Cooperative Navigation Error Model

• Cooperative navigation error models are mathematical frameworks that characterize how shared measurements and network geometry reduce state estimation errors.
• They employ techniques such as Fisher information, Kalman filters, and particle filters to quantify influences of sensing noise, common biases, and communication delays.
• These models guide estimator design and uncertainty-aware control by delineating when cooperation improves accuracy, saturates, or degrades performance.

Cooperative navigation error models are mathematical formulations that quantify how multiple agents reduce navigation uncertainty by sharing measurements, constraints, state estimates, or environmental information, and how residual error depends on sensing noise, common biases, geometry, temporal dynamics, and communication structure. Across the literature, the term covers analytically tractable formulations such as equivalent Fisher information decompositions, centroid-based cooperative map matching, and Kalman or particle-filter covariance propagation, as well as learned uncertainty representations based on map discrepancy, frontier structure, goal ambiguity, or diffusion-based risk models (Shen et al., 2011, Shen et al., 2017, Pöhlmann et al., 14 Aug 2025). The common objective is not merely cooperative state estimation, but explicit characterization of when cooperation improves accuracy, when it saturates, and when network structure or model mismatch causes degradation.

1. Formal scope and canonical problem formulations

A cooperative navigation error model begins by defining a multi-agent state, a set of cooperative measurements or constraints, and a probabilistic or geometric rule for turning those observations into an error quantity. In vehicular cooperative map matching (CMM), the observation model is

$x_i^G = x_i^L + x_i^D + x^C + x_i^N,$

where $x_i^G$ is the GNSS-reported position, $x_i^L$ is the closest point on the lane centerline, $x_i^D$ is deviation from the lane centerline, $x^C$ is a GNSS common error shared by nearby vehicles, and $x_i^N$ is vehicle-specific non-common error (Shen et al., 2017). In cooperative ranging and hybrid PNT, the measurement is typically a range-like quantity,

$$z_{ij}(t) = \|\mathbf{r}_i(t) - \mathbf{r}_j(t)\| + c\,(\delta t_i(t) - \delta t_j(t)) + b_{ij}(t) + n_{ij}(t),$$

which explicitly separates geometry, differential clock error, temporally correlated bias, and random noise (Pöhlmann et al., 14 Aug 2025). In AoA-only vehicular localization, the cooperative factor is angular and jointly constrains relative position and heading,

$z_{ij} = h_{ij}(x_i, x_j) + \eta_{ij},$

with the line-of-sight angle expressed relative to each vehicle heading (Wu et al., 2019).

These formulations differ in representation but share a common structure: a latent navigation state, nuisance variables or biases, and a rule for projecting multi-agent measurements into an error metric or covariance. In the cooperative network navigation framework, this is expressed at the level of the full parameter vector and its Fisher information, with equivalent Fisher information used to eliminate nuisance parameters and isolate the positional states of interest (Shen et al., 2011). In learned multi-agent navigation, the same role is played by latent state embeddings or trajectory distributions conditioned on shared observations and interaction features rather than by explicit closed-form measurement equations (Shizhe et al., 3 Jan 2026, Vemula et al., 2017).

Setting	Cooperative variable	Error representation
V2V cooperative map matching	GNSS common bias $x^C$	centroid error $e$, $x_i^G$0, RMSE (Shen et al., 2017)
UAV/UUV cooperative localization	relative or global pose	EKF covariance, PF likelihood, RSSI/range uncertainty (Yang et al., 2020, Zazo et al., 2016)
Hybrid lunar PNT	stacked user state and bias states	augmented-state Kalman covariance and BCRB (Pöhlmann et al., 14 Aug 2025)
Multi-agent online or social navigation	graph state, goals, trajectory distribution	makespan, frontier utility reduction, CVaR, chance constraints (Shizhe et al., 3 Jan 2026, Wang et al., 2024)

The scope of the field therefore ranges from classical estimation-theoretic models with explicit covariances to control-oriented and learning-based models in which uncertainty is encoded as map inconsistency, latent cooperation mode, or predictive trajectory dispersion. This suggests that “error model” in cooperative navigation is not tied to a single estimator family, but to a broader requirement: explicit accounting of how cooperation changes uncertainty.

