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





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





- 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:




[ Identification of the design variables ]

Identification of the

design variables




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





Activation

dynamics

Contraction

dynamics


u a f

q, q, q



g

Skeletal

dynamics


[1]

. ..






Hypotheses [1]:

1- The musculo-tendon forces are the only

forces that produce joint power

2- All the musculo-tendon forces produce

joint power


1 1 1

1 with f f

m

n p m

f

n m

f

⋅ = < ⋅

M e

L L

M e

⋮ … ⋮





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


1 1 1†

1 with f f

m

n mm n p

f

n m

f ×

⋅ = < ⋅

M e

L L

M e

⋮ … ⋮�� Lf

j




Activation

dynamics

Contraction

dynamics


u a f

q, q, q



g

Skeletal

dynamics


[1]

. ..





Unknowns: a (i.e., muscular activations)

a is linked to the musculo-tendon forces

using a Hill-type based model

Contraction

dynamics

a f


( )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




Activation

dynamics

Contraction

dynamics


u a f

q, q, q



g

Skeletal

dynamics


[1]

. ..





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


( ) ( )1 1jA u A

ja e e×

= − −




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






[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

⋮ ⋯ ⋮ ⋯ ⋮ ⋯ ⋮





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]





[ Objective function(s) and constraints ]

Objective function(s)

and constraints





: Ensure the moment equipollence

: Muscles can only pull, never push


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≥ ∀ ∈ ⋯





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

( )

pm

j j

j

J w f=

= ×∑f

1j

w =

0

1j M

j

wf

=






[1] Crowninshield & Brand, 1981 | [2] Pedersen et al., 1987




( )1

( )

pm

j j

j

J w f=

= ×∑f

: Muscle stress [1,2] 1

j

j

wPCSA

=






[1] Praagman et al., 2006




( )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

× + ×







p = 1

p = 2

p = 3

p = ...

[1] Crowninshield & Brand, 1981 | [2] Challis, 1997 | [3] Rasmussen et al., 2001


Increase the value of p will involve

muscles with a lower force and thus

increase the number of active muscles [1-3]


( )1

( )

pm

j j

j

J w f=

= ×∑f






[1] Praagman et al., 2006

Need for additional constraints preventing

May be the cause of sudden changes = aphysiological discontinuities



0M

j jf f≤





aj (i.e., muscular activations) are the unknows of the optimisation


[1] Kaufman et al., 1991 | [2] Thelen et al., 2003


Similar criteria with a [1,2]

Need for additional constraints (Hill)


( )1

( )

pm

j j

j

J w a=

= ×∑a

1j

w =





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



( )2

1

( ) 1m

j j

j

J w f=

= − − ×∑f

0

1j M

j

wf

=





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



( )( ) maxj j

J w f= ×f





[1] Rasmussen et al., 2001

Influence of the type of criterion & influence of the power p






Influence of the type of criterion


Min/max criterion

p = 2





Influence of the power p


Min/max criterion

p = 2






Min/max criterion

p = 5







Min/max criterion

p = 10







Min/max criterion

p = 100






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× + ×∑






[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

⋮ ⋯ ⋮ ⋯ ⋮ ⋯ ⋮






[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]






RM RML RFML[1] [1] [1]

Introduce further joint reaction forces as unknows reduces the solution space






RML [2]

Introduce joint reaction forces in the minimisation may affects their estimations

RML2 [2]

TFmed

TFlat




[ Mono- vs. multi-objective approaches ]

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






[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






[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





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 !






[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






[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]






[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





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







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]







[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






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




[ Conclusion ]

Conclusion




[ 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




[ 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 !

[email protected]@



F. Moissenet et al.

choice of the objective function in static optimisation

Documents