# Impact of Heterogeneous UWB Sensor Noise on the Optimality and Sensitivity of Mobile Positioning Systems

Mathilde Theunissen, Isabelle Fantoni, Ezio Malis

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## To cite this version:

Mathilde Theunissen, Isabelle Fantoni, Ezio Malis. Impact of Heterogeneous UWB Sensor Noise on the Optimality and Sensitivity of Mobile Positioning Systems. IROS 2025 - IEEE/RSJ International Conference on Intelligent Robots and Systems, IEEE, Oct 2025, Hangzhou, China. pp.982-989, 10.1109/IROS60139.2025.11247384. [hal-05190809](https://hal.science/hal-05190809v1)

HAL Id: hal-05190809

https://hal.science/hal-05190809v1

Submitted on 29 Jul 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# Impact of Heterogeneous UWB Sensor Noise on the Optimality and Sensitivity of Mobile Positioning Systems

Mathilde Theunissen<fnref n="1" />,<fnref n="2" />, Isabelle Fantoni<fnref n="1" />, Ezio Malis<fnref n="2" />

**Abstract**— In this paper, we propose a theoretical framework for designing a multi-robot formation equipped with Ultra-wideband (UWB) sensors to localize a target robot. In the presence of noisy range measurements, the accuracy of the target robot's pose estimation is highly dependent on the chosen formation geometry. Different from existing works, we account for the heterogeneous standard deviations of range measurements across different UWB transmitter-receiver pairs. We establish new optimality conditions for formation geometries and conduct a sensitivity analysis of optimal formations under robot positioning errors. In a 2D setting, we derive necessary and sufficient conditions for both optimality and robustness to robot positioning uncertainty. Experimental results confirm the heterogeneous standard deviations of UWB range measurements and validate the target robot's confidence ellipse model. An experimental comparison of formation geometries, optimized with and without considering heterogeneous noise, emphasizes the importance of accounting for the heterogeneous standard deviations of range measurements. In addition, we experimentally demonstrate that robust formation geometries improve the target robot's confidence ellipse in the presence of positioning errors.

## I. INTRODUCTION

Precise localization is crucial for accurate mobile robot task execution. In obstructed or indoor environments where GNSS is unavailable, localization performance degrades significantly. To address this, wireless sensor networks can be deployed. Among the various available technologies, Ultra-Wideband (UWB) sensors stand out as an attractive solution due to their low cost, robustness to multipath errors and low power consumption [1], [2]. In complex environments, the major drawback of such a positioning system is the need for a dense network to ensure sufficient coverage [3]. This density is even more critical as advanced localization algorithms mitigate anchor ranges based on measurement accuracy [4] and direct line-of-sight criteria [5] to optimize localization reliability. An alternative is to embed the UWB receivers, called anchors, on mobile robots with known positions [6], [7]. Different anchor formations result in different accuracies of the target robot localization.

In this article, we address the problem of the optimal design of a mobile UWB sensor network to maximize the localization accuracy of a target robot.

Localization accuracy is affected by range estimation errors caused by multipath effects, calibration errors, and antenna anisotropy [8]. While mobile anchors help mitigate

<figure><img src="image_3.png" /><figcaption>Fig. 1: Experimental setup composed of 4 ground robots localizing a target robot (in the center) using UWB sensors</figcaption></figure>

non-line-of-sight conditions, optimizing their geometry further reduces localization errors [9]. The Position Dilution of Precision (PDOP) index [10] is commonly used to quantify sensitivity to ranging errors, indicating how measurement noise impacts the estimated position of the target robot. PDOP minimization can be solved analytically, leading to optimal anchor geometries. Such formation geometries have been fully characterized and classified in [11].

The study in [11] demonstrates that an infinite number of optimal formations exist. In the presence of positioning errors, exacerbated by mobile anchors, careful selection of anchor geometry is crucial for designing an effective localization system. The work in [12] investigates cone-shaped formations for underwater applications, experimentally analyzing their robustness against prior target position errors. In our previous work [13], we have analyzed robust anchor placements under mobile anchor location uncertainty.

Most existing studies on optimal formations assume that sensor measurement noise follows a centered Gaussian distribution with a uniform standard deviation across all sensors [12], [14]. However, this assumption does not hold for active UWB nodes, as each antenna has unique characteristics. The use of an oversimplified noise model raises concerns about the true optimality of formation geometries derived under such assumptions. This issue generalizes to other active sensors in network localization.

In this work, we investigate the impact of sensor noise with heterogeneous standard deviations on the optimality of formation geometries and their robustness against robot positioning errors. We model measurement noise as independent Gaussian distributions with distinct standard deviations for each transmitter-receiver pair. This study builds upon our previous work [13]. We relax the assumption of uniform distances between the target robot and the anchors. Therefore, we adopt a Cartesian coordinate framework rather than the

<footnote n="1">¹Nantes Université, École Centrale Nantes, CNRS, LS2N, UMR 6004, F-44000 Nantes, France, {mathilde.theunissen, isabelle.fantoni}@ls2n.fr</footnote>

<footnote n="2">²INRIA Centre at Université Côte d'Azur, Sophia Antipolis, France, ezio.malis@inria.fr</footnote>
polar coordinate approach of [13].

The main contributions are as follows:

* We generalize the necessary and sufficient conditions for optimality of [13] to take into account the measurement noise with heterogeneous standard deviations, which is crucial for real-world deployment.

* We perform a sensitivity analysis of optimal geometries under robot positioning errors and derive conditions for robustness.

* We experimentally demonstrate the importance of accounting for heterogeneous standard deviations of range measurements when designing formation geometries. We show that the robust configurations of [13] loose their robustness properties when heterogeneous standard deviations are not taken into account.

* We validate experimentally our complete approach. The proposed optimal and robust formation geometries are evaluated using UWB sensors embedded on mobile robots.

## II. PROBLEM STATEMENT

### A. Notations

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 2: A range-only positioning system comprises N = 3 robots with heterogeneous standard deviations of their range sensors for localizing a target robot.</figcaption></figure>

The parametrization of the robot formation is presented in Fig. 2. N anchor robots, each equipped with a range sensor, aim to localize a target robot in a two-dimensional global inertial frame $\mathcal{F}_0 = (\mathcal{O}, \mathbf{i}_0, \mathbf{j}_0)$. Without loss of generality, we assume that the position of each range sensor coincides with the center of mass of its respective robot.

