Joint Position–Orientation Deployment Design of UAV-Borne Linear-Array Angle-of-Arrival Sensors for Target UAV Localization

Highlights

What are the main findings?

• For unmanned aerial vehicle (UAV)-borne one-dimensional (1-D) angle-of-arrival (AOA) sensing, the rank-one Fisher information matrix (FIM) structure enables exact onboard array-orientation elimination on the line-of-sight tangent plane.
• The proposed geometry-structured sequential quadratic programming (GS-SQP) method consistently improves the deployment objective and Monte Carlo localization accuracy in both free-flight and constrained hovering-region scenarios.

What are the implications of the main findings?

• Joint UAV position–orientation deployment can be solved through a reduced UAV coordinate optimization problem while preserving the essential sensing geometry of the original formulation.
• For UAV localization planning, waypoint freedom provides most of the performance gain, while orientation control is mainly useful as a low-dimensional refinement when feasible hovering regions are tight.

Abstract

This paper investigates joint deployment of unmanned aerial vehicle (UAV)-borne linear-array angle-of-arrival (AOA) sensors for localizing a target UAV in three-dimensional space. Since each sensing UAV carries a lightweight one-dimensional (1-D) AOA array, each measurement provides only one angular constraint, and its information contribution depends jointly on the UAV waypoint and array pointing direction. This leads to a coupled coordinate–orientation design problem that differs from conventional full-AOA deployment. We formulate a Cramér–Rao lower bound (CRLB)-based framework under A- and D-optimality criteria, covering both free-flight and constrained hovering regions. By exploiting the structure of the 1-D AOA Fisher information matrix, we show that, for fixed UAV coordinates, the orientation block can be exactly eliminated through a low-dimensional eigenproblem. The resulting reduced coordinate problem is then solved by a geometry-structured sequential quadratic programming (SQP) method, whose curvature model captures the radial and tangential sensitivities induced by line-of-sight geometry. Numerical simulations further validate the effectiveness of the proposed approach.

1. Introduction

Unmanned aerial vehicle (UAV) swarms have become flexible sensing platforms for target localization in Global Navigation Satellite System (GNSS)-denied, infrastructure-limited, and emergency scenarios [1,2]. Compared with fixed sensor networks, UAVs can actively adjust their positions to improve the sensing geometry and thereby enhance localization accuracy [3,4,5,6]. Therefore, beyond the measurement model and the estimation algorithm, the deployment geometry of sensing UAVs plays a critical role in determining the attainable localization performance [7,8,9].

For small UAV platforms, linear-array angle-of-arrival (AOA) sensing is an attractive low-complexity solution [10,11]. Here, the term linear array is used in the general array-signal-processing sense and is not restricted to a microphone array or to acoustic-source localization. The same one-dimensional (1-D) AOA model can be realized by a lightweight radio-frequency (RF) receiving array observing a narrowband beacon or communication signal from the target UAV. Compared with planar arrays, linear arrays have simpler hardware structures and lower payload and power requirements, and are easier to calibrate and integrate on compact platforms [12,13]. However, a linear array does not provide a full three-dimensional bearing. Instead, it measures only the angle between the incoming signal direction and the array axis. Consequently, each 1-D AOA measurement imposes a conic constraint on the target position, rather than a bearing-ray constraint as in conventional full-AOA localization [14,15,16].

This conic measurement geometry fundamentally changes the deployment problem. In conventional sensor placement, the information contribution of a sensor is mainly determined by its position relative to the target once the sensing model is fixed [17,18,19,20,21,22,23,24,25]. In contrast, for UAV-borne 1-D AOA sensing, the Fisher information contributed by each UAV depends jointly on the UAV position and the onboard array orientation [26,27,28,29,30]. Hence, deployment design becomes an intrinsically coupled position–orientation optimization problem, which falls outside the coordinate-only placement formulations commonly used in conventional sensor deployment [31,32].

The above discussion reveals a technical gap between existing deployment methods and the UAV-borne 1-D AOA setting considered here. The sensing platform can no longer be modeled merely as a point sensor, because the onboard linear-array orientation directly shapes the information contributed by each measurement. A direct way to address this issue is to optimize UAV positions and array orientations simultaneously. However, such a black-box joint optimization leads to a higher-dimensional nonconvex problem with both deployment constraints and unit-norm orientation constraints, and more importantly, it does not reveal how the array orientation affects the Fisher information structure. These limitations motivate a deployment framework that explicitly models the position–orientation coupling and exploits the specific structure of the 1-D AOA Fisher information matrix (FIM).

To the best of our knowledge, a Cramér–Rao lower bound (CRLB)-driven joint deployment framework that explicitly optimizes both UAV positions and onboard linear-array orientations for 1-D AOA target localization has not been addressed in prior UAV sensor placement or cooperative localization studies. In this paper, we show that the position–orientation coupling admits a more structured treatment: for fixed UAV positions, the array orientations can be optimized through a low-dimensional exact subproblem. This yields an orientation-eliminated CRLB-driven criterion over UAV coordinates, which captures the best orientation response for each UAV placement while preserving the anisotropic FIM structure of the 1-D AOA model.

Based on this orientation-eliminated criterion, we develop a geometry-structured sequential quadratic programming (GS-SQP) method for UAV coordinate updates. Different from black-box joint optimization or isotropic surrogate schemes, the proposed local model captures the radial and tangential sensitivities induced by the line-of-sight (LOS) geometry. As a result, the proposed method is a structured deployment algorithm tailored to UAV-borne 1-D AOA arrays and remains applicable to both free-space and constrained placement regions.

The main contributions of this paper are summarized as follows.

• We formulate a CRLB-driven joint position–orientation deployment problem for multiple UAV-borne 1-D AOA sensors under both A- and D-optimality criteria. The formulation explicitly captures the conic measurement geometry and distinguishes the problem from conventional coordinate-only deployment designs.
• We reveal a structural property of the 1-D AOA FIM: for fixed UAV positions, the array orientations can be optimized through an exact low-dimensional subproblem. This yields an orientation-eliminated deployment criterion over UAV coordinates, rather than a black-box joint optimization over positions and orientations.
• We develop a GS-SQP algorithm for the orientation-eliminated deployment problem. Unlike generic SQP or isotropic surrogate methods, the proposed local model captures the LOS-induced radial and tangential sensitivities of the 1-D AOA FIM, making the algorithm tailored to the anisotropic information geometry of linear-array measurements.

The remainder of this paper is organized as follows. Section 2 reviews related works and clarifies the differences from existing deployment and bearing-only localization studies. Section 3 presents the system model and problem formulation. Section 4 derives the reduced deployment criterion and develops the proposed GS-SQP algorithm; Section 5 provides numerical results and Section 6 concludes the paper.

2. Related Works

2.1. Optimal Deployment and Array Orientation Design

CRLB/FIM-based optimal deployment has been extensively studied for source and target localization. Most existing works formulate the design objective under A-, D-, or E-optimality and optimize sensor coordinates for a given measurement model. For example, ref. [7] developed a unified alternating direction method of multipliers (ADMM) and majorization–minimization (MM) framework for time-of-arrival (TOA), time-difference-of-arrival (TDOA), and received signal strength (RSS) source localization. Ref. [20] proposed a robust CRLB-based placement framework for TOA/RSS localization over an uncertain source region. For multi-target localization, ref. [17] studied TOA sensor placement with shared sensors, while [18] considered coupled sensor clusters with an alternating minimization (AM)-ADMM-MM framework. For AOA-based deployment, ref. [24] proposed a block-MM method for three-dimensional (3-D) AOA sensor placement, and [25] extended 3-D AOA placement to general convex deployment constraints. These studies provide important FIM-based deployment tools, but their design variables are mainly sensor coordinates, and the sensor orientation is usually fixed, idealized, or implicitly absorbed into the measurement model.

