Look Up Table for position solver

Dear all,

I am trying to build a look up table to solve the position. It is basically a voltage-position conversion problem.

I will define the position in cartesian coordinates (x,y,z)

Each point of the position will have 3 volatge measured.

For example, at (0,0,0) the voltage is [5,1,2]

and at (1,0,0) the voltage is [3,2,4]

i will have 1000 points data.

How can i generate a look up table that when the input is [?,?,?] voltage, the output is [x,y,z] coordinate.

For example if the input voltage is [3,2,4] then the output will be (1,0,0)

It is also okay to use simulink or matlab

Could anyone help me? Please~

Thank you ver much

[Torsten]

Is there a unique position vector that corresponds to each voltage vector ?

And for your table lookup, you only choose voltage vectors for which there already exists a position vector in the table or is it necessary to perform interpolation ?

[Chun Wai KO]

Thanks for your reply

Yes, For different position, the voltage will be different

I will need the interpolation.

Or i may talk about my application more detailedly

i am trying to build an electromagnetic tracking system

My sensor can measure 3 voltage value in a point of position

And my tracking volume is 50cm x 50cm x 50cm

i will move the sensor in incremental step along x,y,z for 5 cm

So for example, i place my sensor at a point and define it as (0,0,0) and measure a voltage [v1,v2,v3]

then i move the sensor along x axis for 5 cm, the position will be (1,0,0) and measure the voltage again [v1', v2', v3']

So there are 1000 points in total. And what i want to do is after getting the 1000 points voltage-position data

Then i can solve the position by the real time measured voltage. Not only the position i measured, but also predicting the position like (1.5, 0 , 0)

[Chun Wai KO]

Any thoughts?

[Torsten]

Google

matlab & inverse interpolation

and take the first hit.

[Chun Wai KO]

i still cannot solve the problem.

Is it possible to map a vector to other vector?

[Torsten]

I wonder if the voltage vectors you get depend kind of continuously on position or whether the 6 neighbour voltage vectors to a position P = (i,j,k) (let's assume 1<i,j,k<10) are somewhat random.

[Chun Wai KO]

Yes, the voltage vector is depends on the position

it is basically a constant for each point of position

[Chun Wai KO]

while the voltage has a revese relationship to the distance from the sensor

i.e [10,10,10] will get the smallest voltage for each compoent of voltage [v1,v2,v3]

[Chun Wai KO]

My project is based on one paper. They used 5 voltage compoent and 6 position/orientation compoent to generate the the look up table. I am trying to do simplier model, so my voltage will only have 3 compoents and only x,y,z will be found

[Torsten]

So does it make sense to independently reverse interpolate in x-,y- and z- direction ?

Means : points with x-component 1 have all approximately voltage v1 in the first position, points with x-component 2 have all approximately voltage v2 in the first position and so on. And the same for y- and z- direction ?

[Chun Wai KO]

You mean use the 3 voltage to solve the x independently

and solve the y independently and solve the z independently

like decompose the x y z coordinate ?

[Torsten]

yes.

And in the mentioned paper the authors did not explain how this table look-up was done ?

Neuron networks, AI ?

[Chun Wai KO]

The author didnt explain..

But i think that decomposing the xyz may not work

For example for position [1 0 0] and position [1 0 1]

their x coordinate are both 1

but the voltage compoents for [1 0 0] and [1 0 1] are totally different

[Chun Wai KO]

Or i may try to use inverse interpolation to solve 1 voltage first

using 1 voltage compoent, means f(x,y,z) = v

what is the function to solve 3d reverse interpolation?

i know a function called contour() but it can only solve x and y

[Torsten]

Say you are to interpolate the position for the voltage vector that is half the voltage vector in position 1,0,0 plus half the voltage vector in position 1,0,1. Would it be acceptable to assume that the reverse interpolated position is 1,0,1/2 ?

[Torsten]

Concerning your idea:

Maybe

f(x,y,z) = v1

g(x,y,z) = v2

h(x,y,z) = v3

Three equations for contour in three unknowns x,y,z where f,g and h are the equations for the surface of voltage components 1,2 and 3.

Answer 1

[Torsten]

Use fmincon (or maybe better ga, since the objective function doesn't look differentiable) to solve the following problem:

min:( interp3(xdata,ydata,zdata,v1data,x,y,z) -v1q)^2 + (interp3(xdata,ydata,zdata,v2data,x,y,z)-v2q)^2 + ...

