I have some trouble understanding a formula from a report : https://www.samba.org/tridge/UAV/madgwick_internal_report.pdf
It is formula (20) (Page 7). Could you tell me where it comes from? I can't find anything that resembles in litterature...
Here is an image of the formula: http://www.les-mathematiques.net/phorum/addon.php?4,module=embed_images,url=http%3A%2F%2Fs21.postimg.org%2Ft89ej68k7%2Fpourlesmaths.png
The author claims that "Equation (20) computes the gradient of the solution surface defined by the objective function and its Jacobian"and I don't even understand what he means by gradient since f is a function that goes from R^4 into R^3.
Thanks in advance for your answer
yeah, got the same question :), this is what i think:
we use gradient descent algorithm. A) If your target function is a single function (whatever variables number is) it is simple case described in all literature - you compute gradient without this multiplication from equation (20).
B) If your target function is composed of more functions - it is a vector of function - like in this algorithm by Madgwick :
d - s = [dx dy dx] - [sx sy sz] -> [dx-sx dy-sy dz-yz] [f1 f2 f3]
In this case you also compute gradient for each function, wchich forms a vector, which is a Jacobian matrix.
but how you want to optimize all functions at once??? when you optimzie your step for (dx-sx), you could ruin value of other function e.g. (dz-sz) - (not sure if this is true :)
How can you solve this? you need to multiply each target function component by its value, so the step for those which are already optimized (d-s ~= 0)(x,y or z) would have weight around 0.
Thats mean this multiplication is necessary for this algorithm to produce some solution, but it is actually synthetic, artifical. I feel like this is synthetic.
I didnt found any serious literature that states that ,,if you use vector target function you need to multiplu the gradient vector (jacobian matrix) by target function values vector'', so i cant provide proof for my elaborate but this is how i understand this. I suppose it is preety much fair.
Maybe this is so obvious nobody mention this. It is popular approach.
regards :)