The position of each anchor robot i in this global frame $\mathcal{F}_0$ is denoted by $^0\mathbf{q}_i = [^0x_i, ^0y_i]^\top, \forall i \in \{1, ..., N\}$. The unknown position of the target robot $^0\mathbf{q}_T = [^0x_T, ^0y_T]^T$ is reconstructed using range measurements.

The actual distance $\rho_i$ between anchor robot i and the target robot is given by:

$$ \rho_i = \|^0\mathbf{q}_i - ^0\mathbf{q}_T\| = \sqrt{(^0x_i - ^0x_T)^2 + (^0y_i - ^0y_T)^2} \quad (1) $$

We define the target robot's body frame $\mathcal{F}_T = (\mathcal{O}_T, \mathbf{i}_T, \mathbf{j}_T)$ centered at $^0\mathbf{q}_T$. In $\mathcal{F}_T$, the position of the anchor i, denoted $^T\mathbf{q}_i = [^T x_i, ^T y_i]^\top$ is expressed in polar coordinates $\mathbf{\Theta}_i = [\rho_i, \phi_i]$ as:

$$ ^T\mathbf{q}_i = [^0x_i - ^0x_T, ^0y_i - ^0y_T]^\top = [\rho_i \cos \phi_i, \rho_i \sin \phi_i]^\top \quad (2) $$

where $\phi_i$ is the polar angle of anchor i, defined by:

$$ \phi_i = \arctan \left( \frac{T y_i}{T x_i} \right) \quad (3) $$

For the remainder of this paper, all positions are expressed in the frame $\mathcal{F}_T$. Therefore, the frame specification is omitted: $^T\mathbf{q}_i$ becomes $\mathbf{q}_i = [x_i, y_i]^T$.

### B. Problem Statement

In this paper, we aim to **maximize the localization accuracy of the target robot** using range-only measurements. Such accuracy is quantified by the area of the target robot's confidence ellipse $\mathcal{A}_\epsilon$. In a 2D setting, note that $\mathcal{A}_\epsilon$ is proportional to the Position Dilution of Precision (PDOP) index that is commonly used in satellite navigation [10] and optimal sensor networks [17]. In real-world deployment, this area is a function of the position of anchor robots and the noise of their range sensor. Therefore, we seek the position of the anchor robots that minimize the area $\mathcal{A}_\epsilon$. These positions constitute **optimal formation geometries**.

In the following sections, we derive the formulation of $\mathcal{A}_\epsilon$ to obtain the optimal formation geometries. Then, we conduct a sensitivity analysis of these geometries with respect to the positioning errors of anchor robots to establish the robust conditions. Finally, theoretical formations are validated through experiments using real robots equipped with UWB sensors.

## III. THEORETICAL BACKGROUND

We consider that range sensor measurements are corrupted by additive Gaussian noise. The range measurement $r_i$ between anchor robot i and the target robot is given by:

$$ r_i = \rho_i + \omega_i, \forall i \in \{1, ..., N\} \quad (4) $$

where the noise $\omega_i \sim \mathcal{N}(0, \sigma_i^2)$ is modeled as centered Gaussian noise of variance $\sigma_i^2$. We consider **heterogeneous range sensor precisions**, meaning $\sigma_i^2$ varies among robots. The vector $\omega = [\omega_1, ..., \omega_N]^\top \sim \mathcal{N}(0, \Sigma)$ represents the total measurement noise. Here $\Sigma = \text{diag}(\sigma_1^2, ..., \sigma_N^2)$ is the covariance matrix. Since we consider that the source of noise of each sensor is independent, $\Sigma$ is diagonal.

As explained in [10], the relationship between these noisy distance measurements and the position estimation error $\tilde{\mathbf{q}}_T$ of the target robot is:

$$ \omega = H \tilde{\mathbf{q}}_T \quad (5) $$

where $H$ is the Jacobian matrix of (1). Expressed in polar coordinates, $H$ is equal to:

$$ \mathbf{H} = \begin{bmatrix} \cos \phi_1 & \sin \phi_1 \\ \vdots & \vdots \\ \cos \phi_N & \sin \phi_N \end{bmatrix} \quad (6) $$
In the presence of heterogeneous sensor standard deviations, the optimal solution for $\tilde{q}_T$ is obtained by the weighted least squares estimate (Appendix A of [10]):

$$ \tilde{q}_T = (\mathbf{H}^\top \mathbf{\Sigma}^{-1} \mathbf{H})^{-1} \mathbf{H}^\top \omega \quad (7) $$

From (7) the covariance of $\tilde{q}_T$ is:

$$ \operatorname{cov}(\tilde{q}_T) = \mathbb{E}(\tilde{q}_T \tilde{q}_T^\top) = (\mathbf{H}^\top \mathbf{\Sigma}^{-1} \mathbf{H})^{-1} \quad (8) $$

The uncertainty level of the target robot's position is represented by a **confidence ellipse** $\mathcal{E}$ in $\mathbb{R}^2$. Following [16], the confidence ellipse associated with $q_T$ is described by the inequality:

$$ \tilde{q}_T^\top \operatorname{cov}(\tilde{q}_T)^{-1} \tilde{q}_T \le a^2 \quad (9) $$

where $a = \sqrt{-2 \ln(1 - \eta)}$ and $\eta$ is the probability of being within the ellipse. (9) can be rewritten as:

$$ \tilde{q}_T^\top (a^2 \operatorname{cov}(\tilde{q}_T))^{-1} \tilde{q}_T \le 1 \quad (10) $$

From (10), the area of the confidence ellipse is:

$$ \mathcal{A}_\mathcal{E} = \pi \sqrt{\det(a^2 \operatorname{cov}(\tilde{q}_T))} \quad (11) $$

Using (8), (11) can be rewritten as:

$$ \mathcal{A}_\mathcal{E} = \pi a^2 \frac{1}{\sqrt{\det(\tilde{H}^\top \mathbf{\Sigma}^{-1} \tilde{H})}} \quad (12) $$

From equation (12), it is clear that $\mathcal{A}_\mathcal{E}$ is inversely proportional to $\sqrt{\det(\tilde{H}^\top \mathbf{\Sigma}^{-1} \tilde{H})}$, which depends on the **anchor geometry** parametrized by the polar angles $\phi = [\phi_1, \dots, \phi_N]^\top$ and the **variance of the range sensor noise** $\mathbf{\Sigma}$. From (6), we can notice that $\mathcal{A}_\mathcal{E}$ is independent of anchors-target distances.

