Recognition: unknown
Relative Pose-Velocity Estimation Using Dual IMU Measurements and Relative Position Sensing
Pith reviewed 2026-05-14 19:08 UTC · model grok-4.3
The pith
Relative pose and velocity between a vehicle and moving target can be estimated with global exponential convergence from dual IMUs and relative position measurements when target acceleration satisfies a persistence-of-excitation condition.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
By recasting the nonlinear relative pose-velocity dynamics on SE_2(3) into an exact linear time-varying model in R^15, a deterministic Riccati observer guarantees global exponential convergence of the estimation error in ambient space whenever uniform observability holds; for relative position measurements this reduces to a persistence-of-excitation condition on the target acceleration, while a complementary nonlinear filter on SO(3) separately ensures almost global asymptotic stability of the orientation component.
What carries the argument
The exact recasting of SE_2(3) relative dynamics into a linear time-varying system in R^15 that enables the application of a deterministic Riccati observer whose error convergence is governed by uniform observability conditions.
If this is right
- The estimation error converges globally and exponentially in ambient space for relative position measurements when target acceleration satisfies persistence of excitation.
- The orientation estimate from the SO(3) complementary filter is almost globally asymptotically stable.
- Uniform observability for bearing measurements requires additional conditions beyond persistence of excitation.
- Simulation results confirm the practical performance of the combined observer and filter.
Where Pith is reading between the lines
- The method could be extended to handle cases with occasional loss of relative measurements if the excitation condition is maintained over sufficient intervals.
- Similar LTV embeddings might apply to other estimation problems on Lie groups like SE(3) for rigid body motion.
- Real-world implementation would require checking sensitivity to sensor noise and biases not fully addressed in the basic model.
Load-bearing premise
The nonlinear SE_2(3) relative dynamics can be exactly rewritten as a linear time-varying system in R^15 without any approximation, and the persistence-of-excitation condition on target acceleration holds.
What would settle it
A numerical counterexample in which the target follows a trajectory with non-persistently-exciting acceleration yet the observer error still converges exponentially to zero would falsify the necessity of the condition.
Figures
read the original abstract
This paper addresses the problem of estimating the relative pose (position and orientation) and velocity of a vehicle with respect to a moving target, where both are equipped with Inertial Measurement Units (IMUs), assuming the availability of relative position or bearing measurements. The body-target relative dynamics are formulated on $\mathbf{SE}_2(3)$ and recast into a linear time-varying (LTV) model in the ambient space $\mathbb{R}^{15}$, on which a deterministic Riccati observer is designed. We analyze the uniform observability (UO) conditions required to guarantee global exponential convergence of the estimation error in the ambient space for both measurement cases. In the case of relative position measurements, UO requires only a persistence-of-excitation condition on the target acceleration, whereas for bearing measurements, additional conditions are required. Building on this, a nonlinear complementary filter on $\mathbf{SO}(3)$ is designed to provide a smooth estimate of the orientation component of the state with almost global asymptotic stability. Finally, simulation results are provided to validate the proposed solution.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper addresses relative pose-velocity estimation using dual IMUs and relative position or bearing measurements. It formulates the body-target relative dynamics on SE_2(3), recasts them into an LTV model in R^15, designs a Riccati observer, analyzes uniform observability conditions for global exponential convergence, adds a nonlinear complementary filter on SO(3) for orientation, and validates with simulations.
Significance. If the LTV recasting is exact, this work offers a rigorous framework for globally convergent relative estimation by leveraging classical observer theory on an ambient vector space. The finding that UO for position measurements requires only persistence of excitation on target acceleration is a simplifying and useful result. The combination with the almost globally stable orientation filter enhances practicality. Simulations provide initial validation, but stronger empirical evidence would increase impact.
major comments (1)
- [LTV recasting of SE_2(3) dynamics] The manuscript asserts that the nonlinear relative dynamics on SE_2(3) are recast into an exact LTV model in R^15. This is load-bearing for the convergence claims. Please provide the detailed derivation (e.g., in the section following the kinematics) demonstrating that all nonlinear terms from the group operations and adjoint maps are fully absorbed into the time-varying matrix without leaving state-dependent residuals or approximations. If the equivalence is not exact, the uniform observability analysis and Riccati-based global exponential stability would not transfer directly to the original nonlinear system.
minor comments (3)
- [Abstract] Specify the dimension of the state space more clearly when mentioning R^15.
- [Simulations section] Include plots of estimation errors over time and discuss the observed convergence rates in relation to the theoretical UO conditions.
- [References] Ensure all cited works on Lie group observers and Riccati filters are up to date and directly relevant.
Simulated Author's Rebuttal
We thank the referee for their careful reading of the manuscript and for highlighting the importance of the LTV recasting step. We address the major comment below and will revise the paper to include the requested derivation.
read point-by-point responses
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Referee: The manuscript asserts that the nonlinear relative dynamics on SE_2(3) are recast into an exact LTV model in R^15. This is load-bearing for the convergence claims. Please provide the detailed derivation (e.g., in the section following the kinematics) demonstrating that all nonlinear terms from the group operations and adjoint maps are fully absorbed into the time-varying matrix without leaving state-dependent residuals or approximations. If the equivalence is not exact, the uniform observability analysis and Riccati-based global exponential stability would not transfer directly to the original nonlinear system.
Authors: The recasting is exact with no approximations or residuals. The SE_2(3) relative kinematics are first written in matrix form using the group operation and adjoint representation. Vectorizing the state (relative position, relative velocity, and the nine entries of the rotation matrix) yields a 15-dimensional vector X. All nonlinear contributions arising from the adjoint action on the velocity and from the rotation matrix multiplication are then collected into the time-varying matrix A(t), which depends only on the measured angular velocities from the two IMUs and on the known target acceleration (treated as an exogenous input). The resulting dynamics are precisely linear: dot{X} = A(t)X + B(t)U(t), where U(t) contains the measured accelerations. No state-dependent nonlinear terms remain. We will insert the complete algebraic derivation immediately after the kinematics section in the revised manuscript, explicitly displaying A(t) and B(t) to make the equivalence transparent. This ensures the uniform-observability analysis and the global exponential stability of the Riccati observer carry over directly to the original nonlinear system on SE_2(3). revision: yes
Circularity Check
No significant circularity in derivation chain
full rationale
The paper formulates relative dynamics on the standard SE_2(3) group, recasts them into an LTV system in ambient R^15 coordinates, and applies classical deterministic Riccati observer theory. Uniform observability is analyzed via an independent persistence-of-excitation condition on target acceleration. No load-bearing step reduces by construction to a fitted parameter renamed as prediction, a self-definitional loop, or a self-citation chain that substitutes for external verification. The central claims rest on the exactness of the LTV embedding and standard Riccati convergence results, which are not shown to collapse into the paper's own inputs.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption Relative body-target dynamics can be formulated on SE_2(3)
- domain assumption The SE_2(3) dynamics admit an exact recasting into an LTV model in R^15
Reference graph
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