Anchor-Free Hidden-Target Seeking via Certified Self-Calibration under Correlated Odometry

I. INTRODUCTION

• The vehicle receives range-bearing packets from a single relay but has neither global pose nor direct target sensing, relying only on integrated body-frame odometry.The introduction frames the problem as seeking under simultaneous loss of global localization and direct access to the goal.Source: §I · p. 1
• Earlier approaches address only parts of this setting: source seeking requires sensing the source, while single-beacon navigation anchors the robot rather than relaying a hidden target.Prior work in the globally anchored sensing model did not treat the vehicle trajectory as latent or account for its correlated pose errors.Source: §I · p. 1
• The paper identifies, certifies, and controls on the smallest observable quotient sufficient for the task, rather than reconstructing an unobservable global state.Its architecture excites until the quotient is certified, refuses uncertifiable calibration, seeks in quotient coordinates, and detects relay-frame changes.Source: §I · pp. 1–2
• Two distinct views recover the task quotient, and the estimator’s O(K) uncertainty certificate retains cross-view odometry correlations that independent-pose treatment misses.Treating cross-view poses as independent reduces coverage to 82.1%, versus nominal-level coverage with the correlated treatment.Source: §I · p. 2

Seeking and navigation approaches

• Source seeking reaches a target without position information but requires direct sensing of the source [1],; single-beacon navigation uses a beacon to anchor the robot rather than relay a hidden third party,.The paper’s relay observes both the robot and a hidden target.Source: §II · p. 2

Target and sensor self-calibration

• Prior sensor-calibration and multistatic-radar methods use multiple sensors or global pose priors, unlike this single-relay setting with no global relay-pose prior, [12],.Here, the method estimates and certifies the control-relevant quotient and adopts relay-frame changes only from certified windows.Source: §II · p. 2

Cooperative localization and gauge-aware estimation

• Gauge-aware localization and SLAM respect unobservable directions, while this work estimates directly on the quotient and certifies the variables consumed by the controller,,,,,.Motion supplies the missing calibration that lets a single unknown-pose relay suffice where range-only adaptive source localization requires more beacons than workspace dimensions [18].Source: §II · p. 2
• Unlike general Procrustes uncertainty under arbitrary covariance, this work reduces the odometry random-walk covariance calculation to O(K) for online use and closed-loop certification.The reduction exploits the specific correlation structure of odometry.Source: §II · p. 2

Prior work on this sensing model

• The earlier version of this sensing model assumed a globally known vehicle trajectory [8], and its closed-loop extension used excitation supervision [9]; neither treated the trajectory as latent.Removing the anchor introduces a full SE(2) gauge and correlated pose errors in the calibration uncertainty.Source: §II · p. 2

III. PROBLEM FORMULATION

• The model distinguishes an inaccessible global frame W, the vehicle’s odometry frame O, and relay frame B, with the vehicle, relay, and target represented in W.The vehicle’s odometric pose relates its global position to an unknown SE(2) placement of O.Source: §III · p. 2
• The relay reports range and bearing to both vehicle and target in its own frame, with independent range and bearing noise.The packets use relay-to-odometry yaw θ, defined as relay yaw minus odometry-frame yaw.Source: §III · pp. 2–3
• The control-relevant task vector excludes the global transform and relay position, leaving a single rotation θ as the calibration burden.The reduced model defines local positions in the odometry frame.Source: §III · p. 3
• The sensing and motion assumptions require noisy body-frame odometry, static relay and target over an estimation window, and no sensor measurement of a global-frame quantity.Packets must be associated and timestamped, though delivery may be delayed, jittered, reordered, or dropped.Source: §III · p. 3

A. What no estimator can recover

• Theorem 1’s SE(2) transformation leaves every relay packet and odometry increment invariant, making the global configuration unlocalizable from the available signals.The indistinguishable global states form a three-dimensional orbit.Source: A. What no estimator can recover · p. 3
• Conditioned on the realized odometric trajectory, (θ, xo, po) is a complete set of invariants for the static scene after removal of global placement.The trajectory must remain known context or be included in the invariant tuple.Source: A. What no estimator can recover · p. 3
• At a single view, (θ, ek) is sufficient for control only together with the known odometric pose; it is not by itself the full quotient.The full quotient’s xo and po are recovered from (θ, ek) using the known context.Source: A. What no estimator can recover · p. 3
• Without orientation-bearing metric ego-motion, indistinguishable worlds can require opposite task vectors, preventing any deterministic command from guaranteeing target-distance descent.The residual single-view gauge admits θ and θ + π, with task vectors e and −e.Source: A. What no estimator can recover · p. 3