## IV. OPTIMAL FORMATION GEOMETRIES

We aim to reduce the impact of noisy range measurements on the target robot's localization accuracy by **minimizing the area $\mathcal{A}_\mathcal{E}$ of its confidence ellipse**. From (12), the optimization problem can be formulated as:

$$ \mathcal{O}_N := \arg \max_{q} \det(\tilde{H}^\top \mathbf{\Sigma}^{-1} \tilde{H})(q) \quad (13) $$

The set $\mathcal{O}_N$ represents all formation geometries $q$ that minimize the area $\mathcal{A}_\mathcal{E}$ of the target robot's confidence ellipse. From (6), the matrix $\tilde{H}^\top \mathbf{\Sigma}^{-1} \tilde{H}$ is:

$$ \tilde{H}^\top \mathbf{\Sigma}^{-1} \tilde{H} = \begin{bmatrix} \sum_{i=1}^{N} \frac{1}{\sigma_i^2} \cos^2 \phi_i & \sum_{i=1}^{N} \frac{1}{\sigma_i^2} \cos \phi_i \sin \phi_i \\ \sum_{i=1}^{N} \frac{1}{\sigma_i^2} \cos \phi_i \sin \phi_i & \sum_{i=1}^{N} \frac{1}{\sigma_i^2} \sin^2 \phi_i \end{bmatrix} \quad (14) $$

Following the same derivations as [9], $\det(\tilde{H}^\top \mathbf{\Sigma}^{-1} \tilde{H})$ is given by:

$$ \det(\tilde{H}^\top \mathbf{\Sigma}^{-1} \tilde{H}) = \sum_{j>i} \frac{1}{\sigma_i^2 \sigma_j^2} \sin^2(\phi_i - \phi_j) \quad (15) $$

To find the solutions of (13), we aim to find the conditions on $q$ such that $\det(\tilde{H}^\top \mathbf{\Sigma}^{-1} \tilde{H})$ reaches its lower bound.

**Theorem 1.** A formation geometry $q^*$ belongs to the set $\mathcal{O}_N$, if and only if:

$$ \left\{ \begin{array}{l} \displaystyle \sum_{i=1}^{N} \frac{1}{\sigma_i^2} \cos(2\phi_i^*) = 0 \\ \\ \displaystyle \sum_{i=1}^{N} \frac{1}{\sigma_i^2} \sin(2\phi_i^*) = 0 \end{array} \right. $$

Theorem 1 is obtained by analyzing the critical points of $\det(\tilde{H}^\top \mathbf{\Sigma}^{-1} \tilde{H})$. We provide a demonstration of Theorem 1 in Appendix A. We notice that each robot's polar angle is weighted by the inverse of its variance, meaning that sensors with higher precision contribute more significantly to the formation constraints.

**Remark 1.** In the case of sensors with the same standard deviation the conditions of Theorem 1 simplify. We get back to the optimality conditions of [17] and [9].

**Remark 2.** The properties of optimal formations described in [20] are still valid. Optimal geometries remain invariant to rotation and central symmetry. Furthermore, combining any number of optimal geometries still results in an optimal geometry, regardless of the chosen combination.

## V. SENSITIVITY ANALYSIS OF OPTIMAL FORMATION GEOMETRIES

### A. Local Analysis of Sensitivity

For mobile anchors, the target robot's localization performance depends on the deviation of the formation from the optimal geometry. We assess the sensitivity of optimal geometry by evaluating the variance of the target robot's confidence ellipse under such a deviation.

We assume small perturbations, meaning that anchor deviations from their desired position are small in comparison to their relative positions. Therefore, this analysis focuses on a local neighborhood around an optimal formation geometry.

We consider that the positions of the anchor robots relative to the target robot $q = [q_1, \dots, q_N]^T$, are the summation of the desired positions $q^* = [q_1^*, \dots, q_N^*]^\top \in \mathcal{O}_N$, (that minimize $\mathcal{A}_\mathcal{E}$) and the position errors $\tilde{q} = [\tilde{q}_1, \dots, \tilde{q}_N]^\top$:

$$ q = q^* + \tilde{q} \quad (16) $$

We introduce the bias $\Delta \det$ that represents the difference between the actual and the desired determinant:

$$ \Delta \det = \det(\tilde{H}^\top \mathbf{\Sigma}^{-1} \tilde{H})(q) - \det(\tilde{H}^\top \mathbf{\Sigma}^{-1} \tilde{H})(q^*) \quad (17) $$

Under the assumption of small perturbations, we approximate $\Delta \det$ using the second-order Taylor expansion:

$$ \Delta \det \approx \frac{\partial \det(\tilde{H}^\top \mathbf{\Sigma}^{-1} \tilde{H})}{\partial q} (q^*) \tilde{q} + \frac{1}{2} \tilde{q}^\top \frac{\partial^2 \det(\tilde{H}^\top \mathbf{\Sigma}^{-1} \tilde{H})}{\partial q^2} (q^*) \tilde{q} \quad (18) $$
Since $q^* \in \mathbb{O}_N$ is a maximum of $\det(H^\top\Sigma^{-1}H)$, the gradient at $q^*$, $\frac{\partial \det(H^\top\Sigma^{-1}H)}{\partial q}$ ($q^*$), is zero. We get:

$$ \Delta \det \approx \frac{1}{2} \tilde{q}^\top G (q^*) \tilde{q} \quad (19) $$

where $G = \frac{\partial^2 \det(H^\top\Sigma^{-1}H)}{\partial q^2}$ is the Hessian matrix of size $(2N \times 2N)$. As detailed in Appendix B, $G(q^*)$ can be expressed for $\mathbb{O}_N$ formation geometries as:

$$ G(q^*) = (G_{ij})_{1\le i,j \le N} \quad (20) $$

where each element $G_{ij} = \frac{\partial^2 \det(H^\top\Sigma^{-1}H)}{\partial q_i \partial q_j}$ is a $(2 \times 2)$ matrix defined as:

$$ G_{ij} = \frac{1}{\rho_i \rho_j} \frac{-2}{\sigma_i^2 \sigma_j^2} \cos(2\phi_i - 2\phi_j) \begin{bmatrix} s\phi_i s\phi_j & -c\phi_i s\phi_j \\ -c\phi_j s\phi_i & c\phi_i c\phi_j \end{bmatrix} \quad (21) $$

The symmetric matrix $G$ has the following properties:

1) **Symmetric and negative semi-definite**

2) **Constant trace:** $\mathrm{tr}(G) = \sum_i \mathrm{tr}(G_{ii}) = -2 \sum_i \frac{1}{\rho_i^2 \sigma_i^2}$. The sum of the eigenvalues of $G$ is constant.

