# Hybrid TOA/RSS Range-Based Localization with Self-Calibration in Asynchronous Wireless Networks
## 2 Hybrid TOA/RSS Localization: Formulation and Resolution Approaches
### Resolution Approaches  
As it can be noticed, the function in Equation (9) is highly non-linear and does not admit a closed-form solution. Moreover, due to the large number of nuisance parameters, a direct minimization of Equation (9) would require an impracticable multidimensional numerical optimization. A possible workaround is to split the problem into two separated phases: (i) _calibration_ and (ii) _localization_. Although suboptimal, this approach is more adequate to the limited computational capabilities of common resource-constrained WSN nodes; furthermore, we recall (see Section 1) that assuming \(\mathbf{\beta}\) and \(\mathbf{\theta}\) are known or can be determined once and for all is not reasonable in real operating scenarios due to possible macro-variations of the environments over time; therefore, an automatic self-calibration step at run-time must be provided.  
More specifically, in the calibration phase, we exploit knowledge of the anchors positions and a set of anchor-to-anchor and anchor-to-blind measurements to estimate the unknown nuisance parameters, as will be explained in Section 3.1. Once estimates of the nuisance parameters are available from the calibration phase, the optimization problem in Equation (8) can be reduced to a problem of position-only estimation (localization phase). For the case at hand, however, still the latter task cannot be performed in closed-form. Therefore, one has to resort to a numerical optimization algorithm (e.g., gradient-based iteration) or to a two-dimensional grid search [54]. It should be noticed that both these approaches have some drawbacks. In particular, the first one suffers from the well-known problem of local minima and, depending on the choice of the initial point, wrong or even missing convergence may be experienced. The second one is computationally much more expensive, especially when the area of interest can be large and medium- to fine-grained resolutions are needed; thus, it is indeed not preferable for low-cost WSN nodes due to its super-linear (quadratic) complexity growth.  
The simpler alternative of a range-based localization task is thus considered here, that is, we further split the localization problem into two separated steps: (ii-a) ranging and (ii-b) position estimation. In doing so, we can leverage the hybrid TOA/RSS range estimator derived in [44]--which is available in closed-form--to perform distance measurements between the blind node and each anchor node. Once the set of \(N\) distance estimates is available at the blind node, the ultimate position estimation task can be then accomplished by resorting to a multilateration approach based on the well-established ILS procedure. Differently from other numerical optimization routines, the ILS has received significant attention thanks to its ability to achieve convergence in a small number of iterations, and is indeed widespread adopted in all commercial global navigation satellite system (GNSS) receivers [55]. Moreover, compared to (linearized) multilateration based on pairwise subtraction of the range measurement equations, significantly greater accuracy is achieved without major increase in computational complexity [8]. Indeed, several optimized variants of ILS have also been proposed in literature, which make ILS amenable to practical implementation even on resource-constrained devices, such as low-cost WSN nodes [56; 57]. Although the whole localization procedure is suboptimal with respect to the initial ML estimation problem in Equation (8), we will show that the achievable performance of the proposed algorithm can be very close to the ones obtained via a two-dimensional grid search.