Distributional Trilateration

• Distributional trilateration is a statistical approach that models measurements (range, angle, Doppler) as random variables, unlike classical trilateration based on exact geometric intersections.
• It employs maximum likelihood and weighted least-squares techniques, leveraging convex relaxations and auxiliary variables to fuse noisy sensor data effectively.
• Applications span orbit determination, distributed audio, and motion capture, highlighting the method’s robustness and adaptability to diverse domains with noise-aware estimation.

Searching arXiv for recent and canonical papers relevant to “distributional trilateration,” including the supplied orbit-determination paper and adjacent statistical trilateration formulations. Distributional trilateration denotes a family of formulations in which trilateration is treated not as the exact intersection of ideal geometric loci, but as inference from noisy measurements, noisy path ensembles, or noisy class-conditional representations. In its most explicit current usage, it means modeling range, angle, and Doppler observations as random variables with specified distributions and estimating position–velocity states by maximizing likelihood; in that setting, classical trilateration is recovered as a special case with three range and three range-rate measurements (Ferreira et al., 2024). The broader literature suggests an enlarged “statistical trilateration” (Editor’s term) that also includes multilateration with Gaussian range-difference noise, nonlinear least-squares fusion of noisy sensor outputs, set-based likelihood matching of channel impulse responses, and even distribution reconstruction in Wasserstein space when “trilateration” is used analogically rather than spatially (Kitić et al., 2019, Yang et al., 2022, Li et al., 2022, Dai et al., 23 Jul 2025).

1. Conceptual scope

Classical trilateration in Euclidean space assumes known anchor positions and exact distance measurements. In two dimensions, two circles intersect in up to two points and a third measurement removes the ambiguity; in three dimensions, spheres replace circles, and additional measurements improve redundancy and reduce volumetric uncertainty. Distributional trilateration alters the epistemic status of those measurements: distances, angles, Doppler shifts, range differences, or feature vectors are treated as random variables, and localization becomes an estimation problem over a likelihood, a weighted least-squares surrogate, or a distributional similarity criterion (Ferreira et al., 2024, Mateo et al., 2015, Larsson et al., 25 Feb 2025).

This broader interpretation appears in several distinct literatures. In low-Earth-orbit initial orbit determination, the phrase is explicit and denotes a maximum-likelihood estimator over radar observables. In distributed audio, the phrase is not used explicitly, but the paper gives maximum-likelihood and least-squares estimators for TDoA-derived range differences, together with CRB-based discussion and empirical error distributions. In Kinect-based motion capture, nonlinear trilateration is naturally reinterpreted as Gaussian maximum likelihood. In single-anchor UWB localization, the geometry is replaced by likelihood matching of channel impulse response distributions over discrete areas. In continual learning, “distributional trilateration” is used in a still looser sense: the unknown object is a class-conditional Gaussian distribution rather than a spatial point, and distances are Wasserstein distances between distributions (Kitić et al., 2019, Yang et al., 2022, Li et al., 2022, Dai et al., 23 Jul 2025).

2. Maximum-likelihood trilateration in orbit determination

The clearest formalization is given for initial orbit determination of LEO objects from three monostatic radars at a common epoch. The unknown state is a six-dimensional vector,

$x \in \mathbb{R}^3,\qquad v \in \mathbb{R}^3,$

with radar locations $t_i \in \mathbb{R}^3$. Range is modeled as

$$d_i = \|x - t_i\| + \epsilon_{d_i},\qquad \epsilon_{d_i}\sim\mathcal{N}(0,\sigma_{d_i}^2),$$

angle is modeled as a unit-vector observation with von Mises–Fisher noise on the sphere, and Doppler shift is modeled as a Gaussian perturbation of the radial velocity term (Ferreira et al., 2024).

Under independence across radars and measurement types, the log-likelihood decomposes into range, angle, and Doppler contributions. Minimizing the negative log-likelihood yields three characteristic terms. The range contribution is a weighted least-squares fit,

$$f_{\text{range}}(x)=\sum_{i=1}^N \frac{1}{2\sigma_{d_i}^2}\left(\|x-t_i\|-d_i\right)^2.$$

The angle contribution is a vMF alignment term,

$$f_{\text{angle}}(x)=-\sum_{i=1}^N \kappa_i\, u_i^T \frac{x-t_i}{\|x-t_i\|},$$

so maximizing likelihood is equivalent to maximizing alignment between measured and modeled line-of-sight directions. The Doppler contribution is again a weighted least-squares term in the modeled radial projection of $v$ (Ferreira et al., 2024).

