There was a significant use of trigonometry, geometry, and physics

The given image displays the orientation of the 3 axes of the phone. Clearly, it is important to take the orientation of the phone into account while tracing motion because we dont know how the user is holding the phone. Thus, we must factor in the following rotation matrix

Source : OpenGL Angles to Axes

Till now, we only have information about the acceleration imparted to the phone by the user along each axis. We now need to get information about the velocity, and then the position
As a quick and computationally-light method of integration, we will proceed with using Riemann Sums

curr_vel = (curr_acc + prev_acc)dt/2
curr_pos = (curr_vel + prev_vel)dt/2

Where dt is the sampling interval of the device.

As expected, the data was nowhere close to perfect and hence, neither were the derived results. There were two techniques used to decrease this error

1. APPLYING THRESHOLD ON ACCELERATION
   
   • Intuition
     Even when the device was motionless, the sensors would return a fluctuating residual value. These values must be dampened.
   • Implementation
     

       var THRESHOLD = n; // value chosen after observing several readings
     
       function applyAccelerationThreshold(val) {
       	  if (val < THRESHOLD) {
     		  return val/THRESHOLD
     	  }
     	  return val
       }
     											

2. APPLYING THRESHOLD ON VELOCITY
   
   • Intuition
     A residual velocity was regularly observed due to imperfections in acceleration readings. Hence, maintaining a static baseline value would not work
     Hence, at each step, we calculate an appropriate baseline for velocity
   • Implementation
     

     var VEL_THRESHOLD = n // value chosen after observing several readings
     var vel_shift // new baseline value
     
     function calculateVelocityThreshold(val) {
     	var arr = /* last n values including val */
     	var range_amp = 0
     	arr.forEach(function (vel) {
     		range_amp = abs(vel - val) > range_amp ? abs(vel - val) : range_amp
     	})
     	if (range_amp < VEL_THRESHOLD)
     		vel_shift = range_amp
     }
     											

     The above code calculates how much the baseline shifts after each new velocity is calculated. This shift can then be accomodated into the velocity before using it to calculate the position

This graph shows the acceleration, velocity and position values after a left-right motion. These graphs only show readings along the x-axis




We see a spike in acceleration indicating motion in a given direction and the corresponding deceleration.




The changes in acceleration are reflected in the velocity. In this case, the acceleration readings were clear enough to not warrant a baseline shift for velocity




Clearly, we see the position value going down (indicating motion towards the left) and then come back up (indicating motion towards the right)