B. Motion-induced self-calibration

• Two noiseless views at distinct odometric poses uniquely determine the quotient in closed form.With yaw known, the vehicle and relay positions follow from the view equations, yielding every task vector.Source: B. Motion-induced self-calibration · p. 3
• Differencing the vehicle equations recovers yaw because one unique planar rotation maps the nonzero view-displacement vectors onto each other.The vectors have equal norm, so their orientation difference gives the rotation angle.Source: B. Motion-induced self-calibration · p. 3
• Repeated poses leave a one-parameter gauge with observability rank four of five; one distinct displacement raises the rank to five.Motion is the mechanism that closes this gauge.Source: B. Motion-induced self-calibration · p. 3
• Weighted multi-view registration recovers yaw whenever view spread is positive, reduces to the two-view formula at K = 2, and runs in O(K) per update.The implementation takes 3.6 µs at K = 64.Source: B. Motion-induced self-calibration · p. 4

V. A DEPLOYABLE YAW CERTIFICATE UNDER CORRELATED ODOMETRY

• The deployable yaw-variance certificate is exact to first order for packet noise and integrated translational odometry, including cross-view correlations from shared increments.Its evaluation uses an O(K) suffix-sum form.Source: §V · p. 4
• Conditional on the poses, yaw information is determined by the spread Sv; under isotropic Cartesianized noise, Iθ = Sv/σ2.The spread is zero for repeated poses and positive otherwise.Source: §V · p. 4
• Correlated pose errors arise from shared prefixes of integrated independent increments, with covariance min(vi, vj) I2 before removing the common gauge reference.Automatic centering makes the estimator invariant to a common pose offset.Source: §V · p. 4

0.88 and 1.07, the pooled ratio is 0.91, and the realized 95% coverage lies between 94.0% and 95.6% per condition with a pooled value of 94.8%.

• Variance calibration was evaluated over ten odometry conditions, translation noise of 0.25 to 5 cm per increment, and windows of 8 to 16 views, using 20,000 trials.The passage reports the variance as empirically calibrated over this tested operating envelope.Source: §V · pp. 4–5

V. A DEPLOYABLE YAW CERTIFICATE UNDER CORRELATED ODOMETRY

• Closed-loop guarantees use the deployable certificate; a sequential EKF with matched initialization and bias state can certify an incorrect yaw after unlucky initializations.Its self-reported covariance lacks the certificate’s calibration.Source: §0.88 · p. 5

VI. CERTIFIED ACQUISITION, SEEKING, AND RECOVERY

• The supervisor excites until consecutive windows pass the certified-yaw threshold, refuses uncertified estimates, and has a finite-window signal-to-odometry condition for certification.Certification was observed in 200/200 trials at nominal odometry noise and 188/200 at 1 cm per increment; at 2 to 5 cm, the tested estimators did not extend the boundary.Source: §VI · p. 5
• With certified yaw error below π/2 and bounded estimate disturbance, task error converges exponentially to an ultimate ball whose radius is independent of mission horizon.Continuously re-measured task vectors restrict odometry error to the finite calibration window rather than allowing it to accumulate over the mission.Source: §VI · p. 5
• The implemented controller is a saturated unicycle law; no rigorous gain conditions are claimed for the full heading-error cascade.Its realization residual is bounded when the estimated task vector is bounded, and vanishes as heading error converges.Source: §VI · pp. 5–6
• 0.064 m median station error compares with 0.065 m for the known-yaw oracle, while dead reckoning drifts to a median of 27 m over 600 s.The closed-loop result is reported as statistically indistinguishable from the oracle.Source: §VI · p. 6
• Relay-yaw changes are adopted after persistent certified inconsistency; clean-campaign seeking succeeded in 150/150 trials at each tested step size of 20°, 40°, and 80°.Zero false adoptions occurred in 200 undisturbed trials per policy, with a Wilson 95% upper bound of 1.9%.Source: §VI · p. 6
• Recovery requires motion, while station keeping delays recalibration until the task demands movement because yaw error is task-irrelevant near the target.The reported certification condition is conditional on admissible windows being realized; guaranteeing their realization would require an additional stationarity or ergodicity assumption.Source: §VI · pp. 5–6

Covariance ablation

• 95.0% pooled coverage was achieved by the full certificate, versus 82.1% for the diagonal approximation and 69.3% for packet-only.The full model's pooled empirical-to-predicted yaw-variance ratio was 0.95; the diagonal and packet-only ratios were 3.64 and 9.22, respectively.Source: Figure 4 · p. 8, §VII · p. 8