The objective is nonconvex for three stated reasons: the norm $\|x-t_i\|$, the normalized direction $(x-t_i)/\|x-t_i\|$, and the bilinear coupling of line-of-sight direction with velocity in the Doppler term. The geometric interpretation nevertheless remains classical: range places the object approximately on spheres, angle places it approximately on rays, and Doppler constrains the projection of velocity along the line of sight. What changes is the criterion for fusion: the optimal state is the one making the entire noisy measurement set most probable, rather than the one satisfying an exact intersection construction (Ferreira et al., 2024).

A related but more abstract statistical formulation appears in single-source localization via ranges. There, one starts from

$d_j=\|x-s_j\|+\epsilon_j,$

introduces normalization transforms $\Psi_j$ so that $t_i \in \mathbb{R}^3$0 has Gaussian noise, and derives a weighted least-squares cost in squared-distance residuals,

$t_i \in \mathbb{R}^3$1

In that framework, Gaussian additive distance noise leads to $t_i \in \mathbb{R}^3$2, while log-normal RSS-derived noise leads to logarithmic transforms and different weights. This is distribution-aware trilateration in the strict sense that the measurement distribution explicitly determines the optimization criterion (Larsson et al., 25 Feb 2025).

3. Relaxations, eigenvalue formulations, and asymptotics

For the orbit-determination problem, tractability is obtained by auxiliary variables $t_i \in \mathbb{R}^3$3 and convex relaxations inspired by Soares and Valdeira. The identity

$t_i \in \mathbb{R}^3$4

motivates introducing range vectors $t_i \in \mathbb{R}^3$5 and relaxing the equality constraint to $t_i \in \mathbb{R}^3$6. The resulting surrogate is jointly convex in $t_i \in \mathbb{R}^3$7 and $t_i \in \mathbb{R}^3$8 for fixed $t_i \in \mathbb{R}^3$9. The angle term becomes linear in $$d_i = \|x - t_i\| + \epsilon_{d_i},\qquad \epsilon_{d_i}\sim\mathcal{N}(0,\sigma_{d_i}^2),$$0, and the Doppler term becomes quadratic in $$d_i = \|x - t_i\| + \epsilon_{d_i},\qquad \epsilon_{d_i}\sim\mathcal{N}(0,\sigma_{d_i}^2),$$1 and $$d_i = \|x - t_i\| + \epsilon_{d_i},\qquad \epsilon_{d_i}\sim\mathcal{N}(0,\sigma_{d_i}^2),$$2, though still bilinear jointly. The paper then solves the relaxed approximate MLE by block coordinate descent over two blocks, $$d_i = \|x - t_i\| + \epsilon_{d_i},\qquad \epsilon_{d_i}\sim\mathcal{N}(0,\sigma_{d_i}^2),$$3 and $$d_i = \|x - t_i\| + \epsilon_{d_i},\qquad \epsilon_{d_i}\sim\mathcal{N}(0,\sigma_{d_i}^2),$$4 (Ferreira et al., 2024).

With $$d_i = \|x - t_i\| + \epsilon_{d_i},\qquad \epsilon_{d_i}\sim\mathcal{N}(0,\sigma_{d_i}^2),$$5 fixed, both subproblems have closed form. Position is updated by

$$d_i = \|x - t_i\| + \epsilon_{d_i},\qquad \epsilon_{d_i}\sim\mathcal{N}(0,\sigma_{d_i}^2),$$6

where $$d_i = \|x - t_i\| + \epsilon_{d_i},\qquad \epsilon_{d_i}\sim\mathcal{N}(0,\sigma_{d_i}^2),$$7 is a positive scalar multiple of the identity and $$d_i = \|x - t_i\| + \epsilon_{d_i},\qquad \epsilon_{d_i}\sim\mathcal{N}(0,\sigma_{d_i}^2),$$8 is a weighted sum of $$d_i = \|x - t_i\| + \epsilon_{d_i},\qquad \epsilon_{d_i}\sim\mathcal{N}(0,\sigma_{d_i}^2),$$9. Velocity is updated by weighted least squares,