Array orientation has also been explicitly considered in AOA localization and direct position determination (DPD) studies. For indoor AOA localization, OpArray [26] showed that properly adjusting receiver-array orientations can reduce AOA uncertainty and improve localization accuracy. The authors of [27] further considered the joint optimization of receiver positions and array orientations, but their formulation is mainly developed for a two-dimensional indoor localization scenario with fixed access-point/receiver deployment assumptions. In DPD systems, the authors of [28] derived optimal array geometric structures, studied optimal linear-array orientation design for 3-D DPD via semidefinite relaxation [29], and more recently investigated joint sensor-array path planning and attitude determination for optimal emitter localization [30]. These studies clearly show that array orientation is an important design degree of freedom for improving localization performance. Nevertheless, they do not formulate the UAV-borne 1-D AOA deployment problem as a CRLB-driven joint optimization of multiple UAV coordinates and onboard linear-array axes under A- and D-optimality criteria.

It is also worth noting that some UAV cooperative localization or distributed bearing-only studies focus on information fusion, communication protocols, observability, or estimator design among multiple mobile agents [33,34]. These problems are different from the present work, where the sensing UAVs are used as deployable measurement platforms and the main objective is a centralized CRLB-driven design of their positions and onboard linear-array orientations.

2.2. 1-D AOA Localization

Existing studies on 1-D AOA localization have mainly focused on estimation, calibration, pose recovery, and robustness analysis under given sensing configurations. For example, several works investigated 3-D localization or pose estimation using linear-array AOA measurements, where the main objective is to recover the target position, receiver pose, or relative geometry from the available 1-D angular observations [10,11,16]. Other studies further considered practical array imperfections and uncertainty factors, such as array-spacing perturbations, residual phase bias, calibration errors, and sensor position/orientation uncertainties, and developed corresponding calibration-aware or robust estimation methods [12,13,14,15]. These works demonstrate that linear arrays can provide useful angular information for 3-D localization and that model imperfections must be carefully handled in practical systems. However, they generally focus on estimation or calibration after the sensing configuration has been specified. The design of the sensing configuration itself, namely where multiple UAV-borne linear arrays should be placed and how their array axes should be oriented under a unified CRLB/FIM criterion, remains much less explored.

Notations

Vectors and matrices are denoted by bold lowercase and uppercase letters, respectively. For a vector or matrix, ${(·)}^T$ and ${(·)}^{-1}$ denote transpose and inverse, respectively. The operators $\mathrm{tr}(·)$, $det(·)$, $\parallel ·\parallel$, and $\langle ·,·\rangle$ denote trace, determinant, Euclidean norm, and inner product, respectively. The identity matrix is denoted by $\mathbf{I}$, and $\mathbf{A}⪰\mathbf{0}$ means that the symmetric matrix $\mathbf{A}$ is positive semidefinite. The unit sphere in ${\mathbb{R}}^3$ is denoted by ${\mathbb{S}}^2\triangleq \{\mathit{u}\in {\mathbb{R}}^3:\parallel \mathit{u}\parallel =1\}$. The main symbols used in this paper are summarized in

Table 1. Main notations used in this paper.

Symbol	Description
M	Number of sensing UAVs.
i	Index of the sensing UAV, $i=1,\dots ,M$.
$\mathit{x}$	Unknown target position in ${\mathbb{R}}^3$.
${\mathit{x}}_0$	Nominal target position used for deployment design.
${\mathit{p}}_i$	Position of the ith sensing UAV.
$\mathbf{P}=[{\mathit{p}}_1,\dots ,{\mathit{p}}_M]$	Stacked UAV coordinate matrix.
${\mathit{u}}_i$	Unit orientation vector of the 1-D linear array mounted on the ith UAV.
$\mathbf{U}=[{\mathit{u}}_1,\dots ,{\mathit{u}}_M]$	Stacked array orientation matrix.
${\mathcal{P}}_i$	Feasible coordinate set of the ith UAV.
${\mathcal{U}}_i$	Feasible orientation set of the ith linear array.
${\mathit{r}}_i=\mathit{x}-{\mathit{p}}_i$	Relative position vector from the ith UAV to the target.
$d_i=\parallel {\mathit{r}}_i\parallel$	Range between the ith UAV and the target.
${\mathit{e}}_i={\mathit{r}}_i/d_i$	Unit line-of-sight vector.
${\theta }_i^{\circ }$	Noiseless 1-D AOA measurement of the ith UAV.
${\theta }_i$	Noisy 1-D AOA measurement of the ith UAV.
$n_i$	Additive measurement noise of the ith UAV.
${\sigma }_{\theta ,i}^2$	Variance of the 1-D AOA measurement noise.
$\mathit{h}$	Stacked noiseless measurement vector.
${\mathsf{\Sigma }}_{\theta }$	Measurement-noise covariance matrix.
${\mathit{g}}_i$	Jacobian vector of the ith noiseless measurement.
${\mathbf{P}}_i^{\perp }$	Orthogonal projector onto the LOS tangent plane.
${\mathit{v}}_i$	Effective information-bearing direction of the ith UAV.
${\alpha }_i$	Range- and noise-dependent information weight of the ith UAV.
${\mathbf{J}}_i$	FIM contribution of the ith sensing UAV.
$\mathbf{J}$	Total FIM for estimating the target position.
${\mathbf{J}}_0$	Total FIM evaluated at the nominal target position.
${\mathbf{S}}_i^{\left(t\right)}$	Background FIM formed by all sensing UAVs except the ith UAV.
${\mathsf{\Phi }}_{\mathrm{A}}$, ${\mathsf{\Phi }}_{\mathrm{D}}$	A- and D-optimal deployment objectives.
${\overline{\mathsf{\Phi }}}_{\chi }$	Orientation-eliminated global objective.
$F_{\mathrm{A}}$, $F_{\mathrm{D}}$	Matrix criteria associated with A- and D-optimality.
${\varphi }_{i,\chi }^{\left(t\right)}$	Conditional block objective before orientation elimination.
${\overline{\varphi }}_{i,\chi }^{\left(t\right)}$	Conditional block objective after orientation elimination.
${\mathbf{H}}_{i,\mathrm{D}}^{\left(t\right)}$, ${\mathbf{H}}_{i,\mathrm{A}}^{\left(t\right)}$, ${\mathbf{G}}_{i,\mathrm{A}}^{\left(t\right)}$	Two-dimensional matrices used in the orientation eigenvalue updates.
${\mathbf{B}}_i$	Orthonormal basis of the LOS tangent plane.
${\mathit{q}}_i$	Two-dimensional unit vector in the LOS tangent-plane basis.
${\mathit{g}}_{i,\chi }^{\left(t\right)}$	Gradient of the reduced block objective with respect to ${\mathit{p}}_i$.
$Q_{i,\chi }^{\left(t\right)}$	Local quadratic model in the GS-SQP coordinate update.
$\Delta {\mathit{p}}_i$	Coordinate update step of the ith UAV.
${\mathbf{M}}_{i,\chi }^{\left(t\right)}$	Geometry-structured curvature matrix in GS-SQP.
${\mathbf{P}}_{i,\perp }^{\left(t\right)}$	Tangential projection matrix used in the GS-SQP local model.
${\mu }_{i,r,\chi }^{\left(t\right)}$, ${\mu }_{i,\perp ,\chi }^{\left(t\right)}$	Radial and tangential curvature coefficients in GS-SQP.
$ϵ$	Regularization parameter in the GS-SQP curvature matrix.
$\delta$	Probing step for estimating radial and tangential curvatures.
${\gamma }_i^{\left(t\right)}$	Backtracking step length of the ith UAV update.
$T_{max}$	Maximum number of GS-SQP outer iterations.
${\epsilon }_{\mathrm{tol}}$	Relative convergence tolerance of GS-SQP.
$\Delta$	Target-position mismatch vector in robustness tests.
$\widehat{\mathit{x}}$	Estimated target position.
$N_{\mathrm{MC}}$	Number of Monte Carlo trials.
$\chi$	Optimality criterion index, $\chi \in \{\mathrm{A},\mathrm{D}\}$.
t	Outer iteration index.

