## 4 Simulation Results
### Fixed-Step GD Estimation  
Solving a multilateration problem with fixed-step gradient descent can diverge if the step is too large or slow down convergence considerably if it is too small. At the expense of additional optimization cost, some authors [37; 39] have proposed to use the backtracking line search algorithm, thus solving an additional optimization problem in order to define a suitable step length that satisfies certain conditions. Here, we propose performing a statistical analysis to determine a good approximation of the best fixed-step descent parameter for each formulation of the estimation problems (3)-(6). The convergence results for the formulations (3) and (4) (without explicitly considering position and range measurement noises) are presented in Figure 2. A solution is considered convergent if the positioning error, i.e., the Euclidean distance between the real position of the target and its estimate, is less than 20 cm. Obviously, it can be noted that increasing the fixed descent step length reduces the convergence rate, in particular when the number of neighboring nodes used in the estimation increases. Therefore, by reducing the descent step, the convergence rate is increased. Using problem formulation (3), notice the appearance of a plateau whose value depends on the number of neighboring nodes used in multilateration. This means that below a certain value, it is useless to continue to decrease the step of descent, which will lead to increasing the computation time without improving the convergence success. It should be noted that these \"plateau\" values will be chosen as fixed parameters of the descent step in the comparative simulations. It can also be noted that increasing the number of measurements improves the convergence rates up to a certain threshold (here, 8), beyond which no improvement is detected. Similar observations can be made for the problem formulation (4), noting that convergence for very small step values is extremely slow and the algorithm stops after exceeding the maximum number of iterations. The number of iterations until convergence for both formulations is given in Figure 3 and confirms this observation. It can be concluded that for equivalent performances, problem formulation (4) converges much more slowly than (3). Figure 2: Convergence results on 10,000 simulations using GD with fixed-step descent for estimation problems (3) (**a**) and (4) (**b**), respectively, without explicitly considering positioning and measurement error models and by varying the number of neighboring nodes from 4 to 10. The same tests are carried out by incorporating in the cost functions the position (9) and the measurement (10) error models in the estimation problems (5) and (6). Figures 4 and 5 give the results of convergence and the number of iterations until convergence over 10,000 simulations for the two formulations and for a different number of neighboring nodes, depending the length of the fixed step and the number of distance measurements. The profiles of the curves are similar to the results of the formulations without error models; the only changes are a decrease in performance due to the integration of errors, which impacts situations with a low number of measurements more significantly. It can  
Figure 4: Convergence results on 10,000 simulations using GD with fixed-step descent for estimation problems (5) (**a**) and (6) (**b**), respectively, including position and range measurement error models and by varying the number of neighboring nodes from 4 to 10. Figure 5: Mean number of iterations to convergence with standard deviation on 10,000 simulations using GD with fixed-step descent for estimation problems (5) (**a**) and (6) (**b**), respectively, including position and range measurement error models and by varying the number of neighboring nodes from 4 to 10.