the documentation says that there are several objectives selectable, but they have one thing in commen: Apart from auxillary abjectives they are in general the sum of the power of the muscle activities, while the power § ranges from 1 to 5. In the case of the strict MinMax the power is “infinite” AND the “infinite” root is extracted.
In the case of auxillary objectives the objectives of different powers are added. I wonder why (apart from MinMax) the pth root is NOT extracted. In the case of non-auxillary objectives it is not important, but for objectives with auxillary term, it would be more straight forward to extract the pth root before adding the terms. Is this but a typo?
I know the tutorials on muscle recruitment. The state the same as the documentation.
However, my concern was with regard to “units”. I know that the activity has no “units” like N. But nevertheless, for auxillary objectives, activities are added which are raised to different powers.
2nd concern: The MinMax recruitment behaves different than the other objectives in so far as here - after powering and summing up - the root is extracted (of course not literally). This is - as far as the documentation and tutorials say - different for the other objectives, where only power raising (and summing up) is performed.
Summary: Is the documentation correct? If yes, is this sensible?
You raise a good point. We agree that extraction of the n’th root makes no difference when we are dealing with a pure polynomial because the powersum and the root of the powersum have their minimum the same place.
The min/max case is formulated and solved completely differently. In this case, the max is not raised to any power, but it makes not difference because the max is the max regardless of whether it is raised to a power.
The aux-type criteria are more controversial in the sense that they do not have the same easy interpretation as the aforementioned types. As you mention, activity is dimensionless, so we are not formally violating any dimensions by having things in different powers.
There are some algorithmic reasons why we do not take roots. The algorithms we have are very specialized and work for convex mixtures of polynomial forms but not necessarily when we take roots.
Regardless of whether we take roots or not, there is a problem of weighing the aux term against the max term. This is handled with a weight factor, but the size of this factor depends somewhat on the problem. The trouble is that the squared sum of muscle activities (or its root) increases with the number of muscles in the problem. So of we divide one muscle into two muscles of half size, then we influence the size of the aux term. This is an unsolved problem and it is left to the user to select the weight factor.