Choice of the objective function in static optimisation
Transcript of Choice of the objective function in static optimisation
Choice of the objective function in
static optimisation
F. Moissenet1, L. Chèze2,3,4, R. Dumas2,3,4
1 CNRFR – Rehazenter, Laboratoire d’Analyse du Mouvement et de la
Posture, 1 rue André Vésale, L-2674 Luxembourg, Luxembourg
2 Université de Lyon, F-69622, Lyon, France
3 Université Claude Bernard Lyon 1, F-69622, Villeurbanne, France
4 IFSTTAR, UMR_T9406, LBMC Laboratoire de Biomécanique et de Mécanique des Chocs, F-69675, Bron, France
Second BoHNeS Colloquium, February 3-5 2016, LyonInteruniversity Centre of Bioengineering of the Human Neuromusculoskeletal system
Choice of the objective function
in static optimisation
2F. Moissenet et al.
[ Introduction / Context ]
The use of an optimisation framework is
motivated by the management of the
muscular redundancy [1]
Which muscles participated to the movement
around a degree of freedom and with which
level of contribution ?
[1] Prilutsky & Zatsiorsky, 2002
Choice of the objective function
in static optimisation
3F. Moissenet et al.
[ Introduction / Context ]
The goal of the optimisation will then be to
find the global minimum value of one or
multiple objective functions, if necessary
under constraints
min ( )
subject to:
i
eq eq eq
ineq ineq ineq
J
= + =
= + ≥
X
X
c A X b 0
c A X b 0
Choice of the objective function
in static optimisation
4F. Moissenet et al.
[ Introduction / Context ]
- Identify the potential design variables
- Compare the different types of objective
functions and constraints
- Discuss the mono- vs. multi-optimisation
approaches
min ( )
subject to:
i
eq eq eq
ineq ineq ineq
J
= + =
= + ≥
X
X
c A X b 0
c A X b 0
Objectives of this tutorial:
Choice of the objective function
in static optimisation
5F. Moissenet et al.
[ Identification of the design variables ]
Identification of the
design variables
Choice of the objective function
in static optimisation
6F. Moissenet et al.
Activation
dynamics
Contraction
dynamics
Musculoskeletal model
u a f
q, q, q. ..
[1] Chèze et al., 2015
1- Manage the muscular redundancy
[1]
g
Skeletal
dynamics
[ Identification of the design variables ]
Choice of the objective function
in static optimisation
7F. Moissenet et al.
Activation
dynamics
Contraction
dynamics
Musculoskeletal model
u a f
q, q, q
[1] Chèze et al., 2015
1- Manage the muscular redundancy
g
Skeletal
dynamics
[ Identification of the design variables ]
[1]
. ..
Choice of the objective function
in static optimisation
8F. Moissenet et al.
[1] Chèze et al., 2015
1- Manage the muscular redundancy
Hypotheses [1]:
1- The musculo-tendon forces are the only
forces that produce joint power
2- All the musculo-tendon forces produce
joint power
[ Identification of the design variables ]
1 1 1
1 with f f
m
n p m
f
n m
f
⋅ = < ⋅
M e
L L
M e
⋮ … ⋮
Choice of the objective function
in static optimisation
9F. Moissenet et al.
1- Manage the muscular redundancy
Mi (i.e., net joint moments) calculated with inverse
dynamics analysis (ek : DoF axis)
Lfj (i.e., muscular lever arms) defined by the
muscular geometry
fj (i.e., musculo-tendon forces) are the unknows
of the optimisation (i.e., design variables) Mi
fj
[ Identification of the design variables ]
1 1 1†
1 with f f
m
n mm n p
f
n m
f ×
⋅ = < ⋅
M e
L L
M e
⋮ … ⋮������� Lf
j
Choice of the objective function
in static optimisation
10F. Moissenet et al.
Activation
dynamics
Contraction
dynamics
Musculoskeletal model
u a f
q, q, q
[1] Chèze et al., 2015
1- Manage the muscular redundancy
g
Skeletal
dynamics
[ Identification of the design variables ]
[1]
. ..
Choice of the objective function
in static optimisation
11F. Moissenet et al.
1- Manage the muscular redundancy
Unknowns: a (i.e., muscular activations)
a is linked to the musculo-tendon forces
using a Hill-type based model
Contraction
dynamics
a f
[ Identification of the design variables ]
( )0 ( ) ( ) ( )pl vMM M MM M M M
j j j j j j j j jf f f l f v a f l= ⋅ × × +ɶ ɶ ɶ ɶ ɶɶ
Choice of the objective function
in static optimisation
12F. Moissenet et al.
