Single Anchor Constraint Overview

• Single anchor constraint is defined as a system using one fixed reference, fundamentally altering identifiability and design approaches across diverse applications.
• In object detection, adaptive sample selection replaces multi-anchor redundancy, achieving performance parity using a single square anchor.
• For localization and control, specialized methods like joint AoA-range estimation and sensor fusion overcome the inherent geometric limitations of a single anchor.

A single anchor constraint is a central structural or geometric limitation in diverse research areas, ranging from computer vision detection and indoor/outdoor localization to algebraic graph reductions and control theory. It specifies that only one fixed reference—be it an anchor box, anchor node, anchor graph vertex, or physical anchor—may be used for defining correspondence, estimating position or orientation, or enforcing logic or contact constraints. This paradigm is fundamentally distinct from multi-anchor scenarios, imposing identifiability, expressivity, or observability limitations that require specialized algorithmic or combinatorial strategies.

1. Foundations and Scope of the Single Anchor Constraint

The single anchor constraint is characterized by the presence of exactly one fixed reference (anchor) in a system, as opposed to multiple, often symmetrically placed, anchors. In object detection, the constraint appears as the elimination of multiple anchor boxes per location; in physical localization, it restricts RF or mmWave systems to infer spatial or orientation information from a single point of measurement. In graph theory and computational complexity, it enforces that only one vertex is fixed to represent logical falsehood in graph coloring gadgets, rather than two or more for explicit Boolean assignment. Across these domains, the constraint is not merely a reduction in available measurements or degrees of freedom—it fundamentally alters the identifiability or expressivity landscape of the problem.

2. Object Detection under the Single Anchor Constraint

Anchor-based object detectors such as RetinaNet traditionally rely on tiling multiple anchors (scales, aspect ratios) at each image location, using fixed Intersection-over-Union (IoU) thresholds for positive/negative assignment. However, adaptive training sample selection (ATSS) demonstrates that with per-object statistical thresholding, a single anchor per location suffices for high detection performance (Zhang et al., 2019).

Formally, for each ground-truth box $g$ and feature map level $i$:

• ATSS selects the $k$ nearest anchors (center-wise), combines all $kL$ candidates, and computes their IoU distribution $\mathcal{D}_g$.
• The mean $m_g$ and standard deviation $v_g$ of IoUs define the adaptive threshold $t_g = m_g + v_g$.
• An anchor is positive iff its center is within $g$ and $\mathrm{IoU}(a,g) \ge t_g$.
• Empirically, a single square anchor per location achieves the same AP as nine anchors (AP=39.3% vs 39.2%) on COCO minival, and varying the anchor’s scale or aspect ratio within reasonable bounds yields negligible performance differences.

This operationally collapses the design space: adaptive sample selection eliminates the need for anchor redundancy, matching or exceeding the accuracy of multi-anchor approaches while streamlining architecture and computation.

3. Localization and Positioning: Identifiability with a Single Anchor

Most classical wireless and robotic positioning relies on multilateration, requiring at least three non-collinear anchors in 2D (or more in higher dimensions) for unique localization. The single anchor constraint reduces this to a fundamentally under-determined inverse problem. Several strategies are employed to recover identifiability:

a) Joint AoA and Range Constraints

In single-anchor, array-based localization (e.g. 5G uplink), the intersection of the measured range sphere and AoA ray at the anchor yields a unique 2D/3D position if both constraints are accurate (Spanos et al., 8 Jun 2025). The solution: $x = x_0 + r\cos\theta,\qquad y = y_0 + r\sin\theta$ or, in 3D, including elevation $\phi$,

$$\begin{aligned} x &= x_0 + r\cos\theta\cos\phi,\ y &= y_0 + r\sin\theta\cos\phi,\ z &= z_0 + r\sin\phi. \end{aligned}$$

High precision depends on accurate array calibration and channel order modeling; systematic or multipath errors directly bias the result by $r\Delta\theta$. Simulations and experiments confirm sub-meter accuracy is achievable if these requirements are met.

b) Multipath, Temporal, and Physical Sensor Fusion

In UWB and mmWave settings, identification with one anchor is possible by leveraging temporal diversity (motion), exploiting channel impulse responses (CIR), or fusing additional modalities (IMU, vision). Examples include:

• SARR-LOC: Combines single-antenna, single-anchor RSSI with camera-detected blockage events to recover the ellipse (First Fresnel Zone, FFZ); as each blockage yields a point on the boundary, ellipse fitting retrieves both range and bearing (Sunami et al., 2021). Median error is $\leq1$ m, comparable to three-anchor triangulation.
• CIR fingerprinting: A single anchor's CIR amplitude vector is modeled using multivariate GMMs, classifying area occupancy by maximizing joint likelihood across multiple snapshots (Li et al., 2022). Block-based metrics (MD-GMM–MaxSim) yield area-classification accuracies exceeding 90% in diverse LOS/NLOS conditions, outperforming per-bin or single-snapshot methods.
• UWB+IMU fusion: A robot with a single UWB anchor estimate is made observable by differentiating the range to extract speed, then using IMU heading information; an EKF fuses these to track joint position and orientation (Cao et al., 2020). Observability is guaranteed as long as the robot moves with nonzero velocity and heading varies.

c) Near-Field/Far-Field Limits and Fisher Information Constraints

The single anchor's geometric and information-theoretic limitations are pronounced in electromagnetic localization:

• In the near-field (distance to anchor $\lesssim$ Fraunhofer limit), the spherical wavefront provides both range and angle information, making 3D position and 3D orientation jointly estimable with a single anchor and large arrays (Emenonye et al., 2023).
• In the far-field, only 2D position and orientation are recoverable unless multiple beams are used; orientation becomes unidentifiable without beamforming.
• The Equivalent Fisher Information Matrix (EFIM) quantifies the fundamental performance: $J_\zeta^e$ is full-rank only when the measurement structure provides sufficient geometric diversity (time, angle, Doppler, multipath). NLOS paths function as "virtual anchors"—in static NLOS-only conditions, at least two geometrically diverse NLOS paths are required for 2D positioning (Kakkavas et al., 2018).

