Lie-Group Pose Uncertainty

Definition

Lie-group pose uncertainty is the representation of an uncertain rigid-body pose as a probability distribution carried on the matrix Lie group $\mathrm{SE}(3)$ — a mean group element together with a covariance defined in the Lie algebra $\mathfrak{se}(3)$ — rather than as a Gaussian over a flat vector of position and attitude coordinates. It is the uncertainty counterpart of Exponential Coordinates: the same $\exp /\log$ machinery that gives a minimal chart for a nominal pose gives a consistent chart for the error about that pose.

The motivation for this thesis is that the risk layer plans over pose-valued states — base attitude, end-effector pose, target pose — and a risk measure is only as trustworthy as the covariance it is evaluated against. A pose covariance carried in an arbitrary local chart (Euler angles, a quaternion’s vector part, a stacked ${\mathbb{R}}^6$ vector) is chart-dependent: it inherits the chart’s singularities and its Jacobians depend on where the estimate happens to sit, which is the mechanism behind the standard EKF’s optimism. Carrying the covariance in $\mathfrak{se}(3)$ instead — as the invariant-EKF (IEKF) family does — makes the error propagation independent of the estimate for a class of systems (see Key Equations), which is what buys the consistency guarantee.

Two halves of the problem meet here, and the literature covers them separately:

1. Pose covariance — how a Gaussian on $\mathrm{SE}(3)$ is defined, compounded and propagated (Barfoot & Furgale 2014; Barrau & Bonnabel 2017).
2. Estimated mass-property error — how error in the estimated system centre of mass and inertial parameters propagates into the model the controller uses (Inertial Parameter Identification, ekal2021online).

No verified source carries a mass-property covariance jointly with an $\mathrm{SE}(3)$ pose covariance in one invariant filter for an actuated-base manipulator. That seam is where the thesis contribution would sit.

Inconsistency — do not import the free-floating coupling

It is tempting (and the seed note does it) to say that a biased CoM estimate biases the generalized Jacobian ${\mathbf{J}}_g$ through momentum conservation. That is the free-floating story: ${\mathbf{J}}_g$ exists only because an unactuated base makes momentum a constraint that can be folded out (Free-Flying vs Free-Floating, umetani1989resolved). Our base is fully actuated: momentum is not conserved, there is no constraint to fold, and ${\mathbf{J}}_g$ is not in the loop. A mass-property error still bites — but through a different door: the mass-averaged linear Jacobian ${\bar{\mathbf{J}}}_v$, which is a definitional function of the masses and link CoM offsets and which sits inside the circumcentroidal Jacobian ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ and the transform $\mathbf{\Gamma }$ (giordano2019coordinated, eqs 7, 16). Name the door before attributing the error.

Key Equations

Symbols per notation.md.

Notation flags. (i) $\mathbf{\xi },\mathbf{\rho },\mathbf{\varphi }$ (the $\mathfrak{se}(3)$ error twist and its translation/rotation parts), $\mathbf{\Sigma }$ (a $6\times 6$ pose covariance), $\mathbf{X}$ / $\hat{\mathbf{X}}$ (group state and its estimate) and $\mathbf{\eta }$ (invariant error) are not in notation.md; they are reproduced as in the cited estimation literature, not coined. Registering them centrally is pending (the accent $\widehat{(\cdot )}$ already means the full $12\times 12$ transformed inertia in notation.md — the estimator hat is a distinct, colliding use and must be disambiguated before these symbols enter the registry). (ii) The $\mathfrak{se}(3)$ hat map is written ${}^{\times }$ per the ratified operator, never ${}^{\wedge }$.

A pose as a group element.

$$\mathbf{T}=\left[\begin{array}{cc}\mathbf{R}& \mathbf{p}\\ 0^{\top }& 1\end{array}\right]\in \mathrm{SE}(3)$$

A Gaussian on the group (left-perturbation convention, Barfoot & Furgale 2014). The mean $\bar{\mathbf{T}}$ lives on the group; the covariance lives in the algebra:

$$\mathbf{T}=\exp ({\mathbf{\xi }}^{\times })\bar{\mathbf{T}}\mathbf{\xi }=\left[\begin{array}{c}\mathbf{\rho }\\ \mathbf{\varphi }\end{array}\right]\sim \mathcal{N}(0,\mathbf{\Sigma }){\mathbf{\xi }}^{\times }=\left[\begin{array}{cc}{\mathbf{\varphi }}^{\times }& \mathbf{\rho }\\ 0^{\top }& 0\end{array}\right]$$

The distribution is therefore exactly Gaussian in the 6-dimensional algebra and pushed onto the group by $\exp$ — no small-angle chart is assumed for the mean, only for the error.

