Fisher Information Matrix

Definition

The Fisher Information Matrix (FIM) quantifies how much information a set of
measurements $\tilde{\mathbf{y}}$ carries about an unknown parameter vector
$\mathbf{\theta }$ (e.g. the inertial parameters mass, CoM offset, and moment of inertia).
Under ekal2021online’s assumptions — no process noise on the
parameters (${\mathbf{\theta }}_{k+1}={\mathbf{\theta }}_k$), Gaussian measurement noise, and a
linearized measurement model — the FIM accumulates the squared, noise-weighted measurement
sensitivities over a trajectory. Its inverse is the Cramér–Rao lower bound on the parameter
estimate covariance, so making the FIM “large” (in an optimal-design sense) makes the trajectory
excite the parameters and shrinks the estimation-error ellipsoid. The source uses it as an
information cost inside an online motion planner for an Astrobee free-flyer (a fully-actuated
6-DOF base — the free-flying regime), grappling an uncertain payload; the FIM here is over the
rigid-body inertial parameters, not over a manipulator’s joint kinematics.

Key Equations

Symbols per notation.md.

Over samples $t_0,\dots ,t_N$, with measurement model $h(\cdot )$, measurement-noise covariance
$\mathbf{\Sigma }$, and parameter sensitivity $\mathbf{H}(t_k)$:

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

+ \frac{\partial h}{\partial\boldsymbol x}\,\frac{\partial\boldsymbol x}{\partial\boldsymbol\theta}.$$ The planner injects information by penalizing the trace of the inverse FIM (**A-optimality**) alongside state/input costs, weighted by $\gamma$: $$J = \sum_{k=0}^{N-1}\boldsymbol x_{t+k}^{\top}\boldsymbol Q\,\boldsymbol x_{t+k} + \boldsymbol u_{t+k}^{\top}\boldsymbol R\,\boldsymbol u_{t+k} + \gamma\,\mathrm{tr}\!\left(\boldsymbol F^{-1}\right).$$ > **Notation note.** $\boldsymbol F$, $\boldsymbol H$, $\boldsymbol\Sigma$, $\boldsymbol\theta$ are not yet > in [notation.md](../notation.md); they are used here source-faithfully (Ekal). The bold > $\boldsymbol\theta$ is the **parameter vector** here — **not** the plain scalar $\theta$ (helix > longitude angle) of the guidance pages; bold vs. plain disambiguates. $\gamma$ here is the > information-weighting scalar of this planner — **not** the impedance-derate ramp $\gamma(\sigma_G)$ of > the singularity pages; disambiguate by context. $\boldsymbol Q,\boldsymbol R$ are the planner's > state/input weights, unrelated to the frame quantities $\boldsymbol R_{xy}$. ## Source Support - [ekal2021online](../sources/ekal2021online.md) — defines the FIM (Eq. 9–10) as the information metric in the RATTLE mid-level planner; minimizes $\mathrm{tr}(\boldsymbol F^{-1})$ (A-optimality) to excite the Astrobee free-flyer's inertial parameters during otherwise-useful motion. ## Related Topics - active_parameter_learning — the FIM is the objective that turns passive estimation into *active* learning: the planner shapes the trajectory to maximize information. - [inertial_parameter_identification](inertial_parameter_identification.md) — supplies the parameter vector $\boldsymbol\theta$ (mass, CoM offset, inertia) whose identifiability the FIM measures. - [parameter_estimation](parameter_estimation.md) — the FIM bounds the achievable estimator covariance (Cramér–Rao); in this source an EKF does the actual sequential estimation. - [covariance_propagation](covariance_propagation.md) — $\boldsymbol F^{-1}$ is the lower bound on the parameter covariance, linking the information metric to the propagated uncertainty ellipsoid. - [d_optimality](d_optimality.md) — a *sibling* optimal-design criterion ($\det$-based); Ekal instead uses **A-optimality** ($\mathrm{tr}\,\boldsymbol F^{-1}$). Listed as related as the contrasting information measure, not as the one this source applies. ## Open Questions - [ ] Ekal's FIM is built for a free-*flyer* base **without an arm in the estimated model** (the payload is grappled rigidly). How does the FIM/sensitivity $\boldsymbol H$ change when the parameters include a redundant manipulator's link inertias and the base–arm coupling inertia $\boldsymbol M_{bm}$? - [ ] The linear/Gaussian FIM form (Eq. 9) relies on a linearized measurement model; over the strongly coupled dynamics of our free-flying manipulator, when does that linearization stay valid enough for the Cramér–Rao bound to be meaningful? - [ ] Is A-optimality ($\mathrm{tr}\,\boldsymbol F^{-1}$) or D-optimality (see [d_optimality](d_optimality.md)) the better excitation objective when some inertial parameters are far better observed than others?