$$f_{\text{range}}(x)=\sum_{i=1}^N \frac{1}{2\sigma_{d_i}^2}\left(\|x-t_i\|-d_i\right)^2.$$0

provided the Doppler geometry is sufficient. With $$f_{\text{range}}(x)=\sum_{i=1}^N \frac{1}{2\sigma_{d_i}^2}\left(\|x-t_i\|-d_i\right)^2.$$1 fixed, each $$f_{\text{range}}(x)=\sum_{i=1}^N \frac{1}{2\sigma_{d_i}^2}\left(\|x-t_i\|-d_i\right)^2.$$2-subproblem is a convex trust-region subproblem over a Euclidean ball, solved through KKT conditions and an eigenvalue characterization of the boundary case. Standard BCD results are then invoked to guarantee convergence to a stationary point of the relaxed objective, and each iteration requires only small-dimensional linear algebra with cost scaling linearly in the number of radars or measurements (Ferreira et al., 2024).

The eigenvalue viewpoint is developed further in single-source localization. After suitable normalization, translation, and diagonalization, the stationarity equations are reduced to an ordinary eigenvalue problem of dimension $$f_{\text{range}}(x)=\sum_{i=1}^N \frac{1}{2\sigma_{d_i}^2}\left(\|x-t_i\|-d_i\right)^2.$$3, and the globally optimal solution corresponds to the largest real eigenvalue. The paper gives special treatment to degenerate cases, including multiple and infinitely many solutions, and interprets them geometrically through the rank of $$f_{\text{range}}(x)=\sum_{i=1}^N \frac{1}{2\sigma_{d_i}^2}\left(\|x-t_i\|-d_i\right)^2.$$4. This provides a fast and numerically stable globally optimal solver for the weighted squared-distance surrogate (Larsson et al., 25 Feb 2025).

Asymptotically, the orbit-determination paper treats the relaxed estimator as an approximate MLE. Under the stated regularity conditions—independent noise, correct distributional models, sufficiently large numbers of observations, and reasonable SNR—the unrelaxed MLE has asymptotic unbiasedness and asymptotic efficiency, with covariance approaching the inverse Fisher information. The paper does not derive the Fisher information matrix or CRLB explicitly, but it uses this standard interpretation to explain why more independent measurements reduce both estimation error and variability (Ferreira et al., 2024).

4. Operationalizations in other domains

Several papers instantiate the same pattern outside orbital radar, though often without using the exact phrase.

Domain	Localized object	Distributional mechanism
Distributed audio	Source position from TDoA-derived range differences	ML under Gaussian RD noise; LS as exact ML under i.i.d. Gaussian residuals (Kitić et al., 2019)
Kinect motion capture	Joint position from multiple sensors	Nonlinear least squares on geometric residuals; equivalent to ML under i.i.d. Gaussian distance errors (Yang et al., 2022)
Single-anchor UWB	Discrete area from CIR snapshots	MD-GMM area likelihoods and set-based similarity via summed log-likelihoods / KL interpretation (Li et al., 2022)
Continual learning	Tail-class Gaussian distribution in feature space	Wasserstein-neighbor retrieval and geometric fusion of means and covariances (Dai et al., 23 Jul 2025)

In distributed audio multilateration, the observation model is

$$f_{\text{range}}(x)=\sum_{i=1}^N \frac{1}{2\sigma_{d_i}^2}\left(\|x-t_i\|-d_i\right)^2.$$5

with additive noise on the range differences. The hyperbolic least-squares criterion

$$f_{\text{range}}(x)=\sum_{i=1}^N \frac{1}{2\sigma_{d_i}^2}\left(\|x-t_i\|-d_i\right)^2.$$6

is exact ML when the RD noise is i.i.d. Gaussian, while weighted least squares corresponds to ML under general Gaussian covariance. The paper also surveys constrained spherical LS, conic LS, convex relaxations, and CRB-based analysis, and reports empirical error distributions via boxplots and 2D histograms. This is distributional trilateration in the sense that localization is explicitly tied to a noise law and its induced estimator properties (Kitić et al., 2019).