Activation
dynamics
Contraction
dynamics
Musculoskeletal model
u a f
q, q, q
[1] Chèze et al., 2015
1- Manage the muscular redundancy
g
Skeletal
dynamics
[ Identification of the design variables ]
[1]
. ..
Choice of the objective function
in static optimisation
13F. Moissenet et al.
1- Manage the muscular redundancy
Unknowns: u (i.e., neuromuscular excitations)
u is linked to the muscular activations
using a nonlinear relationship
corresponding to a delay
Activation
dynamics
u a
[ Identification of the design variables ]
( ) ( )1 1jA u A
ja e e×
= − −
Choice of the objective function
in static optimisation
14F. Moissenet et al.
2- Manage other forces
[1] Cleather et al., 2011 | [2] Hu et al., 2013 | [3] Moissenet et al., 2014
Joint reaction forces can be computed
with optimised musculo-tendon forces
(2-step approach) ...
... Or be introduced in the optimisation
process with adapted joint geometries
introducing additional lever arms
(1-step approach) [1-3]
Joint contact forces
Ligament forces
[ Identification of the design variables ]
Choice of the objective function
in static optimisation
15F. Moissenet et al.
2- Manage other forces
[1] Hu et al., 2013
[1]
Musculo-
tendon forces
Joint contact
forces
Ligament
forces
[ Identification of the design variables ]
1 1 1 1 1
1 1 1
c c l l
c l
c l
c l
f f g g g g
m r r
c l
n p m r r
f g g
f g g
⋅ = + + ⋅
M e
L L L L L L
M e
⋮ ⋯ ⋮ ⋯ ⋮ ⋯ ⋮
Choice of the objective function
in static optimisation
16F. Moissenet et al.
2- Manage other forces
Avoid excessive ligament
strains [1]
Interaction between
structures [1-4]
Better joint contact estimations [2]
VALIDATION !
[1] Cleather et al., 2011 | [2] Hu et al., 2013 | [3] Moissenet et al., 2014 | [4] Pandy & Andriacchi, 2010
Estimations
Implant
Minimisation of
joint contact
forces
[3]
[ Identification of the design variables ]
Choice of the objective function
in static optimisation
17F. Moissenet et al.
[ Objective function(s) and constraints ]
Objective function(s)
and constraints
Choice of the objective function
in static optimisation
18F. Moissenet et al.
[ Objective function(s) and constraints ]
: Ensure the moment equipollence
: Muscles can only pull, never push
1- Manage the muscular redundancy
1 1 1
1 with f f
m
n p m
f
n m
f
⋅ = < ⋅
M e
L L
M e
⋮ … ⋮
min ( )
subject to:
i
eq eq eq
ineq ineq ineq
J
= + =
= + ≥
X
X
c A X b 0
c A X b 0
[ ]0, 1 j
f j m≥ ∀ ∈ ⋯
Choice of the objective function
in static optimisation
19F. Moissenet et al.
[ Objective function(s) and constraints ]
a- Polynomial criteria
: Principle of minimal total muscular force [1-4]
: Relative musculo-tendon forces [5,6]
(Physiological meaning)
[1] Seireg & Arvikar, 1975 | [2] Yeo, 1976 | [3] Hardt, 1978 | [4] Patriarco et al., 1981 | [5] Challis, 1997
[6] Pedotti et al., 1978
fj (i.e., musculo-tendon forces) are the unknows of the optimisation
1- Manage the muscular redundancy
( )1
( )
pm
j j
j
J w f=
= ×∑f
1j
w =
0
1j M
j
wf
=
Choice of the objective function
in static optimisation
20F. Moissenet et al.
[ Objective function(s) and constraints ]
a- Polynomial criteria
[1] Crowninshield & Brand, 1981 | [2] Pedersen et al., 1987
(Physiological meaning)
fj (i.e., musculo-tendon forces) are the unknows of the optimisation
1- Manage the muscular redundancy
( )1
( )
pm
j j
j
J w f=
= ×∑f
: Muscle stress [1,2] 1
j
j
wPCSA
=
Choice of the objective function
in static optimisation
21F. Moissenet et al.
