Optimal Control of Isometric Muscle Dynamics


  • Robert Rockenfeller Mathematisches Institut, Universität Koblenz-Landau, Germany
  • Thomas Götz Mathematisches Institut, Universität Koblenz-Landau, Germany




biomechanics, inverse dynamics, muscle model, optimal control, stimulation.


We use an indirect optimal control approach to calculate the optimal neural stimulation needed to obtain measured isometric muscle forces. The neural stimulation of the nerve system is hereby considered to be a control function (input) of the system 'muscle' that solely determines the muscle force (output). We use a well-established muscle model and experimental data of isometric contractions. The model consists of coupled activation and contraction dynamics described by ordinary differential equations. To validate our results, we perform a comparison with commercial optimal control software.


Trltzsch, F., Optimal Control of Partial Differential Equation, 2, Vieweg + Teubner, 2009 (Text in Germany).

Happee, R., Inverse Dynamic Optimization Including Muscular Dynamics, a New Simulation Method Applied to Goal Directed Movements, J. Biomechanics, 27(7), pp. 953-960, 1994.

Kistemaker, D.A., van Soest, A.J. & Bobbert, M.F., Is Equilibrium Point Control Feasible for Fast Goal-Directed Single-Joint Movements?, Journal of Neurophysiology, 95(5),pp. 2898-2912, 2006.

Spagele, T., Kistner, A. & Gollhofer, A., A Multi-Phase Optimal Control Technique for the Simulation of a Human Vertical Jump, J. Biomechanics, 32(1), pp. 87-91, 1999.

Hpler, R., Stelzer, M. & von Stryk, O., Object-Oriented Dynamics Modeling for Legged Robot Trajectory Optimization and Control. Proc. IEEE Conference on Mechatronics and Robotics, pp. 972-977, 2004.

Anderson, F.& Pandy, M., Static and Dynamic Optimization Solutions for Gait Are Practically Equivalent, J. Biomechanics, 34(2),pp. 153-161, 2001.

Maas, R., Siebert, T. & Leyendecker, S.,On the Relevance of Structure Preservation to Simulations of Muscle Actuated Movements, Biomech Model Mechanobiol., 11(3-4), pp. 543-556, 2012.

Buchanan, T. S., Estimation of Muscle Forces and Joint Moments Using a Forward-Inverse Dynamics Model,J. American College of Sports Medicine (ACSM), pp. 1911-1916, 2005.

Stelzer, M., & von Stryk, O., Efficient Forward Dynamics Simulation and Optimization of Human Body Dynamics, ZAMM, 86(10), pp. 828-840, 2006.

Ghobadi, K., On the Discretize then Optimize Approach, Preprint for Industrial and Systems Engineering, 2009.

Heinkenschloss, M., PDE Constrained Optimization, Presentation at SIAM Conference, 2008.

Oberle, H.J., Variational Calculus and Optimal Control, Lecture, Department of Mathematics, University of Hamburg, 2008.

von Stryk, O., Numerical Solving of Optimal Control Problems: Discretizing, Parameter Optimization and Calculation of the Adjoint Variables., 8(441). VDI-Verlag, 1995 (Text in Germany).

RP Optimization Research LLC,http://www.gpops2.com/News/News, (20thJanuary 2015).

Tomlab Optimization Inc., http://www.tomopt.com/scripts/register, (20th January 2015).

Oberle, H.J. & Grimm, W., BNDSCO-A Program for the Numerical Solution of Optimal Control Problems, Institute for Flight Systems Dynamics, DLR, Oberpfaffenhofen, 1989.

Hinze, M., Mathematics of PDE Constrained Optimization - Discrete Concepts, Presentation at Oberwolfach Seminar, 2010.

Pandy, M., Zajac, F., Levine, W. & Sim, E., An Optimal Control Model for Maximum-Height Human Jumping, J. Biomechanics, 23(12),pp. 1185-1198, 1990.

Till, O., Characterization of Isovelocity Extension of Activated Muscle: A Hill-Type Model for Eccentric Contractions and a Method for Parameter Determination, Journal of Theoretical Biology, 225(2), pp. 176-187, 2008.

Bl, M., A Finite Element Approach for the Modeling of Skeletal Muscle Fatigue, GAMM-Mitt., 32(2), pp. 205-220, 2009.

Haufle, D., Hill-Type Muscle Model with Serial Damping and Eccentric Force-Velocity Relation, J. Biomechanics, 47(6), pp. 1531-1536, 2014.

G1/4nther, M., Schmitt, S. & Wank, V., High-Frequency Oscillations as a Consequence of Neglected Serial Damping in Hill-Type Muscle Models, Biological Cybernetics, 97, pp. 63-79, 2007.

Siebert, T. & Rode, C. & Herzog, W., Nonlinearities Make a Difference: Comparison of Two Common Hill-Type Models with Real Muscle, Biol. Cybernetics, 98(2),pp. 133-143, 2008.

van Soest, A. J., Jumping from Structure to Control - A Simulation Study of Explosive Moments, Ph.D Thesis, Faculteit der Bewegingswetens-chappen, Universiteitte Amsterdam, 1992.

Zajac, F. E., Muscle and Tendon: Properties, Models, Scaling, and Application to Biomechanics and Motor Control, Crit. Rev. Biomed. Eng, 17(4), pp. 359-411, 1989.

Hatze, H., A Myocybernetic Control Model of Skeletal Muscle,Biological Cybernetics, 25(2), pp. 103-119, 1977.

Rockenfeller, R., G1/4nther, M., Schmidt, S. & Gtz, T., Comparative Sensitivity Analysis of Muscle Activation Dynamics, Accepted in Hindawi CMMM, 2015.

Hill, A. V., The Heat of Shortening and the Dynamic Constants of Muscle, Proceedings of The Royal Society London B, 126(843), pp. 136-195, 1938.

Wank, V., Muscle Growth and Fiber Type Composition in Hind Limb Muscles During Postnatal Development in Pigs, Cells Tissues Organs, 182(243), pp. 171-81, 2006.

G1/4nther, M., Computer Simulation for the Synthesis of Muscular Generated Human Walking of a Biomechanical Multi-Body-System, Ph.D Thesis, Department of Physics, Universitat T1/4bingen, 1997. (Text in German).

Tomlab Optimization Inc., http://www.tomdyn.com/matlab_optimal_ control_examples.html, (23th January 2015).

Rutquist, P., & Edvall, M., PROPT - Matlab Optimal Control Software. TOMLAB Optimization Inc., 2010.

Kostarina, N., & Eriksson, A., History Effect and Timing of Force Production Introduced in a Skeletal Muscle, Biomech Modell Mechanobiol, 11(7), pp. 947-957, 2012.

Till, O., Siebert, T., & Blickhan, R., Force Depression Decays During Shortening in the Medial Gastrocnemius of the Rat, J. Biomechanics,47(5), pp. 1099-1103, 2014.

Menegaldo, L., A 'Cheap' Optimal Control Approach to Estimate Muscle Forces in Musculoskeletal Systems, J. Biomechanics,39(10), pp.1787-1795, 2006.

Thelen, D., Using Computed Muscle Control to Generate Forward Dynamic Simulation of Human Walking from Experimental Data. J. Biomechanics,39(3), pp. 1107-1115, 2006.

Koziel, S. & Bandler, J., "Accelerated Microwave Design Optimization With Tuning Space Mapping, IEEE Transactions on Microwave Theory and Techniques, 57(2), pp. 383-394, 2009.