Joint Spacetime Optimization Graph

Mechanism

The joint spacetime graph is constructed as a bipartite factor graph in which variable nodes represent the unknown quantities to be estimated and factor nodes represent the constraints imposed by observations. For each peer participating in a given mesh epoch, the variable set comprises a position vector (typically three Cartesian coordinates expressed in a declared reference frame) and a clock state (typically a clock offset and a clock rate against a declared reference timescale). Peers may additionally carry pose variables, velocity variables, or higher-order clock derivatives where the application warrants and the observation set supports their estimation.

Factor nodes are introduced for each observation contributed to the epoch. A two-way ranging observation between peers A and B contributes a factor that constrains the spatial separation of A and B to a measured value, with the measurement uncertainty captured as the factor's information matrix. Because the two-way ranging exchange involves transmission and reception times measured against each peer's local clock, the ranging factor is jointly constrained by the position variables of both peers and by the clock states of both peers; the factor is therefore connected to four variable nodes simultaneously, and its residual function expresses the consistency of the measured round-trip time with the joint position-and-clock hypothesis. One-way time-of-flight observations contribute analogous factors with different connectivity. Time-difference-of-arrival observations contribute factors connecting one transmitter clock to multiple receiver clocks and positions.

The solver finds the variable assignment that maximizes the joint likelihood of all factors simultaneously. In one embodiment, the solver implements iterative non-linear least squares with information-matrix weighting, equivalent to a maximum a posteriori estimate under Gaussian noise assumptions on each factor. In other embodiments, the solver implements a sum-product belief-propagation pass over the factor graph, or a sliding-window smoother that retains a bounded history of recent epochs. The solution carries a full covariance matrix relating all variables, including the cross-covariance between any peer's position and any peer's clock state. Downstream consumers query this joint distribution rather than reading marginalized position and time independently.

Operating Parameters

The graph supports incremental update. As new observations arrive, the corresponding factors are added to the graph and the solver is re-invoked. The solver's iteration count, convergence tolerance, and outlier-rejection threshold are configurable per deployment. Typical operating ranges include convergence tolerances of one part in ten thousand on the residual norm, iteration limits in the range of ten to one hundred per epoch, and outlier thresholds expressed as a multiple of the predicted residual standard deviation. Factor-noise models may be Gaussian, Huber, or Tukey biweight depending on the expected outlier rate of the contributing observation class.

The graph admits heterogeneous factor classes. A single epoch may combine high-precision two-way ranging from terrestrial peers, lower-precision one-way time-of-flight from satellite references, time observations from atomic clock peers, and pseudorange observations from external positioning sources. Each factor class carries its own information matrix and residual function; the solver treats them uniformly. Reference-frame and timescale anchors are introduced as unary factors with very high information weight on the variables corresponding to anchor peers; in deployments without an external anchor, the graph is solved up to a global rigid transform and a global clock offset, with these gauge freedoms either fixed by convention or carried explicitly as additional unobservable directions in the covariance.

Alternative Embodiments

In one embodiment, the solver runs on a designated coordinator peer that aggregates observations from the mesh, computes the joint solution, and distributes the result. In an alternative distributed embodiment, each peer runs a local copy of the solver and the mesh exchanges factor messages and partial solutions until convergence; this embodiment uses gossip-style propagation of factor updates and is appropriate where no single peer is trusted with the full observation set. In a hybrid embodiment, peers form clusters with cluster coordinators, and inter-cluster constraints are exchanged at cluster boundaries.

The variable set is extensible. Embodiments tailored to inertial-aided platforms add velocity and acceleration variables with corresponding inertial-measurement factors. Embodiments tailored to atmospheric-propagation effects add tropospheric and ionospheric delay variables shared across multiple ranging factors. Embodiments tailored to long-baseline operation add Earth-orientation and reference-frame-rotation variables. The solver and the graph structure accommodate these additions without algorithmic change; only the factor library and variable schema are extended.

Lineage support is integral to the architecture. Each estimate produced by the solver carries pointers to the factor nodes that contributed to it, and each factor node carries pointers to the originating observation message and its credentialed author. Downstream audit traverses these pointers to reconstruct the full provenance of any estimate, supporting forensic review of disputed positions or times and supporting credential-revocation propagation when an author's credentials are withdrawn.