[ Objective function(s) and constraints ]
a- Polynomial criteria
[1] Praagman et al., 2006
(Physiological meaning)
fj (i.e., musculo-tendon forces) are the unknows of the optimisation
1- Manage the muscular redundancy
( )1
( )
pm
j j
j
J w f=
= ×∑f
: Minimisation of the
energy-related consumption [1] 0
1 1
2 2
j j
M
j j
f f
PCSA f
× + ×
Choice of the objective function
in static optimisation
22F. Moissenet et al.
[ Objective function(s) and constraints ]
fj (i.e., musculo-tendon forces) are the unknows of the optimisation
a- Polynomial criteria
p = 1
p = 2
p = 3
p = ...
[1] Crowninshield & Brand, 1981 | [2] Challis, 1997 | [3] Rasmussen et al., 2001
(Physiological meaning)
Increase the value of p will involve
muscles with a lower force and thus
increase the number of active muscles [1-3]
1- Manage the muscular redundancy
( )1
( )
pm
j j
j
J w f=
= ×∑f
Choice of the objective function
in static optimisation
23F. Moissenet et al.
[ Objective function(s) and constraints ]
a- Polynomial criteria
[1] Praagman et al., 2006
Need for additional constraints preventing
May be the cause of sudden changes = aphysiological discontinuities
fj (i.e., musculo-tendon forces) are the unknows of the optimisation
1- Manage the muscular redundancy
0M
j jf f≤
Choice of the objective function
in static optimisation
24F. Moissenet et al.
[ Objective function(s) and constraints ]
aj (i.e., muscular activations) are the unknows of the optimisation
a- Polynomial criteria
[1] Kaufman et al., 1991 | [2] Thelen et al., 2003
(Physiological meaning)
Similar criteria with a [1,2]
Need for additional constraints (Hill)
1- Manage the muscular redundancy
( )1
( )
pm
j j
j
J w a=
= ×∑a
1j
w =
Choice of the objective function
in static optimisation
25F. Moissenet et al.
[ Objective function(s) and constraints ]
b- Soft saturation criteria [1]
[1] Siemienski, 1992
Maximisation of a distance from the maximum
load
Ensure distribution of the musculo-tendon forces
No need for additional constraints
fj (i.e., musculo-tendon forces) are the unknows of the optimisation
1- Manage the muscular redundancy
( )2
1
( ) 1m
j j
j
J w f=
= − − ×∑f
0
1j M
j
wf
=
Choice of the objective function
in static optimisation
26F. Moissenet et al.
[ Objective function(s) and constraints ]
c- Min/max criterion [1,2]
[1] Dul et al., 1984 | [2] Rasmussen et al., 2001
Minimise the maximum musculo-tendon force
= Ensure the distribution of musculo-tendon forces
Collaborative criterion
fj (i.e., musculo-tendon forces) are the unknows of the optimisation
1- Manage the muscular redundancy
( )( ) maxj j
J w f= ×f
Choice of the objective function
in static optimisation
27F. Moissenet et al.
[ Objective function(s) and constraints ]
[1] Rasmussen et al., 2001
Influence of the type of criterion & influence of the power p
[1] Rasmussen et al., 2001
Choice of the objective function
in static optimisation
28F. Moissenet et al.
[ Objective function(s) and constraints ]
Influence of the type of criterion
[1] Rasmussen et al., 2001
Min/max criterion
p = 2
Choice of the objective function
in static optimisation
29F. Moissenet et al.
[ Objective function(s) and constraints ]
Influence of the power p
[1] Rasmussen et al., 2001
Min/max criterion
p = 2
Choice of the objective function
in static optimisation
30F. Moissenet et al.
[ Objective function(s) and constraints ]
[1] Rasmussen et al., 2001
Min/max criterion
p = 5
Influence of the power p
Choice of the objective function
in static optimisation
31F. Moissenet et al.
[ Objective function(s) and constraints ]
[1] Rasmussen et al., 2001
Min/max criterion
p = 10
Influence of the power p
Choice of the objective function
in static optimisation
32F. Moissenet et al.
[ Objective function(s) and constraints ]
[1] Rasmussen et al., 2001
Min/max criterion
p = 100
Influence of the power p
Choice of the objective function
in static optimisation
33F. Moissenet et al.
[ Objective function(s) and constraints ]
Influence of the weights wj
[1] Giroux, 2013
[1]
wj fj
fj
fj fj
wj fj 2
wj fj )
The use of weights reduce the difference between criteria ...
and affects the joint contact estimation (better force distribution in this case)
01M
j jw f=
0
1 1
2 2
j j
M
j j
f f
PCSA f× + ×∑
Choice of the objective function
in static optimisation
34F. Moissenet et al.
