Inertial Parameter Identification

Definition

Inertial parameter identification is the estimation of a rigid body’s mass properties — total mass $m$, center-of-mass (CoM) offset, and inertia tensor — from measured motion and applied wrench, exploiting the fact that the rigid-body equations are linear in these parameters. In ekal2021online the system is a NASA Astrobee free-flyer (fully-actuated 6-DOF base, the same actuated-base regime as our free-flying manipulator), and the parameter vector for the planar (3-DOF) demonstration is $\mathbf{\theta }=\{m,c_x,c_y,I_{zz}\}$. Identification matters because grasping an unknown payload or depleting fuel changes these parameters mid-mission, degrading trajectory tracking unless the model is corrected online. The paper performs the identification online and en route, recursively, rather than as an up-front dedicated system-ID maneuver.

Key Equations

Symbols per notation.md.

The body-frame rigid-body dynamics expressed about a body point not at the CoM are linear in the inertial parameters $\mathbf{\theta }$ (paper Eq. 4; here in source notation, with $\mathbf{c}$ the CoM offset and ${\mathbf{I}}_{CM}$ the inertia about the CoM):

$$\left[\begin{array}{c}\mathbf{F}\\ {\mathbf{\tau }}_{C{M}_0}\end{array}\right]=\left[\begin{array}{cc}m{\mathbf{I}}_3& -m[\mathbf{c}{]}_{\times }\\ -m[\mathbf{c}{]}_{\times }& {\mathbf{I}}_{CM}-m[\mathbf{c}{]}_{\times }[\mathbf{c}{]}_{\times }\end{array}\right]\left[\begin{array}{c}\dot{\mathbf{v}}\\ \dot{\mathbf{\omega }}\end{array}\right]+(\text{velocity-product terms}).$$

This linearity makes $\mathbf{\theta }$ identifiable from $(\mathbf{F},\mathbf{\tau })$ and the measured twist/acceleration. The amount of information a trajectory yields about $\mathbf{\theta }$ is quantified by the Fisher Information Matrix (paper Eq. 9), accumulated over measurement times $t_0,\dots ,t_N$ with measurement-sensitivity Jacobian $\mathbf{H}(t_k)$ (paper Eq. 10: the total derivative of $h$ w.r.t. $\mathbf{\theta }$, i.e. the direct term $\partial h/\partial \mathbf{\theta }$ plus the chain term $(\partial h/\partial \mathbf{x})(\partial \mathbf{x}/\partial \mathbf{\theta })$) and noise covariance $\mathbf{\Sigma }$:

$$\mathcal{F}=\sum _{k=0}^N\mathbf{H}(t_k{)}^{\top }{\mathbf{\Sigma }}^{-1}\mathbf{H}(t_k).$$

Notation note: the source writes the FIM as $\mathbf{F}$, which collides with the applied force $\mathbf{F}$; I use $\mathcal{F}$ for the FIM to avoid the clash. Neither the FIM nor $\mathbf{\theta }$, $\mathbf{c}$, ${\mathbf{I}}_{CM}$ are in notation.md (that registry targets the circumcentroidal control stack); they are reproduced source-faithfully here.

The mid-level planner injects excitation by penalizing parameter uncertainty in its cost: the paper minimizes $\mathrm{tr}({\mathcal{F}}^{-1})$ (A-optimality), weighted by a tunable scalar $\gamma$ against state-error and input cost. (This $\gamma$ is the source’s information-weighting scalar; it is unrelated to the $\gamma ({\sigma }_G)$ impedance-derate ramp in notation.md, which belongs to the circumcentroidal control stack.)

Source Support

• ekal2021online — primary. Identifies $\mathbf{\theta }=\{m,c_x,c_y,I_{zz}\}$ for an Astrobee free-flyer grasping an unknown payload; an EKF runs as the online recursive estimator while the planner adds Fisher-information-maximizing excitation (RATTLE). Demonstrated in high-fidelity 3-DOF simulation and on granite-table hardware; reports that intentional rotational excitation sharply reduced $I_{zz}$ variance, whereas CoM offset remained poorly observable.

Related Topics

• parameter_estimation — the umbrella problem; here the recursive estimator is an Extended Kalman Filter producing $\hat{\mathbf{\theta }}$ online.
• active_parameter_learning — RATTLE actively shapes the trajectory to gain information about $\mathbf{\theta }$ rather than passively estimating, trading goal-reaching cost against excitation.
• fisher_information_matrix — the information metric $\mathcal{F}$ that scores how identifiable the parameters are along a candidate trajectory.
• d_optimality — an information-optimality criterion on $\mathcal{F}$; note this source uses A-optimality ($\mathrm{tr}{\mathcal{F}}^{-1}$), a sibling criterion, not D-optimality ($\det$).
• model_predictive_control — the low-level NMPC tracker that continually ingests the updated $\hat{\mathbf{\theta }}$ for more accurate control.

Open Questions

• The source identifies the inertial parameters of a single rigid body (base + rigidly-grasped payload). For our free-flying manipulator, the parameters are configuration-coupled across the base and a moving arm — does the rigid-body linearity and FIM-excitation scheme extend to the articulated, redundant case?
• The paper reports the CoM offset $\mathbf{c}$ is poorly observable even with information-aware planning. What excitation (or arm motion) makes CoM identifiable for an actuated-base manipulator, and does pursuing it conflict with the primary inspection trajectory?