3) **Rank-deficient:** $\mathrm{rank}(G) \le 2$. $G$ has at most 2 non-zero eigenvalues. A demonstration is provided in Appendix C.

## B. Sensitivity of Optimal Geometries to Anchors Position Errors

We define the sensitivity of $\mathbb{O}_N$ formation geometries as the variation of $\Delta \det$ in the presence of anchor robot positioning errors. Our sensitivity index is the variance of $\Delta \det$ caused by the position errors $\tilde{q}$:

$$ \mathrm{Var}(\Delta \det) = \mathrm{Var}(\frac{1}{2} \tilde{q}^\top G (q^*) \tilde{q}) \quad (22) $$

In this study, we consider that all $\tilde{q} \sim \mathcal{N}(0, \Sigma_q)$ are independent random variables with the same diagonal covariance matrix $\Sigma_q = \sigma_q^2 I_{2N}$.

According to Theorem 5.2c of [18], if $y \sim \mathcal{N}(\mu, \Sigma)$ is a multivariate random variable following a Gaussian distribution and $A$ is a symmetric matrix then:

$$ \mathrm{Var}(y^\top A y) = 2 \mathrm{tr}((A\Sigma)^2) + 4\mu^\top A\Sigma A\mu \quad (23) $$

Applying (23) to (22) yields:

$$ \mathrm{Var}(\Delta \det) = \frac{\sigma_q^4}{2} \mathrm{tr}(G^2(q^*)) \quad (24) $$

## C. Robust Formation Geometries to Anchor position errors

We aim to identify the conditions for the optimal formation geometries $q^*$ to minimize sensitivity to anchor robot positioning errors. This involves solving the following optimization problem:

$$ \mathcal{O}'_N := \arg \min_{q^*} \mathrm{Var}(\Delta \det) \\ \text{subject to } q^* \in \mathbb{O}_N \quad (25) $$

The set $\mathcal{O}'_N$ includes all optimal geometries within $\mathbb{O}_N$ that are the least sensitive to anchor positioning errors.

Using the equation (24), the sensitivity of optimal geometries depends on $\mathrm{tr}(G^2)$. Therefore, the optimization problem (25) can be reformulated as:

$$ \mathcal{O}'_N := \arg \min_{q^*} \mathrm{tr}(G^2) \\ \text{subject to } q^* \in \mathbb{O}_N \quad (26) $$

In the light of the properties of $G(q^*)$ listed in section V-A, the minimal value of $\mathrm{tr}(G^2)$ is obtained when the two non-zero eigenvalues of $G(q^*)$ are equal. Therefore, the minimal value of $\mathrm{tr}(G^2)$ is:

$$ 2 \left( \sum_{i}^{N} \frac{1}{\sigma_i^4 \rho_i^2} \right)^2 \quad (27) $$

To solve (26), we aim to find the conditions on $q^*$ such that $\mathrm{tr}(G^2)$ reaches its lower bound.

**Theorem 2.** A PDOP minimizer $q^* \in \mathbb{O}_N$ belongs to the set $\mathcal{O}'_N$, if and only if:

$$ \left\{ \begin{array}{l} \sum_{i=1}^{N} \frac{1}{\sigma_i^4 \rho_i^2} \cos(4\phi_i^*) = 0 \\ \\ \sum_{i=1}^{N} \frac{1}{\sigma_i^4 \rho_i^2} \sin(4\phi_i^*) = 0 \end{array} \right. $$

Theorem 2 is obtained by solving the equation $\mathrm{tr}(G^2) = 2 \left( \sum_{i}^{N} \frac{1}{\sigma_i^4 \rho_i^2} \right)^2$. A complete demonstration is provided in Appendix D.

**Remark 3.** For sensors sharing the same standard deviation and anchor-target distances, Theorem 2 simplifies to the optimality conditions established in [13].

**Remark 4.** Unlike Theorem 1, the conditions in Theorem 2 depend on anchor-target distances. As noted in [13], maintaining homogeneous and sufficiently large anchor-target distances flattens the curvature of $A_\Sigma$. In such cases, the distances can be excluded from the conditions.

**Remark 5.** In the presence of homogeneous distances, Theorem 2 cannot be satisfied for arbitrary sensor standard deviations, as $\mathrm{tr}(G^2)$ does not reach its minimal value. However, using the combination property [11], solutions exist when formation geometries share the same standard deviation within groups of anchors. With heterogeneous distances, it is possible to obtain robust and optimal geometries.

# VI. EXPERIMENTAL RESULTS

In this section, we present the validation results of our sensitivity analysis.

## A. Experimental set-up

The experimental setup is shown in Fig. 1.

It consists of four anchor robots, each equipped with a UWB receiver, and one target robot equipped with a UWB transmitter. The sampling frequency of the UWB transmitter is set to 6 Hz. UWB sensors are placed in direct line-of-sight
and on the same horizontal plane. This is to ensure that the noise model hypothesis and the 2D condition are respected.

We obtain the ground truth position of all anchor and target robots using a motion capture system. It is worth noticing that the ground truth position of anchor robots is used to place them in desired positions. Besides, the ground truth position of the target robot is used to compute the trilateration error.

The procedure of all our experiments is the following:

1) Anchor robots are positioned around the target robot according to a predefined formation geometry.

2) Once the anchor robots are in place, the target robot computes its position using the optimal trilateration algorithm proposed in [19]. We use the weighted version of the algorithm to take into account the heterogeneous standard deviations of the range measurements.

3) We record the ground truth positions of every robot, the range measurements, and the trilateration results for a duration of 100 seconds.

