## 4 Simulation Results
### Variable-Step GD Estimation  
An alternative approach to fixed-length step descent is to adapt the rate step to the normalized gradient, as defined by (8). The same scenarios from the previous tests are reconsidered using the variable-step descent algorithm. The simulation results showed a very weak convergence (around 20%) with all the formulations of cost functions, with and without explicit incorporation of error models. This observation motivated us to investigate the problem and led us to reformulate the gradient step by introducing a damping parameter \(\beta\), as defined in (8). The results are shown in Figure 6, with and without position and measurement errors. Note that \(\beta=1\) corresponds to the classical formulation of the variable-step descent (i.e., without damping). First, one can notice the increase in convergence rates with the number of relative measurements used in the estimation. This is particularly noticeable for a small number of measurements. Moreover, when the number of measurements exceeds seven, the convergence rate varies very little. Another point to note is the impact of the value of the damping parameter \(\beta\). Indeed, using a value close to 0.1 for the optimization problem (3) gives better results than the classical variable step (\(\beta=1\)) for any number of measurements. The analysis of the number of iterations until convergence (second row of Figure 6) shows that reducing the damping parameter improves the convergence results but, consequently, also increases the number of iterations to convergence. For instance, a value \(\beta=0.1\) leads to 95% of convergent estimates in less than 500 iterations for optimization problem (3). Similar results are shown for the optimization problem (4) with a slight decrease in performance. It therefore appears more interesting to use the cost function (3) with value \(\beta=0.1\) in the case of an adaptive descent with free-error models. While integrating positioning and measurement error models into the formulations, the curves also confirm these trends with a slight drop in performance. This confirms that the cost function (5) seems to give better results than (6), and that a damping value of \(\beta=0.1\) is an appropriate choice, even with positioning and measurements errors. Further analysis of the adaptive step descent is carried out in Figure 7, revealing different cases of convergence of the algorithm: _(i)_ the algorithm converges towards the correct estimate; _(ii)_ the algorithm converges towards an erroneous estimate corresponding to a local minimum; _(iii)_ the algorithm does not have time to converge and reaches the maximum number of iterations; and finally, _(vi)_ the algorithm diverges. The results in Figure 7 are given for the estimation problem (5), including position and measurement errors, based on measurements acquired from 4 and 10 neighboring nodes, but similar results have been observed for the other cases. They clearly indicate the importance of increasing the number of measurements to improve the convergence of the algorithm to correct solutions. Convergence to local minima solutions is greatly reduced when using a significant number of relative measurements in the estimation. A clear improvement can be noticed when the damping parameter is used in the gradient descent step. It should be noted that the number of situations where the slow convergence of the algorithm does not lead to any solution is very limited except for very small values of the damping parameter which will obviously not be used. These results clearly show the relevance of introducing a damping parameter whose value could be suitably chosen according to the number of measurements available or desired for the estimation. Finally, more detailed and comparative results revealing the average behavior over 10,000 simulations of each multilateration formulation scenario are summarized in Tables 2-8.