OK, so we need to recreate the varying radial velocities that are currently harvesting gain in v1.
Every other possibility has been methodically eliminated; this unequal radial velocity between inbound vs outbound masses is
evidently carving out a nice fat slice of free pie..
It
may be that the
optimum radial speed variation for harnessing this gain is different from the one currently delivering the goods; for now, however, let's just try and refine the technique we know works..
So, we
now know that this MoI-control trick was only initially useful when using a single pair of masses on a vMoI; once a second pair are mounted orthogonally in reciprocal phase, the MoI remains constant
regardless of the individually-varying MoI of each pair, phasing in perfect cancellation...
..thus in the v1 gain config, this MoI-control technique
was entirely redundant for its
intended purpose; the net MoI would've remained constant without it, regardless...
..however,
without that time-wasting, CPU-consuming oversight, we'd once again have come up empty of any gain from the s&b cycles, so, classic 'serendipitous mistake' there..
The priority now is to refine that technique; the most practical solution i've settled on thus far is 'reactive feedback' - where the instantaneous MoI of each pair is constantly recalculated on the fly, applying corrective changes in velocity, force, acceleration or displacement to increment the radial excursions by
however much is required to maintain a given target MoI.
These techniques are a bit of a black art - for any working solution, i've little idea
how it's working..
it just does. Kind of abrogates the whole problem over to the CPU so i can just clear
that hurdle and stumble on to the next... but it's not 'clean' or transparent enough. A simpler, entirely passive solution
is possible, with a little thought, so let's have at it..
This simple example encapsulates the problem:
The actuator controls are exposed so you can see how it works:
• actuators are being controlled for 'length'
• the wheel radius is 1 meter, the masses are 1 kg, and the MoI of the wheel itself is ignored - we're only looking at the net MoI of the two masses
• 'point[10]' is the base end of the orange actuator, connecting to the wheel rim, and effectively being used as an angle sensor
• we could use 'wheel angle' instead for this purpose (ie. derived from 'body[1].p.r') - it may even be better to do so - but this works for now
• you can see the combined MoI of the pair goes from a peak value of '1', down to '0.5' as they cross the center, and back up to '1' on the other side..
• ..the objective is to
smoothly increase the distance between the masses as they approach the center, decreasing it again as they cross back out
• the distance needs to vary by an amount equal to the square root of two; that is, the current fixed distance between them needs to rise from 1 m, up to 1.414 m, and then drop back down again
So, the 'best' solution here has got to be the one that takes the bull by the horns; first off,
why does the MoI vary this way in the first place?
• We know MoI = mass * radius squared
• thus outbound MoI is changing faster than inbound MoI
• this rate of change is a function of instantaneous radius
Thus, in principle, this should be a very simple problem to solve cleanly!
However if i was any good at maths i'd've solved it already, instead of just shorting the 'length' control to the MoI calc.. but no more hillbilly ingenuity, now we need to reduce the solution to a simple equation..
So, what is that equation, then?
There's no 'wrong' answer here - in fact the more you think about it, the more ways you start to consider of approaching it..
It's obviously an eminently tractable problem.
For just a few of the possibilities:
• just one of the current actuators could move
both masses; with the second actuator now placed
between them; so one moves them across the wheel, while the other regulates their gap
This may not be ideal, since the two sets of calcs could end up producing resonances, over/under compensating for one another.. OTOH it may work fine, just seems less than ideal tho..
• since the wheel radius is 1 m, the peak distance between the weights has to be reached as they pass the 0.5 m mark; hence we could produce a multiplier that increases when moving
towards that 0.5 m threshold, and decreases when moving
away from it
This however leaves the issue of non-linearity of the MoI variation wrt radius - if the MoI contour's non-linear, then a linear variation in gap distance is still going to see MoI changes..
I think the best option's this one:
• generate a multiplier that is a function of radius, per
mr², applied on a per-actuator basis
So each actuator's just doing its own thing, oblivious to what the other's doing or their combined MoI..
Basic objective for each mass:
• change radius as a function of wheel angle
• accelerate when inbound, as a function of mr²
• decelerate when outbound, as a function of mr²
If both actuators are observing these rules individually, then the combined MoI should remain constant, even as their radial speeds vary respectively..
..thus no feedback control required, no third actuator / rotary solenoids, and we can harvest the gain with the
same actuators powering the GPE inputs..
This would obviously leave us with a big head-start on harvesting the gain with GPE lifts using radial springs..
No simming knowledge required either; all we need is the general form of the solution..
