Android Question Solves a formulation of n-D space trilateration problem using a nonlinear least squares optimizer. Input: positions, distances Output: centroid with

float ra = beacon1.filteredDistance;
float rb = beacon2.filteredDistance;
float rc = beacon3.filteredDistance;

Sub Trilaterate(positions() As Float, distances() As Float) As Float()
    If positions.Length <> 6 Then Log("!!! s/b 3 positions")
 
    Dim xa As Float = positions(0)
    Dim ya As Float = positions(1)
    Dim xb As Float = positions(2)
    Dim yb As Float = positions(3)
    Dim xc As Float = positions(4)
    Dim yc As Float = positions(5)

    If distances.Length <> 3 Then Log("!!! s/b 3 distances")
 
    Dim ra As Float = distances(0)
    Dim rb As Float = distances(1)
    Dim rc As Float = distances(2)
    
    'var S = (Math.pow(xc, 2.) - Math.pow(xb, 2.) + Math.pow(yc, 2.) - Math.pow(yb, 2.) + Math.pow(rb, 2.) - Math.pow(rc, 2.)) / 2.0;
    Dim S As Float = (xc * xc - xb * xb + yc * yc - yb * yb + rb * rb - rc * rc) / 2
 
    'var T = (Math.pow(xa, 2.) - Math.pow(xb, 2.) + Math.pow(ya, 2.) - Math.pow(yb, 2.) + Math.pow(rb, 2.) - Math.pow(ra, 2.)) / 2.0;
    Dim T As Float = (xa * xa - xb * xb + ya * ya - yb * yb + rb * rb - ra * ra) / 2
 
    'var y = ((T * (xb - xc)) - (S * (xb - xa))) / (((ya - yb) * (xb - xc)) - ((yc - yb) * (xb - xa)));
    Dim y As Float = ((T * (xb - xc)) - (S * (xb - xa))) / (((ya - yb) * (xb - xc)) - ((yc - yb) * (xb - xa)))
 
    'var x = ((y * (ya - yb)) - T) / (xb - xa);
    Dim x As Float = ((y * (ya - yb)) - T) / (xb - xa)
 
    Return Array As Float(x, y)
End Sub

Dim Positions(6) As Float = Array As Float(0, 0, 100, 0, 0, 100)

Dim NumBeacons As Int = Positions.Length / 2
If Positions.Length <> NumBeacons * 2 Then Log("!!! odd number of beacons")

Log("Random X,Y" & TAB & "Distances A, B, C" & TAB & TAB & "Reconstructed X,Y")

For Burl = 1 To 10
    Dim X As Float = Rnd(10, 90)
    Dim Y As Float = Rnd(10, 90)

    Dim Distances(NumBeacons) As Float
    For b = 0 To NumBeacons - 1
        Dim dx As Float = X - Positions(b * 2)
        Dim dy As Float = Y - Positions(b * 2 + 1)
        Distances(b) = Sqrt(dx * dx + dy * dy)
    Next

    Dim EstLoc(Positions) As Float = Trilaterate(Positions, Distances)

    Log( _
        X & TAB & _
        Y & TAB & _
        NumberFormat2(Distances(0), 1, 3, 3, False) & TAB & _
        NumberFormat2(Distances(1), 1, 3, 3, False) & TAB & _
        NumberFormat2(Distances(2), 1, 3, 3, False) & TAB & _
        NumberFormat2(EstLoc(0), 1, 4, 4, False) & TAB & _
        NumberFormat2(EstLoc(1), 1, 4, 4, False) _
    )
Next

Sub KalmanFilter(distance As Double) As Double
    ' Initialize variables
    Dim q As Double = 0.001 ' Process noise
    Dim r As Double =  0.1     ' Measurement noise
    Dim x As Double = 0 ' Initial estimate
    Dim p As Double = 1 ' Initial error estimate
    Dim k As Double ' Kalman gain
   
