min/max criterion


This is a question to the optimization criterion used in AnyBody.

In your paper:

  1.         Rasmussen J, Damsgaard M, Voigt M. Muscle recruitment by the min/max criterion – a comparative numerical study. J Biomech 34: 409-415, 2001

you say, that AnyBody uses the min/max criterion id est,

Minimize G(f(M)) = max? (fi(M)/Ni)…

This criterion is to be distribute the collaborate muscle forces in such a way that the maximum? relative muscle force is as small as possible.

But isn’t that a contradiction itself: Minimize the maximum? relative muscle force?

Shouldn’t it be called minimize the sum of muscle forces or physical fatigue?

i.e. G(f(M)) = min (fi(M)/Ni)

I do not understand the term maximum? relative muscle force.


Best regards,


I might be way out my league here, but anyway…

The above “equation” is not the same as “minimize the sum of muscle forces”, because you minimize the objective function (G). This objective function only consider maximal activity of the muscles. Hence, a min/max formulation. To solve for the sub-maximally activated muscles they have implemented an iterative scheme that is described in the (unfortunately hard to come by) article:

[li]M. Damsgaard, J. Rasmussen & S.T. Christensen (2001): Inverse dynamics of musculo-skeletal systems using an efficient min/max muscle recruitment model. Proceedings of IDETC: 18-th Biennial Conference on Mechanical Vibration and Noise, Pittsburgh, September 9-13, 2001.[/li][/ul]

But I guess you simply could say that you minimize the normalized muscle force, then it might be easier to understand the concept.


thank you for your reply.

But what does it mean: maximal muscle activity

It is stated, that maximal muscle activity corresponds to a minimum fatigue criterion.

However, in my opinion muscles with a larger lever arm reduces the muscle activity and will lead to a minimum fatigue of muscles.

So finally I do not understand what maximal muscle activity means?

Best regards,


Hi Thomas,

The Min/Max muscle recruitment will try to minimize the activation of the maximal activated muscle in the model. This will lower the activation envelope, which is the maximum relative load of any muscle in the system.

Here is an explanation from the muscle recruitment tutorial:
“This would physiologically be a minimum fatigue criterion because fatigue is likely to happen first in the muscle with the maximum relative load, and it makes physiological sense that the body might work that way. It would mean that the body would maximize its endurance and precisely this criterion might decide survival of the fittest in an environment where organisms are competing with each other for limited resources”

With the introduction of Ver.4 several new criterias can now be used, polynomial, quadratic, please see http://www.anybodytech.com/index.php?id=923, for the details, there is a description of each of them.

Concerning the muscle lever arm, the min/max reqruitment will try to make use of all muscles with a usefull momentarm, as a soon as a muscle has a usefull moment arm it may be activated and help lower the activations of the other muscles. This will happen even if the momentarm is very small.

If this behaviour is undesired a polynomial criteria can be used, this is a sum of muscle activations lifted in a user defined power. With a very high power it will resemble the min/max criteria but when lowing the power, muscles with smaller momentarms will in general gradually be left unused, because the objective is a summation and it will have cost to have a muscle activated, so if it does not help significantly it will be left inactive.

Please ask again if you have further questions

Best regards