1. 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

2. How to estimate muscle forces in musculoskeletal systems ?

3. 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

4. Main features of IDSO Low computational cost Relative simplicity
Robustness to dynamical and numerical instabilities

5. 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

6. b) Forward dynamics optimal control (FDOC)
Formulate a forward dynamics model (state-space representation)
Formulate an Optimal Control Problem
Solution of the Optimal Control Problem gives the displacements (states) and muscle excitations (controls)

7. 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

8. Main features of FDOC High numerical and analytical complexity
High computational cost
Cannot be used to analyze real movements, only simulated

9. 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

10. 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

11. 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

12. Main features of IDOC
Numerical difficulties associated to FDOC dynamical instability are greatly reduced:
choice of the algorithms
discretization level
tolerances etc.

13. Main features of IDOC Mild computational costs can lead to:
Clinical Applications in functional surgery simulation
Increase of biomechanical model and task complexity

14. d) IDOC formulation Collect musculoskeletal kinematics
Previous FDOC solution for posture (Menegaldo et al., 2003, J. Biomech. 36, )

15. 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)

16. 3. Calculate [rFom] matrix for each time-step
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

17. 4. Find polynomial expressions that fits:
Joint moments x time
Moment arms x time

18. 5. Formulate the cost function

19. 6. Formulate the optimal control problem
No endpoint constraints required !
No multi-body equations required !

20. e) Results
Consistent approximations algorithms from Polak and Schwartz
RIOTS: Recursive Integration Optimal Control Solver
SQP NPSOL optimization solver

21. Comparative analysis - FDOC - IDOC
- IDSO: using a similar static cost function subjected to
- IDSO_CB: (Crowninshield and Brand, 1981)

22. Test problem Human posture model 10 Muscles 3-link inverted pendulum
1 sec., 0.5 sec.

23. Normalized force [0,1] x time (s), 0.5 sec.
Continuous line = FDOC Dashed = IDOC
Dotted = IDSO Dash-dot = IDSO_CB

24. Normalized force [0,1] x time (s), 1 sec.
Continuous line = FDOC Dashed = IDOC
Dotted = IDSO Dash-dot = IDSO_CB

25. Torque reconstruction 0.5 sec
Continuous line = FDOC Dashed = IDOC
Dotted = IDSO Dash-dot = IDSO_CB

26. Moment reproduction error:
TOR: Moment generated by the FDOC solution
MOM: Moment reconstructed from IDOC or IDSO solution

28. 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

29. 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

30. 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