We estimate the area of the confidence ellipse offline using the equation (11) with a probability of $\eta = 95\%$. The covariance matrix of the target robot's position $cov(\tilde{q}_T)$ is computed with $\tilde{q}_T$ being the difference between the estimation by trilateration and the ground truth position of the target robot. The theoretical area of the confidence ellipse is computed using equation (12). This formula is based on the assumption of a Gaussian noise model. The diagonal covariance matrix $\Sigma$ is determined using the standard deviations of the range sensor measurements from the experiment.

## B. The Heterogeneous Standard Deviation of UWB Sensors

Here, we show that UWB sensors, despite belonging to the same model, possess different standard deviations.

<figure><img src="image_6.png" /><figcaption>Fig. 3: UWB ranging error</figcaption></figure>

We can observe from Fig. 3 that the UWB measurements have accuracy at the centimeter level. As highlighted in [8], calibration errors and antenna anisotropy are the main source of errors. They lead to a directional bias, which depends on the orientation of the antennas. In this article, this bias is estimated and removed by comparing the mean of range measurements with the ground truth distance obtained from the motion capture system. For real-world deployment, this

offset can be estimated during trilateration by adding extra anchors [19] or using filtering techniques [8].

Removing the directional bias enables us to model the range measurement errors as a centered Gaussian noise. This Gaussian approximation and its comparison with the histogram of range measurements are shown in Fig. 3. Although the Gaussian approximation models the spread of measurement errors well, gaps in the histograms reveal deviations in the error distribution. We will see in the next section that this simplified model remains a good approximation for modeling the area of the target robot's confidence ellipse.

## C. Comparison between Formation Geometries in the Case of Homogeneous Range Sensors' Standard Deviation

The objective of this experiment is to compare the performance of a robust formation geometry $C'_4 \in O'_4$ (defined by equation (26)) against an optimal formation geometry $C_4 \in O_4$. The two formation geometries are shown in Fig. 4. In addition, we validate the derivation of the area of the confidence ellipse $A_\epsilon$ under the assumption that the noise of UWB measurements follows a Gaussian distribution. To this end, we compare the theoretical value of $A_\epsilon$ in equation (12) against its experimental value calculated from trilateration errors using (11).

<figure><img src="image_7.png" /><figcaption>Fig. 4: Compared formation geometries</figcaption></figure>

We introduce to each anchor robot $i$ a positioning error $\tilde{q}_i$. The $C_4$ and $C'_4$ formations, shown in Fig. 4, are represented by the faded color. The amplitude of $\tilde{q}_i$ increases gradually while its orientation is chosen to maximize $\text{Var}(\det(H^\top\Sigma^{-1}H))$. The final formation geometries, where the positioning errors $\tilde{q}_i$ reach their maximum value, are denoted by the bright color. It is worth noting that these final formations remain consistent with the small perturbations assumption. For each amplitude of $\tilde{q}_i$, we compute the area of the confidence ellipse $A_\epsilon$.

Here, we consider a homogeneous standard deviation for the range sensor measurements. Therefore, we **standardize** (standard normalization) the UWB measurements by setting their standard deviation to a common value of $\sigma_{exp} = 0.015\text{m}$.

The mean results over five experiments are shown in Fig. 5.

From Fig. 5, we first notice that the mean confidence ellipse area is coherent with the theoretical area computed with the Gaussian noise assumption. Error bars bound the theoretical predictions. It demonstrates that, although the
<figure><img src="image_8.png" /><figcaption>Fig. 5: Comparison of the area of the target robot's confidence ellipse between $O_4$ and $O'_4$ formation geometries in the presence of anchor robot positioning errors.</figcaption></figure>

Gaussian noise approximation is rough, it remains a valid assumption.

For both formations, the larger the positioning error is, the larger the confidence ellipse of the target robot, evidenced by the upper trend in Fig. 5. Such an increase is expected, since deforming the geometries leads to a gradual deviation from optimal geometries. However, we can notice that the increase is smaller for the $O'_N$ geometries. The target robot position estimation is then computed with a better precision.

### D. Importance of Taking into Account Heterogeneous Range Sensors' Standard Deviation

Processing data obtained from experiments in section VI-C, without standardization of the UWB measurements, yields the results shown in Fig. 6. Compared with the rising trend of Fig. 5, a fluctuation is observed. The match between the theoretical calculations (dashed lines) and the experimental results (solid lines) confirms a correlation between this fluctuation and the variations of the UWB sensors' standard deviations in equation (12). We observe that these variations stem from the fact that the UWB standard deviation depends on the sensor calibration and the orientation between antennas. As a result, these variations outweigh the positioning errors of the anchor robots, causing the tested formation geometries to lose their optimality. Taking into consideration these standard deviations is then crucial.

<figure><img src="image_9.png" /><figcaption>Fig. 6: Experimental confidence ellipse of the target robot for the two formation geometries presented in VI-C, without standardizing the UWB range measurements.</figcaption></figure>

The experiment results in Fig. 7 presents the comparison of our robust formation geometries against the state of the art. To demonstrate the impact of heterogeneous standard deviations on formation design, we place the robots according to three formation geometries $C_1$, $C_2$ and $C_3$. The rectangle geometry $C_1$ is a non-optimal geometry. The optimal formation geometry $C_2$ does not account for heterogeneous standard deviations while $C_3 \in O'_4$ does. We use the standard deviation presented in Fig. 3 to compute $C_3$. Results over 5 experiments are shown in Fig. 7.

<figure><img src="image_10.png" /><figcaption>Fig. 7: Experimental area of the target robot's confidence ellipse for the three formation geometries in the presence of heterogeneous standard deviations</figcaption></figure>

We observe similar performance between the two optimal geometries, $C_2$ and $C_3$. The rectangular formation, $C_1$, has a higher average value because its shape is not designed to minimize $A_\epsilon$. $C_3$ has a significantly lower interquartile range than $C_2$. This is due to the fact that $C_3$ is designed using Theorem 1, which accounts for heterogeneous standard deviations of range measurements, leading to more consistent results.

## VII. CONCLUSIONS

This article explores optimal anchor robot formations for range-only collaborative localization, considering measurement noise with heterogeneous standard deviations. We conduct a sensitivity analysis on optimal formations in the presence of anchor positioning errors and establish the necessary and sufficient conditions for optimality and robustness.

We carry out a complete experimental analysis of our proposed formations using UWB sensors. We confirm the presence of heterogeneous standard deviations in range measurements. Results demonstrate that accounting for these variations improves target robot localization accuracy and validate the benefits of optimized formations.