[ Objective function(s) and constraints ]
2- Manage other forces
[1] Hu et al., 2013
[1]
Musculo-
tendon forces
Joint contact
forces
Ligament
forces
1 1 1 1 1
1 1 1
c c l l
c l
c l
c l
f f g g g g
m r r
c l
n p m r r
f g g
f g g
⋅ = + + ⋅
M e
L L L L L L
M e
⋮ ⋯ ⋮ ⋯ ⋮ ⋯ ⋮
Choice of the objective function
in static optimisation
35F. Moissenet et al.
[ Objective function(s) and constraints ]
2- Manage other forces
[1] Hu et al., 2013 | [2] Moissenet et al., 2014 | [3] Moissenet et al., 2012
Unknows
f, gc
min J(f)
Constraints
moment
equipollence
only
RM
Unknows
f, gc, gl
min J(f)
Constraints
moment
equipollence
only
RML
Unknows
f, gc, gl
min J(f)
Constraints
force & moment
equipollence
only
RFML
Unknows
f, gc, gl
min J(f, )
Constraints
moment
equipollence
& link between
Lagrange multipliers
RML2[1] [1] [1][2]
λ
λ[3]
Choice of the objective function
in static optimisation
36F. Moissenet et al.
[ Objective function(s) and constraints ]
2- Manage other forces
RM RML RFML[1] [1] [1]
Introduce further joint reaction forces as unknows reduces the solution space
Choice of the objective function
in static optimisation
37F. Moissenet et al.
[ Objective function(s) and constraints ]
2- Manage other forces
RML [2]
Introduce joint reaction forces in the minimisation may affects their estimations
RML2 [2]
TFmed
TFlat
Choice of the objective function
in static optimisation
38F. Moissenet et al.
[ Mono- vs. multi-objective approaches ]
Mono- vs. multi-objective
approaches
Choice of the objective function
in static optimisation
39F. Moissenet et al.
[ Mono- vs. multi-objective approaches ]
Hypotheses:
1- The musculo-tendon forces are the only
forces that produce joint power
2- All the musculo-tendon forces produce
joint power
1- The historical choice of the mono-objective approach
Traditional mono-objective
optimisation min J(f)
1 1 1
1 with f f
m
n p m
f
n m
f
⋅ = < ⋅
M e
L L
M e
⋮ … ⋮1 1 1
†
1 f f
m
n mm n p
f
f ×
⋅ = ⋅
M e
L L
M e
⋮ … ⋮�������
( )1
( )
pm
j j
j
J w f=
= ×∑f
Choice of the objective function
in static optimisation
40F. Moissenet et al.
[ Mono- vs. multi-objective approaches ]
1- The historical choice of the mono-objective approach
[1] Glitsch & Baumann, 1997
[1]
Allows setting different representations of the joint dynamics
wck = 0 : Enforces first the joint contact forces to ensure the moment
equipollence
wck = 105 : Enforces first the musculo-tendon forces to ensure the moment
equipollence
( ) ( )1 1
( )
c ppm rc c
j j k k
j k
J w f w g= =
= × + ×∑ ∑f
Choice of the objective function
in static optimisation
41F. Moissenet et al.
[ Mono- vs. multi-objective approaches ]
1- The historical choice of the mono-objective approach
[1] Cleather & Bull, 2011
[1]
(BW)
The choice of the weights highly affects the outputs
How to set these weights ?
( ) ( ) ( )1 2 3
1 1 1
( )
c lp ppm r rc c l l
j j k k k k
j k k
J k w f k w g k w g= = =
= × + × + ×∑ ∑ ∑f
Choice of the objective function
in static optimisation
42F. Moissenet et al.
[ Mono- vs. multi-objective approaches ]
2- Selection / definition of the weights
[1] Moissenet et al., 2014
RML [1] RML2 [1]
TFmed
TFlat
Learn from measurements(e.g., implant measurements)
An a priori selection of the weights is
possible to reach an objective
Ex: If tibiofemoral contact forces are
overestimated, we can try to
introduce them in the minimisation
process and tune weights !
Subjective methodology !
Choice of the objective function
in static optimisation
43F. Moissenet et al.