.

3. Problem Formulation

This section presents the UAV-borne 1-D AOA localization model and formulates the CRLB-driven joint position–orientation deployment problem. We consider a three-dimensional localization network consisting of one target UAV, and M sensing UAVs, as shown in Figure 1. The target position is denoted by $\mathit{x}\in {\mathbb{R}}^3$. The ith sensing UAV is located at ${\mathit{p}}_i\in {\mathbb{R}}^3$ and carries a 1-D linear-array AOA sensor whose array axis is represented by a unit vector ${\mathit{u}}_i\in {\mathbb{S}}^2$, where ${\mathbb{S}}^2\triangleq \{\mathit{u}\in {\mathbb{R}}^3:\parallel \mathit{u}\parallel =1\}$. The vector ${\mathit{u}}_i$ is expressed in the global deployment frame. In practice, it can be realized by the UAV attitude or gimbal command through a body-to-global rotation, though this implementation-level transformation is not explicitly optimized in this paper. For deployment design, a nominal target position ${\mathit{x}}_0$ is assumed available, as commonly adopted in CRLB-based sensor placement.

For the ith sensing UAV, define the relative vector, range, and line-of-sight (LOS) direction as ${\mathit{r}}_i\left(\mathit{x}\right)\triangleq \mathit{x}-{\mathit{p}}_i,d_i\left(\mathit{x}\right)\triangleq \parallel {\mathit{r}}_i\left(\mathit{x}\right)\parallel ,{\mathit{e}}_i\left(\mathit{x}\right)\triangleq {\displaystyle \frac{{\mathit{r}}_i\left(\mathit{x}\right)}{d_i\left(\mathit{x}\right)}}.$ In this model, the incident signal is treated as a generic narrowband source observed by the UAV-borne linear receiving array. It may correspond, for example, to an RF beacon or communication signal rather than an acoustic emission. Since a linear array provides only one angular degree of freedom, the noiseless 1-D AOA measurement is modeled as the angle between the incoming direction and the array axis, i.e.,

$$h_i(\mathit{x};{\mathit{p}}_i,{\mathit{u}}_i)=arccos\left({\mathit{u}}_i^T{\mathit{e}}_i\left(\mathit{x}\right)\right).$$

(1)

The corresponding noisy measurement is

$${\theta }_i=h_i(\mathit{x};{\mathit{p}}_i,{\mathit{u}}_i)+n_i,$$

(2)

where $n_i$ is zero-mean Gaussian noise with variance ${\sigma }_{\theta ,i}^2$. The measurement noises are assumed mutually independent, and hence $\mathit{n}={[n_1,\dots ,n_M]}^T\sim \mathcal{N}(\mathbf{0},{\mathsf{\Sigma }}_{\theta })$ with ${\mathsf{\Sigma }}_{\theta }=\mathrm{diag}\{{\sigma }_{\theta ,1}^2,\dots ,{\sigma }_{\theta ,M}^2\}$.

Let $\mathit{h}(\mathit{x};\mathit{P},\mathit{U})={[h_1,\dots ,h_M]}^T$ denote the stacked noiseless measurement vector, where $\mathit{P}=[{\mathit{p}}_1,\dots ,{\mathit{p}}_M]$ and $\mathit{U}=[{\mathit{u}}_1,\dots ,{\mathit{u}}_M]$ collect the sensing UAV coordinates and array orientations, respectively. Under the independent Gaussian measurement model, the FIM for estimating $\mathit{x}$ is

$$\mathbf{J}(\mathit{x};\mathit{P},\mathit{U})={\left({\displaystyle \frac{\partial \mathit{h}}{\partial \mathit{x}}}\right)}^T{\mathsf{\Sigma }}_{\theta }^{-1}\left({\displaystyle \frac{\partial \mathit{h}}{\partial \mathit{x}}}\right)={\displaystyle \sum _{i=1}^M}{\displaystyle \frac{1}{{\sigma }_{\theta ,i}^2}}{\mathit{g}}_i\left(\mathit{x}\right){\mathit{g}}_i^T\left(\mathit{x}\right),$$

(3)

where ${\mathit{g}}_i\left(\mathit{x}\right)\triangleq {\nabla }_{\mathit{x}}{h}_i(\mathit{x};{\mathit{p}}_i,{\mathit{u}}_i)$ is the Jacobian vector of the ith noiseless measurement. For the 1-D AOA model in (1), a direct differentiation gives

$${\mathit{g}}_i\left(\mathit{x}\right)=-{\displaystyle \frac{1}{d_i\left(\mathit{x}\right)sin{h}_i}}\left(\mathbf{I}-{\mathit{e}}_i\left(\mathit{x}\right){\mathit{e}}_i^T\left(\mathit{x}\right)\right){\mathit{u}}_i.$$

(4)

Equivalently, by defining the orthogonal projector

$${\mathbf{P}}_i^{\perp }\left(\mathit{x}\right)\triangleq \mathbf{I}-{\mathit{e}}_i\left(\mathit{x}\right){\mathit{e}}_i^T\left(\mathit{x}\right),$$

(5)

and the effective information-bearing direction

$${\mathit{v}}_i\left(\mathit{x}\right)\triangleq {\displaystyle \frac{{\mathbf{P}}_i^{\perp }\left(\mathit{x}\right){\mathit{u}}_i}{\parallel {\mathbf{P}}_i^{\perp }\left(\mathit{x}\right){\mathit{u}}_i\parallel }},$$

(6)

the ith FIM contribution can be written as the rank-one form

$${\mathbf{J}}_i(\mathit{x};{\mathit{p}}_i,{\mathit{u}}_i)={\displaystyle \frac{1}{{\sigma }_{\theta ,i}^2{d}_i^2\left(\mathit{x}\right)}}{\mathit{v}}_i\left(\mathit{x}\right){\mathit{v}}_i^T\left(\mathit{x}\right).$$

(7)

Thus,

$$\mathbf{J}(\mathit{x};\mathit{P},\mathit{U})={\displaystyle \sum _{i=1}^M}{\mathbf{J}}_i(\mathit{x};{\mathit{p}}_i,{\mathit{u}}_i).$$

(8)

For any unbiased estimator $\widehat{\mathit{x}}$, if $\mathbf{J}(\mathit{x};\mathit{P},\mathit{U})$ is nonsingular, the CRLB satisfies

$$\mathrm{Cov}\left(\widehat{\mathit{x}}\right)⪰{\mathbf{J}}^{-1}(\mathit{x};\mathit{P},\mathit{U}).$$

(9)

where $\mathrm{Cov}\left(\widehat{\mathit{x}}\right)$ denotes the covariance matrix of the estimator $\widehat{\mathit{x}}$. In the deployment stage, the CRLB is evaluated at the nominal target position ${\mathit{x}}_0$. For compactness, define

$${\mathbf{J}}_0(\mathit{P},\mathit{U})\triangleq \mathbf{J}({\mathit{x}}_0;\mathit{P},\mathit{U}).$$

(10)

We consider two standard CRLB-based design criteria. The A-optimality criterion minimizes the trace of the CRLB, namely

$${\mathsf{\Phi }}_{\mathrm{A}}(\mathit{P},\mathit{U})\triangleq \mathrm{tr}\left({\mathbf{J}}_0^{-1}(\mathit{P},\mathit{U})\right),$$

(11)

while the D-optimality criterion minimizes the logarithmic volume of the CRLB ellipsoid, namely

$${\mathsf{\Phi }}_{\mathrm{D}}(\mathit{P},\mathit{U})\triangleq logdet\left({\mathbf{J}}_0^{-1}(\mathit{P},\mathit{U})\right)=-logdet\left({\mathbf{J}}_0(\mathit{P},\mathit{U})\right).$$

(12)

Let ${\mathcal{P}}_i\subseteq {\mathbb{R}}^3$ denote the feasible deployment region of the ith sensing UAV, and let ${\mathcal{U}}_i\subseteq {\mathbb{S}}^2$ denote the admissible orientation set of its onboard array. The joint deployment problem under criterion $\chi \in \{\mathrm{A},\mathrm{D}\}$ is formulated as

$$\begin{array}{cc}\hfill \underset{{\{{\mathit{p}}_i,{\mathit{u}}_i\}}_{i=1}^M}{min}& {\mathsf{\Phi }}_{\chi }(\mathit{P},\mathit{U})\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& {\mathit{p}}_i\in {\mathcal{P}}_i,i=1,\dots ,M,\hfill \\ \hfill & {\mathit{u}}_i\in {\mathcal{U}}_i,i=1,\dots ,M.\hfill \end{array}$$

(13)

Equivalently, choosing $\chi =\mathrm{A}$ yields the A-optimal deployment problem, and choosing $\chi =\mathrm{D}$ yields the D-optimal deployment problem.

