---
title: Uncertainty & Disturbance Estimator (UDE)
url: https://www.emergentmind.com/topics/uncertainty-and-disturbance-estimator-ude
type: topic
---

# Uncertainty & Disturbance Estimator (UDE)

Searching arXiv for recent and foundational papers on Uncertainty and Disturbance Estimator (UDE) and closely related observer formulations.
Search query: "Uncertainty and Disturbance Estimator UDE control observer robotics arXiv"
Uncertainty and Disturbance Estimator (UDE) denotes a real-time filter-based scheme that reconstructs lumped disturbances in a control system and then uses estimation/cancellation to compensate plant uncertainty, modeling error, nonlinearities, and external disturbance. In the cited arXiv literature, UDE appears in several closely related forms: a standard low-pass-filter estimator for rigid-body manipulators, a discrete-time observer-based robust controller for robot manipulators, an extended architecture that separately estimates dynamic coupling and other unmodeled terms for mobile manipulators, and a modular disturbance-compensation block combined with a neural control contraction metric (CCM) controller and an almost-globally asymptotically stable (AGAS) attitude loop for multi-drone slung-payload transportation [2404.00443] [2203.03805] [2011.06383] [2510.01489]. Across these formulations, the central idea is consistent: disturbances are treated as a lumped signal to be estimated online and canceled through an additive control channel.

## 1. Core definition and canonical formulation

In the standard robotic formulation, the plant is written as
\[
M(q)\,\ddot q + C(q,\dot q)\,\dot q + g(q) = \tau + d(t),
\]
where \(q\in\mathbb R^n\) is the joint vector, \(M(q)\) is the positive-definite inertia matrix, \(C(q,\dot q)\dot q\) collects Coriolis and centrifugal terms, \(g(q)\) is the gravity vector, \(\tau\in\mathbb R^n\) is the control torque, and \(d(t)\in\mathbb R^n\) lumps all unmodeled dynamics and external disturbances. The standard UDE law estimates the disturbance by applying a low-pass filter \(Q(p)\), with \(p\equiv d/dt\), to the model residual:
\[
\hat d(t) = Q(p)\bigl[\tau(t)-M(q)\ddot q(t)-C(q,\dot q)\dot q(t)-g(q)\bigr].
\]
A common choice is
\[
Q(p)=\frac{\lambda}{p+\lambda},\qquad \lambda>0,
\]
so that disturbances below the cutoff frequency \(\omega_c\approx \lambda\) are passed with unity gain and zero phase lag, while higher-frequency components are attenuated [2404.00443].

This formulation makes UDE a filter-based disturbance observer rather than a full disturbance model. The disturbance is not parameterized explicitly; instead, it is reconstructed from the mismatch between measured or inferred plant behavior and the nominal dynamics. In the cited work, this property is used to handle unmodeled dynamics, friction, environmental interaction, external forces, and coupled base-induced effects within a single compensation framework.

## 2. Estimation–cancellation principle and relation to ESO and EDO

A broader observer-theoretic interpretation is given by the Extended Dynamics Observer (EDO) formulation for the SISO LTI plant
\[
\dot x(t)=A\,x(t)+B[\,d(t)+u(t)\,],\qquad y(t)=C\,x(t),
\]
with disturbance decomposed as
\[
d(t)=d_1(t)+d_2(t).
\]
In that setting, \(d_1(t)\) lies in a known disturbance sub-space \(\Omega(G)\), while \(d_2(t)\) is an unknown residual satisfying a bounded-derivative condition. The EDO estimates both the state and the disturbance simultaneously by combining an internal-model block with a high-gain observer block. The working mechanism consists of two parts: the disturbance with known dynamics is canceled completely by its dynamics, and the disturbance with unknown dynamics is absorbed by high-gain. The main advantage is that prior information about both the control plant and the disturbance can be utilized as much as possible; the more prior information is available, the better performance the observer would be [2011.06383].

Within this framework, UDE is not isolated from neighboring observer constructions. If no disturbance model is available, one may set \(G=0\), and the EDO reduces to an extension of the well-known extended state observer or high-gain observer. The cited paper states that the EDO family smoothly interpolates between pure ESO/UDE \((G=0)\) and the full internal model principle when the disturbance lies entirely in \(\Omega(G)\) [2011.06383]. This suggests that UDE is best understood as one point in a continuum of disturbance-observer designs ranging from purely lumped estimation to exact cancellation of known disturbance sub-dynamics.