Robustness

• 200/200 success held through communication degradation and relay noise sweeps up to 0.4 m range and 4° bearing noise.Gross packet outliers degraded accuracy gracefully, reaching 0.204 m in the hardest case; dropout, delay, and jitter left accuracy essentially flat.Source: Figure 5 · p. 8, §VII · p. 8

Operating boundary

• 188/200 trials succeeded at 0.01 m translational odometry noise per step, while the certificate refused release at 0.02 and 0.05 m per step.Bias, scale error, and heading drift were tested empirically but lie outside the analytical certificate's translational random-walk model.Source: §VII · pp. 8–9

Baselines

• Across 200 paired geometries, the method achieved 0.064 m median station error versus 0.065 m for the known-yaw oracle, with 200/200 successes.The EKF reached 0.087 m; endpoint registration succeeded in 194/200, while a roughly 50× slower fixed-lag smoother was statistically indistinguishable from the proposed estimator.Source: §VII · p. 7, Figure 6 · p. 9

Physics-based validation

• 30/30 success held in every Gazebo scenario; station RMSE was 0.077 m nominal and 0.164 m under combined communication degradation.The 60° relay rotation was adopted at a median 7.3 s, with task recovery at a median 10.3 s and success in every trial.Source: §VII · p. 8, Table I · p. 9

Long horizon and disturbance

• At 5 cm/s body bias over 600 s, median dead-reckoning error reached 27 m while task error stayed near 0.06 m.Large relay-frame steps recalibrated once motion supplied views; small at-rest steps could remain unadopted near the certified tolerance without affecting the task.Source: §VII · p. 9

Estimator comparison and scope

• The fixed-lag smoother matched the proposed estimator's median and 90th-percentile station errors, 0.064 and 0.117 m, at about 50× the estimator update time.This conclusion is scoped to the present static-relay, static-target measurement and odometry model, not loop-rich SLAM problems where smoothing may recover additional state information.Source: §VIII · pp. 9–10

Refusal boundary

• Above the supported translational-odometry regime, the certificate withholds seeking rather than treating accumulated pose uncertainty as a precise relay-frame rotation.Heading-drift results remain empirical; deploying the full SE(2) recursion is identified as the route to certifying heading uncertainty.Source: §VIII · p. 10

Generalization and limits

• The quotient construction generalizes when an unknown rigid local frame reports a static task vector and repeated ego-relative vectors, with metric motion increments and time-associated packets.Moving targets, unassociated packets, nonrigid relay frames, and persistent frame drift are outside the present guarantees.Source: §VIII · p. 10

Long-horizon behavior

• In a representative 600-second trial, task error stayed within the theorem's ultimate ball while dead-reckoned pose error reached 6.95 m.The trial used a 1 cm/s bias.Source: Figure 8 · p. 10

IX. CONCLUSION

• The full vehicle-relay-target configuration remains unobservable up to a global SE(2) transformation, but two distinct vehicle views recover relay-to-odometry yaw and target displacement in closed form.The conclusion frames this identifiable quotient as sufficient for control without reconstructing global pose.Source: §IX · p. 10
• The certificate accounts for integrated-odometry cross-view correlations in O(K), supporting excitation until calibration is informative, refusal of uncertifiable estimates, quotient-based seeking, and relay-frame change detection.Zero false adoptions were observed across 200 control trials.Source: §IX · p. 10
• Across 200 randomized geometries, the paired median difference from an oracle given the true relay yaw is 0.001 m, with a bootstrap 95% interval including zero; ROS 2/Gazebo trials retain 30/30 success across three operating conditions.The ultimate task-error bound is independent of mission horizon, although global dead-reckoning error continues to accumulate.Source: §IX · p. 10

APPENDIX A QUOTIENT GEOMETRY AND LOCAL RANK

• A second distinct view supplies the missing yaw direction, giving the stacked Jacobian full rank 5; one view has rank 4, so two-view local rank matches the constructive identifiability result.The two vehicle relay vectors need only be different, not differently directed.Source: Appendix A · p. 11

APPENDIX B NATIVE RANGE-BEARING INFORMATION

• Native range-bearing noise contributes radial and tangential information, and bearing-derived position information decays as 1/∥u∥^2 with range.Range noise can also contribute to first-order yaw uncertainty; its radial contribution vanishes only in the special geometry stated in the appendix.Source: Appendix B · pp. 11–12

APPENDIX C JOINT QUOTIENT COVARIANCE UNDER GENERAL SE(2)

• The generalized SE(2) covariance construction retains increment anisotropy, within-increment translation-heading correlation, and long-range position correlation from integrated heading error, while its backward recursion is O(K).When heading error is absent, the result reduces to the translation-only suffix sum.Source: A. Translation and heading increments · p. 12

