Cislunar navigation is an observability problem: arc-length thresholds, Gramian conditioning, and the Doppler lever

This paper examines whether dilution of precision (DOP), a standard measure of navigation geometry, is appropriate for determining cislunar spacecraft states from inter-satellite ranging. Its central argument is that instantaneous range measurements depend on position but not velocity and therefore cannot determine a spacecraft’s complete dynamical state, regardless of the number of available range links. For this full-state, range-only snapshot problem, DOP is undefined rather than merely unfavorable. The author instead proposes evaluating navigation through dynamical observability: whether measurements accumulated over time determine the state and how well conditioned that determination becomes. This distinction does not invalidate conventional DOP when the transmitting satellites’ states are assumed known. The principal experiment uses four differentially corrected planar distant retrograde orbits in the Earth–Moon circular restricted three-body model. One spacecraft is the unknown “chief,” while three reference spacecraft have known states. The estimated state comprises two position and two velocity components. Measurement information is accumulated through an observability Gramian that combines range-measurement Jacobians with state transition matrices describing the evolution of initial-state perturbations. A square-root information filter supplies a second numerical calculation based on the same underlying Jacobians. The orbits have perilune radii of 18,345–44,120 km and nondimensional periodicity residuals ranging from approximately 3.8 × 10⁻¹¹ to 4.7 × 10⁻⁹. Instantaneous range-only measurements yield rank two for the four-component state, leaving velocity unresolved. As measurements accumulate, the dynamics make subsequent ranges sensitive to the initial velocity. Under the numerical rank criterion used, the first sampled full-rank arc occurs at 2.09 hours, following a rank-deficient sample at 1.83 hours on a 0.26-hour grid. The filter becomes well posed at the same sampled arc, and the condition numbers agree to six significant figures. Because both calculations use the same information, this agreement demonstrates computational consistency rather than independent physical validation. Full rank does not necessarily imply a useful estimate. The Gramian condition number decreases from approximately 1.20 × 10¹¹ at three hours to 2.05 × 10⁹ at six hours, 4.83 × 10⁷ at twelve hours, and 1.36 × 10⁵ at forty-eight hours. Adding range-rate measurements increases the instantaneous rank from two to four throughout the tested configuration, enabling a local snapshot solution. A separate joint-estimation experiment estimates a lunar surface station’s position and clock offset alongside satellite positions and clock offsets. With three satellites and six Earth stations, the information matrix has rank fifteen out of sixteen without an external interferometric delay observable and full rank with it across twenty realizations. Median station errors are 900 m without the observable and 13.3 m with it. The median paired improvement factor is 83, although individual factors range from 0.9 to 2,395. In the reported satellite-count sweep, the rank deficiency persists through four satellites and disappears with five or more. The practical conclusion is that navigation assessments should specify tracking duration, information conditioning, and measurement types rather than rely on full-state, range-only snapshot DOP. Several qualifications limit the findings’ generalizability. The principal dynamical experiment is planar, uses simplified three-body dynamics and ideal geometric measurements, and considers only one orbit family and phasing arrangement. Its numerical threshold depends on the configuration and rank criterion; it is not a universal minimum tracking time. Condition numbers do not represent achieved position accuracies, and the rank improvement from Doppler measurements does not establish operational performance or cost. The text also contains inconsistencies requiring clarification. Its limitations section describes joint estimation as future work despite reporting such an experiment, and its deterministic-data statement does not explain the twenty-realization experiment. These issues should be resolved before the findings are generalized to mission requirements.

Why Cislunar Navigation Needs More Than a Geometry Score

How do you determine a spacecraft’s position and velocity near the Moon using distances to other spacecraft? A familiar navigation tool, dilution of precision, or DOP, might seem like the natural starting point. But Chakshu Baweja’s paper, “Cislunar navigation is an observability problem: arc-length thresholds, Gramian conditioning, and the Doppler lever,” argues that this question requires a different framework.

The central distinction is between measuring geometry at one instant and learning a dynamical state over time. For a complete spacecraft state estimated from range-only snapshots, the problem is not simply poor geometry. Some of the information needed to solve it is absent.

