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 (state-space 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 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 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

5. Formulate the cost function

6. Formulate the optimal control problem 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 subjected to - IDSO_CB: (Crowninshield and Brand, 1981)

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. Continuous line = FDOC Dashed = IDOC Dotted = IDSO Dash-dot = IDSO_CB

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

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

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

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