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Hello, I'm trying to understand static optimization results for the arm26 model for different elbow angles. The results are depicted in the figure below.


These extensor results are not what I was expecting. I understand that the solution depends on the muscles' force-length relation, moment arms, and other properties like maximum isometric force, but given that the extensors generate extension moments I was expecting the triceps forces to be much smaller.


Here's what I did:

• Opened the arm26 model

• Created a motion file with a static posture of r_shoulder_elev=0 and r_elbow_flex=X for 100 ms, where X is an arbitrary angle between 0 and 90 deg

• Ran the Static Optimization tool with the default settings.

• Repeated the three steps above for elbow angles ranging from 0 to 90.

Why are the extensor forces so high?

The extensor forces are predominately due to the passive forces of the muscles.

That sounds plausible, but if that were true why do the extensor forces return to zero at about 150 deg flexion as depicted in the figure below? Shouldn't there be greater passive extensor force at 150 deg than at 90 deg?

Youre right, not all of these forces are from passive structures.

Even in this simple motion, activating the biceps to generate elbow flexion results in unintended motion at the shoulder, without any other force to counteract it. Without the triceps muscle(s), or a coordinate lock/constraint, biceps activation will generate unintended accelerations in accelerations r_shoulder_eve. In a linked system, we know that muscles can produce moments across joints they dont cross. In this case, the triceps are being activated to balance the accelerations caused by the biceps.

You can test this by removing all the triceps muscles from the model and running static optimization. You will get a bunch of constraint violations because the optimizer wont be able to generate a great solution, but if you take the output controls and use them in FWD, you will see that motion made in the r_shoulder_eve coordinate.

I understand that the solution depends on the muscles' force-length relation, moment arms, and other properties like maximum isometric force, but given that the extensors generate extension moments I was expecting the triceps forces to be much smaller.

Once you actually multiply the force by the force-length and moment arm the effective moment that muscle force is producing could easily by rather small. You could use scripting to get the force-length multiplier, moment-arm, and force at every time point and assess what the moment generate by the muscle force is.

Hi James,

Thank you very much for your answers. Your explanations make sense, but I'm still having difficulty understanding these static optimization results...

I've done what you suggested in the figure below:

(a) extracted the moment arms for each muscle,
(b) computed muscle moments,
(c) computed moment sums for the flexors and extensors and calculated the gravity moment,
(d) computed the residual moment.

Aside from a small residual moment everything looks fine. I presume the residual moment is negligible and can be reduced with stricter static optimization settings.




The above results appear fine at first glance, but I still don't understand the following:

If the primary role of the extensors in this task is to maintain the shoulder angle at 0 deg, then why do they produce such a large moment?

For an elbow angle of 90 deg the net flexor moment only has to be about 2.7 Nm to offset the gravitational moment. In these results the flexion moment is about 7.2 Nm, well over twice the magnitude of the gravity moment. I understand that a slight increase in moment is necessary to maintain posture (shoulder = 0 deg, elbow = 90 deg), but I don't understand why the moment should be as large as this.


To try to understand the situation I've moved on to the shoulder and repeated moment analyses, yielding the results below. Here there is a big problem: the net shoulder moment is non-zero.

• Is there a shoulder moment I'm not considering?

• Are my moment arm data correct? They seem to be fine for the elbow but potentially incorrect for the shoulder. I obtained the shoulder moment arms by plotting "r_shoulder_elev moment arm" vs. elbow angle for each muscle using the OpenSim UI, then exported these data.)

• Your response cites "force-length", which I presume is the force-length relation for each muscle? I understand why the force-length relation can affect the optimum solution when the optimization goal is to minimize muscle activations, but I don't understand why it is optimum for the elbow extensors to produce a greater-than-gravity moment.

> Why are the extensor forces so high?

Perhaps because generating flexion moments at the elbow requires: 1) bi-articular biceps which flex both shoulder and elbow and 2) even uni-articular elbow flexors will induce shoulder flexion acceleration. In reality, there are numerous other muscles that cross the glenohumeral joint which that are not represented in this model. To maintain a stationary shoulder requires activating the shoulder extensors, but this model has none other than the bi-articular triceps and these in turn generate elbow extension. The solution is some balance of these requirements/limitations. If you want to isolate the effect of muscles only on elbow flexion in this toy model, you can try locking the shoulder to account for the action of unrepresented muscles. My guess is the subsequent results will be more inline with your expectations.

At what point does a model transition from a "toy model" to a realistic one whose results can be regarded as a reasonably realistic? To me three flexors and three extensors seem sufficient for elbow flexion control, but I appreciate that shoulder flexion/extension muscles are underrepresented. How can one judge how many additional shoulder and/or elbow muscles are required to achieve reasonably realistic results?

At what point does a model transition from a "toy model" to a realistic one whose results can be regarded as a reasonably realistic? How can one judge how many additional shoulder and/or elbow muscles are required to achieve reasonably realistic results?

It depends how "reasonably realistic" is defined, which is problem dependent. There's some relevant discussion in the following paper:
Hicks, J.L., Uchida, T.K., Seth, A., Rajagopal, A., Delp, S.L. Is my model good enough? Best practices for verification and validation of musculoskeletal models and simulations of movement. ASME Journal of Biomechanical Engineering 137(2):020905, 2015. https://nmbl.stanford.edu/publications/ ... ks2015.pdf