The force-producing properties of muscle are complex, highly nonlinear, and can have substantial effects on movement (see McMahon, 1984 for a review). For simplicity, lumped-parameter, dimensionless muscle models that are capable of representing a variety of muscles with different architectures are commonly used in the dynamic simulation of movement (Zajac, 1989). In this exercise, you will explore the differential equations that describe muscle activation and muscle–tendon contraction dynamics when using a Hill-type muscle model. You will use OpenSim to implement a simple muscle–tendon model and conduct simulations to investigate how various model parameters affect the dynamic response of the actuator. The lab will conclude with a Virtual Muscle Tug-of-War in which you will design an optimal muscle and compete against other user-generated muscles. May the best muscle win!

By working through this lab, you will:


The original lab was designed by Jeff Reinbolt, B.J. Fregly, Kate Saul, Darryl Thelen, Silvia Blemker, Clay Anderson, and Scott Delp. The lab was refined by Hoa Hoang and Daniel Jacobs.


The model in this exercise consists of a cube with a single translational degree of freedom along the Z-axis. A Tug_of_War model has been included in the OpenSim distribution (Models/Tug_of_War/) that uses two Thelen 2003 Muscles to move the cube. In this exercise, we will use two Millard 2012 Muscles instead of the Thelen muscles. The muscles are arranged to pull on opposite sides of the cube. The cube has a mass of 20 kg and sides of length 0.1 m, and the distance between the fixed ground supports is 0.7 m. Thus, each muscle–tendon actuator is 0.3 m long when the block is centered.


Millard 2012 Muscle Model

Activation dynamics


Design Challenge

The competition is a single-elimination tournament between pairs of muscles. Each match is a one-second forward dynamic simulation where each muscle starts with minimal activation (a = 0.01). The match is won by the muscle that has moved the block to its side of the arena at the end of the simulation. Your challenge is to specify the muscle–tendon parameters and excitation time history necessary to overcome the other challengers.


  1. Anderson, F.C. and Pandy, M.G. (1999). A dynamic optimization solution for vertical jumping in three dimensions. Computer Methods in Biomechanics and Biomedical Engineering, 2(3):201–231.
  2. Hatze, H. (1976). The complete optimization of a human motion. Mathematical Biosciences, 28(1–2):99–135.
  3. McMahon, T.A. (1984). Muscles, Reflexes, and Locomotion. Princeton University Press, Princeton, New Jersey.
  4. Millard, M., Uchida, T., Seth, A., Delp, S.L. (2013). Flexing computational muscle: modeling and simulation of musculotendon dynamics. ASME Journal of Biomechanical Engineering, 135(2):021005.
  5. Schutte, L.M. (1993). Using Musculoskeletal Models to Explore Strategies for Improving Performance in Electrical Stimulation-Induced Leg Cycle Ergometry. PhD Dissertation, Mechanical Engineering Department, Stanford University.
  6. Thelen, D.G. (2003). Adjustment of muscle mechanics model parameters to simulate dynamic contractions in older adults. ASME Journal of Biomechanical Engineering, 125(1):70–77.
  7. Winters, J.M. (1990). Hill-based muscle models: a systems engineering perspective, in Multiple Muscle Systems: Biomechanics and Movement Organization, edited by Winters, J.M. and Woo, S.L., Springer-Verlag, New York.
  8. Zajac, F.E. (1989). Muscle and tendon: properties, models, scaling, and application to biomechanics and motor control. Critical Reviews in Biomedical Enginering, 17(4):359–411.

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