Composition With Mesh Operation

The joint spacetime graph composes with the broader mesh by serving as the canonical time reference for all mesh events. Coordination events, settlement events, and dispute events each carry timestamps drawn from the solver's clock-state estimates, and time-bounded operations are evaluated against the joint solution rather than against any single peer's local clock. Where a mesh event's correctness depends on relative timing between two peers, the relevant query consults the joint covariance to obtain a properly cross-coupled timing uncertainty.

The graph also functions as a diagnostic substrate. When the solver fails to converge, or converges to a solution with anomalously large residuals on a subset of factors, the graph structure identifies the implicated factor classes and contributing peers, enabling targeted investigation rather than wholesale mesh re-initialization. The factor-residual signature additionally serves as an integrity check against adversarial observation injection: factors whose residuals are persistently inconsistent with the consensus solution are downweighted or rejected under the configured outlier policy, with the rejection itself recorded in lineage.

Implementation Details

The factor-graph data structure is implemented as an in-memory adjacency representation supplemented by serialized lineage records persisted to the governance-chain. Variable nodes are keyed by a tuple comprising peer identifier, variable class (position, clock, velocity, or extension), and epoch identifier; factor nodes are keyed by the originating observation's lineage hash. The solver operates on a working copy of the graph held in dense or sparse linear-algebra representations as appropriate to the deployment scale. For mesh sizes in the range of tens of peers, dense representations are typically adequate; for larger deployments, sparse Cholesky or sparse QR factorizations are employed, with structural sparsity preserved across incremental updates by reusing symbolic factorizations.

Information-matrix construction follows from the observation noise model. For two-way ranging, the residual is the round-trip-time measurement minus the model-predicted round-trip time, and the information weight is the inverse of the variance contributed by clock-frequency stability, transceiver latency uncertainty, and propagation-medium variability. For one-way time-of-flight, the residual additionally absorbs any modeled clock offset between transmitter and receiver, and the information weight reflects the additional uncertainty introduced by the asymmetric measurement geometry. For atmospheric-propagation factors, the residual incorporates a parameterized delay model whose parameters are themselves additional variables in the graph.

Outlier handling is implemented through robust kernels applied to factor residuals during solver iterations. Where a residual exceeds the kernel's transition threshold, the effective information weight on that factor is reduced according to the kernel's weighting function, preventing a single anomalous observation from dominating the solution. Persistently down-weighted factors are flagged for offline review and may, under deployment policy, be excluded from future epochs pending credential re-evaluation of the contributing peer. The flagging itself is recorded as a lineage event, providing an audit trail of the solver's outlier-handling decisions.

Marginalization is supported for sliding-window operation. As epochs age beyond the configured retention horizon, their variables are marginalized out of the active graph and replaced by equivalent factors connecting the remaining active variables. The marginalization preserves the joint information content of the discarded variables in compressed form, allowing long-horizon estimation without unbounded memory growth. Marginalization order and strategy are configurable; typical deployments marginalize oldest first, with exceptions for variables that anchor the global gauge or that participate in unresolved disputes.

Distinction Over Prior Art

Prior approaches to mesh localization and synchronization treat position and time as separable problems. GPS-class systems compute position from pseudoranges using a fixed satellite-clock reference, then propagate the receiver's clock estimate as a byproduct. Network Time Protocol and Precision Time Protocol synchronize clocks using assumed-symmetric path delays, ignoring the spatial geometry that produces those delays. Simultaneous-localization-and-mapping factor-graph approaches solve for position and pose but typically assume synchronized clocks. The disclosed mechanism differs by treating ranging and timing as observations against a shared joint variable set, producing cross-coupled uncertainty that prior approaches do not surface, and supporting query semantics that depend explicitly on the joint distribution.

Disclosure Scope

The joint spacetime graph mechanism is disclosed as a feature of the mesh-time subsystem within Provisional Application 64/049,409. The factor-graph construction, the joint position-and-clock variable schema, the cross-coupled covariance output, the incremental-update solver, the lineage attachment, and the alternative centralized, distributed, and clustered solver embodiments are each disclosed as independently claimable subject matter. The mechanism applies to surgical-robotic guidance requiring jointly precise spatial and temporal control, defense engagement against moving targets, autonomous vehicle coordination, and any other application in which spatial-temporal coupling materially affects operational correctness.