    ' Prediction
    x = x ' No prediction step in this example
    p = p + q
   
    ' Update
    k = p / (p + r)
    x = x + k * (distance - x)
    p = (1 - k) * p
   
    Return x
End Sub

Here is a complete example of how to use the routine to calculate the accurate distance to an iBeacon:

```basic4android
' RSSI smoothing
The value of Alpha in the RSSISmooth() function determines the amount of weight given to the current RSSI value compared to the previous RSSI value. A higher Alpha value will give more weight to the current RSSI value, while a lower Alpha value will give more weight to the previous RSSI value.

In general, an Alpha value between 0.5 and 0.8 is a good starting point. An Alpha value less than 0.5 can cause a response that is too slow to the variation of the RSSI, while an Alpha value greater than 0.8 can cause a response that is too sensitive to the variation of the RSSI.

To choose the optimal value of Alpha, you need to run some tests with your data. You can start with an Alpha value of 0.5 and then increase or decrease it based on your test results.

Here are some tips for choosing the value of Alpha:

If the RSSI is relatively stable, a higher Alpha value can be used.
If the RSSI varies rapidly, a lower Alpha value can be used.
If the RSSI is subject to interference, a lower Alpha value can be used.
Sub RSSISmooth(RSSI, Alpha)

     If RSSI < 0 Then
         ' RSSI invalid
         RSSISmooth = RSSI
     Else
         RSSISmooth = (Alpha * RSSI) + ((1 - Alpha) * RSSISmooth)
     End If

     Return RSSISmooth
End Sub

' Particle filter

RSSIThreshold is a variable that is used in the routine to calculate the accurate distance to an iBeacon. This variable determines the minimum RSSI value that will be considered valid for distance calculation.

In general, an RSSIThreshold value between -60 and -80 is a good starting point. An RSSIThreshold value less than -60 may cause some iBeacons to be missed, while an RSSIThreshold value greater than -80 may cause inaccurate distance estimates.

To choose the optimal RSSIThreshold value, you need to run some tests with your data. You can start with an RSSIThreshold value of -60 and then increase or decrease it based on your test results.

Here are some tips for choosing the value of RSSIThreshold:

If the environment in which you are measuring distance is noisy, you can use a higher RSSIThreshold value.
If the iBeacon is placed in a location where it is subject to interference, a higher RSSIThreshold value can be used.
If you need greater accuracy in distance estimation, you can use a lower RSSIThreshold value.

Sub ParticleFilter(RSSI, RssiThreshold, ParticleCount, Alpha)

     ' Create a particle vector
     Dim Particles(ParticleCount) As Particle

     ' Initialize the particles
     For i As Integer = 0 To ParticleCount - 1
         Particles(i).RSSI = RSSI
         Particles(i).Distance = GetDistance(Particles(i).RSSI)
     EndFor

     ' Perform particle update
     For i As Integer = 0 To ParticleCount - 1
         Particles(i).RSSI = RSSISmooth(Particles(i).RSSI, Alpha)
         Particles(i).Distance = GetDistance(Particles(i).RSSI)
     EndFor

     ' Returns the average distance of the particles
     Dim DistanceMean As Double
     DistanceMean = 0
     For i As Integer = 0 To ParticleCount - 1
         DistanceMean = DistanceMean + Particles(i).Distance
     Next
     DistanceMean = DistanceMean / ParticleCount

     Return DistanceMean
End Sub

' Main function

Sub Main()

     ' Settings
     Dim RssiThreshold As Double = -50
     Dim ParticleCount As Integer = 100
     Dim Alpha As Double = 0.7

     ' Read the RSSI value of the iBeacon
     Dim RSSI As Double
     RSSI = InputBox("Enter the RSSI value of the iBeacon: ")

     ' Calculate the accurate distance
     Dim Distance As Double
     Distance = ParticleFilter(RSSI, RssiThreshold, ParticleCount, Alpha)

     ' Display the distance
     MsgBox(Distance)

End Sub
```

This example works like this:

1. The `RSSISmooth()` function is used to apply an RSSI filter to the RSSI value provided by the user.
2. The `ParticleFilter()` function is used to apply a particle filter to the filtered RSSI value.
3. The `GetDistance()` function is used to calculate the distance of the iBeacon in meters based on its RSSI.

To run this example, you need to set the values of the `RssiThreshold`, `ParticleCount`, and `Alpha` variables according to your needs.

Here's an example of how to use the example:

```
Enter the RSSI value of the iBeacon: -60

Distance: 2.4 meters
```

In this example, the iBeacon distance is 2.4 meters. This is more accurate than the distance calculated without applying RSSI smoothing and particle filtering.