Dr Andy H. Kemp A.H.Kemp@leeds.ac.uk 08/05/2018 Tracking Technology Dr Andy H. Kemp A.H.Kemp@leeds.ac.uk 08/05/2018

On-going position estimates 08/05/2018 Tracking Technology Localization data Position estimate Initial position estimate On-going position estimates User input Localization data Time-based Methods: ToA/ToF, TDoA Angle of Arrival: AoA Signal strength: RSSI Obtain position estimate Trilateration, Triangulation, Statistical Technique Connectivity Tracking algorithm Tracking Algorithm Tracking algorithms, e.g. Kalman fitlers, particle filters…

Basic localization methodology r1 r0 1 r2 2 So where are we? 08/05/2018

Basic localization methodology So how do we actually do this? Trilateration (cf. triangulation). Assume: 3 anchors Ai i.e. 3 devices at known positions (xi, yi, i = 0..2). We have some way of determining the distance/range, ri, from the anchors to our unknown position (xs, ys). From Pythagoras’ theorem we can readily see that: (ri)2 = {(xs – xi)2 + (ys – yi)2} i x y s (xi,, yi) (xs,, ys) ri (xs – xi) (ys – yi) 08/05/2018

Analytical methodology Solve system of equations (using Pythagoras!) (xi, yi) : coordinates of anchor point Ai, ri distance to anchor Ai from s which is at (xs, ys). (ri)2 = {(xs – xi)2 + (ys – yi)2} for i = 0..2 We now construct 3 equations with 2 unknowns (xs, ys). By subtracting 3rd equation from 1st & 2nd we make the equations linear in our unknowns: (r02 – r22) - (x02 – x22) - (y02 – y22) = 2xs(x2 – x0) +2ys(y2 – y0) (r12 – r22) - (x12 – x22) - (y12 – y22) = 2xs(x2 – x1) +2ys(y2 – y1) 08/05/2018

Analytical methodology This can be expressed in matrix form as: (r02 – r22) - (x02 – x22) - (y02 – y22) = 2 x2 – x0 y2 – y0 xs (r12 – r22) - (x12 – x22) - (y12 – y22) x2 – x1 y2 – y1 ys In matrix form this is: [b] = [A][s] With [b] = 1 (r02 – r22) - (x02 – x22) - (y02 – y22) 2 (r12 – r22) - (x12 – x22) - (y12 – y22) [A] = x2 – x0 y2 – y0 and [s] = xs x2 – x1 y2 – y1 ys 08/05/2018

(xs, ys) = (2, 5) Practice Time for an exercise: 08/05/2018 Practice Time for an exercise: We have 3 anchors A0..A2 on a flat x, y, grid measured out in metres, at: A0(10, 5), A1(2, 10) and A2(10, 15). The range from each anchor to the subject is r0 = 8m, r1= 5m, and r2 = 12.8m. Where is the subject? So what do you think is the best way to learn how to use this maths? Practice it right? (Answer should not be on student slides… And should only come up here after a key press… x, y of subject is 6.25, 6.25 Probably useful to draw this on an x, y coordinate system.) (xs, ys) = (2, 5) 08/05/2018

08/05/2018 Problem What happens if we can’t get an accurate range measurement from our satellite? There will always be PDP on range ESTIMATES. 08/05/2018

Ambiguity r1 r0 r2 1 So where are we? 08/05/2018 Ambiguity r1 1 r0 Range estimate: most probable position. r2 2 So where are we? 08/05/2018

GDOP 08/05/2018

Probabilistic ranges From our ranging mechanism we obtain a range estimate Provided this is unbiased, estimates will have a mean of the correct range Each estimate will have a random error

Resolution of some current technologies

Conclusion The tracking methodology relies on some kind of positioning method The positioning method inherently displays probabilistic inaccuracy Averaging will enhance an unbiased system Tracking algorithms can enhance positioning performance

(Source: Da Zhang, Localization Technologies for Indoor Human Tracking,2010, IEEE)