Description
Requested by Brian Umberger. Here are two use cases he describes:
We are currently testing the effectiveness of control effort raised to various powers in predictive simulations of walking. Based on prior research, it is unlikely that effort^1 will be the best performing objective function, but it would be a useful point of comparison and its absence makes the analysis incomplete.
I have found that minimizing muscle activations or excitations weighted by individual muscle volumes yields simulation results that are similar to what we get from minimizing metabolic cost (as in Ackermann & van den Bogert 2010), with the advantage of converging better and faster than when minimizing metabolic cost using a Bhargava/Umberger-type energy model. The actual metabolic energy consumption can still be calculated post-hoc using a metabolic cost model. In Moco, I think this can currently only be done for activation or excitation raised to a power of 2 or higher. That may have some merit, but muscle volume weighted activation raised to the power 1 (and also divided by displacement) would be closer to a typical metabolic cost of transport measure that people use.
The key challenge is that with an exponent of 1, controls and states can go negative. This could be addressed by using an absolute value function, although this is non-smooth. I believe CasADi has a differentiable implementation of abs(x) that we might be able to leverage.