(interp3(xdata,ydata,zdata,v3data,x,y,z)-v3q)^2

for x,y,z under the constraints

0<=x,y,z<=10

Here, v1data,v2data and v3data are your voltages at (xdata,ydata,zdata) and (v1q,v2q,v3q) is the voltage combination for which you want to find the coordinates (x,y,z).

[Chun Wai KO]

Thank you very much for your effort

Is it a function similar to lsqnonlin?

I dont have much knowledge about it

but in my understnading, lsqnonlin can also be used to find the minimize value

[Torsten]

lsqnonlin and fmincon are optimizers that work if the objective function is differentiable. You can try both of them, but I think ga is better suited in your case because of the nonnecessary differentiability assumption.

[Chun Wai KO]

Is it the basic idea? Sorry i don't have the voltage data yet

seems i need to adjust the lsqnonlin parameter (like upper bound lower bound? or tolerance?)

Sorry i don't have much experience on it.

[Torsten]

xdata,ydata and zdata are vectors of length 11^3 since x,y and z coordinates run from 0 to 10.

v1,v2 and v3 are the associated voltage vectors, thus also of length 11^3.

If you take them randomly, you won't succeed with lsqnonlin because of differentiability considerations

Additionally, you must fix them since as you wrote above, they will be generated anew in every call to "position", and the solver will have a newly generated problem each time.

Upper and lower bounds for the solution are 0 and 10 for x,y and z.

[Chun Wai KO]

i am trying to do a small dimenson model to verify the code first

let's say it is a 3x3x3 dimenson

i know that the voltage v1,v2,v3 will be also 3x3x3

but you mean xdata,ydata,zdata are also 3x3x3?

i thought for each, they are 3x1 only

[Torsten]

Look at the documentation of interp3.

xdata,ydata and zdata used in interp3 must be generated by the meshgrid command from the corresponding 1x3 vectors.

In your case, calling interp3 as interp3(vidata,x,y,z) where vidata are the 11x11x11 matrices of voltage data might be the easiest way.

[Chun Wai KO]

Dear Sir, i finally completed my measurement system and got a set of data.

i am trying to do a 1 degree of freedom system first.

However, when i use the lsqnonlin, it shows 'the relative norm of the current step

is less than options.StepTolerance'.

For example, my data is

[4.3827, 4.2624, 3.2585, 0.9934, 0.5985, 0.1775, 0.2299, 0.0952, 0.0935, 0.0954, 0.08]'

The lsqnonlin can solve the position when the v1q is large (>0.5).

When the v1q is small, the lsqnonlin cannot solve the position.

Even if i set the StepTolerance = 1e-100, the Optimization stopped and show relative norm(step) = 0e00

i may show you the code here.

Thank you very much

[Torsten]

You didn't specify how to extrapolate values that are outside [1:10]. interp1 will return NaN in the case that the suggested "currentPandO" value is outside this interval.

I gave you the advice to

a) use ga

b) constrain the solution

You use lsqnonlin (wrong, since you have to specify out = interp1(y,x) - vq) and don't constrain the solution. So how can I help you ?

[Chun Wai KO]

Dear Sir,

i tried to constrain the solutuon to [1:10] for the lsqnonlin

but it still cannot compute the correct solution

y = [4.3827, 4.2624, 3.2585, 0.9934, 0.5985, 0.1775, 0.2299, 0.0952, 0.0935, 0.0954, 0.08]'

when my vq= 3, it can compute a solution x = 3.1161

but when the vq is small, for example vq = 0.7, it shows that x = 7.00

i expected the value should be bewteen 4 to 5 from the interpoation.

It only happens when the vq is small. (it is okay when the vq is larger than 2)

I don't know why it happens.

I will try ga also

Thank You very much

[Torsten]

vq = 0.7;
y = [4.3827, 4.2624, 3.2585, 0.9934, 0.5985, 0.1775, 0.2299, 0.0952, 0.0935, 0.0954, 0.08];
fun=@(x) interp1(1:numel(y),y,x,'pchip') - vq;

x0 = 1;
[solution,res] = lsqnonlin(fun,x0,1,numel(y))

But you should really try ga instead of lsqnonlin.

[Chun Wai KO]

Dear sir,

I am trying to use the ga to build the 2D system

It can compute a solution but the output keeps varying. (not very accurate)

So i tried to set the options:

options = optimoptions(@ga,'FunctionTolerance',1e-16,'MaxGenerations',150,'MaxStallGenerations',150,'ConstraintTolerance',1e-16)

The output becomes more accurate

however, i found that the computation time for the ga is quite long for each output (1-2 seconds)

As my application is a real time tracking system, this situation is not ideal

is there any way to keep the accuracy while shortening the run time?

thank you very much