An Equivariant Filter for Spacecraft Nonlinear Relative Navigation with Range and Bearing Measurements

I. INTRODUCTION

• Accurate and robust relative-state estimation is essential for autonomous proximity operations, including future On-Orbit Servicing and Active Debris Removal missions.The introduction connects interest in these missions to increasing satellite deployments,.Source: §I · p. 1
• Even with no perturbations or a circular target orbit, relative orbital motion follows nonlinear equations; CW and Tschauner–Hempel models instead linearize or linearize in time the relative dynamics under stated assumptions.CW assumes a circular orbit, a spherically symmetric Earth, and small spacecraft separation; Tschauner–Hempel equations are used for elliptical orbits.Source: §I · p. 1
• The Equivariant Filter uses a semidirect-product Lie-group symmetry to lift observer dynamics to the group and linearize error dynamics around the origin on the state manifold.The paper describes this geometry-aware approach as leveraging equivariance for more consistent and robust estimates than filters that disregard state-space geometry.Source: §I · p. 1
• Equivariant filtering for relative orbital navigation remains largely unexplored, motivating identification of the nonlinear dynamics’ symmetry and derivation and simulation validation of an EqF.The stated contributions are identifying the equivariant structure, deriving the filter, and validating it in simulation.Source: §I · p. 1

II. PROBLEM STATEMENT

• The problem considers an unactuated target and actuated chaser, with relative position defined as the chaser position minus the target position.The dynamics used throughout are expressed in the target’s rotating LVLH frame; the chaser’s control input is treated as expressed in that frame when the spacecraft separation is small relative to target distance.Source: §II · pp. 1–2
• The chaser is assumed to measure bearing and range to the target, with the bearing defined using the normalized relative position and a negligible angular difference between the chaser and target LVLH frames.The target’s local-frame angular velocity and acceleration and its position are assumed available from ground stations.Source: §II · p. 2

A. Group Symmetry

• The proposed symmetry group is G = (SO(3) × MR(1)) ⋉ R3, with Lie algebra g = (so(3) × mr(1)) ⋉ r3.The paper identifies this group as the similarity group SIM(3).Source: A. Group Symmetry · p. 2
• An element X = (R, s, z) comprises a rotation R, positive scalar s, and three-dimensional vector z.The components belong respectively to SO(3), MR(1), and R3.Source: A. Group Symmetry · p. 2
• The group identity is given as id = (I, 1, 0).Source: A. Group Symmetry · p. 2
• The formulation specifies a group product for pairs of elements X2 and X1.The supplied passage introduces the product but does not state its expression.Source: A. Group Symmetry · p. 2
• The formulation also defines the adjoint AdX on Lie-algebra elements (Ω, τ, η).The supplied passage does not state the adjoint’s expression.Source: A. Group Symmetry · p. 2
• The group is chosen for compatibility with range-and-bearing measurements and to support future work on a bearing-only approach.Source: A. Group Symmetry · p. 2

B. System Equivariance

• The state, input, and output manifolds are specified, with S2 identified as the unit 2-sphere.The input manifold is given as a product of five R3 factors and one R factor.Source: B. System Equivariance · p. 2
• The extended system includes virtual inputs uv and uw, which are required to prove system equivariance.Source: B. System Equivariance · p. 2
• The original system is recovered from the extended system by setting uv = 0 and uw = 1.The extension includes these virtual inputs as part of the system used in the equivariance argument.Source: B. System Equivariance · p. 2
• Equivariance is formulated using right-transitive actions on the state and input manifolds.The state and input actions describe how the group acts on their respective manifolds.Source: B. System Equivariance · pp. 2–3
• The proposed state and input actions satisfy the equivariance condition, so the extended system is equivariant.The proof develops the condition using the proposed actions.Source: B. System Equivariance · p. 3
• The state action is not transitive over all of R3 × R3; it is transitive only on R3 \ {0} × R3.Source: B. System Equivariance · p. 3

C. Output Equivariance

• The output is equivariant because applying the right output action ρ to h(ξ) equals measuring the state after its action ϕ.Source: C. Output Equivariance · p. 3