Problem (13) is nonconvex in general. The nonconvexity comes from both the feasible set and the 1-D AOA information structure. First, the orientation constraint ${\mathit{u}}_i\in {\mathbb{S}}^2$ is a unit-norm equality constraint and is therefore nonconvex. Second, even if ${\mathcal{P}}_i$ is convex, the mapping from $({\mathit{p}}_i,{\mathit{u}}_i)$ to ${\mathbf{J}}_i$ is nonconvex because of Equation (7). Hence, ${\mathbf{J}}_0(\mathit{P},\mathit{U})$ is not an affine or convex mapping of the deployment variables. Although $\mathrm{tr}\left({\mathbf{J}}^{-1}\right)$ and $-logdet\left(\mathbf{J}\right)$ are convex functions of a positive definite matrix $\mathbf{J}$, their composition with the nonconvex mapping $(\mathit{P},\mathit{U})\mapsto {\mathbf{J}}_0(\mathit{P},\mathit{U})$ is not convex in general. Therefore, the difficulty is intrinsic to the 1-D AOA position–orientation geometry rather than to a particular parametrization or solver.

4. Proposed Method

The joint deployment problem in (13) involves all UAV positions and array orientations through the total FIM. Directly optimizing all variables simultaneously is possible in principle, but it leads to a high-dimensional nonconvex problem with mixed coordinate and unit-norm orientation constraints. To exploit the additive structure of the FIM, we adopt a block-wise update strategy, where one sensing UAV is updated at a time while the information contributions of the other UAVs are fixed.

For notational simplicity, the FIM in this section is evaluated at the nominal target position ${\mathit{x}}_0$, and the dependence on ${\mathit{x}}_0$ is omitted. Recall that the total FIM can be written as

$$\mathbf{J}(\mathit{P},\mathit{U})={\displaystyle \sum _{j=1}^M}{\mathbf{J}}_j({\mathit{p}}_j,{\mathit{u}}_j).$$

(14)

At the tth iteration, when updating the ith UAV, the information contributed by all other UAVs is fixed and collected as

$${\mathbf{S}}_i^{\left(t\right)}\triangleq \sum _{j\ne i}{\mathbf{J}}_j({\mathit{p}}_j^{\left(t\right)},{\mathit{u}}_j^{\left(t\right)}).$$

(15)

Then the total FIM associated with the ith block update becomes

$${\mathbf{J}}_i^{\mathrm{tot}}({\mathit{p}}_i,{\mathit{u}}_i)={\mathbf{S}}_i^{\left(t\right)}+{\mathbf{J}}_i({\mathit{p}}_i,{\mathit{u}}_i).$$

(16)

Define the matrix criterion $F_{\mathrm{A}}\left(\mathbf{J}\right)=\mathrm{tr}\left({\mathbf{J}}^{-1}\right),F_{\mathrm{D}}\left(\mathbf{J}\right)=-logdet\left(\mathbf{J}\right),$ corresponding to A- and D-optimality, respectively. The conditional block objective for the ith UAV under criterion $\chi \in \{\mathrm{A},\mathrm{D}\}$ is then written as

$${\varphi }_{i,\chi }^{\left(t\right)}({\mathit{p}}_i,{\mathit{u}}_i)\triangleq F_{\chi }\left({\mathbf{S}}_i^{\left(t\right)}+{\mathbf{J}}_i({\mathit{p}}_i,{\mathit{u}}_i)\right).$$

(17)

Accordingly, the block update is formulated as

$$\begin{array}{cc}\hfill \underset{{\mathit{p}}_i,{\mathit{u}}_i}{min}& {\varphi }_{i,\chi }^{\left(t\right)}({\mathit{p}}_i,{\mathit{u}}_i)\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& {\mathit{p}}_i\in {\mathcal{P}}_i,\hfill \\ \hfill & {\mathit{u}}_i\in {\mathcal{U}}_i.\hfill \end{array}$$

(18)

It should be emphasized that (18) is not an independent optimization problem for the ith UAV. The coupling among UAVs is retained through the background FIM ${\mathbf{S}}_i^{\left(t\right)}$, which depends on the current positions and orientations of all other sensing UAVs. The role of the block reformulation is to isolate the only part of the total FIM affected by the current update, namely ${\mathbf{J}}_i({\mathit{p}}_i,{\mathit{u}}_i)$. This conditional structure provides the basis for the orientation elimination developed next.

4.1. Orientation Elimination on the LOS Tangent Plane

We now use the rank-one 1-D AOA FIM structure in (7) to eliminate the array orientation variable from the conditional block problem (18). For a fixed UAV position ${\mathit{p}}_i$, the range $d_i$, the LOS direction ${\mathit{e}}_i$, and the tangent plane orthogonal to ${\mathit{e}}_i$ are all fixed. Therefore, the only remaining effect of the array orientation is to select the information-bearing direction ${\mathit{v}}_i$ on this tangent plane. In other words, the orientation design of the full array axis ${\mathit{u}}_i\in {\mathbb{S}}^2$ can be equivalently replaced by the selection of a unit vector ${\mathit{v}}_i$ satisfying ${\mathit{v}}_i^T{\mathit{e}}_i=0,\parallel {\mathit{v}}_i\parallel =1.$ This observation removes the redundant component of ${\mathit{u}}_i$ along the LOS direction and converts the orientation block into a low-dimensional direction-selection problem on the two-dimensional LOS tangent plane.

Let ${\mathbf{B}}_i\in {\mathbb{R}}^{3\times 2}$ be an orthonormal basis of this tangent plane, satisfying ${\mathbf{B}}_i^T{\mathbf{B}}_i={\mathbf{I}}_2,$ ${\mathbf{B}}_i^T{\mathit{e}}_i=\mathbf{0}.$ Then any feasible information-bearing direction can be parameterized as ${\mathit{v}}_i={\mathbf{B}}_i{\mathit{q}}_i,\parallel {\mathit{q}}_i\parallel =1.$ For this fixed coordinate, define ${\alpha }_i\triangleq 1/\left({\sigma }_{\theta ,i}^2{d}_i^2\right)$ as the scalar information weight. The conditional orientation design for fixed ${\mathit{p}}_i$ becomes

$$\underset{\parallel {\mathit{q}}_i\parallel =1}{min}{F}_{\chi }\left({\mathbf{S}}_i^{\left(t\right)}+{\alpha }_i{\mathbf{B}}_i{\mathit{q}}_i{\mathit{q}}_i^T{\mathbf{B}}_i^T\right),\chi \in \{\mathrm{A},\mathrm{D}\}.$$

(19)

This transformation reduces the orientation search from the three-dimensional unit sphere to a one-dimensional direction search on the LOS tangent plane. More importantly, it reveals the effective information-bearing degree of freedom of a 1-D AOA array and enables an exact low-dimensional solution of the orientation block.