Our theoretical framework is more general and scalable than our previous work, paving the way for extensions to 3D localization, which will be the focus of future research.
A. *Proof of Theorem 1*

Since $\det(H^\top\Sigma^{-1}H)$ is differentiable, $q^*$ is a critical point if and only if:

$$ \frac{\partial \det(H^\top \Sigma^{-1} H)}{\partial q_k}(q^*) = 0, \forall k \in \{1, \dots, N\} \quad (28) $$

Using the chain rule, we have:

$$ \frac{\partial \det(H^\top \Sigma^{-1} H)}{\partial q_k} = \frac{\partial \det(H^\top \Sigma^{-1} H)}{\partial \phi_k} \frac{\partial \phi_k}{\partial q_k} \quad (29) $$

Given that $\frac{\partial \phi_k}{\partial q_k} \neq 0$, $q^*$ is a critical point if and only if $\forall k \in \{1, \dots, N\}$:

$$ \frac{\partial \det(H^\top \Sigma^{-1} H)}{\partial \phi_k} = \sum_{i=1}^{N} \frac{1}{\sigma_i^2} \frac{1}{\sigma_k^2} \sin(2\phi_k - 2\phi_i) = 0 \quad (30) $$

Equation (30) can be rewritten as:

$$ \frac{\sin(2\phi_k)}{\sigma_k^2} \sum_{i=1}^{N} \frac{\cos(2\phi_i)}{\sigma_i^2} - \frac{\cos(2\phi_k)}{\sigma_k^2} \sum_{i=1}^{N} \frac{\sin(2\phi_i)}{\sigma_i^2} = 0 \quad (31) $$

Equivalently, (31) can be expressed using the cross product:

$$ \left( \frac{1}{\sigma_k^2} \begin{bmatrix} \cos(2\phi_k) \\ \sin(2\phi_k) \\ 0 \end{bmatrix} \wedge \sum_{i=1}^{N} \frac{1}{\sigma_i^2} \begin{bmatrix} \cos(2\phi_i) \\ \sin(2\phi_i) \\ 0 \end{bmatrix} \right) \cdot e_3 = 0 \quad (32) $$

$q$ satisfies (32) if all vectors $\frac{1}{\sigma_k^2}[\cos(2\phi_k), \sin(2\phi_k), 0]^\top$ are aligned or equal to zero, or when $\sum_{i=1}^N \frac{1}{\sigma_i^2}[\cos(2\phi_i), \sin(2\phi_i), 0]^\top$ is equal to zero. The two first solutions correspond to minimal values of $\det(H^\top\Sigma^{-1}H)$. Therefore, a formation geometry $q$ satisfies (13) if and only if:

$$ \sum_{i=1}^{N} \frac{1}{\sigma_i^2} \sin(2\phi_i) = 0 \quad \text{and} \quad \sum_{i=1}^{N} \frac{1}{\sigma_i^2} \cos(2\phi_i) = 0 \quad (33) $$

B. *Computation of $G(q^*)$*

The Hessian matrix of $\det(H^\top\Sigma^{-1}H)$ in polar coordinates can be computed by differentiating (30):

$$ \left\{ \begin{aligned} \frac{\partial^2 \det(H^\top \Sigma^{-1} H)}{\partial \phi_i^2} &= \sum_{j=1}^{N} \frac{2}{\sigma_i^2 \sigma_j^2} \cos(2\phi_i - 2\phi_j) - \frac{2}{\sigma_i^4} \\ \frac{\partial^2 \det(H^\top \Sigma^{-1} H)}{\partial \phi_i \partial \phi_j}(q^*) &= \frac{-2}{\sigma_i^2 \sigma_j^2} \cos(2\phi_i - 2\phi_j) \end{aligned} \right. \quad (34) $$

At the critical point $q^*$, where (33) holds, (34) becomes:

$$ \left\{ \begin{aligned} \frac{\partial \det(H^\top \Sigma^{-1} H)}{\partial \phi_i}(q^*) &= 0 \\ \frac{\partial^2 \det(H^\top \Sigma^{-1} H)}{\partial \phi_j^2}(q^*) &= \frac{-2}{\sigma_i^4} \\ \frac{\partial^2 \det(H^\top \Sigma^{-1} H)}{\partial \phi_i \partial \phi_j}(q^*) &= \frac{-2}{\sigma_i^2 \sigma_j^2} \cos(2\phi_i^* - 2\phi_j^*) \end{aligned} \right. \quad (35) $$

Next, we compute the gradient of $\det(H^\top\Sigma^{-1}H)$ in Cartesian coordinates using the chain rule:

$$ \frac{\partial \det(H^\top \Sigma^{-1} H)}{\partial q_i} = \frac{\partial \det(H^\top \Sigma^{-1} H)}{\partial \phi_i} \frac{\partial \phi_i}{\partial q_i} \quad (36) $$

By differentiating (3), we obtain:

$$ \frac{\partial \phi_i}{\partial q_i} = \frac{1}{\rho_i} [-\sin \phi_i \cos \phi_i] \quad (37) $$

Note that different from the gradient in polar coordinates, $\nabla \det(H^\top\Sigma^{-1}H)$ in Cartesian coordinates depends on the anchor-target distance $\rho_i$. The second derivatives of $\det(H^\top\Sigma^{-1}H)$ are expressed using the chain rule as well:

$$ \begin{aligned} \frac{\partial^2 \det(H^\top \Sigma^{-1} H)}{\partial q_i^2} = & \frac{\partial \det(H^\top \Sigma^{-1} H)}{\partial \phi_i} \left( \frac{\partial \rho_i}{\partial q_i} \frac{\partial^2 \phi_i}{\partial \rho_i \partial q_i} + \frac{\partial \phi_i}{\partial q_i} \frac{\partial^2 \phi_i}{\partial q_i \partial q_j} \right) \\ & + \frac{\partial^2 \det(H^\top \Sigma^{-1} H)}{\partial \phi_i^2} \frac{\partial \phi_i}{\partial q_i} \frac{\partial^2 \phi_i}{\partial q_i \partial q_j} \end{aligned} \quad (38) $$