2. Geometric structure, information decomposition, and covariance

The most distinctive feature of cooperative navigation error models is that error is typically geometry-dependent. In V2V CMM, each vehicle contributes a half-plane lane constraint

$x_i^G$1

and the simplified estimator takes the centroid of the feasible intersection of these constraints. Under first-order linearization, the CMM error admits the decomposition

$x_i^G$2

with covariance

$x_i^G$3

Here $x_i^G$4 is the deterministic geometric centroid error, $x_i^G$5 is a sensitivity matrix, and $x_i^G$6 stacks the normal projections of composite non-common errors (Shen et al., 2017). The resulting mean-squared error depends simultaneously on road-angle diversity and per-vehicle variance.

The same geometric dependence appears in the more general EFI framework for cooperative network navigation. There, the EFIM for positional states decomposes into mobility, temporal cooperation, and spatial cooperation terms,

$x_i^G$7

and CRLBs follow from inversion of the corresponding EFIM blocks (Shen et al., 2011). In 2-D, temporal cooperation from velocity-like measurements contributes along-track and cross-track information, while spatial cooperation from inter-node ranging contributes information along the inter-node direction. This decomposition supplies a direct geometrical interpretation: each cooperative source sharpens the information ellipse only in the directions it actually observes.

In AoA-only vehicular localization, the local geometry is especially transparent. The Jacobian of the AoA measurement at node $x_i^G$8 contains the position term $x_i^G$9, which means each AoA contributes position information in the direction perpendicular to the line of sight, and a coefficient $x_i^L$0 on heading, which contributes direct heading information (Wu et al., 2019). The paper’s approximate Fisher-information interpretation therefore explains why diverse neighbor bearings matter at least as much as raw neighbor count.

Distributed vehicular RBPF fusion can also be represented in covariance form. The network-level common-bias estimate evolves according to a linearized dynamical approximation,

$x_i^L$1

with error covariance recursion

$x_i^L$2

or, in steady state, $x_i^L$3 (Shen et al., 2017). This converts nonlinear particle-filter interactions into a tractable surrogate in which fusion weights, graph sparsity, and effective measurement strength directly determine stability and variance.

A recurrent implication across these models is that cooperation is only partly a matter of sample size. The rest is conditioning: lane normals, neighbor bearings, motion directions, anchor geometry, and measurement anisotropy determine whether additional measurements reduce uncertainty isotropically or merely reinforce already well-observed directions.

3. Communication topology, distributed fusion, and network effects

Cooperative navigation error models are also network models. They must specify how information moves through a graph, how missing or delayed messages are handled, and how fusion weights affect robustness. In semi-interpenetrating vehicular localization, each node runs an RBPF using its own and neighbors’ GNSS measurements, then fuses its RBPF output with neighbors’ outputs, so that information becomes reachable beyond direct communication range by repeated local fusion (Shen et al., 2017). The associated distributed optimization minimizes a variance surrogate,

$x_i^L$4

subject to graph sparsity and nonnegative fusion weights, making robustness a function of both estimation quality and consensus design (Shen et al., 2017).

In underwater radio localization, packet loss is modeled explicitly as Bernoulli loss with probability $x_i^L$5, and when a packet is lost, neighbors use the last known estimate of that node (Zazo et al., 2016). The cooperative least-squares cost is

$x_i^L$6

which is optimized in distributed successive convex approximation form. In that setting, the expected FIM scales approximately as $x_i^L$7, so packet loss affects error not just operationally but structurally through reduced information accumulation (Zazo et al., 2016).

For pairwise-only UAV cooperation with ranging and magnetic anomaly measurements, communication is scheduled as a time-multiplexed perfect matching over three edge sets $x_i^L$8, and information propagation obeys the bound $x_i^L$9 steps to disseminate an item of information to all vehicles (Yang et al., 2020). The most recent EKF update may therefore be $x_i^D$0 steps old, and dead reckoning must bridge the delay: $x_i^D$1 Here, the communication model enters the error model through covariance growth between delayed updates, not merely as an implementation detail (Yang et al., 2020).

