Excitation-Supervised Closed-Loop Self-Calibration and Target Seeking for an Unknown-Pose Range-Bearing Relay

A vehicle seeking a hidden target through a rangebearing relay of unknown position and yaw must decide, online, whether its own motion has already made the relay calibration trustworthy, and what to do when it has not. Two distinct vehicle-relative observations are known to remove the calibration gauge and make the target’s relay-local packet globally actionable [1], but that identifiability statement is static: it classifies a stored trajectory window only after the fact. This paper supplies the closed-loop layer: we show that the trajectory-spread margin Sv that governs identifiability is simultaneously a finite-noise seed accuracy bound, a local-vector variance decomposition, and a circle-geometry excitation budget, and we use it to supervise an excitation-reset controller. An excitation-supervised algorithm retriggers exploratory motion whenever the spread certificate is insufficient, projecting the target-seeking input away from the excitation’s push direction while it does so, and otherwise proceeds to unrestricted target seeking; under explicit sampling assumptions the supervision rule provably acquires any required excitation within a finite time, while, in the noiseless local regime with positive excitation decay, estimator convergence yields target-seeking convergence after certification.

The trajectory-spread margin Sv that governs identifiability conditioning in [1] turns out to carry three additional roles once the estimator is placed inside a control loop: it bounds the constructive seed’s sensitivity to measurement noise, it decomposes the target-estimate covariance into an averaging term and a calibration-propagation term, and, for circular excitation, it has a closed form in the excitation radius and packet count. Together these give a controller a certifiable, checkable stopping rule for exploratory motion.

The paper's contributions are:

1. "a quantitative excitation theory showing that the spread margin Sv simultaneously bounds constructive-seed error, decomposes target-estimate variance into averaging and calibration-propagation terms, and gives an explicit excitation budget for circular motion, together with an accuracy-driven rule selecting the spread threshold from a desired calibration-uncertainty level;"

2. an excitation-supervised estimation and target-seeking algorithm that formalizes the informal reset rule of [1] into an explicit online supervision loop triggered by the spread certificate alone;

3. "a finite excitation-acquisition proposition showing that, under explicit sampling assumptions, the supervision rule itself accumulates any required spread within an explicit finite time, with every hypothesis satisfied by the controller as implemented through an underexcited-phase projection of the seeking input, and a local recovery proposition showing that, in the noiseless model and for positive excitation decay, entry into the Gauss–Newton convergence basin yields target-estimate and target-seeking convergence;"

4. a ROS 2/Gazebo software-in-the-loop experiment in which the supervised loop, running as a ROS 2 node, steers a vehicle integrated by the Gazebo physics engine through real message transport and sensing delay;

5. "a 100-trial paired Monte Carlo decay-rate sweep and a spread-threshold ablation that validate the accuracy-driven threshold rule as a population root-mean-square error target, quantify its excitation cost, and expose its domain of validity, together with a target-seeking scenario showing that ordinary closed-loop motion can supply the needed excitation naturally, so that supervision matters most when such incidental excitation is absent."

The paper introduces an estimation and target-seeking algorithm (Algorithm 1) that uses the trajectory-spread margin Sv to decide between exploratory motion and unrestricted target seeking. The algorithm involves:

(i) Estimation:

"Estimation uses the Gauss–Newton implementation of [1], with analytic Jacobians for the residuals in (2) and Levenberg damping. The target estimate remains at its prior until the stored window contains two distinct views; the implementation then applies the constructive seed once and warm-starts every later packet from the previous refined estimate, with damping initialized at 10−2 and at most 25 iterations per packet."

(ii) Motion Control:

"The vehicle obeys q̇ = u with u = useek (t) + uexp (t), where useek is the nominal target-seeking law −k(q − p̂), projected while the stored window is underexcited so that it can never cancel the excitation’s push:

(ΠH(t) − k(q − p̂), Sv (W) < S̄, seek u = (−k(q − p̂), Sv (W) ≥ S̄, with H(t) = [v: v T n(t) ≥ −Ae−λT̄ /π] and n(t) = (−1⌊ωt/π⌋ [0, 1]T the push direction of the current excitation half-period (T̄ is the inter-packet bound of Proposition 2)."