For the D-optimality criterion, by the matrix determinant lemma, we have

$$\begin{array}{cc}\hfill F_{\mathrm{D}}\left({\mathbf{S}}_i^{\left(t\right)}+{\alpha }_i{\mathit{v}}_i{\mathit{v}}_i^T\right)& =-logdet\left({\mathbf{S}}_i^{\left(t\right)}+{\alpha }_i{\mathit{v}}_i{\mathit{v}}_i^T\right)\hfill \\ \hfill & =-logdet{\mathbf{S}}_i^{\left(t\right)}-log\left(1+{\alpha }_i{\mathit{v}}_i^T{\left({\mathbf{S}}_i^{\left(t\right)}\right)}^{-1}{\mathit{v}}_i\right).\hfill \end{array}$$

(20)

Thus, minimizing the D-optimal block objective is equivalent to maximizing a Rayleigh quotient on the tangent plane:

$$\underset{\parallel {\mathit{q}}_i\parallel =1}{max}{\mathit{q}}_i^T{\mathbf{H}}_{i,\mathrm{D}}^{\left(t\right)}{\mathit{q}}_i,$$

(21)

where ${\mathbf{H}}_{i,\mathrm{D}}^{\left(t\right)}={\mathbf{B}}_i^T{\left({\mathbf{S}}_i^{\left(t\right)}\right)}^{-1}{\mathbf{B}}_i.$ Therefore, the optimal projected direction is obtained from the principal eigenvector of the $2\times 2$ matrix ${\mathbf{H}}_{i,\mathrm{D}}^{\left(t\right)}$.

For the A-optimality criterion, applying the Sherman–Morrison formula yields

$$\begin{array}{cc}\hfill F_{\mathrm{A}}\left({\mathbf{S}}_i^{\left(t\right)}+{\alpha }_i{\mathit{v}}_i{\mathit{v}}_i^T\right)& =\mathrm{tr}\left[{\left({\mathbf{S}}_i^{\left(t\right)}+{\alpha }_i{\mathit{v}}_i{\mathit{v}}_i^T\right)}^{-1}\right]\hfill \\ \hfill & =\mathrm{tr}\left({\left({\mathbf{S}}_i^{\left(t\right)}\right)}^{-1}\right)-{\displaystyle \frac{{\alpha }_i{\mathit{v}}_i^T{\left({\mathbf{S}}_i^{\left(t\right)}\right)}^{-2}{\mathit{v}}_i}{1+{\alpha }_i{\mathit{v}}_i^T{\left({\mathbf{S}}_i^{\left(t\right)}\right)}^{-1}{\mathit{v}}_i}}.\hfill \end{array}$$

(22)

Therefore, minimizing the A-optimal block objective is equivalent to maximizing the following generalized Rayleigh quotient:

$$\underset{\parallel {\mathit{q}}_i\parallel =1}{max}{\displaystyle \frac{{\mathit{q}}_i^T{\mathbf{H}}_{i,\mathrm{A}}^{\left(t\right)}{\mathit{q}}_i}{{\mathit{q}}_i^T{\mathbf{G}}_{i,\mathrm{A}}^{\left(t\right)}{\mathit{q}}_i}},$$

(23)

where ${\mathbf{H}}_{i,\mathrm{A}}^{\left(t\right)}={\alpha }_i{\mathbf{B}}_i^T{\left({\mathbf{S}}_i^{\left(t\right)}\right)}^{-2}{\mathbf{B}}_i,{\mathbf{G}}_{i,\mathrm{A}}^{\left(t\right)}={\mathbf{I}}_2+{\alpha }_i{\mathbf{B}}_i^T{\left({\mathbf{S}}_i^{\left(t\right)}\right)}^{-1}{\mathbf{B}}_i.$ The optimal ${\mathit{q}}_i$ is given by the dominant generalized eigenvector of the matrix pair $({\mathbf{H}}_{i,\mathrm{A}}^{\left(t\right)},{\mathbf{G}}_{i,\mathrm{A}}^{\left(t\right)})$.

Let ${\mathit{q}}_{i,\chi }^{★}\left({\mathit{p}}_i\right)$ denote the optimal solution of (21) or (23), and define

$${\mathit{v}}_{i,\chi }^{★}\left({\mathit{p}}_i\right)={\mathbf{B}}_i\left({\mathit{p}}_i\right){\mathit{q}}_{i,\chi }^{★}\left({\mathit{p}}_i\right).$$

(24)

An admissible array orientation realizing this information-bearing direction can be chosen as ${\mathit{u}}_{i,\chi }^{★}\left({\mathit{p}}_i\right)={\mathit{v}}_{i,\chi }^{★}\left({\mathit{p}}_i\right)$ when the full unit sphere is allowed. Consequently, the orientation-eliminated block objective is defined as

$${\overline{\varphi }}_{i,\chi }^{\left(t\right)}\left({\mathit{p}}_i\right)\triangleq F_{\chi }\left({\mathbf{S}}_i^{\left(t\right)}+{\alpha }_i\left({\mathit{p}}_i\right){\mathit{v}}_{i,\chi }^{★}\left({\mathit{p}}_i\right){\mathit{v}}_{i,\chi }^{★T}\left({\mathit{p}}_i\right)\right).$$

(25)

The original block update over $({\mathit{p}}_i,{\mathit{u}}_i)$ is thus converted into a coordinate-only problem

$$\underset{{\mathit{p}}_i\in {\mathcal{P}}_i}{min}{\overline{\varphi }}_{i,\chi }^{\left(t\right)}\left({\mathit{p}}_i\right).$$

(26)

This reduced problem still depends on the orientation response through ${\mathit{v}}_{i,\chi }^{★}\left({\mathit{p}}_i\right)$, but the orientation variable no longer needs to be optimized together with the UAV coordinate. The remaining task is to efficiently solve (26), which is addressed by the structured coordinate solver in the next subsection.

4.2. GS-SQP for the Reduced Coordinate Problem

After the orientation elimination in (25), the block update of the ith sensing UAV is reduced to the coordinate-only problem in (26). It should be noted that this reduced problem is still nonconvex. This is because the range weight ${\alpha }_i\left({\mathit{p}}_i\right)$, the LOS direction ${\mathit{e}}_i\left({\mathit{p}}_i\right)$, the tangent-plane basis ${\mathbf{B}}_i\left({\mathit{p}}_i\right)$, and the optimal information-bearing direction ${\mathit{v}}_{i,\chi }^{★}\left({\mathit{p}}_i\right)$ all vary nonlinearly with the UAV coordinate ${\mathit{p}}_i$. Therefore, a dedicated local optimization strategy is still required for efficiently updating ${\mathit{p}}_i$ under the deployment constraint ${\mathcal{P}}_i$.

At iteration t, we construct a local quadratic model of the orientation-eliminated block objective around the current coordinate ${\mathit{p}}_i^{\left(t\right)}$. Let

$${\mathit{g}}_{i,\chi }^{\left(t\right)}={\nabla }_{{\mathit{p}}_i}{\overline{\varphi }}_{i,\chi }^{\left(t\right)}\left({\mathit{p}}_i\right){|}_{{\mathit{p}}_i={\mathit{p}}_i^{\left(t\right)}}$$

(27)

denote the gradient of the reduced block objective. For a coordinate increment $\Delta {\mathit{p}}_i$, the local model is written as

$$Q_{i,\chi }^{\left(t\right)}(\Delta {\mathit{p}}_i)={\overline{\varphi }}_{i,\chi }^{\left(t\right)}\left({\mathit{p}}_i^{\left(t\right)}\right)+{\left({\mathit{g}}_{i,\chi }^{\left(t\right)}\right)}^T\Delta {\mathit{p}}_i+{\displaystyle \frac{1}{2}}\Delta {\mathit{p}}_i^T{\mathbf{M}}_{i,\chi }^{\left(t\right)}\Delta {\mathit{p}}_i,$$

(28)

where ${\mathbf{M}}_{i,\chi }^{\left(t\right)}⪰\mathbf{0}$ is a curvature matrix. A standard SQP-type update can be obtained by minimizing (28) subject to the local feasibility condition

$${\mathit{p}}_i^{\left(t\right)}+\Delta {\mathit{p}}_i\in {\mathcal{P}}_i.$$

(29)