## 3. Structural variants

Across the cited literature, UDE is implemented in several structurally distinct ways.

| Variant | Distinguishing structure | Source |
|---|---|---|
| Standard UDE | Single low-pass filter \(Q(p)\) on model residual | [2404.00443] |
| Extended-UDE | Two parallel first-order filters \(Q_c(p)\), \(Q_u(p)\) for coupling and unmodeled terms | [2404.00443] |
| DT-UDE | First-order digital filter \(G(\gamma)=1/(1+\tau\gamma)\) with observer-based auxiliary error | [2203.03805] |
| Slung-payload UDE | Disturbance decomposition into \(\delta_T\) and \(\delta_{\perp,j}\) with additive thrust compensation | [2510.01489] |

For mobile manipulators, the extended form separates two disturbance sources in task space:
\[
d_c(t)=-\bigl[M_0(\ddot\eta+\dot d)+C_0(\dot\eta+d)\bigr]
\]
for dynamic coupling, and
\[
d_u(t)=\mu_d(t)
\]
for other unmodeled terms. Two parallel first-order filters are then used,
\[
Q_c(p)=\frac{\omega_{c,c}}{p+\omega_{c,c}},\qquad
Q_u(p)=\frac{\omega_{c,u}}{p+\omega_{c,u}},
\]
with \(\omega_{c,c}\gg \omega_{c,u}\), so that fast base-induced coupling is tracked by \(\hat d_c\) while slower uncertainties are assigned to \(\hat d_u\). The stated purpose is to reduce interference between coupling and unmodeled terms and to allow independent tuning of the two estimators [2404.00443].

For robot manipulators in sampled-data form, the continuous-time system
\[
\dot x(t)=A\,x(t)+B\,u(t)+L(x(t),t)
\]
is discretized as
\[
x(k+1)=F_n\,x(k)+G_n\,u(k)+D_d(x(k),k),
\]
with
\[
F_n=e^{A T_s},\qquad
G_n=\int_{0}^{T_s}e^{A\tau}B\,d\tau.
\]
The corresponding DT-UDE uses the \(z\)-domain filter
\[
G(\gamma)=\frac{1}{1+\tau\gamma},\qquad \gamma=\frac{z-1}{T_s},
\]
so that \(\widehat L_d(k)=G(\gamma)L_d(k)\). In this design, the control law is generated using an auxiliary error \(\hat e(k)=\hat x(k)-x_m(k)\) rather than the unavailable state-tracking error \(e(k)=x(k)-x_m(k)\) [2203.03805].

For the three-drone slung-payload system, the disturbance is first decomposed into a payload-effective component and three perpendicular components. With cable vector \(l_j\in\mathbb R^3\) and \(\|l_j\|=l\),
\[
\delta_{\parallel,j}=\frac{l_j l_j^\top}{l^2}\,\delta_j,\qquad
\delta_{\perp,j}=\delta_j-\delta_{\parallel,j},
\]
and
\[
\delta_T=\delta_p+\sum_{j=1}^3\delta_{\parallel,j}.
\]
The estimator then constructs \(\hat\delta_T\) and \(\hat\delta_{\perp,j}\), which are used to allocate a disturbance-canceling thrust increment on each quadrotor [2510.01489].

## 4. Integration with closed-loop control architectures

The role of UDE in closed-loop design is not uniform across applications, but the cited work consistently places it in an estimation/cancellation loop that is modular with respect to a nominal controller.

In the multi-drone slung-payload system, the full dynamics are written in control-affine form,
\[
\dot x = f(x)+G(x)\,\zeta+G_\delta(x)\,\delta,
\]
where \(x\in\mathbb R^{12}\) collects payload position, three cable directions and velocities, \(\zeta\in\mathbb R^{12}\) is the vector of thrust control increments, and \(\delta\in\mathbb R^{12}\) collects the unknown disturbance forces on payload and quadrotors. The baseline CCM controller is trained offline under \(\delta\equiv 0\) to produce a smooth neural control law \(\zeta_{nn,sat}(x,x^\*,k^\*)\), enforcing hard thrust-saturation via a \(\tanh(\cdot)\) at its output. In closed loop, the total thrust command becomes
\[
\zeta_c=\zeta_{nn,sat}+f_\delta,
\]
where the compensation increment is determined from
\[
f_{\delta,j}=-\,n_j\,\hat\delta_{T,j}-\hat\delta_{\perp,j},\qquad
n_j=\frac{l_j}{l}.
\]
The UDE enters as an additive disturbance-compensation block outside the learned CCM law, and no modification of the CCM contraction conditions is needed. The inner-loop attitude controller then tracks \(\zeta_c\) via an AGAS attitude controller [2510.01489].