In cooperative sensing for connected autonomous vehicles, the fusion architecture is explicitly tiered. Local JPDA–EKF fusion uses observation covariances parameterized by measured distance, while global fusion augments each track with localization covariance parameterized by platform velocity and heading (Andert et al., 2022). The global covariance used for weighting is

$x_i^D$2

so communication does not merely relay detections; it transfers a state estimate together with a scenario-specific uncertainty description (Andert et al., 2022).

A distinct network formulation appears in MAV-swarm navigation by landmark recognition and advice exchange. There, recognition and advice errors are modeled as Bernoulli processes with probabilities $x_i^D$3 and $x_i^D$4, and majority voting reduces the recognition error to

$x_i^D$5

with guaranteed improvement when $x_i^D$6; the advice process obeys the analogous threshold $x_i^D$7 (Barbeau et al., 2019). This is a cooperative navigation error model without continuous states or covariances, showing that networked error reduction can also be discrete and decision-theoretic.

Taken together, these formulations show that communication assumptions are part of the estimator, not an external systems concern. Perfect sharing, delayed pairwise exchange, packet loss, majority voting, and optimized fusion weights all define different error propagation laws.

4. Uncertainty-aware planning, control, and learned cooperation

Not all cooperative navigation error models are estimator-centric. In several recent formulations, error is embedded directly into planning objectives, control laws, or learned policies. ORION models uncertainty in partially known environments through prior utility $x_i^D$8, current utility $x_i^D$9, verified signals $x^C$0, guideposts $x^C$1 and $x^C$2, and a dual-encoder cross-attention architecture that learns to down-weight inconsistent priors and emphasize online-consistent structure (Shizhe et al., 3 Jan 2026). The policy does not explicitly parameterize sensor noise; instead, uncertainty is encoded through “verified” versus “unverified” regions and frontier utilities. The critic uses privileged ground-truth map $x^C$3 during training to reduce value-estimation error and provide long-horizon credit assignment (Shizhe et al., 3 Jan 2026).

The option-critic formalism in ORION further converts uncertainty into a mode-selection problem. High-level options represent self-directed navigation and cooperative assistance, with learned termination and masked option selection. The decision to help another agent is therefore treated as an uncertainty-aware control variable whose effect is measured through makespan reduction rather than through posterior covariance alone (Shizhe et al., 3 Jan 2026). This suggests a broader interpretation of cooperative navigation error: the relevant error quantity may be team-level completion time under map discrepancy, not only localization RMSE.

In socially aware navigation, NaviDIFF defines deterministic tracking, safety, speed, acceleration, and time-to-collision errors, then embeds them in a port-Hamiltonian energy function and a diffusion-based probabilistic model (Wang et al., 2024). The deterministic part uses

$x^C$4

while the probabilistic part represents policy-induced uncertainty through diffusion samples and evaluates tail risk using

$x^C$5

together with chance constraints on safety and tracking (Wang et al., 2024). Here, cooperation error is not a measurement residual but a mismatch between robot and human cooperative behavior distributions, regularized by a distribution-coupling objective.

A different learned formulation appears in dense human crowds, where the navigation model is the joint posterior over future trajectories conditioned on past observations and inferred goals. Per-agent velocity is modeled by Gaussian-process conditionals $x^C$6, and total predictive uncertainty decomposes into goal inference error and interaction prediction error through a mixture over goals (Vemula et al., 2017). This is a cooperative navigation error model in which crowd behavior itself is the latent source of uncertainty.

Sequential aiding in underwater navigation returns to a more classical formulation. A single Cooperative Navigation Aid vehicle assists multiple agents using a scalar discrete-time Kalman filter, with agent variance growing linearly between aiding events and shrinking at intercept times according to the measurement variance $x^C$7 (Wolek, 2024). The CNA’s own uncertainty enters the aided agents’ error model directly, and surfacing resets that uncertainty to $x^C$8, thereby improving future aiding quality (Wolek, 2024). This links cooperative planning and error modeling at the level of task scheduling.

These examples indicate that cooperative navigation error models increasingly serve as control abstractions. They do not only predict estimator performance; they determine which cooperative action should be taken, when exploration is worth the detour, and how much uncertainty can be tolerated in socially or operationally constrained environments.