The key point of the proposed GS-SQP method is the construction of ${\mathbf{M}}_{i,\chi }^{\left(t\right)}$. Instead of using an isotropic regularization or a generic Hessian approximation, we exploit the anisotropic geometry of 1-D AOA sensing. A perturbation of ${\mathit{p}}_i$ along the LOS direction mainly changes the range-dependent information weight ${\alpha }_i\left({\mathit{p}}_i\right)$, whereas a perturbation on the plane perpendicular to the LOS direction mainly changes the LOS direction, the tangent plane, and the associated optimal projected direction. Therefore, the coordinate perturbation is decomposed into radial and tangential components:

$$\Delta {\mathit{p}}_i=\Delta {\mathit{p}}_{i,r}+\Delta {\mathit{p}}_{i,\perp },\Delta {\mathit{p}}_{i,r}={\mathit{e}}_i^{\left(t\right)}{\left({\mathit{e}}_i^{\left(t\right)}\right)}^T\Delta {\mathit{p}}_i,\Delta {\mathit{p}}_{i,\perp }={\mathbf{P}}_{i,\perp }^{\left(t\right)}\Delta {\mathit{p}}_i,$$

(30)

where ${\mathbf{P}}_{i,\perp }^{\left(t\right)}=\mathbf{I}-{\mathit{e}}_i^{\left(t\right)}{\left({\mathit{e}}_i^{\left(t\right)}\right)}^T.$ Accordingly, the geometry-structured curvature matrix is chosen as

$${\mathbf{M}}_{i,\chi }^{\left(t\right)}={\mu }_{i,r,\chi }^{\left(t\right)}{\mathit{e}}_i^{\left(t\right)}{\left({\mathit{e}}_i^{\left(t\right)}\right)}^T+{\mu }_{i,\perp ,\chi }^{\left(t\right)}{\mathbf{P}}_{i,\perp }^{\left(t\right)}+ϵ\mathbf{I},$$

(31)

where ${\mu }_{i,r,\chi }^{\left(t\right)}\ge 0$ and ${\mu }_{i,\perp ,\chi }^{\left(t\right)}\ge 0$ characterize the local radial and tangential curvatures of the reduced objective, respectively, and $ϵ>0$ is a small regularization parameter used to ensure positive definiteness. This model assigns different curvatures to the LOS direction and its orthogonal plane, thereby reflecting the anisotropic sensitivity of the 1-D AOA FIM with respect to UAV coordinate perturbations.

In implementation, the radial and tangential curvature coefficients can be obtained from directional second-order differences in the reduced objective. Let $\delta >0$ be a small probing step and let $\{{\mathit{b}}_{i,1}^{\left(t\right)},{\mathit{b}}_{i,2}^{\left(t\right)}\}$ be an orthonormal basis of the tangent plane perpendicular to ${\mathit{e}}_i^{\left(t\right)}$. Then one possible choice is

$${\mu }_{i,r,\chi }^{\left(t\right)}=\left|{\displaystyle \frac{{\overline{\varphi }}_{i,\chi }^{\left(t\right)}({\mathit{p}}_i^{\left(t\right)}+\delta {\mathit{e}}_i^{\left(t\right)})-2{\overline{\varphi }}_{i,\chi }^{\left(t\right)}\left({\mathit{p}}_i^{\left(t\right)}\right)+{\overline{\varphi }}_{i,\chi }^{\left(t\right)}({\mathit{p}}_i^{\left(t\right)}-\delta {\mathit{e}}_i^{\left(t\right)})}{{\delta }^2}}\right|,$$

(32)

$${\mu }_{i,\perp ,\chi }^{\left(t\right)}={\displaystyle \frac{1}{2}}{\displaystyle \sum _{\ell =1}^2}\left|{\displaystyle \frac{{\overline{\varphi }}_{i,\chi }^{\left(t\right)}({\mathit{p}}_i^{\left(t\right)}+\delta {\mathit{b}}_{i,\ell }^{\left(t\right)})-2{\overline{\varphi }}_{i,\chi }^{\left(t\right)}\left({\mathit{p}}_i^{\left(t\right)}\right)+{\overline{\varphi }}_{i,\chi }^{\left(t\right)}({\mathit{p}}_i^{\left(t\right)}-\delta {\mathit{b}}_{i,\ell }^{\left(t\right)})}{{\delta }^2}}\right|.$$

(33)

Other curvature estimates can also be used, but the essential idea is to preserve the radial–tangential anisotropy induced by the LOS geometry.

The coordinate update is then obtained by solving the following convex quadratic program (QP):

$$\begin{array}{cc}\hfill \Delta {\mathit{p}}_i^{★}=arg\underset{\Delta {\mathit{p}}_i}{min}& Q_{i,\chi }^{\left(t\right)}(\Delta {\mathit{p}}_i)\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& {\mathit{p}}_i^{\left(t\right)}+\Delta {\mathit{p}}_i\in {\mathcal{P}}_i.\hfill \end{array}$$

(34)

After obtaining $\Delta {\mathit{p}}_i^{★}$, the coordinate is updated as

$${\mathit{p}}_i^{(t+1)}={\mathit{p}}_i^{\left(t\right)}+{\gamma }_i^{\left(t\right)}\Delta {\mathit{p}}_i^{★},$$

(35)

where ${\gamma }_i^{\left(t\right)}\in (0,1]$ is selected by a backtracking line search to ensure a sufficient decrease in the orientation-eliminated objective. Finally, the array orientation is recovered by the orientation response derived in the previous subsection:

$${\mathit{u}}_i^{(t+1)}={\mathit{u}}_{i,\chi }^{★}\left({\mathit{p}}_i^{(t+1)}\right).$$

(36)

The above procedure yields a structured block update: the orientation variable is optimized exactly for each candidate coordinate, while the coordinate itself is updated through a geometry-aware SQP step. Compared with a black-box joint optimizer, the proposed GS-SQP method reduces the explicit search dimension and exploits the radial–tangential sensitivity structure of the 1-D AOA FIM.

The complete GS-SQP procedure is summarized in Algorithm 1.

4.3. Convergence and Complexity Discussion

We briefly discuss the convergence behavior of the proposed GS-SQP method. Since the original deployment problem is nonconvex, the proposed algorithm is not intended to guarantee global optimality. Instead, the goal is to obtain a high-quality stationary deployment by exploiting the 1-D AOA information structure. In each block update, the orientation variable is optimized exactly for the current UAV coordinate through the Rayleigh-type subproblem. Therefore, the remaining update is performed on the orientation-eliminated objective ${\overline{\varphi }}_{i,\chi }^{\left(t\right)}\left({\mathit{p}}_i\right)$.

Algorithm 1 Geometry-structured SQP for UAV-borne 1-D AOA deployment

1: Input: Criterion $\chi \in \{\mathrm{A},\mathrm{D}\}$, feasible sets, initial deployment $({\mathbf{P}}^{\left(0\right)},{\mathbf{U}}^{\left(0\right)})$, $T_{\max }$, and ${\epsilon }_{\mathrm{tol}}$. 2: Output: Optimized deployment $({\mathbf{P}}^{★},{\mathbf{U}}^{★})$. 3: Initialize $t=0$ and evaluate ${\mathsf{\Phi }}_{\chi }^{\left(0\right)}$. 4: while $t<T_{max}$do 5:     for $i=1,\dots ,M$ do 6:         Fix the other UAVs, form ${\mathbf{S}}_i^{\left(t\right)}$, and eliminate orientation by solving (21) or (23). 7:         Build the reduced objective (25) and the geometry-structured curvature matrix (31). 8:         Solve the reduced QP (34), apply backtracking, and update $({\mathit{p}}_i,{\mathit{u}}_i)$. 9:     end for 10:   Evaluate ${\mathsf{\Phi }}_{\chi }^{(t+1)}={\mathsf{\Phi }}_{\chi }({\mathbf{P}}^{(t+1)},{\mathbf{U}}^{(t+1)})$. 11:   if $|{\mathsf{\Phi }}_{\chi }^{(t+1)}-{\mathsf{\Phi }}_{\chi }^{\left(t\right)}|/max\left\{\right|{\mathsf{\Phi }}_{\chi }^{\left(t\right)}|,1\}<{\epsilon }_{\mathrm{tol}}$ then 12:         break 13:    end if 14:    $t\leftarrow t+1$. 15: end while 16: Return the latest accepted deployment as $({\mathbf{P}}^{★},{\mathbf{U}}^{★})$.

