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The Inverse Dynamics Optimal
Control method to estimate muscle
forces in musculoskeletal systems
Luciano Luporini Menegaldo
Agenor de Toledo Fleury
São Paulo State Institute for Technological Research
University of São Paulo, Brazil
Hans Ingo Weber
Pontifical Catholic University of Rio de Janeiro, Brazil
How to estimate muscle forces in
musculoskeletal systems ?
a) Inverse dynamics and static
optimization (IDSO)
-
Measure the kinematics of the body
-
Calculate joint moments using a inverse
dynamics model
-
Formulate and solve a static optimization
problem to find the muscle forces that
produce the estimated joint moments
Main features of IDSO
Low computational cost
Relative simplicity
Robustness to dynamical and numerical
instabilities
Main features of IDSO
Need of kinematical measurements
(noise, filtering, calculus of velocities and
accelerations etc.)
Muscle dynamics is not taken into account in
the formulation
Optimality is stated for each instant of time, not
to the overall task
b) Forward dynamics optimal control
(FDOC)
Formulate a forward dynamics model (statespace representation)
Formulate an Optimal Control Problem
Solution of the Optimal Control Problem gives
the displacements (states) and muscle
excitations (controls)
Main features of FDOC
No kinematics measurement required
Muscle dynamics considered in the formulation
Optimality is stated for all the time-span of the
movement
Main features of FDOC
High numerical and analytical complexity
High computational cost
Cannot be used to analyze real movements,
only simulated
c) Inverse Dynamics Optimal Control
(IDOC)
Joint moments are found by inverse dynamics
(or FDOC using torque-actuated models)
Optimal Control problem is formulated:
– Without Multi-Body equations
– Cost function in augmented with a
moment-tracking error function
Main features of IDOC (this paper)
The features are quite similar to IDSO,
but:
• Muscle dynamics is taken into account
• Optimality is stated to the overall
movement
Main features of IDOC
• Eliminate Multi-body equations
No more dynamical instability of FDOC
• If the joint moments are estimated
using torque-actuated models, muscle
forces can be estimated with a
inexpensive optimal control approach
Main features of IDOC
Numerical difficulties associated to FDOC
dynamical instability are greatly
reduced:
– choice of the algorithms
– discretization level
– tolerances etc.
Main features of IDOC
Mild computational costs can lead to:
• Clinical Applications in functional
surgery simulation
• Increase of biomechanical model and
task complexity
d) IDOC formulation
1. Collect musculoskeletal kinematics
Previous FDOC solution for posture (Menegaldo
et al., 2003, J. Biomech. 36, 1701-1712)
2. Calculate joint moments using inverse
dynamics model
(In this paper, joint moments were
evaluated using the moment arm
matrix and the muscle forces
calculated in FDOC solution)
3. Calculate [rFom] matrix for each time-step
 Fom1,1  Fom1,10   r1,1 (1 ,  2 ,  3 )  r1,10 (1 ,  2 ,  3 ) 
rFom  Fom 2,1  Fom 2,10   *r2,1 (1 ,  2 ,  3 )  r2,10 (1 ,  2 ,  3 )
 Fom 3,1  Fom 3,10   r3,1 (1 ,  2 ,  3 )  r3,10 (1 ,  2 ,  3 ) 
Fomi,j: optimal (maximum) force
ri,j: moment arm for the musculotendon actuator i in the
joint j, evaluated with regression equations
(Menegaldo et al., 2004, J. Biomech., in press)
θ1, θ2 and θ3 ankle, knee and hip joint angles
4. Find polynomial expressions that
fits:
– Joint moments x time
– Moment arms x time