What a Range Snapshot Cannot Tell You

A range measurement tells you the distance between two objects. At a particular instant, that distance depends on their positions, not their velocities. Adding more simultaneous range links can strengthen position information, but it cannot directly reveal velocity.

In the paper’s planar example, the unknown spacecraft has four state components: two position components and two velocity components. Instantaneous range-only measurements provide rank two, leaving velocity unresolved. For this complete-state snapshot problem, DOP is therefore undefined, rather than merely unfavorable.

This is a targeted criticism, not a rejection of conventional DOP. DOP remains appropriate for conventional positioning applications in which transmitting satellites’ states are assumed known. The issue arises when a geometry-based snapshot metric is expected to characterize recovery of an entire dynamical state from ranges alone.

Letting Motion Supply the Missing Information

The alternative is dynamical observability: asking whether measurements collected over time can determine the initial state, and how robustly they do so. An initial velocity perturbation changes the spacecraft’s later position, which changes later ranges. Motion therefore creates information that a single snapshot lacks.

Baweja tests this using four differentially corrected planar distant retrograde orbits in the Earth–Moon circular restricted three-body model. One spacecraft is the unknown “chief”; three reference spacecraft have known states. The orbits have perilune radii from 18,345 to 44,120 kilometers, with small nondimensional periodicity residuals of approximately 3.8 × 10⁻¹¹ to 4.7 × 10⁻⁹.

The main calculation uses an observability Gramian. It combines range-measurement sensitivities with state transition matrices, which describe how initial-state perturbations evolve. A square-root information filter provides a second numerical calculation using the same underlying measurement information.

A Sampled Threshold, Not a Universal Deadline

Under the study’s numerical rank criterion, the first sampled full-rank tracking arc occurs at 2.09 hours. The preceding sample, at 1.83 hours, remains rank deficient; the sampling interval is 0.26 hours. The result identifies a transition in this tested configuration, not a universal minimum tracking duration.

The filter becomes well posed at the same sampled arc, and the condition numbers agree to six significant figures. That is a useful consistency check. Because both calculations use the same underlying Jacobians, however, their agreement is not independent physical validation.

Solvable Does Not Necessarily Mean Useful

Full rank indicates that the linearized information can distinguish all state components under the adopted criterion. It does not guarantee a stable or operationally useful estimate. Conditioning reveals how unevenly the available information constrains different combinations of state components.

The reported Gramian condition number improves substantially as the tracking arc grows:

• Three hours: approximately 1.20 × 10¹¹.
• Six hours: approximately 2.05 × 10⁹.
• Twelve hours: approximately 4.83 × 10⁷.
• Forty-eight hours: approximately 1.36 × 10⁵.

These values show why “observable” should not be treated as a simple performance guarantee. Longer arcs improve conditioning here, but condition numbers are not achieved position accuracies.

The Doppler Lever and an Additional Experiment

Adding range-rate measurements changes the instantaneous picture. Throughout the tested configuration, instantaneous rank rises from two to four, enabling a local snapshot solution. Range rate, associated with Doppler tracking, supplies velocity-sensitive information. This establishes a structural benefit, not demonstrated operational accuracy or cost-effectiveness.

A separate experiment jointly estimates a lunar surface station’s position and clock offset alongside satellite positions and clock offsets. With three satellites and six Earth stations, the information matrix has rank fifteen out of sixteen without an external interferometric delay observable, and full rank with it across twenty realizations.

Median station errors are 900 meters without that observable and 13.3 meters with it. The median paired improvement factor is 83, although individual factors range from 0.9 to 2,395. In a satellite-count sweep, the deficiency persists through four satellites and disappears with five or more.

What Mission Planners Should Take Away

The practical message is to specify tracking duration, measurement types, and information conditioning rather than rely on full-state, range-only snapshot DOP. Still, the main experiment uses planar, simplified dynamics, ideal geometric measurements, and one orbit family and phasing arrangement.

Two reporting issues also need clarification: joint estimation is described as future work despite a reported experiment, and the deterministic-data statement does not explain the twenty realizations. Before translating these findings into mission requirements, those inconsistencies and the model’s limits should be addressed. The central insight remains compelling: navigation depends not only on where measurements come from, but also on what they reveal as time passes.