In Kinect-based motion capture, the per-sensor residual is

$$f_{\text{range}}(x)=\sum_{i=1}^N \frac{1}{2\sigma_{d_i}^2}\left(\|x-t_i\|-d_i\right)^2.$$7

and the nonlinear objective is

$$f_{\text{range}}(x)=\sum_{i=1}^N \frac{1}{2\sigma_{d_i}^2}\left(\|x-t_i\|-d_i\right)^2.$$8

The paper solves this by a Gauss–Newton-type iteration and states that the formulation is naturally interpretable as Gaussian maximum likelihood with independent, equal-variance distance errors. Hard occlusion rejection then acts as a quality gate on the measurement set before trilateration (Yang et al., 2022).

Single-anchor UWB localization replaces explicit geometry by learned area-conditioned distributions of channel impulse response amplitudes. For area $$f_{\text{range}}(x)=\sum_{i=1}^N \frac{1}{2\sigma_{d_i}^2}\left(\|x-t_i\|-d_i\right)^2.$$9, the main model is an $$f_{\text{angle}}(x)=-\sum_{i=1}^N \kappa_i\, u_i^T \frac{x-t_i}{\|x-t_i\|},$$0-dimensional Gaussian mixture,

$$f_{\text{angle}}(x)=-\sum_{i=1}^N \kappa_i\, u_i^T \frac{x-t_i}{\|x-t_i\|},$$1

and localization is either snapshot-wise maximum likelihood,

$$f_{\text{angle}}(x)=-\sum_{i=1}^N \kappa_i\, u_i^T \frac{x-t_i}{\|x-t_i\|},$$2

or set-based maximization of

$$f_{\text{angle}}(x)=-\sum_{i=1}^N \kappa_i\, u_i^T \frac{x-t_i}{\|x-t_i\|},$$3

which the paper interprets as empirical KL-divergence minimization between a test-set distribution and area models. Multipath is thereby treated as an information source rather than a nuisance (Li et al., 2022).

The most non-geometric usage appears in ViRN, where each class is modeled as a Gaussian $$f_{\text{angle}}(x)=-\sum_{i=1}^N \kappa_i\, u_i^T \frac{x-t_i}{\|x-t_i\|},$$4 in feature space, 2-Wasserstein distance selects neighbor classes, and a tail-class distribution is refined by convex fusion:

$$f_{\text{angle}}(x)=-\sum_{i=1}^N \kappa_i\, u_i^T \frac{x-t_i}{\|x-t_i\|},$$5

with an analogous covariance update that includes an additional mean-dispersion term. The paper explicitly calls this distributional trilateration because the unknown entity is a distribution located in Wasserstein space by its distances to better-estimated neighbor distributions (Dai et al., 23 Jul 2025).

5. Relation to classical trilateration, multilateration, and ambiguity resolution

The statistical formulations do not abolish the geometry of trilateration; they generalize it. In the orbit-determination setting, the geometric core is unchanged: ranges define approximate spheres, angles define approximate rays, and Doppler defines line-of-sight velocity projections. What changes is the estimator: exact intersection is replaced by likelihood maximization, and adding more measurements increases Fisher information. In the minimal regime of three radars with one range and one Doppler measurement each, the approximate MLE is reported to be equivalent in practice to classical trilateration under Gaussian, Cauchy, and Laplace perturbations, while additional measurements produce more accurate and less uncertain state estimates (Ferreira et al., 2024).

In distributed audio, the same relation appears between multilateration and statistical estimation. Hyperbolic LS is exact ML under i.i.d. Gaussian RD noise, weighted LS is ML under general Gaussian covariance, and constrained or relaxed formulations can be read as approximate ML under transformed models. The paper explicitly notes that least-squares objectives are often “artificial,” which is important because mismatch between assumed and actual noise produces bias, threshold effects, or robustness failures (Kitić et al., 2019).