For the coordinate update, the local quadratic model in (28) uses a positive definite curvature matrix ${\mathbf{M}}_{i,\chi }^{\left(t\right)}\succ \mathbf{0}$. The step length ${\gamma }_i^{\left(t\right)}$ is selected by backtracking line search to ensure a sufficient decrease in the orientation-eliminated objective. Consequently, each accepted block update satisfies

$${\overline{\varphi }}_{i,\chi }^{\left(t\right)}\left({\mathit{p}}_i^{(t+1)}\right)\le {\overline{\varphi }}_{i,\chi }^{\left(t\right)}\left({\mathit{p}}_i^{\left(t\right)}\right).$$

(37)

After updating the corresponding orientation response, the global CRLB-based objective is non-increasing over block sweeps. Under standard assumptions commonly used in recent nonconvex MM, alternating minimization, and coordinate-descent analyses, including compact feasible deployment sets, local smoothness of the reduced objective, and uniformly bounded positive definite curvature matrices, any accumulation point of the generated sequence satisfies the first-order stationarity condition of the orientation-eliminated deployment problem [35,36,37]. Similar descent and stationarity arguments have also been adopted in recent CRLB/FIM-based sensor-placement algorithms using block-MM or successive approximation strategies [20,21,25].

The computational cost of each block update is low. The orientation elimination only requires solving a $2\times 2$ eigenvalue or generalized eigenvalue problem on the LOS tangent plane. The coordinate update solves a three-dimensional convex QP with the local deployment constraint ${\mathcal{P}}_i$. Therefore, for fixed-size local constraints, the per-sweep complexity scales linearly with the number of sensing UAVs, i.e., $\mathcal{O}\left(M\right)$, apart from the cost of evaluating the reduced objective and its local derivatives. This makes the proposed method suitable for repeated deployment updates and constrained UAV placement scenarios.

5. Numerical Simulations

In this section, we evaluate the proposed GS-SQP framework from three complementary perspectives: convergence behavior, optimized UAV deployment geometry under different feasible regions, and Monte Carlo localization-error behavior of the resulting UAV-borne sensing configurations. We also include robustness experiments to examine the sensitivity of the proposed design to random initialization and target-position mismatch.

5.1. Simulation Settings

Unless otherwise specified, we use $M=8$ sensing UAVs and the nominal target position ${\mathit{x}}_0={[0,0,200]}^{\top }$ m. Each sensing UAV carries a lightweight 1-D AOA linear array. The uniform-noise case sets ${\sigma }_{\theta ,i}=5^{\circ }$ for all UAVs, while the nonuniform case sets ${\sigma }_{\theta ,i}\sim \mathcal{U}[1^{\circ },5^{\circ }]$. These values are representative post-estimation angular-error standard deviations, not guaranteed hardware accuracies. Lower signal-to-noise ratio (SNR) links can be modeled by assigning larger ${\sigma }_{\theta ,i}^2$ without changing the proposed FIM/CRLB formulation.

Two deployment scenarios are used. In the free-flight case, each sensing UAV can move in ${\mathcal{P}}_i=[-250,250]\times [-250,250]\times [-50,50]{\mathrm{m}}^3.$ In the constrained hovering-region case, the feasible set is a collection of local hovering cubes around the vertices of the same bounding box, with side length $L_{\mathrm{v}}=60$ m. All coordinates in Figure 2 and in the following simulations are relative coordinates with respect to a chosen reference point in the deployment space, rather than absolute altitudes measured from the ground plane. Therefore, a negative z coordinate only indicates a position below the selected reference height.

Practical attitude disturbances, stabilization errors, and position-tracking errors can perturb the realized ${\mathit{u}}_i$ and ${\mathit{p}}_i$, causing AOA bias or larger effective angular noise. Robust UAV control and disturbance-observer-based compensation methods [38,39] are therefore relevant for realizing the planned sensing geometry; robust joint control–deployment design is left for future work.

The solver comparison includes GS-SQP, isotropic SQP (Iso-SQP), Manopt manifold optimization [40], trust-region SQP (TR-SQP) [41], particle swarm optimization (PSO) [42], and differential evolution (DE) [43]. GS-SQP is initialized from random feasible deployments satisfying $\mathbf{J}({\mathbf{P}}^{\left(0\right)},{\mathbf{U}}^{\left(0\right)})\succ \mathbf{0}$. We use at most 100 global iterations, where each global iteration updates all sensing UAVs once, with $\mu =10$ and ${ϵ}_s={10}^{-2}$. The algorithm terminates when the relative change in ${\overline{\mathsf{\Phi }}}_{\chi }$ is below ${10}^{-6}$. Each experiment is repeated over 100 independently generated feasible initializations. All solvers start from the same feasible deployment in each Monte Carlo trial, and this deployment is included in the initial PSO/DE populations.

The settings in

Table 2. Algorithmic parameter settings used in the solver comparison.

Method	Main Settings
GS-SQP	Maximum global iterations: 100. One global iteration updates all sensing UAVs once. Coordinate subproblem: MATLAB quadprog. Maximum iterations inside each coordinate subproblem: 20. Optimality/constraint tolerances: ${10}^{-6}$/${10}^{-8}$.
Iso-SQP	Maximum global iterations: 100. Same coordinate-subproblem settings as GS-SQP. Curvature model: isotropic.
Manopt	Maximum iterations: 50. Gradient-norm tolerance: ${10}^{-6}$. Finite-difference step: ${10}^{-5}$.
TR-SQP	Maximum trust-region iterations: 80. Each trust-region step is solved by fmincon with SQP. Maximum inner fmincon iterations: 20. Maximum function evaluations per inner call: 800. Initial/minimum/maximum trust-region radius: 60/${10}^{-5}$/300 m.
PSO	Swarm size: 40. Maximum iterations: 80. Inertia weight: 0.72. Cognitive/social coefficients: 1.49/1.49.
DE	Population size: 50. Maximum iterations: 80. Mutation factor: 0.8. Crossover rate: 0.9.

are fixed across all simulation cases. All simulations are implemented in MATLAB R2022b on a Windows 11 desktop with 32-GB RAM.

5.2. Algorithmic Effectiveness

We first evaluate whether the proposed GS-SQP solver can reliably reduce the deployment objective under different feasible regions and design criteria. The comparison includes structured local solvers, generic manifold and SQP solvers, and population-based global-search methods. The objective histories and runtime results are reported together to assess both optimization accuracy and computational efficiency.

Figure 3 shows that GS-SQP achieves the best or near-best final objective values in all four tested cases. Iso-SQP also improves the deployment from the same initializations, but its isotropic quadratic model cannot fully capture the anisotropic radial–tangential LOS sensitivity of a 1-D AOA sensor, leading to slower and more conservative convergence. Manopt can approach GS-SQP in some cases, but it treats the problem as a generic manifold optimization problem and relies on finite-difference gradients. TR-SQP directly optimizes the full position–orientation variables and is therefore more sensitive to the nonconvex search space. PSO and DE provide useful derivative-free baselines, but their final objective values are generally limited by the fixed population and iteration budgets.

Figure 4 compares the corresponding runtime. GS-SQP is slightly slower than PSO and DE in some cases, but the difference is modest and is accompanied by better final objective values. Compared with the generic local solvers, GS-SQP is usually faster because each block update only requires a closed-form orientation refinement and a three-dimensional QP.