5. Asymptotics, performance metrics, and empirical regimes

One of the main achievements of the literature is the derivation of explicit error scaling laws. For cooperative map matching with i.i.d. uniformly distributed road directions and small non-common error, the expected square error of the GNSS common-error correction satisfies

$x^C$9

with

$x_i^N$0

whereas under the orthogonal Bernoulli road-direction model it satisfies

$x_i^N$1

(Shen et al., 2017). The distinction is fundamental: cooperation benefits scale much faster under angular diversity than under axis-aligned clustering.

The more general optimization framework for V2V CMM refines this observation by showing that the geometric centroid term obeys

$x_i^N$2

for small nearest-neighbor angular gaps $x_i^N$3, and that for road angles drawn from a continuous density $x_i^N$4,

$x_i^N$5

which is minimized by the uniform distribution (Shen et al., 2017). Real-world evaluation on the Ann Arbor Safety Pilot dataset reported sub-meter RMS during daytime, approximately $x_i^N$6–$x_i^N$7 m, and approximately $x_i^N$8–$x_i^N$9 m at night, reflecting changes in vehicle density and angular diversity (Shen et al., 2017).

In AoA-only vehicular localization using PLBP, both positional RMSE and heading RMSE decrease significantly and converge to a low value in a few iterations, with convergence essentially reached after $$z_{ij}(t) = \|\mathbf{r}_i(t) - \mathbf{r}_j(t)\| + c\,(\delta t_i(t) - \delta t_j(t)) + b_{ij}(t) + n_{ij}(t),$$0 linearization iterations and $$z_{ij}(t) = \|\mathbf{r}_i(t) - \mathbf{r}_j(t)\| + c\,(\delta t_i(t) - \delta t_j(t)) + b_{ij}(t) + n_{ij}(t),$$1 inner BP iterations (Wu et al., 2019). The same study reports that increasing communication radius sharply reduces RMSE by increasing neighbor count and connectivity, while larger prior heading uncertainty rapidly degrades both position and heading accuracy (Wu et al., 2019). This identifies communication radius, network geometry, and heading prior as primary error drivers.

Hybrid lunar PNT demonstrates a different performance regime. By augmenting Kalman filters with temporally correlated satellite and cooperative bias states, the model shows that hybrid navigation enables optimal performance when using a lunar reference station, achieving sub-meter accuracy with only two visible satellites (Pöhlmann et al., 14 Aug 2025). The paper further shows that an augmented IEKF or EKF-2 tracks close to the BCRB when at least three sources are available, whereas a standard EKF that ignores correlated errors is unstable (Pöhlmann et al., 14 Aug 2025).

Cooperative sensing in connected autonomous vehicles quantifies the value of parameterized covariance generation rather than fixed covariances. In a four-vehicle cooperative fusion scenario, the method yields average and max improvement in RMSE of $$z_{ij}(t) = \|\mathbf{r}_i(t) - \mathbf{r}_j(t)\| + c\,(\delta t_i(t) - \delta t_j(t)) + b_{ij}(t) + n_{ij}(t),$$2 and $$z_{ij}(t) = \|\mathbf{r}_i(t) - \mathbf{r}_j(t)\| + c\,(\delta t_i(t) - \delta t_j(t)) + b_{ij}(t) + n_{ij}(t),$$3 over a typical fixed error model (Andert et al., 2022). This is a clear example in which better error modeling, rather than additional sensing hardware, changes fusion accuracy.

Range–magnetic cooperative localization for UAV groups provides a longer-horizon operational metric. For a group of $$z_{ij}(t) = \|\mathbf{r}_i(t) - \mathbf{r}_j(t)\| + c\,(\delta t_i(t) - \delta t_j(t)) + b_{ij}(t) + n_{ij}(t),$$4 UAVs in a GNSS-denied environment, the average estimated position error is approximately $$z_{ij}(t) = \|\mathbf{r}_i(t) - \mathbf{r}_j(t)\| + c\,(\delta t_i(t) - \delta t_j(t)) + b_{ij}(t) + n_{ij}(t),$$5 meters after flying about $$z_{ij}(t) = \|\mathbf{r}_i(t) - \mathbf{r}_j(t)\| + c\,(\delta t_i(t) - \delta t_j(t)) + b_{ij}(t) + n_{ij}(t),$$6 kilometers in $$z_{ij}(t) = \|\mathbf{r}_i(t) - \mathbf{r}_j(t)\| + c\,(\delta t_i(t) - \delta t_j(t)) + b_{ij}(t) + n_{ij}(t),$$7 hour, and robustness improves as more UAVs are added, even under high and low resolution magnetic maps (Yang et al., 2020). The same work shows that cooperative likelihood aggregation across the group sharpens the particle weights and resolves the rotational ambiguity left by range-only localization.

