Passive force in the two element muscle model

Hello and good day to all,

I had a couple of questions about the two element muscle model (the class
AnyMuscleMODEL2ELin). Specifically my question is related to the tutorial and what the tutorial says for how the two element model takes into account passive muscle forces. In the muscle modeling tutorial, the following three sentences are present:

“This is passive elasticity. The two-element muscle model of the preceding section handles the presence of this elasticity by increasing the strength of the muscle, and this works fine if the muscle is supposed to be active in the sense that the model in such a state would predict a high force with a low muscle activity. But the passive muscle force cannot be switched off, so it will still be present even if it is disadvantageous, and the two-element model will not predict this.”

I have two questions about this statement. The first is what does the tutorial mean when it says that the two element model accounts for passive forces by increasing the strength of the model? Is there some function of passive force related to the length of the muscle which computes maximum possible force and adds that to the strength of the muscles? Could it be some constant value that is added to the strength regardless of muscle length?

The second question is with regards to the last statement “But the passive muscle force cannot be switched off, so it will still be present even if it is disadvantageous, and the two-element model will not predict this”. Could someone explain how it’s even possible to switch off passive force? I thought passive force was related to the length of the muscle such that it is either present or not. If the length of the muscle is a certain value, there is going to be a passive force associated with that. Maybe the answer to the first question will help with my second one.

Thanks for the help,

Brandon Brown


  1. Passive elasticity in the 2 element model: To understand this, let’s first take a look at the equation for the muscle strength which is given a little further up on the same page in the tutorial. You can see that the strength is a product of a muscle length based term and a velocity term.

Let’s focus on the first, length based term while assuming the muscle velocity to be zero. The passive force in a muscle is generated by the fiber elasticity, i.e. force will increase with muscle stretch. In a way this behaviour is what this term captures rather crudely. When Lm is <= 0.5*Lf, the strength will be 0 since both active and passive force cannot be generated at this fiber length.

But as you stretch the muscle further, the strength increases and this implies that the muscle will need lower activation to produce the same forces. This is what would have happened in a real muscle stretched to that extent, where the passive forces would take the bulk of the load and thus necessitate only small activations.

  1. Switching off passive forces: You are exactly right, passive forces cannot be switched off. But the rather simple nature of the 2 element model means that this rule can be disobeyed, and this is what that sentence explains.

So if you had a 2 element muscle in your model which was super-redudant, the simulation has the option of setting its activation to zero. This would be irrespective of how stretched the muscle was kinematically. In a 3 element model, the active and passive force components and separate, so even if active force is zero, the passive force would be substantial in the above case. Hence a 3 element model would be more realistic. Hope this helps :slight_smile:

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