A summary table of estimability as a function of field regime is provided:

Unknown	Near Field: 3D Pos/Ori	Far Field (beams): 2D	Far Field (no beams): Ori unidentifiable
Agent pos+ori	Yes	Yes	No
Anchor pos+ori	Yes	Yes	No

Uncertainty in anchor calibration directly penalizes the agent's localization accuracy via the EFIM Schur complement, possibly rendering the problem unidentifiable if the anchor's state is poorly known.

4. Graph-Theoretic and Logical Formulations

In computational complexity and graph reductions, the single anchor constraint arises as a mechanism for logical gadget construction. In the single-anchor 3-coloring framework (Güngör, 9 Nov 2025):

• Only one vertex (the anchor $a_0$) is fixed to color 0 (logical false), with all logical "truth" encoded via the other two colors {1,2}.
• A logical gadget (ladget) is a tuple $(G, a_0, I, \theta)$, specifying the underlying graph, anchor, inputs, and output vertex. Boolean implementability is defined via universality (all Boolean input assignments are colorable) and consistency (output assigns uniquely).
• The constraint yields an asymmetric system: only "false" is pinned by the anchor, "true" must be internally enforced or swapped as needed. Structural constraints (non-adjacency, degree, universality) become strict, resulting in a combinatorially rare set of valid primitives. For XNOR (2-input), exactly two non-isomorphic gadgets are minimal among $2.95 \times 10^{10}$ candidate configurations.
• Embedding the system into $k$-coloring for $k>3$ is done by isolating $k-3$ colors in a clique, ensuring only the 3-color ladget's logical structure remains; this generalizes the single-anchor constraint.

This contrasts sharply with the two-anchor approach (two fixed colors for true/false), which is structurally and combinatorially more forgiving but less expressive in the single-anchor sense.

5. Control and Simulation Relations

In control-theoretic formulations, the single anchor constraint appears as physical contact preservation—e.g., ensuring a humanoid robot remains in valid ground (anchor) contact during template-based trajectory planning (Kurtz et al., 2019). Specifically:

• Approximate simulation relations between high-DOF anchor models and template models (e.g., Linear Inverted Pendulum, LIP) use a simulation function $\mathcal{V}$ and interface map $u_\mathcal{V}$ to ensure $\epsilon$-closeness of outputs.
• The ground-contact (single-anchor) constraint is expressed as a set of linear inequalities enforcing that the net spatial force on the contact foot lies within a frictional cone (contact-wrench cone, CWC).
• This is encoded as

$$A_{cwc} \begin{bmatrix}x_{task} \ u_{task}\end{bmatrix} \leq b_{cwc}$$

and integrally imposed within a finite-horizon quadratic program (QP) that couples the template and anchor (task) dynamics.

• The solution guarantees (a) $\epsilon$-close tracking of template plans and (b) strict satisfaction of the single-contact support condition throughout the planned motion.

6. Implications, Limitations, and Extensions

The single anchor constraint creates strong identifiability and expressivity bottlenecks. Essential implications and trade-offs include:

• In detection, it greatly simplifies design via adaptive assignment mechanisms but requires sophisticated statistical decision rules to compensate for lost geometric coverage.
• In localization, it forces the exploitation of auxiliary modalities (IMU, vision, temporal dynamics) or rich multipath structure. Near-field operation, high array apertures, careful calibration, or fusion with additional sensors become necessary for observability.
• Logical and combinatorial problems exhibit a marked reduction in the set of constructible gadgets; asymmetry and non-trivial output constraints become prevalent.
• In dynamic control, using single anchor support constraints ensures both physical feasibility (contact) and provable tracking bounds.
• Limitations often stem from the underdetermined geometry, direct sensitivity to calibration or modeling errors, and for fingerprints, non-stationarity in environment or measurement noise. For practical systems, these necessitate domain-specific fusion or compensation mechanisms.
• Extensions include multi-modal sensor integration, enriched environmental modeling (multi-path, clutter), active anchor relocation, and temporal averaging to mitigate single-shot degeneracy.

7. Comparative Summary Across Domains

Domain	Nature of Single Anchor Constraint	Key Algorithmic Response
Object Detection	One anchor per location; adaptively assigned	Statistical thresholding (ATSS)
Physical Localization	One reference RF/mmWave anchor	Joint AoA+range, multipath exploitation, sensor fusion
Logical Graph Gadgets	One fixed-color vertex (anchor) for falsehood	Asymmetric universality/consistency, careful gadget design
Control and Robotics	Single ground contact support per plan	Linearized contact wrench cone constraints in MPC

The single anchor constraint is thus a pervasive structural or geometric challenge, requiring domain-specific innovations for practical tractability or performance, and often serving as a valuable lens into the observational or expressivity limits of systems.