5.3. Position–Orientation Ablation

We further perform an ablation study to clarify the roles of UAV positions and array orientations. Figure 5 compares three schemes: joint position–orientation optimization (“all”), coordinate-only optimization (“p-only”), and orientation-only optimization (“u-only”).

The results show that UAV position is the dominant design variable. In the free-flight case, the “p-only” curves are close to those of “all,” while “u-only” gives much smaller improvement. In the constrained hovering-region case, orientation optimization becomes more useful because the UAV coordinates cannot move freely, but “p-only” still provides most of the gain. Therefore, the proposed structure is consistent with the observed behavior: coordinate updates provide the main improvement, and orientation updates act as a low-dimensional refinement.

5.4. Robustness to Nominal-Target Mismatch

In this subsection, we evaluate the robustness of the optimized UAV deployment with respect to nominal target mismatch. Specifically, the sensing UAV configuration is designed for the nominal target UAV position ${\mathit{x}}_0$, while the actual target position is perturbed as $\mathit{x}={\mathit{x}}_0+\Delta$, where the mismatch vector $\Delta$ has a fixed radius ${\parallel \Delta \parallel }_2$ but a uniformly random direction. For each mismatch radius, the reported result is obtained by averaging over 1000 Monte Carlo trials.

Figure 6 shows the mean normalized degradation of the optimized objective as the mismatch radius ${\parallel \Delta \parallel }_2$ increases. It is observed that, for both A-optimal and D-optimal designs, the degradation increases monotonically with the mismatch radius, which is expected since the UAV deployment is optimized locally around the nominal target position. Nevertheless, the overall degradation remains moderate in the considered range. In particular, the D-optimal design exhibits noticeably stronger robustness than the A-optimal design: even at ${\parallel \Delta \parallel }_2=40$ m, the mean normalized degradation is only about 1.006 in the free-flight scenario and remains almost unchanged in the constrained hovering-region scenario. By contrast, the A-optimal design is more sensitive to target mismatch, especially in the free-flight case, where the degradation increases to about 1.088 at ${\parallel \Delta \parallel }_2=40$ m. Moreover, the constrained hovering-region deployment consistently shows smaller degradation than the free-flight deployment under both criteria, indicating that stronger UAV flight-region constraints lead to a more conservative but more mismatch-tolerant sensing geometry.

5.5. Localization Performance Before and After Optimization

We first compare the deployment objective before and after GS-SQP optimization. Figure 7 and

Table 3. Objective values before and after GS-SQP optimization under different UAV deployment scenarios and design criteria.

Scenario	Criterion	Random	Optimized	Improvement
Free-flight	A	861.47	265.44	69.19%
Free-flight	D	16.47	13.28	19.31%
Constrained hovering-region	A	1931.32	1119.22	42.05%
Constrained hovering-region	D	18.93	17.77	6.16%

show that the optimized UAV configurations consistently improve the A-optimal and D-optimal objectives. In the free-flight case, the A-optimal objective decreases from 861.47 to 265.44, and the D-optimal objective decreases from 16.47 to 13.28. In the constrained hovering-region case, the A-optimal objective is reduced from 1931.32 to 1119.22, and the D-optimal objective is reduced from 18.93 to 17.77. For both criteria, the optimized sensing UAVs move toward target-facing feasible locations while maintaining a wide angular spread around the target. In constrained hovering regions, this appears as movement toward the closest corner or face within each local cube. In free-flight deployment, the UAVs move upward toward the upper boundary of the admissible volume, closer to the target height, while spreading in the horizontal plane to improve the spatial conditioning of the FIM.

We then examine whether this design-stage gain can be translated into actual localization accuracy. To this end, we compare the maximum likelihood estimation (MLE) performance of random feasible and optimized UAV-borne 1-D AOA sensing configurations over 1000 Monte Carlo runs.

Table 4. Comparison of the MLE performance for random and optimized UAV-borne 1-D AOA sensing configurations over 1000 Monte Carlo runs.

Test Scenario	Angular-Noise Model	Tr(CRLB) (m2)	UAV Deployment	Total MSE (m2)	Bias (m)
Constrained hovering regions, $M=8$	Nonuniform	115.93	Random	548.58	0.33
			Optimized	118.89	0.16
	Uniform	434.37	Random	1205.95	0.83
			Optimized	445.74	0.73
Constrained hovering regions, $M=12$	Nonuniform	30.80	Random	69.96	0.19
			Optimized	31.29	0.15
	Uniform	193.15	Random	324.15	0.72
			Optimized	198.95	0.67
Free-flight region, $M=8$	Nonuniform	129.98	Random	675.32	0.54
			Optimized	132.55	0.26
	Uniform	769.76	Random	2232.80	1.36
			Optimized	794.79	0.74
Free-flight region, $M=12$	Nonuniform	117.02	Random	369.58	0.58
			Optimized	121.15	0.39
	Uniform	516.72	Random	1144.31	0.95
			Optimized	522.19	1.11

reports the corresponding total mean squared error (MSE) and bias, together with the theoretical lower bound $\mathrm{tr}\left({\mathbf{J}}^{-1}\left({\mathit{x}}_0\right)\right)$ achieved by the optimized UAV deployment. The empirical total MSE is defined as

$$\mathrm{MSE}={\displaystyle \frac{1}{N_{\mathrm{MC}}}}{\displaystyle \sum _{n=1}^{N_{\mathrm{MC}}}}{\parallel {\widehat{\mathit{x}}}^{\left(n\right)}-{\mathit{x}}_0\parallel }^2,$$

(38)

and the bias magnitude is defined as

$$\mathrm{Bias}=∥{\displaystyle \frac{1}{N_{\mathrm{MC}}}}{\displaystyle \sum _{n=1}^{N_{\mathrm{MC}}}}{\widehat{\mathit{x}}}^{\left(n\right)}-{\mathit{x}}_0∥.$$

(39)

Here, $N_{\mathrm{MC}}$ is the number of Monte Carlo trials. The empirical MSE is compared with the theoretical bound $\mathrm{tr}\left({\mathbf{J}}^{-1}\left({\mathit{x}}_0\right)\right)$.

Table 4 shows that the optimized deployment reduces localization error in all tested cases, with total MSE reductions ranging from 38.62% to 80.37%. For example, the MSE decreases from 548.58 to 118.89 in the constrained hovering-region case with $M=8$ and nonuniform noise, and from 2232.80 to 794.79 in the free-flight case with $M=8$ and uniform noise. The optimized MSE is also close to the corresponding theoretical bound $\mathrm{tr}\left({\mathbf{J}}^{-1}\left({\mathit{x}}_0\right)\right)$, while the random feasible configurations remain much farther from this bound. The bias is reduced in most cases as well; even in the only case where it slightly increases, the total MSE is still reduced from 1144.31 to 522.19. These results confirm that GS-SQP improves actual localization accuracy, not only the analytical CRLB objective.

6. Conclusions

This paper investigated CRLB-driven deployment design for UAV-borne 1-D AOA sensing in target UAV or aerial-emitter localization. The considered linear-array setting differs from conventional full-AOA or range-based deployment because each sensing UAV contributes a single orientation-dependent angular constraint. We formulated a joint position–orientation deployment problem under A- and D-optimality criteria, showed that the orientation block can be exactly eliminated for fixed UAV coordinates through a low-dimensional eigenproblem, and developed a geometry-structured SQP method for the resulting reduced coordinate problem. Numerical results showed that the proposed framework improves the deployment objective in both free-flight and constrained hovering-region scenarios, while Monte Carlo localization tests confirmed that the optimized deployments reduce estimation error relative to random feasible configurations.

Future work will focus on practical validation and more realistic mission constraints. In particular, hardware-in-the-loop experiments, real-world RF/linear-array measurements, and UAV flight experiments are needed to verify the achievable AOA accuracy, synchronization, calibration, and onboard implementation under realistic platform motion.