Carlos RENGIFO Franck PLESTAN Yannick AOUSTIN … · Carlos RENGIFO Franck PLESTAN Yannick AOUSTIN Institut de Recherche en Communications et Cyberntique de Nantes May 12, 2008 1.

Control of a neuromusculoskeletal system

Carlos RENGIFOFranck PLESTANYannick AOUSTIN

Institut de Recherche en Communications et Cyberntique de Nantes

May 12, 2008

1

Section I.

Introduction

2

Anthropomorphic arm

q2

m1

m3

m5

m2

m6q1

m4The system is composed by:

2 Joints.

4 Monoarticular muscles.

2 Biarticular muscles.

3

Robot controlled by virtual muscles

Robot

(2 DOF)

Control

System

Virtual

Muscles

Computer

UΓqr , qr

L

q, q

Virtual Robot(2 DOF)Muscles

Anthropomorphic arm

UΓ q, q

4

Section II.

Mathematical model of a muscle

5

Hill’s model of a musculotendon unit (Physicalrepresentation)

ft α

lt lm cos α

l = lt + lm cos α

fm

fa

ft : Tendon force.

fp: Passive muscular force.

fa: Active muscular force.

lt : Tendon fiber length.

lm: Muscle fiber length.

α: Pinnation angle.

6

Expressions for the forces

Tendon force:ft = ft (lt)

Passive muscular force:

fp = fp (lm)

Active muscular force:

fa = fa

(

lm, lm, a)

Total muscular force:

fm = fp + fa

7

Hill’s model of a musculotendon unit (Block diagram)

a

l

fm = ft

lm

lm

l

ddtlt

+

+ −

−

lt

f −1t (.)

f −1t (.)

fp (.) + fa (.)

This model could be rewriten also using a clasical representation:

lm := fc (lm, l , a) = f −1a ( lm, a, ft (l − lm) − fp (lm) )

8

Section III.

Anthropomorphic arm mathematical model

9

Anthropomorphic arm (subsystems interaction)

DynamicsActivation

Dynamics

SkeletalGeometry

SkeletalDynamics

Contraction Jacobian

AF Γ q, q

U

L

Muscular excitations

U =[

u1 . . . u6

]T

Muscular activations

A =[

a1 . . . a6

]T

Fiber length

L =[

l1 . . . l6]T

Muscular forces

F =[

f1 . . . f6]T

Joint couples

Γ =[

Γ1 Γ2

]T

Joint positions

q =[

q1 q2

]T

10

Activation Dynamics

JacobianDynamics

Contraction

SkeletalGeometry

SkeletalDynamics

ActivationDynamics

U AF Γ q, q

L

Mathematical model

dai

dt= (ui − ai )

[

ui

τai

+1 − ui

τdi

]

i = 1, . . . 6

11

Contraction Dynamics

JacobianDynamicsActivation

SkeletalGeometry

SkeletalDynamics

ContractionDynamics

U AF Γ q, q

L

Mathematical model

lmi= fci

(lmi, li , ai )

fi = fti (li − lmi)

, i = 1, . . . 6

12

Jacobian

DynamicsActivation

DynamicsContraction

SkeletalGeometry

DynamicsSkeletalJacobian

U AF Γ q, q

L

Torque-force relationship

Γ = R(q)F

13

Skeletal Dynamics

JacobianDynamicsActivation

DynamicsContraction

SkeletalGeometry

SkeletalDynamics

U AF Γ q, q

L

Mathematical model

D (q) q + C (q, q) q + G (q) = Γ

14

Skeletal Geometry

JacobianDynamicsActivation

DynamicsContraction Skeletal

Dynamics

SkeletalGeometry

U AF Γ q, q

L

Mathematical model

L = Lrest − RT (q) (q − Qrest)

15

The whole model

State space representation

X = F (X ) + G (X )U,

With:

X =

q

q

A

Lm

16×1

U =

u1

...u6

6×1

Yc =

[

q

q

]

4×1

Ym =

q

q

L

10×1

The system is overactuated!

16

Section IV.

Proposed control strategy

System

Anthropomorphic

Arm

Control

Ym

Ycqr , qr U

17

The strategy

DynamicsJacobianContraction Skeletal

Dynamics

SkeletalGeometry

ActivationDynamics

DesiredDesiredTorques Activations

Desired ExcitationsForces

Control system

Anthropomorphic arm

ΓFA

L

U

Γr Fr Ar

L

q, q

qr , qr

q, q

18

Desired torques

DesiredActivations

DesiredForces

ExcitationsDesiredTorques

q, q

FrΓrqr Ar U

L

Problem statement

minη

∫ tf

0

[

(q − qr )T (q − qr ) et + ΓT

r QΓr

]

dt

Subject to:

Γr = D (q) η + C (q, q) q + G (q)

η := fc (qr , qr , q, q)

19

Desired forces

DesiredActivationsTorques

Desired ExcitationsDesiredForces

q, q

FrΓrqr Ar U

L

Problem statement

minfri >0

6∑

i=1

(

frifmaxi

)2

Subject to:Γr = R(q)Fr .

20

Desired activations

DesiredForcesTorques

Desired ExcitationsDesiredActivations

q, q

FrΓrqr Ar U

L

Problem statement

lmi= solve

lm[fti (li − lm) = fri ]

ari = solvea

[

fci

(

lmi, li , a

)

= 0]

21

Excitations

DesiredActivations

DesiredForcesTorques

Desired Excitations

q, q

FrΓrqr Ar U

L

Excitations

ui =

{

u |dai

dt= 0

}

=

{

u | (u − ari )

[

u

τai

+1 − u

τdi

]

= 0

}

= ari

22

Section V.

Simulation results

23

Joint positions

0 5 10 15−90

−80

−70

−60

−50

−40

−30

−20

−10

0

10

Time [sec]

deg

rees

Positions - Shoulder

0 5 10 150

10

20

30

40

50

60

70

80

90

100

Time [sec]

Positions - Elbow

24

Torques

0 5 10 15−2

0

2

4

6

8

10

12

14Torque - Shoulder

Time [sec]

N.m

0 5 10 150

1

2

3

4

5

6Torque - Elbow

Time [sec]

N.m

25

Muscular excitations

0 5 10 150

0.2

0.4

u1(t

)

Time [sec]0 5 10 15

0

0.1

0.2

u2(t

)

Time [sec]

0 5 10 150

0.1

0.2

u3(t

)

Time [sec]0 5 10 15

0

0.01

0.02

u4(t

)

Time [sec]

0 5 10 150

0.1

0.2

u5(t

)

Time [sec]0 5 10 15

0

0.02

0.04

u6(t

)

Time [sec]

26

Conclusion.

We have proposed a technique to compute robot joint torquesusing a virtual muscles system.

27

Perspective

Control

System

Virtual

Muscles

Computer

Robot

Humanoide

Muscular parametersPerturbation of

PathologicalGait

UΓqr , qr

L

q, q

28

Bye.

Thank you very much!

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