI 2 1 m 2 L 1 l 2 cos θ 2

(4.85)

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

1

Explicit gradients of torque with respect to state variables are derived as

follows:
@τ 1
@θ 1 52

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

(4.86)

@τ 1
@θ 2 5 ð 2

θ 2 Þθ 1 1 ð 2 θ 2 Þθ 2 2 2 m 2 L 1 l 2 θ 1 θ 2 cos

2 m 2 L 1 l 2 sin m 2 L 1 l 2 sin θ 2

(4.87)

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 52

2 m 2 L 1 l 2 θ 2 sin

θ 2 (4.88)

@τ 1
@θ 2 52

2 m 2 L 1 l 2 θ 1 sin 2 m 2 L 1 l 2 θ 2 sin

θ 2 2 θ 2 (4.89)

@τ 1
@θ 1 5

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 (4.90)

@τ 1
@θ 2 5

m 2 l 2 1 I 2 1 m 2 L 1 l 2 cos θ 2 (4.91)

@τ 2
@θ 1 52

m 2 gl 2 sin ðθ 1 1 θ 2 Þ 2 fL 2 sin ðθ 1 1 θ 2 Þ (4.92)

@τ 2
@θ 2 5 ð 2

2
1 cos

θ 2 Þθ 1 1 m 2 L 1 l 2 θ

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

(4.93)