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Multibody dynamics and numerical models of muscles LORENZO GRASSI

Why multibody dynamics?

Skeleton  of  a  baseball  pitcher  during  the  different  phases  of  a  pitch  

3D  musculotendinous  model  to  simulate  the  biomechanical  effects  of    

rectus  femoris  transposi9on  

(3)  

Kinema9c  analysis  for  the  rehabilita9on  planning  1.  Chao, E.Y. Med Eng Phys, 2003. 25(3) 2.  Asakawa, D.S., et al. J Bone Joint Surg Am, 2004. 86-A(2) 3.  Leardini, A. et. al. Gait & Posture 26 (2007)

1) 2)

3)

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Why multibody dynamics?

Taddei et al., Clin Biomech 27(3) 2012:273-280

•  10 years old patient with high grade Osteosarcoma at the distal left femur

•  Osteotomy, and femur reconstructed by means of an intercalary massive bone allograft from fibula

•  Many muscles had to be excised/removed/moved

This kid now can play karate!

But it’s a long way to go to get these results…

…let’s start from the beginning!

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Schematic of a multibody system

The  human  body  is  modelled  as  a  number  of  rigid  bodies  connected  by  

ideal  joints...          

...remember  assignments  1  &  2?  

Different types of joints in our body

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Your (very) first tastes of multibody dynamics

Assignment 1…

…and 2

Assignment 1: from motion to forces In  our  assignment  1,  the  human  leg  was  modelled  with:  -­‐  2  rigid  bodies  (upper  and  

lower  leg)  -­‐  2  hinges  à  2  dof  -­‐  movements  limited  to  

the  sagi9al  plane...  

   

Kinema9cs  data  were  used  to  calculate  joint  

forces  (but  muscles  were  not  considered)  

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…3D is way more complicated!

Ball  and  socket  (3  dof)  

Hinge  (1  dof)  

Hinge  (1  dof)  

l = number of degrees of freedom of a system

n = # rigid bodies lk = degrees of freedom

for the kth joint

∑ =−−−=

n

k klnl1

)6()1(6(Gruber)  

104*53*236

)6()17(6 6

1

=−−

=−−−= ∑ =k kll

For  our  3D  model:  

Solution of assignment 1

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Assignment 2: redundancy & recruiting

External  forces  were  known,  and  we  used  a  sta9c  op9miza9on  

approach  to  calculate  muscular  forces  

Several  muscles  act  on  the  same  dof:  1.  Performing  the  same  joint  mo>on  (synergist)  2.  Neutralizing  each  other  (antagonists)    

Features:  1.  Repeated  movements  produce  similar  aDva>on  

pa9erns  à  pre-­‐defined  control  strategies  exist?  2.  When  the  ar>cular  load  increases,  so  does  the  

muscular  ac>va>on,  up  to  the  tetanic  limit    

Why are we redundant? 1.  Increase articular stability

Weight lifting, execution of new motor tasks, instability.

2.  Transferring forces/moments between joints Co-contractions at the hip can produce an increment of the bending moment of the knee (e.g., co-contraction of gluteus maximus and rectus femoris produces knee extension).

3.  High accuracy movements Highly accurate and precise finger movements require complex activation patterns

4.  Improve movements that require changes in direction

5.  Protect the joints in extreme articular positions

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Static optimization

max

21

int

0

),...,,(

0

)(

FFFFFff

Ff

FqRM

im

nmmm

i mi

MT

≤≤

=

⎪⎩

⎪⎨

⎧

=∂

∂

×=

∑

∑ ⎟⎟⎠

⎞⎜⎜⎝

⎛=

i

n

imi

PCSAFf

n  =  1    not  effec9ve  in  predic9ng  synergies,  especially  for  low  load  magnitudes  n  >  1    synergies  are  predicted,  but  addi9onal  constraints  are  necessary  to  avoid  muscular  overloads  n    ∞  synergies  are  maximized,  and  effort  is  minimized    All  the  exponents  n  >  1  predict  synergies,  but  fail  at  predic9ng  antagonisms  

Mint = moment equilibrium equations f = cost function

Assignments 1 & 2 were just the first taste of the multibody dynamics

problem…now let’s have the main course

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Generation of the body motion

0),()()()()( 2 =+×+++ qqMeFqRqGqqCqqM MT

1.   Excita9on  2.   Ac9va9on  3.   Force  

4.   Joint  torques  5.   Dynamics  of  the  rigid  body  system  

BODY  MOTION  

Joint  moments  due  to  muscle  forces  

Joint  moments  due  to  external  forces  (e.g.  ground  reac9on)  

Mass  matrix  

Gravita9onal  effects  

Centrifugal  and  Coriolis  effects  

Different approaches are possible

Assignment 1

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Forward dynamics

[ ] { }),()()()()( 21 qqMeFqRqGqqCqMq MT +×++= −

Looks like a very nasty equation to solve…and it is! But computers can help us with its solution!

