How to implement Shrinking-Circle Method Based on Time Difference of Arrival (TDoA) in MatLab?

Im working on a robotics SWARM problem, and need to localized the robots using acoustic modems. Its a Source Localization Based on Time Difference of Arrival (TDoA) Measurements. I have found this approach Two New Shrinking-Circle Methods for Source Localization Based on TDoA Measurements . Here the link of the approach :

Article Two New Shrinking-Circle Methods for Source Localization Bas...

And they described the approach and mentioned its implementation in MatLab. The approach can be described in this steps

Step 1. Compute the maximum radius Rmax and minimum radius Rmin of r1.

Step 2. Compute Rmid, where Rmid = (Rmax + Rmin)/2.

Step 3. Calculate the solution X when r1 equals Rmid, and then calculate f(r1).

Step 4. Update the value of Rmax and Rmin. If f(r1) > 0, Rmin = Rmid; otherwise, Rmax = Rmid.

Step 5. Compare the value of |f(r1)| with a threshold TH1 and compare the value of |Rmax − Rmin| with a threshold TH2. If |f(r1)| < TH1 or |Rmax − Rmin| < TH2, terminate the procedure; otherwise, return to Step 2. When the procedure is finished, X is the coordinate of the target.

So please can help to find some possible solution implementation in Matlab or some help to point it out how can be done that in MatLab?

Thanks

Answer 1

[Abhishek Chakram]

Hi Bojan Andonovski,

I see that you want to implement Shrinking-Circle Method Based on Time Difference of Arrival (TDoA) in MATLAB. The approach you are referring to is an iterative method for source localization using Time Difference of Arrival (TDoA) measurements. The shrinking-circle method refines the estimated position of the source by iteratively adjusting the radius of a circle centred on a receiver until the circle passes through the source.

To implement this in MATLAB, you will need to establish a few things:

• TDoA Measurements: The time differences of arrival at various receivers from the source.
• Receiver Positions: The known positions of the receivers.
• Initial Estimates: Initial estimates for Rmax and Rmin. These could be based on the maximum possible distance in your environment and a minimum distance that is greater than zero, respectively.

Here is a sample code for the same:

function [X] = shrinking_circle(TDoA, receiver_positions, initial_Rmax, initial_Rmin, TH1, TH2)
    % TDoA: Time Difference of Arrival measurements
    % receiver_positions: Known positions of receivers
    % initial_Rmax: Initial maximum radius estimate
    % initial_Rmin: Initial minimum radius estimate
    % TH1: Threshold for the function f(r1)
    % TH2: Threshold for the difference between Rmax and Rmin

    % Step 1: Initialize Rmax and Rmin
    Rmax = initial_Rmax;
    Rmin = initial_Rmin;

    % Start the iteration process
    while true
        % Step 2: Compute Rmid
        Rmid = (Rmax + Rmin) / 2;

        % Step 3: Calculate the solution X when r1 equals Rmid
        % This involves solving the equations based on TDoA measurements
        % and the known positions of the receivers.
        X = calculate_position(TDoA, receiver_positions, Rmid);
        f_r1 = calculate_f_r1(X, TDoA, receiver_positions);

        % Step 4: Update Rmax and Rmin
        if f_r1 > 0
            Rmin = Rmid;
        else
            Rmax = Rmid;
        end

        % Step 5: Check termination conditions
        if abs(f_r1) < TH1 || abs(Rmax - Rmin) < TH2
            break;
        end
    end

    % The final position of the target
end

function X = calculate_position(TDoA, receiver_positions, Rmid)
    % Implement the position calculation using TDoA and Rmid
    % This typically involves solving a set of nonlinear equations
    % which can be done using optimization techniques or algebraic methods.
    % Placeholder for actual implementation:
    X = [0, 0]; % Replace with actual calculation
end

function f_r1 = calculate_f_r1(X, TDoA, receiver_positions)
    % Implement the calculation of the function f(r1) based on the
    % estimated position X, TDoA measurements, and receiver positions.
    % Placeholder for actual implementation:
    f_r1 = 0; % Replace with actual calculation
end

In the above code, “calculate_position” and “calculate_f_r1” are placeholder functions where you would implement the specific calculations required to determine the source position X and the evaluation of the function f(r1) based on the TDoA measurements and receiver positions.

I hope this helps!

Best Regards,

Abhishek Chakram