Using the same procedure, the mixed partial derivatives of $\det(H^\top\Sigma^{-1}H)$ are:

$$ \frac{\partial^2 \det(H^\top \Sigma^{-1} H)}{\partial q_i \partial q_j} = \frac{\partial^2 \det(H^\top \Sigma^{-1} H)}{\partial \phi_i \partial \phi_j} \frac{\partial \phi_j}{\partial q_j} \frac{\partial \phi_i}{\partial q_i} \quad (39) $$

Finally, the coefficient $G_{ij}$ of the Hessian matrix $\frac{\partial \det(H^\top\Sigma^{-1}H)}{\partial q}$ at point $q^*$ are:

$$ \left\{ \begin{aligned} \frac{\partial^2 \det(H^\top \Sigma^{-1} H)}{\partial q_i^2}(\phi^*) &= \frac{-2}{\sigma_i^4} \frac{\partial \phi_i}{\partial q_i} \frac{\partial \phi_i}{\partial q_j} \\ \frac{\partial^2 \det(H^\top \Sigma^{-1} H)}{\partial q_i \partial q_j}(\phi^*) &= \frac{-2}{\sigma_i^2 \sigma_j^2} \cos(2\phi_j^* - 2\phi_i^*) \frac{\partial \phi_j}{\partial q_j} \frac{\partial \phi_i}{\partial q_i} \end{aligned} \right. \quad (40) $$

which lead in the developed equation we see in (21).

C. *Rank of $G(q^*)$*

We denote $\frac{\partial^2 \det(H^\top\Sigma^{-1}H)}{\partial \Theta}$ as the Hessian matrix of $\det(H^\top\Sigma^{-1}H)$ with respect to the polar coordinates $\Theta_i = [\rho_i, \phi_i]$. The matrix $\frac{\partial^2 \det(H^\top\Sigma^{-1}H)}{\partial q}$ and $\frac{\partial^2 \det(H^\top\Sigma^{-1}H)}{\partial \Theta}$ are congruent as:

$$ \frac{\partial^2 \det(H^\top \Sigma^{-1} H)}{\partial q} = P^\top \frac{\partial^2 \det(H^\top \Sigma^{-1} H)}{\partial \Theta} P \quad (41) $$

where $P = \text{diag}(A_1, \dots, A_N)$ is an invertible block-diagonal matrix and $A_i, \forall i \in \{1, \dots, N\}$ is:

$$ A_i = \begin{bmatrix} \frac{-\sin \phi_i}{\rho_i} & \frac{\cos \phi_i}{\rho_i} \\ \frac{\cos \phi_i}{\rho_i} & \frac{\sin \phi_i}{\rho_i} \end{bmatrix} \quad (42) $$

Therefore, $\frac{\partial^2 \det(H^\top\Sigma^{-1}H)}{\partial q}$ and $\frac{\partial^2 \det(H^\top\Sigma^{-1}H)}{\partial \Theta}$ have the same rank. Let us then compute the rank of $\frac{\partial^2 \det(H^\top\Sigma^{-1}H)}{\partial \Theta}$.

To determine the rank of $\frac{\partial^2 \det(H^\top\Sigma^{-1}H)}{\partial \Theta}$ at point $q^*$, we look for the number of independent equations of:

$$ \frac{\partial^2 \det(H^\top \Sigma^{-1} H)}{\partial \phi^2}(q^*) \cdot \mathbf{v} = \mathbf{0} \quad (43) $$

where $\mathbf{v} = [v_1, \dots, v_N]^\top$ is the eigenvector of $\frac{\partial^2 \det(H^\top\Sigma^{-1}H)}{\partial \phi^2}(q^*)$ associated with the null eigenvalue. This equation is equivalent to:

$$ \sum_{i=1}^{N} \frac{1}{\sigma_i^2 \sigma_j^2} c_{ji} v_i = 0, \forall j \in \{1, \dots, N\} \quad (44) $$
where $c_{ij} = \cos(2\phi_i^* - 2\phi_j^*)$. By applying to the $(N-1)$ last equations of (44) the trigonometric relation:

$$c_{ji} = c_{j1}c_{1i} - s_{j1}s_{1i} \quad (45)$$

where $s_{ij} = \sin(2\phi_i^* - 2\phi_j^*)$, we obtain:

$$\left\{
\begin{array}{l}
\displaystyle\sum_{i=1}^{N} \frac{1}{\sigma_i^2} c_{1i} v_i = 0 \\[2ex]
\displaystyle\sum_{j=1}^{N} \frac{1}{\sigma_j^2} s_{1j} v_j = 0, \quad j \in \{2, \dots, N\}
\end{array}
\right.
\quad (46)$$

On one hand, if $\forall j \in \{2, \dots, N\}$, $\sin(2\phi_j^* - 2\phi_1^*) = 0$, then $\cos(2\phi_1^* - 2\phi_j^*) = 1$. The system (46) simplifies to:

$$\sum_{i=1}^{N} \frac{1}{\sigma_i^2} v_i = 0 \quad (47)$$

One equation remains. As a result, the eigenvalue 0 has a multiplicity of $N-1$, hence:

$$\mathrm{rank}(\mathbf{G}(\mathbf{q}^*)) = 1 \quad (48)$$

On the other hand, if it exists $j \in \{2, \dots, N\}$, such that $\sin(2\phi_j^* - 2\phi_1^*) \neq 0$, (46) becomes:

$$\left\{
\begin{array}{l}
\displaystyle\sum_{i=1}^{N} \frac{1}{\sigma_i^2} \cos(2\phi_1^* - 2\phi_i^*) v_i = 0 \\[2ex]
\displaystyle\sum_{i=1}^{N} \frac{1}{\sigma_i^2} \sin(2\phi_1^* - 2\phi_i^*) v_i = 0
\end{array}
\right.
\quad (49)$$

Two equations remain. The eigenvalue 0 has a multiplicity of $N-2$, giving:

$$\mathrm{rank}(\mathbf{G}(\mathbf{q}^*)) = 2 \quad (50)$$

From (48) and (50), we conclude that:

$$\mathrm{rank}(\mathbf{G}(\boldsymbol{\phi}^*)) \le 2 \quad (51)$$

### D. Proof of Theorem 2

We solve $\mathrm{tr}(\mathbf{G}^2) = 2 \left(\sum_i \frac{1}{\sigma_i^4 \rho_i^2}\right)^2$, which leads to:

$$\sum_{i,j}^{N} \left( \frac{-2}{\sigma_i^2 \sigma_j^2} \frac{1}{\rho_i \rho_j} \cos(2\phi_i - 2\phi_j) \right)^2 = 2 \left( \sum_i \frac{1}{\sigma_i^4 \rho_i^2} \right)^2 \quad (52)$$