Numerical modeling of the tendon and muscle mechanics

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The modeling approach There  are  two  possible  numerical  descrip9ons:  a  phenomenological  one  (Hill,  1886-­‐1977),  and  a  mechanicis9c  one,  based  on  physiology  and  the  biochemistry  of  muscular  contrac9on  (Huxley,  1917-­‐1963)  

Simple  mathema9cal  expressions,  based  on  measurable  parameters  

Differen9al  equa9ons,  with    several  parameters  hard  to  quan9fy  

Muscle model (Thelen, 2003)

CE = contractile element αM = pennation angle

The muscle force generated is a function of three factors: the activation value (a), the normalized length of the muscle unit, and the normalized velocity of the muscle unit.

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Generalized Hill model

)()(

)()(

wucwukFvcvkucukFFF

aaa

aappap

−+−=

+++=+=

Muscle model (Thelen, 2003) a)  The  rela>on  between  ac>ve  force  versus  length  can  

be  described  as  a  Gaussian.  The  rela>on  between  passive  force  and  length  has  a  first  exponen>al  phase,  followed  by  a  second  linear  phase  

b)  scarico.    

b)  The  rela>on  between  ac>ve  force  and  velocity  can  be  scaled  in  order  to  reduce  the  contrac>on  velocity  in  sub-­‐tetanic  condi>ons

c)  The  force-­‐strain  rela>on  for  the  tendon  has  a  first  exponen>al  phase,  followed  by  a  linear  phase  

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Muscle activation dynamics

Where  τa(a,u)  is  a  9me  factor  which  varies  with  the  ac9va9on  level,  a  is  the  muscle  ac9va9on,  and  u  è  is  the  excita9on  signal  (Thelen,  2003).      A  more  refined  model  could  include  different  τ  for  rise  and  fall  

;),( uaau

dtda

aτ−

=

( ) ( )auuaudtda

fallrise

−+−⎟⎟⎠

⎞⎜⎜⎝

⎛=

ττ11 2

Active muscle force

 Where:  

•  fl    è  is  a  scale  factor  •   LM  is  the  normalized  muscle  

length    •  γ  is  a  shape  factor  

γ2)1( −−

=ML

l ef

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Passive muscle force

⎪⎩

⎪⎨

⎧

≥+−

≤−−=

∗

Ttoe

TTtoe

Ttoe

Tlin

Ttoe

Tk

k

TtoeT

Fk

eeF

FT

toe

Ttoe

toe

εεεε

εεεε

;)(

);1(1

Where:    1.             is  the  tendon  force  

normalized  by  the  max  isometric  force  

2.             is  the  tendon  deforma>on  3.                 is  the  limit  elonga>on  over  

which  it  behaves  linearly    4.  ktoe    is  a  shape  factor  5.  klin  is  a  scale  factor.    6.                                           is  the  limit  

normalized  force  over  which  the  tendon  behaves  linearly  

( )33.0=TtoeF

TF

TεT

toeε

Ttoeε

Muscle force Vs. velocity

                       is  the  max  contrac>on  velocity,  and  b  is  calculated    differently  whether  the  muscle  fiber  is  shortening  (FM<afl)  or  lengthening  (FM>  afl)  

bafFVaV l

MMM −

+= max)75.025.0(

⎪⎪⎩

⎪⎪⎨

⎧

≥−

−−

≤+

=

lM

Mlen

MMlenlf

lM

f

Ml

afFF

FFafA

afFAFaf

b;

)1())(/22(

;

             is  the  max  normalized  muscle  force  when  the  fiber  is  elongated    af    is  a  shape  factor  

MlenF

MVmax

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So, let’s get back to our forward dynamics problem…

Forward dynamics

Now we have all the theoretical background to start playing with our multibody dynamics software…

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 Open-­‐source  sohware  for  the  mul9body  

simula9on  of  the  neuromuscular  system  and  the  mo9on  dynamics  simula9on  

 (numerical  methods  for  the  coupled  solu9on  of  the  mul9body  

dynamic  problem  and  the  op9mal  distribu9on  of  the  muscle  forces)    

     

Website: https://simtk.org/home/opensim

There you can download and install the software for, and find a lot of tutorials and instructions

SimTK and SimBios are trademarks of Stanford University

     

The GUI

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Our test case: simulation and prevention of ankle sprain

Research questions of our test case

You will examine and address how the following factors may affect angle inversion sprain injury: •  Muscle reflexes

•  Muscle co-activation

•  Introduction of a passive orthosis

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When & Where

Monday, October 6th , 10-12 (group 1) and 15-17 (group 2). Room INA 3,4 in the M:house