In the mobile-manipulator case, UDE is embedded in a feedforward–feedback architecture designed to enforce a hybrid impedance model,
\[
M_d(\ddot x-\ddot x_d)+C_d\dot e+K_d e = K_{f,d}e_f.
\]
The feedforward term is
\[
f_{FF}=-M_0(\ddot\eta+\dot d)-C_0(\dot\eta+d)+G_0-f_e+C_0\dot x_d,
\]
and the UDE-based feedback uses the extended estimator to implement
\[
f_{UDE}=\hat d_u(t).
\]
The total control is
\[
f=f_{FF}+f_{UDE}.
\]
According to the cited formulation, the feedforward component cancels the coupling \(d_c\), while the UDE-based feedback cancels \(d_u\), so that the resulting error dynamics match the desired impedance behavior [2404.00443].

In the DT-UDE observer–controller design, the total control law is
\[
u(k)=-K_d\,\hat e(k)+u_d(k),
\]
with \(u_d(k)\) generated by the discrete disturbance estimator and \(\hat e(k)\) formed from the state observer. The distinguishing feature emphasized in the cited work is that the filter and state-feedback term act on \(\hat e(k)\) instead of the true \(e(k)\) [2203.03805].

## 5. Stability, convergence, and robustness results

The stability claims associated with UDE are strongly architecture-dependent and assumption-dependent.

For the multi-drone CCM+UDE system, Assumption III.1 requires that all \(\delta_p\) and \(\delta_j\) are bounded, and that near-hover and calm-wind operation imply \(\dot\delta_T\approx 0\) and \(\dot\delta_j\approx 0\). Under this quasi-static assumption, the first-order UDE yields simplified estimation-error dynamics and exponential convergence of the estimation errors. The stated closed-loop result is:

> Consider the control-affine model (10) under (i) the neural CCM baseline law (16),(19), (ii) the UDE (17),(23),(24), (iii) the AGAS attitude-tracker (27), and (iv) Assumption III.1. Then all system trajectories \(\eta_1(t)\) converge to the nominal reference \(\eta_0(t)\) as \(t\to\infty\), and the total thrust \(\zeta\) remains uniformly bounded by a known \(f_n\).

The proof sketch uses a composite Lyapunov function \(V_e\) for the UDE estimation errors, shows \(\dot V_e\le 0\), applies Barbalat’s lemma to obtain error convergence, and then invokes a contraction-robustness result stating that contracting nominal dynamics with bounded additive disturbance lead to exponentially convergent tracking error to a residual ball of size \(O(\|d\|_\infty)\) [2510.01489].

For the mobile-manipulator extended-UDE architecture, substituting the control into the task-space dynamics yields closed-loop error dynamics that match the desired impedance model. A Lyapunov-passivity argument with
\[
V=\frac{1}{2}\dot e^\top M_0 \dot e+\frac{1}{2}e^\top K_d e
\]
shows global asymptotic stability in full-motion mode \((K_{f,d}=0)\) and passivity, hence stable force/motion interaction, in hybrid mode [2404.00443].

For the DT-UDE observer–controller, qualitative asymptotic stability is obtained when
\[
\rho(F_n-G_nK_d)<1,\qquad
\rho(F_n-\beta C)<1,\qquad
\left|1-\frac{T_s}{\tau}\right|<1,
\]
that is, when the controller and observer poles lie inside the unit circle and \(\tau>T_s/2\). The quantitative Lyapunov analysis shows that, for slowly varying disturbance \(\|\Delta L_d\|\) small, the closed-loop errors converge exponentially into a small ball around zero [2203.03805].