2
 10 ~ 2
~
f IDOC (u, x)    w1  Fi (u, x)  w 2 rFom F   dt
i 1

0
tf
5. Formulate the cost function
tf

2
 10 ~ 2
~
f IDOC (u, x )    w1  Fi (u, x )  w2 rFom F    dt
i 1

0
~
Fi muscle force
[ ]3 x1 moments vector
w1, 2 relative weights
[rFom] matrix of polynomial s
[ ] vector of polynomial s
~
F ( x, u )
min f IDOC (u, x)
u
6. Formulate the optimal control problem
min  f IDOC (u, x )
u
min f IDOC (u, x)
u
s.t. 0  ui (t )  1, i  1,..., 10
a i  a (u i , a i )
~
~ T ~ T ~T
Fi  F(a i , F , K , L s ,...)
No endpoint constraints required !
No multi-body equations required !
e) Results
• Consistent approximations algorithms from
Polak and Schwartz
• RIOTS: Recursive Integration Optimal Control
Solver
• SQP NPSOL optimization solver
Comparative analysis
- FDOC
- IDOC
- IDSO: using a similar static cost function
2
~
subjected to
rFomk Fk  k
0
- IDSO_CB: (Crowninshield and Brand, 1981)
1/ 3
10


~ IDSO 3  
min f IDSO (u k , x )    Fi k
 
uk

 i 1
 


Test problem
- Human posture model
- 10 Muscles
- 3-link inverted
pendulum
- 1 sec., 0.5 sec.
Normalized force [0,1] x time (s), 0.5 sec.
Gluteus medius
Biceps femoris l. head
0.16
0.25
0.14
0.2
0.12
0.1
0.15
0.08
0.1
0.06
0.04
0.05
0.02
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0
0
gastrocnemius
0.14
0.12
0.12
0.1
0.1
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Rectus femoris
0.14
0
0.05
0.5
Continuous line = FDOC
Dotted = IDSO
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Dashed = IDOC
Dash-dot = IDSO_CB
Normalized force [0,1] x time (s), 1 sec.
Gluteus medius
Biceps femoris l. head
0.16
0.25
0.14
0.2
0.12
0.1
0.15
0.08
0.1
0.06
0.04
0.05
0.02
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0
gastrocnemius
0.14
0.06
0.12
0.05
0.1
0.04
0.08
0.03
0.06
0.02
0.04
0.01
0.02
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Rectus femoris
0.07
0
0.1
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Continuous line = FDOC
Dashed = IDOC
Dotted = IDSO
Dash-dot = IDSO_CB
Torque reconstruction 0.5 sec
Joint Torques, 0.5 s
200
ankle
knee
hip
150
joint torque (N.M)
100
50
0
-50
-100
0
0.05
0.1
0.15
0.2
0.25
0.3
time (s)
0.35
0.4
0.45
0.5
Continuous line = FDOC
Dashed = IDOC
Dotted = IDSO
Dash-dot = IDSO_CB
Moment reproduction error:
1/ 2
 (TORankle  MOM ankle )  
tf 

2

TFE   (TORknee  MOM knee )   dt

0
2
 (TORhip  MOM hip )



2
TOR: Moment generated by the FDOC solution
MOM: Moment reconstructed from IDOC or IDSO
solution
Cost
Final Torque fitting
function time error
IDOC
0.5 s 3.0447
IDOC
1.0 s 0.3361
IDSO
0.5 s 0.42166
IDSO
1.0 s 0.01603
IDSO_CB 0.5 s 0.42166
IDSO_CB 1.0 s 0.01603
FDOC
0.5 s FDOC
1.0 s * Pentium IV 1.4 MHz
CPU time*
8.77 minutes
37.02 minutes
1.39 minutes
2.62 minutes
1.57 minutes
3.17 minutes
2.30 days
13.38 days
f) Concluding remarks
• Force patterns obtained with classical static
optimization (IDSO) were unlike those of FDOC
“standand” solution
• The patterns obtained by IDOC follows quite closely
those obtained by FDOC
• Some muscles have shown a better agreement:
gmed, bifemlh, gmax, vasti
• In others, the differences were relatively grater: rf,
gas, sol
• The differences in FDOC and IDOC coordination patterns
are greater for 0.5 than 1.0 sec.
• Reconstruction of the torque curves is satisfactory for all
methods, but the error is greater for IDOC than IDSO
• The CPU time was greater for IDOC when compared to
IDSO , 3 to 14 times, but
• The reduction in the CPU time between FDOC and IDOC
was of 520 times for 1.0 s and 378 times for 0.5 s
• The Inverse Dynamics Optimal Control
method is reliable, numerically robust, fast
and give results much closer to those
obtained with Forward Dynamics Optimal
Control, when compared to classical Static
Optimization