Algorithm II. Inertial-Based Shifting Trilateration.

Input: Asynchronous readers’ detections, raw inertial sensor data

Output: Updated Absolute Tag Location (ATL)

1: initialize tag memory: empty Detections, IS, TDV

2: while (tag is not detected)

3:   Do   record data from inertial sensors in IS table

4: End While

5: Read Detections, IS, and TDV tables // a tag is detected

6: Update Detections table with current detection

7: If (Detections table has 2 entries)

8:  Calculate displacement vector(s) $v_{0\to 1}$ based on IS table

9:  $C_0$ = Circle around $AT{L}_0$ coordinates in $l_0^n$ by a radius of $\Vert v_{0\to 1}\Vert$

10:  $C_1$ = the circle around the current reader by a radius of RSSI mapped detection range

11:  Calculate the set of intersection points between $C_0$ and $C_1$

12:  All intersection points $I$ are reported to a central server as $AT{L}_{1\_i},$ $i\in I$

13: Else if (Detections table has 3 entries)  // check for enough detections to perform IBST

14:  Calculate displacement vector(s) $v_{1\to 2}$ based on IS table

15:  $C_0$ = Circle around $AT{L}_0$ coordinates in $l_0^n$ by a radius of $\Vert v_{0\to 2}\Vert$

16:  $C_1$ = Circle around $AT{L}_1$ coordinates in $l_1^n$ by a radius of $\Vert v_{1\to 2}\Vert$

17:  $C_2$ = the circle around the current reader by a radius of RSSI mapped detection range

18:  Calculate the set of intersection points A’s between $C_0$ and $C_1$

19:  Calculate the set of intersection points B’s between $C_1$ and $C_2$

20:   If (there are 2 intersections between $C_1$ and $C_2$)

21:    $AT{L}_2$ = $B_{.}$ of min $\left(\Vert A_1-B_1\Vert ,\Vert A_1-B_2\Vert ,\Vert A_2-B_1\Vert \Vert A_2-B_2\Vert \right)$,

22:   Else if (there are 4 intersections between $C_1$ and $C_2$)

23:    $AT{L}_{2\_1}$ = $B_{.}$ of min $\left(\Vert A_1-B_1\Vert \Vert A_1-B_2\Vert \Vert A_2-B_1\Vert \Vert A_2-B_2\Vert \right)$,

24:    $AT{L}_{2\_2}$ = $B_{.}$ of min $\left(\Vert A_3-B_3\Vert \Vert A_3-B_4\Vert \Vert A_4-B_3\Vert \Vert A_4-B_4\Vert \right)$,

25:    Update Detections table with $l_2^n$ and TDV table with $v_{1\to 2}$, go to 2

26:   Else (there is no intersection between $C_1$ and $C_2$)

27:    Find vector from ${\overrightarrow{D}}_i=A_i\to R_2,$ find $D_i=\Vert A_i-R_2\Vert$ for $i$=1 to 4

28:    $AT{L}_2$ = $R_2+d_2.{\overrightarrow{D}}_i$ of min $D_i$,

29:   End if

30:  Report ATLi to a central database server

31:  Update Detections table with $l_3^n$ and TDV table with $v_{1\to 2}$, go to 2

32: Else

33:  go to 2

34: End If