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I am trying to optimice a model for balance in a crouch position, but when using the BalanceMeasure function the first thing the model does is stand up. I have tried to use the DofMeasure function on the pelvis, but when I use it the model falls apart regardless of the weight I assign to the function.

I need the model to be able to move the pelvis (reducing its displacement as much as possible) and also to be able to stretch and contract the knees, they cannot be locked since the final objective is to evaluate the model's balance on a laterally rotating platform.

Is there any way to use the BalanceMeasure function while keeping the COM in a lower position? If no, is there any other function I can use?

The best approach in your case would be to either use the MimicMeasure, to try to keep the model in a specific pose (you do need a .sto file with the crouching pose), or to use the ScriptMeasure. For inspiration for the latter, please look at Tutorial 6a.