D. Equivariant Lift

• The lift maps system dynamics from the state manifold to the Lie algebra of the symmetry group.Source: D. Equivariant Lift · p. 3
• The proposed lift satisfies both required conditions, establishing it as an equivariant lift of the system.Source: D. Equivariant Lift · p. 4
• Λs relates to the time derivative of inverse range, while ΛR relates to rotational kinematics through bearing angular velocity.The bearing angular velocity is (r × v)/∥r∥2.Source: D. Equivariant Lift · p. 4

E. Filter Derivation

• The filter state evolves on the group, with correction terms applied to its group-state components before producing the state-manifold estimate.The correction terms are denoted Δ = (ΔR, δs, δz).Source: E. Filter Derivation · p. 4
• The manifold coordinate origin is chosen as (γe3, 0), with γ ≠ 0, because the group action is not transitive over the entire state manifold.Source: E. Filter Derivation · p. 4
• The filter defines errors on the group and state manifold, then maps the manifold error into tangent-space coordinates using a local chart based on normal coordinates.Source: E. Filter Derivation · p. 4
• The pre-observer tangent-space error dynamics are linearized around ε = 0, yielding a first-order model with a second-order remainder.Source: E. Filter Derivation · p. 4
• The output residual is linearized around ε = 0 using output equivariance, while the correction terms depend on a Riccati state.The explicit state and output matrices are given in the Appendix.Source: E. Filter Derivation · p. 4

F. Convergence Analysis

• Range and bearing measurements make relative position and velocity observable, under which the EqF is locally exponentially stable.Source: F. Convergence Analysis · p. 4

IV. SIMULATIONS

• Monte Carlo simulations in MATLAB are used to verify the proposed EqF’s performance.Source: §IV · p. 5

A. Simulation setup

• The simulations used a 1000 km circular LEO target orbit, ran each trajectory 100 times, and updated measurements at 1 Hz.Filter prediction ran every 0.01 s; bearing noise was modeled with angular deviation θ ∼ N(0, 0.01²), and range measurements were within 1% of the true range.Source: A. Simulation setup · p. 5

B. Circumnavigation

• Circumnavigation used an ellipse with 500 m and 250 m semi-major and semi-minor axes; mean relative position and velocity errors were 0.05% and 0.28%.The chaser completed one orbit around the target; estimated and true states are shown in Fig. 3.Source: B. Circumnavigation · p. 5, Figure 3 · p. 5

C. R-bar approach

• The r-bar case was a constant-velocity, continuously actuated straight-line approach from 500 m below the target; mean relative position and velocity errors were 0.04% and 1.50%.The chaser took one orbital period to rendezvous; estimated and true states are shown in Fig. 5.Source: C. R-bar approach · pp. 5–6, Figure 5 · p. 6

V. DISCUSSION

• Noisy 1 Hz range-and-bearing measurements yield accurate relative position and velocity estimates in both scenarios, with convergence in less than one-tenth of an orbit under identical tuning and initialization.Source: §V · p. 6
• The state-manifold equations use Cartesian coordinates, whereas the group equations are more akin to a polar-coordinate parametrization because of the chosen symmetry group.Source: §V · p. 6
• The simulations omit orbital disturbances, sensor constraints and environmental effects, and assume synchronous measurements plus complete knowledge of ω, α, and r_t.Source: §V · p. 6
• For the R-bar approach, Fig. 6 presents the norm of averaged manifold errors.Source: Figure 6 · p. 6
• A disproportionate choice of origin r̄ may cause a large transient and compromise numerical stability; terms depending on 1/∥r̂∥ may also cause computational issues as rendezvous approaches.The authors suggest parallel filtering as a possible implementation measure and call for additional testing.Source: §V · p. 6

VI. CONCLUSIONS

• The authors describe the results as promising validation of the equivariant structure and filter, while identifying measurements, different orbits, perturbations, and bearing-only operation as future-work areas.Source: §V · p. 6, §VI · p. 6

• For readability, the appendix decomposes Ā_t into nine components.Source: APPENDIX · p. 6
• The nine components of Ā_t are specified individually.Source: APPENDIX · p. 6
• The appendix also gives the matrix C_t.Source: APPENDIX · p. 7