(iii) Supervision Loop:

"The excitation epoch t0 resets whenever the stored packet window W is not yet excited enough; Algorithm 1 makes this check explicit using the spread certificate Sv (W) alone, so both the reset trigger and the projection window are exactly the quantity covered by the acquisition guarantee of Proposition 2."

The paper demonstrates that the supervision rule itself acquires any required spread within an explicit finite time, and in a noiseless model with positive excitation decay, estimator convergence yields target-seeking convergence after certification. The validation includes a ROS 2/Gazebo software-in-the-loop experiment where the supervised loop steers a physics-integrated vehicle through real message transport and sensing delay. The results show that "supervision matters when the schedule’s decay outruns the unknown time-to-adequate-excitation: over 100 paired trials the fixed baseline’s yaw RMSE rises from 0.010 to 0.065 rad and success falls to 56%, while target-tracking error remains insensitive; supervision keeps yaw RMSE between 0.0095 and 0.0191 rad with 100% success. The final conclusion is that supervision is an online adequacy mechanism, not a universal source of lower tracking error when the nominal trajectory is already informative."

The paper also provides quantitative analysis for the spread margin Sv:

(Lemma 1):

For any fixed yaw perturbation δψ, minδx k∥δx + RSlvk δψ∥2 =Pδψ2 Sv, where S = 01 −1.

(Proposition 1 - Constructive-seed accuracy):

"Let the two selected vehicle packets satisfy∥l̃vj − lvj ‖ ≤ϵ, j ∈ a, b, and let each of the K target packets satisfy∥l̃tk − lt,∗ ‖ ≤ϵ. Define P̄l̃ t = K−1 k l̃tk, let dv =∥lvb − lva ‖ with 4ϵ ≤ dv, and form the target seed as p̂ = x̂ + R(ψ̂)l̃ t. Then the resulting constructive estimates satisfy

2πra/2πϵ,∥x̂ − x∥ ≤ϵ1+,ψ̂ − ψ ≤dv/dv

2π(ra + rt)/∥p̂ − p∥ ≤ϵ2+,dv"

(Corollary 1 - Local-vector variance decomposition):

"Marginalizing p removes the target rows’ Fisher information about (x, ψ); the information on (x, ψ) comes from the vehicle packets alone, and the yaw Schur complement equals Sv /σ 2. Consequently var(ψ̂) = σ 2 /Sv in the linearized model."

(Corollary 2 - Excitation budget):

Let the stored window contain K vehicle positions on a circle P of radius ρ, qk = c + ρuk with∥uk∥ = 1, and let ū = K−1 k uk. Then, in the noiseless model, Sv = Kρ2 (1 −∥ū∥2).

(Remark 1 - Accuracy-driven threshold selection):

"A principled S̄ follows instead from an accuracy requirement: Corollary 1 gives var(ψ̂) = σ 2 /Sv, so a required yaw standard deviation ϵψ induces σ2/σ2 hence K≥ 2 (6) S̄ ≥ 2,ϵψ/ρ,ϵψ under equally spaced circular excitation by Corollary 2."

(Table I - Excitation Budget of Corollary 1):

This table shows the relationship between the excitation radius ρ (m), packet count K, and the required spread threshold S̄. For example: K for S̄ = 9.04 corresponds to a radius ρ = 1.5 m, and K = 3."

The paper concludes that Algorithm 1 does not need to know the time-to-adequate-excitation in advance; it measures Sv directly and keeps resetting until the certificate clears, at the excitation cost quantified next. Furthermore, the supervised controller’s excitation acquisition is nearly independent of the mismatch parameter, and S̄ = 0.16 targets a population yaw RMSE of 0.05 rad and the supervised policy remains below it at every λ (at most 0.0192 rad). The study validates that ordinary closed-loop motion can deliver excitation naturally; supervision matters most when that incidental excitation is absent or insufficient, as in the no-transient sweep. The final comparison shows that the measured price of supervision is arrival time, not accuracy.

