Parameter estimation is the online identification of unknown or time-varying model
quantities — here, the lumped coupling terms a space robot inherits after grasping an
unknown (non-cooperative) target — by fitting a regression model to measured signals.
In ye2019research the unknowns are not raw inertias but the estimation errors of the base–arm angular-momentum coupling products
(K1,K2,K3), stacked into a parameter block W and
updated recursively so that an Adaptive Reaction Null-Space (reaction_null_space)
planner can stay reactionless despite the changed dynamics. The cited source assumes the free-FLOATING regime (uncontrolled base, conserved linear/angular momentum); the
estimation machinery, however, is regime-agnostic and transfers to our free-flying coordinated_control setting.
The lumped coupling unknowns enter as a linear regression (Eq. 21; the compact yT=ΦW restatement is unnumbered in the source — Eq. 19 is the known-parameter RNS planner, not this regression):
\boldsymbol{\Phi}=\begin{bmatrix}\mathbf{1} & \boldsymbol{\omega}_0 & \dot{\boldsymbol{\xi}}\end{bmatrix}^{T},\qquad
\mathbf{W}=\begin{bmatrix}\mathbf{K}_1 & \mathbf{K}_2 & \mathbf{K}_3\end{bmatrix}^{T}$$
estimated by **Variable-Forgetting-Factor Recursive Least Squares** (VFF-RLS, Eq. 20), which
weights recent data more heavily to track the time-varying post-capture parameters:
$$\hat{\mathbf{W}}_{(k)}=\hat{\mathbf{W}}_{(k-1)}+\mathbf{N}_{(k)}\,\varepsilon_1,\qquad
\mathbf{N}_{(k)}=\frac{\mathbf{Q}_{(k-1)}\boldsymbol{\Phi}_{(k)}}{\lambda_{(k)}\mathbf{E}+\boldsymbol{\Phi}_{(k)}^{T}\mathbf{Q}_{(k-1)}\boldsymbol{\Phi}_{(k)}}$$
with a priori residual $\varepsilon_1=\mathbf{y}_{(k)}^{T}-\boldsymbol{\Phi}_{(k)}\hat{\mathbf{W}}_{(k)}$
and an **exponentially relaxing forgetting factor** that gives fast early convergence and
late-phase stability (Eq. 26):
$$\lambda(k)=\lambda_{\max}-\sigma_1 e^{-\sigma_2 k}\qquad(\lambda_{\max}=0.99,\ \sigma_1=0.1,\ \sigma_2=0.04)$$
> Symbol note (source-local, absent from [notation.md](../notation.md)): $\boldsymbol{\Phi}$ =
> regressor, $\mathbf{W}$ = parameter block, $\mathbf{Q}$ = inverse autocorrelation (covariance)
> matrix, $\mathbf{N}_{(k)}$ = recursive-LS update gain, $\lambda$ = forgetting factor,
> $\boldsymbol{\omega}_0$ = base angular velocity, $\dot{\boldsymbol{\xi}}$ = null-space velocity
> term ($\in\mathbb{R}^n$, "any non-zero vector" in Ye et al.). These are *not* the canonical
> glyphs: in particular $\mathbf{N}_{(k)}$ here is Ye's RLS gain, **not** the canonical null-space
> projector $\boldsymbol{N}$ of [notation.md](../notation.md), and $\lambda$ is the forgetting
> factor, not the Tikhonov damping $\lambda_\Gamma$ — kept as Ye et al. write them.
## Source Support
- [ye2019research](../sources/ye2019research.md) — primary: formulates online estimation of the
unknown post-capture coupling terms as a $\mathbf{y}^T=\boldsymbol{\Phi}\mathbf{W}$ regression and
identifies $\mathbf{W}$ via VFF-RLS, fixing the data-saturation failure of plain RLS on
time-varying parameters; estimate feeds the Adaptive RNS planner (free-floating regime).
## Related Topics
- [inertial_parameter_identification](inertial_parameter_identification.md) — the physical-parameter
special case: estimating mass/inertia/CoM of a grasped body, of which this page's lumped-coupling
regression is one realization.
- active_parameter_learning — closes the loop by *commanding* motions
that excite the regressor $\boldsymbol{\Phi}$; Ye et al. estimate passively from whatever motion the
RNS planner produces.
- [reaction_null_space](reaction_null_space.md) — the consumer of the estimate: $\hat{\mathbf{W}}$
corrects the null-space projection so arm motion stays reactionless after the target's parameters
change.
- [fisher_information_matrix](fisher_information_matrix.md) — quantifies how informative the excitation
is; the VFF-RLS gain $\mathbf{N}_{(k)}$ and covariance $\mathbf{Q}$ are the recursive-LS analogue of
inverse Fisher information.
- [time_delay_estimation](time_delay_estimation.md) — an alternative model-free route to the same lumped
uncertainty (estimate the disturbance from delayed signals instead of regressing physical parameters).
## Open Questions
- The source assumes a free-FLOATING base, so the regressor is built from conserved-momentum quantities
($\boldsymbol{\omega}_0,\dot{\boldsymbol{\xi}}$). For our free-flying (actuated 6-DOF base) system,
momentum is not conserved — what is the correct regressor when base wrench is an input rather than zero?
- Ye et al. also assume the joint inertia $\mathbf{M}$ is constant and diagonal, which decouples the
estimation joint-wise; does VFF-RLS still converge once full configuration-dependent off-diagonal
coupling is restored for a redundant arm?
- The forgetting-factor schedule $\lambda(k)$ relaxes purely in iteration count $k$, not on a measured
parameter-drift signal — is a fixed exponential schedule robust to a second, later contact event that
re-changes the parameters?