Compounding two uncertain poses. With $\mathbf{T}={\mathbf{T}}_1{\mathbf{T}}_2$ and independent errors, the Adjoint transports the second error into the first frame:

$$\mathrm{Ad}(\mathbf{T})=\left[\begin{array}{cc}\mathbf{R}& {\mathbf{p}}^{\times }\mathbf{R}\\ 0& \mathbf{R}\end{array}\right]\in {\mathbb{R}}^{6\times 6}\mathbf{\xi }\approx {\mathbf{\xi }}_1+\mathrm{Ad}({\bar{\mathbf{T}}}_1){\mathbf{\xi }}_2$$ $$\mathbf{\Sigma }\approx {\mathbf{\Sigma }}_1+\mathrm{Ad}({\bar{\mathbf{T}}}_1){\mathbf{\Sigma }}_2\mathrm{Ad}({\bar{\mathbf{T}}}_1{)}^{\top }$$

This is the second-order (leading) term; the contribution of Barfoot & Furgale 2014 is precisely that the neglected fourth-order terms are not negligible once rotational uncertainty is large, which is the regime a tumbling target sits in.

Group-affine dynamics and the log-linear error (Barrau & Bonnabel 2017). Let the state $\mathbf{X}$ evolve on a matrix Lie group with identity $\mathbf{E}$. The dynamics are group-affine if

$$f(\mathbf{X}\mathbf{Y})=f(\mathbf{X})\mathbf{Y}+\mathbf{X}f(\mathbf{Y})-\mathbf{X}f(\mathbf{E})\mathbf{Y}$$

for all $\mathbf{X},\mathbf{Y}$. Under that condition the right-invariant error is autonomous — it obeys a differential equation that does not depend on the trajectory being estimated:

$$\mathbf{\eta }=\hat{\mathbf{X}}{\mathbf{X}}^{-1}\mathbf{\xi }=\log (\mathbf{\eta }{)}^{\vee }\dot{\mathbf{\xi }}={\mathbf{A}}_t\mathbf{\xi }$$

with ${\mathbf{A}}_t$ independent of the estimate (the noise-free “log-linear” property). Consequence for the covariance recursion of Covariance Propagation: the propagation Jacobian is not a linearization about a possibly-wrong estimate, so the filter cannot manufacture information along directions the standard EKF wrongly believes it has observed. That is the whole consistency argument, and it is the reason to prefer an invariant filter over the plain Extended Kalman Filter for pose states.

Where a mass-property error enters our free-flying system. The mass-averaged linear Jacobian is a definitional function of the mass distribution (giordano2019coordinated eqs 6–7; project master, eqs 1.5–1.6):

$${\bar{\mathbf{J}}}_v=\frac{1}{m}\sum _{j=0}^n{m}^{(j)}{\mathbf{R}}_{jb}^{\top }{\mathbf{J}}_{v_j}m=\sum _{j=0}^n{m}^{(j)}{\mathbf{J}}_{{\nu }_e}^{\oplus }={\mathbf{J}}_{{\nu }_e}-\left[\begin{array}{c}{\mathbf{R}}_{eb}\\ 0\end{array}\right]{\bar{\mathbf{J}}}_v$$

An error in the estimated masses $m^{(j)}$ or link CoM offsets therefore perturbs ${\bar{\mathbf{J}}}_v$, hence ${\mathbf{J}}_{{\nu }_e}^{\oplus }$, hence $\mathbf{\Gamma }$ and the transformed inertia $\hat{\mathbf{M}}={\mathbf{\Gamma }}^{-\top }\mathbf{M}{\mathbf{\Gamma }}^{-1}$ — the CoM loop and the coupled $9\times 9$ block alike. This is a chain of definitions, not a result: how a mass-property covariance maps to a base-attitude or EE-pose covariance is derived nowhere in the corpus (see Open Questions).

Source Support

In the corpus:

• ekal2021online — the closest verified anchor for the second half. An Astrobee free-flyer (fully-actuated 6-DOF base — our regime) runs an EKF over $\mathbf{\theta }=\{m,c_x,c_y,I_{zz}\}$ while the planner injects Fisher-information-maximizing excitation. Load-bearing finding: the CoM offset stays poorly observable even under information-aware excitation, whereas $I_{zz}$ variance drops sharply. A poorly-observable CoM is exactly a large, badly-shaped CoM covariance — the input to the propagation problem this page poses.
• giordano2019coordinated — supplies ${\bar{\mathbf{J}}}_v$, ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ and $\mathbf{\Gamma }$: the objects a mass-property error actually corrupts in the free-flying regime.
• umetani1989resolved · papadopoulos1993dynamic — the free-floating ${\mathbf{J}}_g$ / ${\mathbf{J}}^{\ast }$ line, cited here to exclude it: their momentum-conservation fold is unavailable to an actuated base.
• zheng2024informed — the flat-vector-space counterpart of the propagation step; useful as the contrast case, since its linearize-and-recurse split $P_k={\hat{P}}_k+{\tilde{P}}_k$ is what the Lie-group treatment replaces for pose states.