A different generalization occurs in reconstruction from unlabeled path or loop lengths. There the measurements form a multiset rather than a labeled distance matrix, and trilateration is embedded in algebraic identifiability theory. For generic configurations in $$f_{\text{angle}}(x)=-\sum_{i=1}^N \kappa_i\, u_i^T \frac{x-t_i}{\|x-t_i\|},$$6, if unlabeled path or loop data can be explained by a trilateration-supporting ensemble, then the configuration is uniquely determined up to congruence or, in the full path/loop setting, up to integer scale and relabeling. This suggests a distributional reading in which a “distance distribution” can certify geometry when enough trilaterative structure can be extracted from it (Gkioulekas et al., 2020).

Ambiguity resolution can also be expanded beyond probability models. In 2D wireless sensor localization under the unit disk graph property, the absence of an edge implies a lower bound on distance, so a candidate bilateration or trilateration position is rejected if it falls within sensing range of a localized non-neighbor. The paper uses that logic to eliminate flip ambiguities and to localize some graphs that are not trilateration graphs in the classical sense. Although this is not itself a probabilistic formulation, it shows that generalized trilateration can combine equality constraints from distances with inequality constraints derived from connectivity structure (Cagirici, 2016).

6. Robustness, limitations, and open directions

The main strength of distributional trilateration is that it makes the uncertainty model part of the estimator rather than an afterthought. In the radar IOD paper, Gaussian range and Doppler noise together with vMF angular noise yield a likelihood whose estimator improves as the number of measurements increases; the numerical experiments show decreasing error and shrinking spread with increasing effective radar count. In the eigenvalue-based single-source formulation, different noise laws are accommodated through normalization transforms and corresponding weights. In the UWB setting, joint modeling of delay-bin amplitudes and set-based similarity improve robustness relative to independent-bin and snapshot-by-snapshot baselines. These results collectively suggest that explicit modeling of the measurement distribution, rather than merely the measurement mean, is the decisive step in moving from geometric trilateration to distributional trilateration (Ferreira et al., 2024, Larsson et al., 25 Feb 2025, Li et al., 2022).

The limitations are equally consistent across domains. In orbit determination, the one-shot method assumes measurements taken approximately at the same instant, fixed and known monostatic radar locations, synchronized data, and simplified measurement models without explicit atmospheric or ionospheric effects; the relaxed problem remains nonconvex, and the paper does not provide a closed-form FIM or CRLB. The same experiments also show limited robustness to Cauchy noise, as expected for a Gaussian-based MLE. In audio, RD errors are often non-Gaussian because of reverberation and quantization, so Gaussian ML and LS can become suboptimal. In Kinect fusion, equal-variance Gaussian residuals and hard occlusion rejection are simplifications. In single-anchor UWB, the learned area models require environmental stationarity and sufficient training coverage, and layout changes remain a source of performance loss (Ferreira et al., 2024, Kitić et al., 2019, Yang et al., 2022, Li et al., 2022).

A complementary line of work provides deterministic robustness theory. Perturbation bounds for trilateration show that localization error depends explicitly on landmark conditioning through the half-width $$f_{\text{angle}}(x)=-\sum_{i=1}^N \kappa_i\, u_i^T \frac{x-t_i}{\|x-t_i\|},$$7, on target and landmark radii, and on perturbations in both landmark locations and squared-distance inputs. Those bounds were developed in the context of manifold learning pipelines, but they formalize an important general point: any distributional trilateration method inherits a geometric condition number, and uncertainty modeling does not eliminate sensitivity to poor anchor geometry (Arias-Castro et al., 2018).

Taken together, the literature presents distributional trilateration less as a single algorithm than as a unifying shift in viewpoint. The common move is to replace exact geometric intersection by inference over noisy observations, noisy ensembles, or neighboring distributions. In some settings that shift yields an approximate or exact MLE; in others it yields KL-based matching, Wasserstein barycentric refinement, or algebraic certification from unlabeled length multisets. The phrase is therefore narrow in its strictest usage—statistical radar trilateration for IOD—but broad in its methodological significance: trilateration becomes a problem of distribution-aware estimation rather than only one of Euclidean construction (Ferreira et al., 2024, Dai et al., 23 Jul 2025).