Extended Kalman Filter

Definition

The Extended Kalman Filter (EKF) is the standard recursive estimator for a nonlinear stochastic system, obtained by linearizing the dynamics and measurement models about the current estimate at each step. In ekal2021online’s RATTLE stack the EKF is the in-the-loop estimator for the free-flyer’s uncertain inertial parameters $\theta =\{m,c_x,c_y,I_{zz}\}$ — mass, center-of-mass offset, and moment of inertia — fed poses, twists, and applied forces/torques, with its updated model pushed to the planner and controller (every 16 s, to avoid transient-driven instability).

Key Equations

The estimated system has nonlinear stochastic dynamics and measurement models (Eqs. 1–2):
$\dot{x}=f(x,u,\theta )+w_x,\tilde{y}=h(x,u,\theta )+w_y,$
with $w_x\sim \mathcal{N}(0,{\Sigma }_Q)$, $w_y\sim \mathcal{N}(0,{\Sigma }_R)$, and parameter prior ${\theta }_0\sim \mathcal{N}({\hat{\theta }}_0,{\Sigma }_{\theta ,0})$. Treating parameters as a frozen random walk ${\theta }_{k+1}={\theta }_k$ with a linear measurement model, the parameter information is the Fisher Information Matrix
$F={\sum }_{k=0}^{N}H(t_k{)}^{\top }{\Sigma }^{-1}H(t_k),H(t_k)=\frac{\partial h}{\partial \theta }+\frac{\partial h}{\partial x}\frac{\partial x}{\partial \theta },$
whose inverse (the EKF parameter covariance) the planner shrinks via the A-optimality cost $\mathrm{tr}(F^{-1})$ — gloss: information-aware excitation actively reduces the EKF’s parameter uncertainty.

Source Support

• ekal2021online — EKF as the online inertial-parameter estimator inside RATTLE (fed poses/twists + applied wrench), the nonlinear dynamics/measurement models, and the FIM/A-optimality link by which planned excitation drives down the filter’s parameter covariance (I_zz dropped 25–38%).

Related Topics

• inertial_parameter_identification — the quantities ($m,c,I$) the EKF estimates.
• fisher_information_matrix — the inverse FIM is the parameter-covariance the EKF reports and the planner minimizes.
• parameter_estimation — the general estimation problem the EKF instantiates recursively.
• covariance_propagation — the predict/update covariance recursion at the EKF’s core.

Open Questions

• ekal2021online notes the CoM offset $(c_x,c_y)$ stays poorly observable; what EKF tuning or excitation would make it observable for a grappled, off-center payload on a free-flying manipulator?