A. Translation and heading increments

• At 0.5°/s heading drift, all 200/200 trials complete and median station RMSE rises from 0.064 to 0.080 m; this is a robustness result, not coverage validation for the deployed certificate.The online implementation and main-text coverage statements use the translation-only specialization; the general SE(2) certificate is an analytic extension.Source: A. Translation and heading increments · p. 13

B. Complete yaw–task covariance

• The complete first-order yaw–task covariance yields a joint certificate, while the deployed yaw gate is a scalar projection of that quotient-level chance certificate.The appendix also gives a confidence ellipse for task error and a three-dimensional ellipsoid for joint yaw-task uncertainty.Source: B. Complete yaw–task covariance · p. 13

• While the window fills, geometric spread is nondecreasing and the packet-noise upper bound decreases toward a positive constant set by saturated-arc spread and the weight ratio.The odometry term reaches its exact saturated-arc value.Source: A. Expanded Proof of Proposition 2 · p. 13
• Certification need not improve monotonically once the window slides, and guaranteeing L consecutive certified windows requires an additional stationarity or ergodicity condition.The stated guarantee is conditional on those realized windows; their frequency is left to empirical validation.Source: A. Expanded Proof of Proposition 2 · pp. 13–14
• The reference acquisition arc reaches a certifiable window in finite time when the exact saturated total variance is strictly below the confidence threshold.A saturated reference window is refused if its odometry variance floor alone reaches or exceeds the threshold.Source: A. Expanded Proof of Proposition 2 · p. 13

B. Expanded Proof of Proposition 3

• Small steps satisfying the combined yaw-error threshold need no adoption, while sufficiently large relay-frame steps are detected with probability approaching one and adoption completes within L windows plus one turnover.The small-step case preserves contraction to the stated ultimate ball.Source: B. Expanded Proof of Proposition 3 · p. 14

A. Certificate Conditioning

• The 0.25 m-chord, two-view cell overstates precision by 74% and achieves 91.6% coverage; every tested cell with at least four views or a 0.5 m chord is within a few points of nominal coverage.At a 1 m chord, the variance ratio is within 10% of unity at every tested view count.Source: Table III · p. 14, A. Certificate Conditioning · p. 14

B. Correlated-Odometry Certificate: Full Coverage Table

• Across both tested window sizes and translation noise from 0.25 to 5 cm per increment, coverage stays within about two points of the nominal 95% level.The authors report certificate quality depends more on conditioning than on absolute odometry noise within this tested range.Source: Table IV · p. 14, B. Correlated-Odometry Certificate:… · pp. 14–15

C-D. Odometry Refusal Boundary; Relay Sensing and Communication Stress, Full Sweeps

• Refusal rows never certify or release seeking, while relay-sensing and communication sweeps report 200/200 successful trials per cell.The refusal rows’ residual distance is the initial geometry, not tracking error.Source: Table V · p. 15, Table VI · p. 15, Table VII · p. 15

E. Paired Baselines and Runtime Distributions

• The sequential EKF and endpoint-only registration have q90 errors of 0.172 m and 0.169 m, respectively, versus 0.117 m for the proposed estimator.The fixed-lag smoother’s q90 is statistically indistinguishable from the proposed estimator’s at roughly 50× the runtime cost.Source: Table VIII · p. 15, E. Paired Baselines and Runtime… · p. 15

F. Disturbance Recovery: Denominators and Wilson Intervals

• Every disturbance-recovery cell retains 100% task success; adoption denotes re-registration after persistent certified inconsistency, whereas task recovery requires task error to return below tolerance.The undisturbed control row had zero adoptions in 200 trials, giving a 1.9% Wilson upper bound on empirical false adoption.Source: Table IX · p. 15, F. Disturbance Recovery: Denominators… · p. 15

G. Long-Horizon Distributions

• At 5 cm/s bias, q90 station error is 0.108 m versus 0.108 m bias-free, while median dead-reckoning error grows from 1.4 m to 27.2 m.The campaign ran for 600 s; the authors attribute the stable station error to the finite calibration window rather than accumulated drift.Source: Table X · p. 16, G. Long-Horizon Distributions · p. 16

H. Gazebo Physics-Based Validation, Expanded

• The expanded Gazebo table reports 30 launches per scenario; combined stress includes 30% packet dropout, 0.2 s delay, 0.1 s jitter, and 5% gross outliers.Its station RMSE of 0.164 m exceeds any single offline stress cell despite milder per-factor levels.Source: Table XI · p. 16, H. Gazebo Physics-Based Validation,… · p. 16