Not yet in the corpus (no .bib entry, no source page; cited by title/DOI, unverified against the raw text — ingest before any of this is cited in the thesis):

• A. Barrau, S. Bonnabel, “The Invariant Extended Kalman Filter as a Stable Observer,” IEEE TAC 62(4), 2017, DOI 10.1109/TAC.2016.2594085 (arXiv 1410.1465) — group-affine dynamics, log-linear autonomous error, the stability result.
• T. D. Barfoot, P. T. Furgale, “Associating Uncertainty With Three-Dimensional Poses for Use in Estimation Problems,” IEEE T-RO 30(3), 2014, DOI 10.1109/TRO.2014.2298059 — the pose-compounding and covariance-propagation algebra, with the fourth-order terms.
• A. Barrau, S. Bonnabel, “The Geometry of Navigation Problems,” IEEE TAC 67(2), 2022, DOI 10.1109/TAC.2022.3144328 — group structures that admit body-frame quantities (biases, lever arms) in the state; the natural route to putting a CoM offset inside an invariant filter.
• S. Mathavaraj, E. Butcher, “Consensus SE(3)-Constrained EKF for Close Proximity Orbital Relative Pose Estimation,” Aerospace 11(9):762, 2024, DOI 10.3390/aerospace11090762 — nearest verified spacecraft analog for relative target pose.
• H. Chen et al., “Relative Pose Determination for Noncooperative Spacecraft Under Noninertial Observation Frame,” IEEE T-AES, 2025, DOI 10.1109/TAES.2025.3562539 — the only located spacecraft work that jointly estimates relative pose and target inertia ratios.
• Y. Murotsu, K. Senda, M. Ozaki, S. Tsujio, “Parameter Identification of Unknown Object Handled by Free-Flying Space Robot,” JGCD 17(3), 1994, DOI 10.2514/3.21225 — the free-FLYING mass/CoM/inertia identification classic.
• T. Zhang et al., “Continuous-discrete extended Kalman filter based parameter identification method for space robots in postcapture,” Nonlinear Dynamics 112, 2024, DOI 10.1007/s11071-024-10079-y — an EKF that carries a covariance on the identified inertial parameters (most identification work returns point estimates only).

Related Topics

• Extended Kalman Filter — the baseline this displaces for pose states; its Jacobians are evaluated at the estimate, which is the source of the inconsistency the invariant formulation removes.
• Covariance Propagation — the predict/update recursion; on a Lie group the same recursion runs with the algebra covariance $\mathbf{\Sigma }$ and Adjoint-built Jacobians.
• Exponential Coordinates — the same $\exp /\log$ chart, applied to the nominal pose rather than the error; its chart-singularity caveat at rotation angle $\pi$ applies to the error twist too.
• Inertial Parameter Identification — supplies the mass-property estimate (and, via ekal2021online, the finding that its CoM component is the badly-observed one).
• Free-Flying vs Free-Floating — the regime gate: it decides which Jacobian a CoM error corrupts, and rules out the momentum-conservation route.
• Center of Mass Jacobian · Circumcentroidal Motion — the mass-weighted objects that carry the error into the control law.

Open Questions

• The joint filter. Can the CoM offset and inertial parameters be carried as body-frame quantities in the same invariant filter as the pose, so that one $\mathbf{\Sigma }$ covers both? The Barrau & Bonnabel 2022 construction is built for exactly this class of body-frame states (biases, lever arms), but no located work does it for an actuated-base manipulator.
• The propagation map. What is $\partial {\mathbf{J}}_{{\nu }_e}^{\oplus }/\partial \mathbf{\theta }$, and hence the map from a mass-property covariance to an EE-pose covariance? The chain above is definitional; the sensitivity is not derived anywhere in the corpus. If it stays underived, it is a thesis contribution rather than a citation to find.
• Observability of the CoM under arm motion. ekal2021online finds CoM offset poorly observable for a rigid free-flyer plus payload. Our arm moves the system CoM continuously — does articulated motion make the CoM more observable (free excitation), and does the excitation that helps conflict with the inspection trajectory?
• No gravity anchor. The well-developed terrestrial invariant-filter literature (INS/GNSS, legged robots) leans on a measured gravity vector and near-constant-velocity priors. A 6-DOF actuated base in microgravity has neither. Which of the IEKF guarantees survive the transfer, and which were quietly resting on gravity?
• Interaction with the singularity machinery. Near a singular configuration the controller derates its gains via $\gamma ({\sigma }_G)$, so the closed loop is no longer the design loop. Does a pose covariance propagated through a derated closed loop stay calibrated, or does the derate open a gap between the carried $\mathbf{\Sigma }$ and the true error?