Across these results, performance is reported through heterogeneous metrics—MSE, RMSE, makespan, average travel distance, utility reduction, collision rate, SPEB, and BCRB—but the common analytical pattern is the same: cooperation changes error through geometry, diversity, and bias correlation, not simply through raw sensor count.

6. Limitations, misconceptions, and open directions

A common misconception is that adding more agents necessarily improves cooperative navigation accuracy monotonically. The CMM literature shows a more conditional picture: uniformly distributed road directions yield $$z_{ij}(t) = \|\mathbf{r}_i(t) - \mathbf{r}_j(t)\| + c\,(\delta t_i(t) - \delta t_j(t)) + b_{ij}(t) + n_{ij}(t),$$8 error decay, whereas orthogonal or clustered directions yield only $$z_{ij}(t) = \|\mathbf{r}_i(t) - \mathbf{r}_j(t)\| + c\,(\delta t_i(t) - \delta t_j(t)) + b_{ij}(t) + n_{ij}(t),$$9 decay, and nearly parallel roads make the feasible polygon elongated and poorly conditioned (Shen et al., 2017, Shen et al., 2017). Likewise, distributed vehicular fusion with random weights can diverge on sparse networks, while optimized weights remain robust (Shen et al., 2017). Cooperation helps only when the added participants contribute independent or complementary information.

Another misconception is that cooperative navigation error models are always covariance-based. Several important formulations instead represent error through map discrepancy, frontier uncertainty, goal uncertainty, or behavioral mismatch. ORION assumes reliable sensing and perfect global communication in simulation, and encodes uncertainty through verified versus unverified regions rather than explicit sensor-noise parameters (Shizhe et al., 3 Jan 2026). NaviDIFF represents uncertainty through diffusion distributions and risk measures rather than through a classical Kalman covariance alone (Wang et al., 2024). Dense-crowd trajectory models propagate uncertainty through mixtures over latent goals and Gaussian-process predictive variance (Vemula et al., 2017). These approaches broaden the concept of navigation error beyond state-estimation residuals.

Closed-form analytic models also have clear validity limits. CMM linearization requires small non-common errors, expressed as

$z_{ij} = h_{ij}(x_i, x_j) + \eta_{ij},$0

and assumes locally straight lane segments and accurate map geometry (Shen et al., 2017). Underwater radio localization assumes a linear RSSI path-loss model and independent Gaussian dB-domain noise, while packet loss is treated as Bernoulli and handled by using the last known neighbor estimate (Zazo et al., 2016). Lunar hybrid PNT assumes LOS satellite links and models cooperative multipath with a specular two-ray reflection; terrain-induced NLOS and richer multipath are not explicitly modeled (Pöhlmann et al., 14 Aug 2025).

The relationship between analytic tractability and estimator optimality is also contested. Consensus, graph-SLAM, and factor-graph CMM often yield superior accuracy but lack tractable closed-form error expressions and require heavier computation (Shen et al., 2017). Conversely, analytic models provide design guidance, sensitivity analysis, and membership optimization, even when the deployed algorithm is a Rao-Blackwellized particle filter or a learned policy (Shen et al., 2017, Shen et al., 2017). This suggests complementarity rather than replacement.

Open directions follow directly from the surveyed limitations. The literature repeatedly points to communication-aware policies under bandwidth constraints or intermittent connectivity, explicit modeling of non-Gaussian and temporally correlated errors, adaptive fusion weights, soft or weighted map constraints for large-noise regimes, and tighter integration of physical error models with learned cooperation policies (Shizhe et al., 3 Jan 2026, Pöhlmann et al., 14 Aug 2025, Shen et al., 2017). A plausible implication is that future cooperative navigation error models will increasingly combine the interpretability of EFI- or covariance-based analysis with the representational capacity of learned map, behavior, and interaction models.