The paper also details a rolling-window implementation (Corollary 3) and a statistical validation protocol involving 100 paired Monte Carlo trials. The results confirm that the same selfexcitation mechanism that makes the nominal scenario of Section V-A easy: any trajectory with a substantial transit accumulates Sv incidentally, and the certificate clears without dedicated exploration. Ordinary closed-loop motion can deliver excitation naturally; supervision matters most when that incidental excitation is absent or insufficient, as in the no-transient sweep. The final results confirm that the supervised policy remains below it [the accuracy target] at every λ (at most 0.0192 rad).

The summary is concluded by stating: Algorithm 1 does not need to know the time-to-adequate-excitation in advance; it measures Sv directly and keeps resetting until the certificate clears, at the excitation cost quantified next. and the supervision rule provably acquires any required spread within an explicit finite time, with every hypothesis of that guarantee satisfied by the controller as implemented. The paper also notes that "The control contribution is the certificate-supervised hybrid law, particularly the underexcited-phase projection (9) that makes the online trigger and finite-acquisition proof refer to the controller as implemented, without requiring online trajectory optimization or a matrix-valued observability objective."

(Note: The summary above is constructed by quoting and synthesizing all relevant parts of the paper as requested.)

Based on the provided research, here are specific improvements that can be made to existing AI systems, along with what those improved systems would be capable of:

The core contribution is a novel control-design layer that replaces static identifiability checks with a dynamic, online supervision loop based on the trajectory-spread margin, denoted as the spread certificate (Sv).

Here are the specific improvements and capabilities:

1. Improve real-time state estimation and target tracking for unknown-pose range-bearing relays by replacing fixed exploration schedules with an excitation-supervised control law.

2. Enable autonomous vehicles to robustly recover accurate position, yaw, and target estimates even when the initial calibration is poor or when the environment lacks sufficient incidental motion (e.g., in static or low-excitation scenarios).

3. Develop a control system that dynamically manages the trade-off between necessary exploration (to gather new information) and high-precision target seeking (to minimize tracking error), ensuring that exploration is only triggered when current motion has not sufficiently excited the measurement set to be trustworthy.

4. Create an AI/Control system capable of performing online, certificate-driven adaptation: the system will automatically re-trigger exploratory maneuvers whenever the stored data's spread margin falls below a dynamically selected accuracy threshold, effectively monitoring its own calibration quality without needing explicit ground-truth state knowledge for every step.

5. Implement a mechanism that quantifies and bounds estimation uncertainty by decomposing it into two orthogonal terms: an averaging term (from target packet statistics) and a calibration-propagation term (governed by the spread margin). This allows the system to explicitly understand whether its current uncertainty stems from target noise or from poor frame calibration, leading to more informed control decisions.

6. Design a controller that utilizes an underexcited-phase projection of the nominal seeking law. This ensures that during periods of low confidence, the vehicle's motion is steered away from directions that would actively oppose the current excitation push (i.e., it prevents the exploration from canceling itself out), while still allowing for controlled, directed motion toward a better estimate.

7. Create an AI system with guaranteed finite-time acquisition: The system will provably acquire any required level of excitation within a finite time, meaning that if the environment requires more information than currently available, the supervision rule is mathematically guaranteed to eventually provide it through repeated actions.

8. Enable closed-loop navigation in complex sensing environments (e.g., ROS 2/Gazebo simulations with real message transport and latency) where the control logic remains stable and accurate despite realistic communication delays and sensor noise, as validated by the software-in-the-loop experiments.

In summary, the improved AI system would be a robust navigation agent capable of reliably finding a hidden target using an unknown beacon, even when its initial orientation estimate is highly uncertain or when external motion is minimal. It achieves this by intelligently self-calibrating its reference frame in real-time based on a quantifiable measure of data richness (the spread margin), ensuring it only explores when necessary and never letting the exploration move in the wrong direction.