Writing the doc

docs/LinearHallEncoder.md

@@ -39,9 +39,36 @@ Ratiometric means that the output signal is proportional to the voltage supply t
 
 These oscilloscope traces are the sensor output when rotating the rotor forth and back. (a bit less than 180º on the 3rd motor)
 The channel 0 (Yellow) is the Hall 1 and the Channel 1 (Green) is the Hall 2
-![hall sensors traces](LinearHallTrace_3rd_motor_120deg.png)
+![hall sensors traces](courbes.png)
 
-### 3. Analysis
+We can see that in the first movement (positive rotation), the green is out of phase of π/2.`
+
+![Sinwave figure](cosSinEncoderDiagram.png)
+### 3. Encoding the position 
+1. Get the angle of one periodic signal
+
+Since the 2 signals correspond to a cos and sin signals, it is possible to compute the angle inside the period using arctan2 function. 
+
+2. Incremental position
+- increment periods 
+- start from known position (one end)
+
+## Coding the solution
+1. Get the angle in the perdiod
+
+In order to compute the angle from the cos and sin with atan, it is necessary to remap the values of the analog readings from -1 to 1. 
+Beforehand, the maximum and minimum peak of the signals need to be found. It can be done by swiping the motor on startup in open-loop mode.
+Then the arctan function can be applied. It is preferable to use arctan2 as it will give an angle within the 4 quadrants (-π,π). Whereas arctan give an angle between (-π/2,π/2). [Wikipedia](https://en.wikipedia.org/wiki/Atan2)
+
+```C++
+float norm(float x, float in_min, float in_max)
+{
+    return (float)(x + 1.0) * (2.0) / (float)(in_max - in_min) -1.0;
+}
+
+A = norm(analogRead(CH1),minCh1, maxCh1);
+B = norm(analogRead(CH2),minCh2, maxCh2);
+
+angle = atan2(A,B);
+``` 
 
-We can see that in the first movement (positive rotation), the green is out of phase and opposite 
-### 4. Encoder strategy