By applying trigonometric relation $\cos^2 \theta = \frac{1}{2}(\cos 2\theta + 1)$ on (52), can be simplified as:

$$4 \sum_{i,j}^{N} \frac{1}{\sigma_i^4 \sigma_j^4} \frac{1}{\rho_i^2 \rho_j^2} \frac{\cos(4\phi_i - 4\phi_j) + 1}{2} = 2 \left( \sum_i \frac{1}{\sigma_i^4 \rho_i^2} \right)^2 \quad (53)$$

Leading to:

$$2 \sum_{i}^{N} \sum_{j}^{N} \frac{1}{\sigma_i^4 \sigma_j^4} \frac{1}{\rho_i^2 \rho_j^2} \cos(4\phi_i - 4\phi_j) = 0 \quad (54)$$

Finally, using the sum formula for cosines, we get:

$$\left(\sum_{i}^{N} \frac{1}{\sigma_i^4} \frac{1}{\rho_i^2} \cos(4\phi_i)\right)^2 + \left(\sum_{i}^{N} \frac{1}{\sigma_i^4} \frac{1}{\rho_i^2} \sin(4\phi_i)\right)^2 = 0 \quad (55)$$

## ACKNOWLEDGMENT

This project is funded and carried out within the framework of the SAMURAI project (ANR-21-CE33-0014) and partially supported by ROBOTEX 2.0 (Grants ROBOTEX ANR-10-EQPX-44-01 and TIRREX ANR-21-ESRE-0015).

## REFERENCES

[1] W. Zhao, A. Goudar and A. P. Schoellig. "Finding the Right Place: Sensor Placement for UWB Time Difference of Arrival Localization in Cluttered Indoor Environments," In IEEE Robotics and Automation Letters, vol. 7, Issue 3, pp. 6075-6082, 2022.

[2] L. Santoro, M. Nardello, D. Brunelli and D. Fontanelli. "UWB-Based Indoor Positioning System With Infinite Scalability," In IEEE Transactions on Instrumentation and Measurement, vol. 72, pp. 1-11, 2023.

[3] L. Génevé, O. Kermorgant and É. Laroche. "Limits of Trilateration-Based Sensor Placement Algorithms," In IEEE Sensors Applications Symposium, Glassboro, United States, Mar 2017.

[4] B. Xiao, L. Chen, Q. Xiao and M. Li. "Reliable Anchor-Based Sensor Localization in Irregular Areas," In IEEE Trans. on Mobile Computing, vol. 9, no. 1, p. 60-72, Jan 2010.

[5] Y. Chen, J. Wang and J. Yang. "Exploiting anchor links for NLOS combating in UWB localization," In ACM Trans. on Sensor Networks, vol. 20, no 3, p. 1-22, 2024.

[6] Y. Xianjia, L. Qingqing, J. P. Queralta, J. Heikkonen, and T. Westerlund, T. "Cooperative UWB-Based Localization for Outdoors Positioning and Navigation of UAVs aided by Ground Robots," In IEEE International Conference on Autonomous Systems, Montreal, Canada, pp. 20211-5, Aug 2021.

[7] J. Cano and J. Le Ny. "Ranging-Based Localizability Optimization for Mobile Robotic Networks," In IEEE Transactions on Robotics, vol. 39, no. 4, pp. 2842-2860, Aug 2023.

[8] T. Kim, B. Yoon and D. Lee. "UWB-Based Localization System Considering Antenna Anisotropy and NLOS/Multipath Conditions," In IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Abu Dhabi, United Arab Emirates, pp. 8810-8816, Oct. 2024.

[9] S. Martínez and F. Bullo. "Optimal sensor placement and motion coordination for target tracking," In Automatica, vol. 42, Issue 4, p.661-668, Apr 2006.

[10] C. Hegarty and E. Kaplan. "Understanding GPS Principles and Applications, Second Edition," In Artech, 2006.

[11] S. Xue and Y. Yang. "Positioning Configurations with the Lowest GDOP and Their Classification," In Journal of Geodesy, vol. 89, no 1, Jan 2015.

[12] Y. Liu, S. Cai, M. Zou, Z. Ning, Z. Xiong, R. Yu and P. Jiang, "Optimal configuration design of heterogeneous swarm for 3-D cooperative detection based on DOP analysis," In Ocean Engineering, vol. 302, Issue 4, p.117555, March 2024.

[13] M. Theunissen, I. Fantoni, E. Malis and P. Martinet. "Robustness Study of Optimal Geometries for Cooperative Multi-Robot Localization," In IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Abu Dhabi, United Arab Emirates, pp. 3409-3416, Oct 2024.

[14] L. Zhang, Z. Zhang, R. Siegwart and J. J. Chung. "Distributed PDOP Coverage Control: Providing Large-Scale Positioning Service Using a Multi-Robot System," In IEEE Robotics and Automation Letters, vol. 6, no. 2, pp. 2217-2224, April 2021.

[15] M. Kok, J. D. Hol and T. B. Schön. "Indoor Positioning Using Ultrawideband and Inertial Measurements," In IEEE Transactions on Vehicular Technology, vol. 64, no. 4, pp. 1293-1303, April 2015.

[16] L. Jaulin. "Mobile robotics: Kalman Filter," Online Course ENSTA Bretagne, Available: https://www.ensta-bretagne.fr/kalmococ/, 2021.

[17] A.N. Bishop, B. Fidan, B.D.O. Anderson, K. Doğançay, P.N. Pathirana. "Optimality analysis of sensor-target localization geometries," In Automatica, vol. 46, Issue 3, p.479-492, Mar 2010.

[18] A. C. Rencher and G. B. Schaalje. "Linear models in statistics (2nd edition)," In John Wiley & Sons, 2008.

[19] M. Larsson, V. Larsson, K. Astrom and M. Oskarsson. "Optimal Trilateration Is an Eigenvalue Problem," In IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Brighton, UK, pp. 5586-5590, May 2019.

[20] S. Zhao and B. M. Chen. "Optimal Sensor Placement for Target Localisation and Tracking in 2D and 3D," In International Journal of Control, vol. 86, no 10, p. 1687-704, Oct 2013.