For the EDO, after an invertible transformation to cascade form, the block-triangular generator has two Hurwitz diagonal blocks. The cited result states that there exists \(M>0\) independent of \(\omega_0\) such that
\[
\limsup_{t\to\infty}\|(\epsilon_x,\epsilon_v)(t)\|\le M\,\|\dot e\|_S/\omega_0.
\]
With the output-feedback controller \(u(t)=-\hat d(t)+F_{\omega_c}\hat x(t)\), the ultimate bound becomes
\[
\limsup_{t\to\infty}\|x(t)\|=O(\|\dot e\|/(\omega_0\omega_c)).
\]
The cited text interprets this as a rigorous LTI foundation with explicit convergence rates [2011.06383].

## 6. Empirical evidence, assumptions, and interpretive issues

The simulation and experimental evidence in the cited papers is application-specific but consistently presented as support for disturbance rejection and improved tracking.

For the three-drone slung-payload system, a \(63\) s figure-8 trajectory is commanded under a thrust disturbance \(\delta=\delta_c+\delta_s\), with
\[
\delta_c=[0.3,-0.2,0.5,\ldots]^\top,\qquad
\delta_s\sim \mathrm{Uniform}[0,0.3].
\]
Two cases are compared: UDE OFF and UDE ON. With UDE OFF, the position error remains \(O(0.3\,\mathrm{m})\). With UDE ON, the error decays rapidly to zero once the stochastic component is switched off at \(t=31.5\,\mathrm{s}\). Figure 8.b is reported to show the payload tracking error and the UDE estimation error both converging to zero, in perfect agreement with Theorem III.5 [2510.01489].

For the mobile manipulator, model verification under sinusoidal base motions reports that the coupling model \(M_0(\ddot\eta+\dot d)+C_0(\dot\eta+d)\) predicts the measured wrench with weighted MAPE of \(81.5\%\) in \(X\) and \(80.1\%\) in \(Y\) directions. In simulation on rough terrain, three controllers are compared: C1 (Extended-UDE + feedforward coupling), C2 (feedforward coupling only), and C3 (standard impedance control). In full-motion control, C1’s Cartesian tracking error converges within \(0.01\,\mathrm{m/rad}\), whereas C2 and C3 exhibit steady-state offsets due to friction and unmodeled coupling. In hybrid motion/force tasks, C1 achieves RMSE \(=2.90\,\mathrm{N}\), MAD \(=0.60\,\mathrm{N}\), and SSE \(=1.15\,\mathrm{N}\) in Simulation 1, and RMSE \(=2.71\,\mathrm{N}\), MAD \(=0.63\,\mathrm{N}\), and SSE \(=1.05\,\mathrm{N}\) in Simulation 2. In a physical wall-wiping experiment on a TT15+SR3 mobile manipulator at \(0.16\,\mathrm{m/s}\), C1 versus C3 yields motion errors \(<0.01\,\mathrm{m/rad}\) throughout, force RMSE \(=5.34\,\mathrm{N}\) versus \(8.34\,\mathrm{N}\), MAD \(=1.93\,\mathrm{N}\) versus \(2.09\,\mathrm{N}\), and SSE \(\approx 0\,\mathrm{N}\) versus \(1.40\,\mathrm{N}\) [2404.00443].

For the DT-UDE on a two-link robot manipulator, only joint positions are measured. The reported simulation uses \(T_s=0.01\,\mathrm{s}\) and \(\tau=0.01\). The results state that joint positions track the reference with negligible error, estimated velocities coincide with true velocities, torques remain moderate, and disturbance estimates accurately follow the true disturbances. In a comparative study against sliding-mode control, gravity-compensated PD, and continuous-time UDE, the UDE-based designs give far better tracking than SMC or PD, and under nonzero initial conditions the DT-UDE requires up to \(25\times\) less peak torque than CT-UDE [2203.03805].

These results also delimit what UDE does and does not assume. The cited work does not present UDE as assumption-free. Standard UDE reconstructs lumped disturbance without an explicit disturbance model, but the estimator is embedded in plant dynamics; EDO requires observability of the augmented system; DT-UDE relies on a matching-condition assumption and filter stability condition; and the slung-payload formulation requires bounded disturbances together with a near-hover, calm-wind, quasi-static disturbance assumption [2404.00443] [2011.06383] [2203.03805] [2510.01489]. This suggests that the strongest claims—zero steady-state tracking error, arbitrarily small ultimate bounds, or exact cancellation of known sub-dynamics—are conditional on structural assumptions, estimator bandwidth selection, and the separation between modeled and residual disturbance channels.

Source: https://www.emergentmind.com/topics/uncertainty-and-disturbance-estimator-ude