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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!
29