Time Delay Estimation

Definition

Time-delay estimation (TDE) is a model-free control technique that estimates the
unknown (lumped) nonlinear dynamics of a system from its most recent past — one
control input and the measured acceleration delayed by a single sampling period —
instead of computing those terms online from a model. Introduced by Hsia & Gao (1990)
for fixed-base industrial manipulators, zhang2020adaptive
applies it to a multi-arm space robot: a constant diagonal gain matrix $\bar{\mathbf{H}}$
reformulates the coupled dynamics so that a single lumped nonlinear term $\mathbf{N}$
(Coriolis, base–arm and arm–arm coupling, disturbances) absorbs everything not captured
by $\bar{\mathbf{H}}\ddot{\mathbf{q}}$; $\mathbf{N}$ is then approximated by its
delayed value $\mathbf{N}(t-\delta )$. The source controls the base actively (it drives
base force/torque to keep the base stable), so it operates in the free-flying regime,
not the free-floating regime of the RNS motion-planning literature it contrasts against.
TDE supplies the feedforward; a sliding-mode term (see sliding_mode_control)
adds robustness against the residual estimation error.

Key Equations

Symbols per notation.md.

TDE-specific symbols are not in the registry; they are reproduced source-faithfully here
and flagged: $\bar{\mathbf{H}}$ = constant diagonal gain matrix (note: distinct from the
canonical coupled inertia $\mathbf{M}$ — the source writes inertia as $\mathbf{H}$);
$\mathbf{N}$ = lumped unknown nonlinear term; $\hat{\mathbf{N}}$ = its TDE estimate;
$\delta$ = control sampling period (the time-delay length); $\mathbf{F}$ = generalized
control force/torque; $\mathbf{\epsilon }=\mathbf{N}-\hat{\mathbf{N}}$ = TDE error;
$\mathbf{e}$ = joint tracking error ($\mathbf{e}={\mathbf{q}}_d-\mathbf{q}$; the
source’s joint-space error, distinct from the registry’s task-space $\tilde{\mathbf{x}}$);
$\mathbf{\Lambda }$ = sliding-surface matrix (diagonal, positive-definite — note: distinct
from the registry’s helix set $\Lambda =\{{\mathbf{\lambda }}_i\}$ and operational-space task
inertia $\mathbf{\Lambda }$; local to this page); $\mathbf{G}$ = switching-gain matrix
(diagonal, positive-definite — note: distinct from the registry’s velocity maps
${\mathbf{G}}_{v_c},{\mathbf{G}}_{{\omega }_b}$); $\mathbf{s}$ = sliding-surface vector.

Reformulated dynamics with the lumped term, and the one-step-delay estimate (Eqs. 36–38):

\hat{\boldsymbol N} \approx \boldsymbol N(t-\delta) = \boldsymbol F(t-\delta) - \bar{\boldsymbol H}\ddot{\boldsymbol q}(t-\delta)$$ TDE-based sliding-mode control law on the sliding surface $\boldsymbol s=\dot{\boldsymbol e}+\boldsymbol\Lambda\boldsymbol e$ (Eqs. 39–40): $$\boldsymbol F = \underbrace{\bar{\boldsymbol H}(\ddot{\boldsymbol q}_d + \boldsymbol\Lambda\dot{\boldsymbol e}) + \hat{\boldsymbol N}}_{\boldsymbol F_{\mathrm{TDEeq}}} + \boldsymbol G\,\mathrm{sgn}(\boldsymbol s)$$ Stability holds element-wise when the switching gain dominates the bounded TDE error, $G_i > E_i$ with $|\varepsilon_i|\le E_i$, giving $\dot V<0$ (global asymptotic stability proven in the source). ## Source Support - [zhang2020adaptive](../sources/zhang2020adaptive.md) — primary and only source: derives the TDE reformulation, the TDE-based SMC decoupling law for a multi-arm (dual-arm) space robot, proves global asymptotic stability via the bounded-TDE-error argument, and studies sensitivity to the delay length $\delta$ ($0.5$–$4$ ms), concluding TDE replaces online model computation (CTC) with cheaper online estimation that is "almost immune" to disturbances and uncertainties. ## Related Topics - [sliding_mode_control](sliding_mode_control.md) — the robustifying companion: SMC's switching term $\boldsymbol G\,\mathrm{sgn}(\boldsymbol s)$ suppresses the residual TDE error $\boldsymbol\varepsilon$, while TDE removes SMC's reliance on an accurate model; the two "complement and reinforce" each other. - [parameter_estimation](parameter_estimation.md) — TDE is the *model-free* alternative: rather than identifying physical inertial/Coriolis parameters online (the CTC route), it estimates the lumped term directly from delayed I/O, trading parameter knowledge for a small sampling period. - [high_gain_observer](high_gain_observer.md) — a related estimation primitive: both reconstruct unmodeled dynamics, but TDE uses a one-step time delay of measured signals where a high-gain observer uses a fast continuous-time estimator; relevant when comparing how each recovers $\boldsymbol N$. - [dynamic_coupling](dynamic_coupling.md) — the problem TDE targets: the base–arm (and, for multi-arm, arm–arm) coupling force/torque is exactly what the lumped term $\boldsymbol N$ absorbs and the controller decouples. ## Open Questions - TDE accuracy hinges on the delay length $\delta$ being "sufficiently small" relative to the dynamics; the source uses $\delta\approx1$ ms. For our free-flying inspection system with a fully-actuated 6-DOF base and a redundant arm, is a 1 ms loop rate attainable, and does the delayed-estimate approximation $\hat{\boldsymbol N}\approx\boldsymbol N(t-\delta)$ remain valid through the high-coupling configurations near a [dynamic_singularity](dynamic_singularity.md)? - The lumped term $\boldsymbol N$ includes the *acceleration* feedback $\ddot{\boldsymbol q}(t-\delta)$, which the source reconstructs from delayed signals; how sensitive is TDE to acceleration measurement/estimation noise, and does that interact with the [circumcentroidal_motion](circumcentroidal_motion.md) coordinate split we use for decoupling? - The source's stability proof assumes the TDE error is bounded element-wise ($|\varepsilon_i|\le E_i$); is that bound preserved when the base is a controlled 6-DOF flyer (extra actuated DOFs and reference acceleration) rather than the dual-arm topology analyzed here?