[ Mono- vs. multi-objective approaches ]
2- Selection / definition of the weights
[1] Mombaur et al., 2009
Upper level
Lower level
Minimize the root mean square
error of predicted tibiofemoral
contact forces from
measurements
W f
Solve the static optimization
problem using the updated
weight matrix W
Inverse optimal control approach [1]
Find the set of weights allowing to find
the closest estimations to the
measurements
Ex: Estimated tibiofemoral contact forces
vs. implant measurements
Choice of the objective function
in static optimisation
44F. Moissenet et al.
[ Mono- vs. multi-objective approaches ]
2- Selection / definition of the weights
[1] Moissenet et al., unpublished
0
1
2
3
4
5
6
7
8
Subject 1 Subject 2 Subject 3 Subject 4
Cycle 1 Cycle 2 Cycle 3
Cycle 4 Cycle 5
0
2
4
6
8
10
Subject 1 Subject 2 Subject 3 Subject 4
Cycle 1 Cycle 2 Cycle 3
Cycle 4 Cycle 5
w_TBmed w_TBlat
[1]
Choice of the objective function
in static optimisation
45F. Moissenet et al.
[ Mono- vs. multi-objective approaches ]
2- Selection / definition of the weights
[1] Moissenet et al., unpublished
Inverse optimal control approach
- Best solution regarding measurements
- Can only be done a posteriori
- High intra-subject variability
- High inter-subject variability
[1]
Define manually weights without (invasive)
measurements analysis is not feasible (?) !
w_TBmed
Choice of the objective function
in static optimisation
46F. Moissenet et al.
[ Mono- vs. multi-objective approaches ]
3- Need for a multi-objective approach
Few studies
Bottasso et al., 2006
Bensghaier et al., 2012
Chihara et al., 2014
Dumas et al., 2014
Moissenet et al., 2014
[1] Moissenet et al., 2014
Choice of the objective function
in static optimisation
47F. Moissenet et al.
[ Mono- vs. multi-objective approaches ]
3- Need for a multi-objective approach
Objective functions:
( )
( )
( )
( )
2
1
1
2
2
1
2
3
1
2
4
1
1, 43
1, 5
1, 5
1, 2
c
l
b
m
i
j
rc c
ick
rl l
ilk
rb b
ibk
J f mm
J g rr
J g rr
J g rr
=
=
=
=
= =
= =
= =
= =
∑
∑
∑
∑
A priori articulation of preference:
“Weighted sum method”
No articulation of preference:
“Unweighted min-max method”
1 1 2 2 3 3 4 4sU w J w J w J w J= + + +
1 2 3 4max( , , , )mU J J J J=
Optimisation problem:
min
constraint to eq eq
ineq ineq
U
=
≥
x
A x b
A x b
Inequality constraints:
Muscles and
ligaments are in
traction, joint
contacts and bones
are in compression
[1] Moissenet et al., 2014
[1]
Choice of the objective function
in static optimisation
48F. Moissenet et al.
[ Mono- vs. multi-objective approaches ]
3- Need for a multi-objective approach
[1] Moissenet et al., 2014
[1]
Gait cycle (%) Gait cycle (%)
Co
nta
ct f
orc
e (
N)
Co
nta
ct f
orc
e (
N)TBmed TBlat
Implant measurements Weighted sum method Min-max method
Choice of the objective function
in static optimisation
49F. Moissenet et al.
[ Mono- vs. multi-objective approaches ]
3- Need for a multi-objective approach
Weighted sum method and unweighted min-max method provide similar results
Weighted sum method
- Subjective weights
- But allows increase or decrease
the minimisation of a group
of force (e.g., pathology)
Unweighted min-max method
- No arbitrary adjustment of
the weights
Choice of the objective function
in static optimisation
50F. Moissenet et al.
[ Conclusion ]
Conclusion
Choice of the objective function
in static optimisation
51F. Moissenet et al.
[ Conclusion ]
The optimisation problem that we defined is
sensitive to every parameter!
The need for validation tends to suggest using
advanced joint models (e.g., two compartment
tibiofemoral joint model) introducing joint contact
(and ligament) lever arms
The management of the interaction between
the musculoskeletal forces can be done
through the optimisation process
Choice of the objective function
in static optimisation
52F. Moissenet et al.
[ Conclusion ]
Need for an efficient multi-objective
optimisation framework !
The management of the interaction between
the musculoskeletal forces can be done
through the optimisation process
53
THANK YOU FOR YOUR ATTENTION !
Choice of the objective function
in static optimisation
F. Moissenet et al.