Variable-Gain ESOs for Aerial Manipulator Control

• Variable-Gain ESOs are advanced observer architectures that dynamically adjust gain to estimate fast, time-varying disturbances, balancing noise suppression with rapid convergence.
• They employ a nonlinear gain scheduling law to effectively decouple unmodeled base-arm coupling in aerial manipulators, ensuring tight tracking error bounds.
• Empirical benchmarks in scenarios like aerial cart-pulling and staff-twirling demonstrate rapid convergence and enhanced tracking performance under severe nonlinear dynamics.

A variable-gain Extended State Observer (ESO) is an advanced observer architecture designed to estimate fast, time-varying disturbances in control systems—particularly those characterized by strong, unmodeled coupling and aggressive nonlinear dynamics. In the context of aerial manipulators, which integrate a multirotor platform with a serial robotic arm, the variable-gain ESO enables real-time reconstruction of dynamic coupling effects, allowing precise and robust motion control even under rapid arm movements. Within the PreGME (Prescribed Performance Control of Aerial Manipulators based on Variable-Gain ESO) framework, the observer’s variable-gain mechanism supports both noise suppression and fast convergence, ensuring tight tracking error bounds as articulated through prescribed-performance constraints (Ji et al., 28 Dec 2025).

1. Mathematical Formulation and System Dynamics

Consider an aerial manipulator consisting of a quadrotor base (mass $m_B$) and a serial manipulator arm (mass $m_R$). In the inertial North–East–Down frame $\Sigma_I$, the base’s position $p\in\mathbb{R}^3$, velocity $v=\dot p\in\mathbb{R}^3$, attitude $R\in SO(3)$, and angular velocity $\omega\in\mathbb{R}^3$ describe the system state. Propulsion generates thrust $T n$ ($n=[0\;0\;1]^T$), and rotor torque $\tau\in\mathbb{R}^3$. The coupling of the arm onto the base—manifested as force disturbance $\Delta_v\in\mathbb{R}^3$ and torque disturbance $\Delta_\omega\in\mathbb{R}^3$—is unmodeled and time-varying, demanding real-time estimation:

$$\begin{cases} \dot p = v \ \dot v = -\dfrac{T R n}{m_B + m_R} + g n + \Delta_v \end{cases} \qquad \begin{cases} \dot R = R [\omega]_\times \ \dot\omega = I^{-1}(\tau - \omega\times I\omega) + \Delta_\omega \end{cases} \tag{1}$$

Here, $I$ is the quadrotor inertia, $g$ the gravitational constant, and $[\cdot]_\times$ the skew-symmetric mapping.

2. ESO State-Space Structure and Estimation Dynamics

To reconstruct unmeasured disturbances, the variable-gain ESO adopts a canonical “integrator + disturbance” augmented system for each channel of $\dot v_i$ or $\dot\omega_i$:

$$\dot y_1 = y_2 + u,\qquad \dot y_2 = \dot\Delta \tag{2}$$

Where $y_1$ is the measured output, $u$ the control input, and $\Delta$ the unknown disturbance. The internal observer state $h$ is updated to minimize the observation error $e = y_1 - h$:

$$\dot h = \frac{\alpha}{\varepsilon} g(e) + u,\qquad \hat{\Delta} = \frac{\alpha}{\varepsilon} g(e) \tag{3}$$

The complete observer for $n$ channels is vectorized as:

$$\dot h = \mathrm{diag}\left(\frac{\alpha_i}{\varepsilon_i}\right) g(e) + u,\qquad \hat{\Delta} = \mathrm{diag}\left(\frac{\alpha_i}{\varepsilon_i}\right) g(e)$$

3. Variable-Gain Scheduling Law

The variable-gain law $g(e)$ modulates the observer bandwidth dynamically as a function of $|e|$, promoting noise immunity near zero error and rapid response when errors become large:

$$g(e) = \frac{e (\exp(e) + \exp(-e))}{w (\exp(e) + \exp(-e)) + d},\qquad w > 0,\, d > 0 \tag{4}$$

Properties:

• For $|e| \to 0$, $g(e) \approx \frac{2e}{2w + d} \to 0$ (noise suppression).
• For $|e| \gg 0$, $g(e) \approx \frac{e}{w}$ (high gain, fast convergence).

Parameters $w$ and $d$ determine the high-gain ceiling and low-gain floor, enabling tailored response characteristics for aggressive or noisy environments.

4. Stability and Error Boundedness Analysis

A quadratic Lyapunov function $V(e) = \frac{1}{2} e^2$ is selected to analyze the error dynamics:

$$\dot e = y_2 - \frac{\alpha}{\varepsilon} g(e),\qquad \tilde\Delta = \Delta - \hat\Delta = y_2 - \frac{\alpha}{\varepsilon} g(e)$$

$$\dot V = e\,\dot e = e (\Delta - \frac{\alpha}{\varepsilon} g(e)) = e\,\tilde\Delta - \frac{\alpha}{\varepsilon} e\,g(e) \tag{5}$$

There exists a continuous function $\rho(e) > 0$ such that $e\,g(e) \ge \rho(e)\,e^2$. For bounded $\dot\Delta$ (rate of disturbance variation), Input-to-State Stability (ISS) arguments yield that for any ultimate error bound $\delta_f > 0$ and time $t_f > 0$, suitable $\varepsilon$ guarantees:

$$|\tilde\Delta(t)| \le \delta_f,\qquad \forall\,t \ge t_f$$

This ensures the observer’s estimation error remains bounded and converges rapidly.

5. Gain Tuning and Scheduling Methodology

Parameter selection directly governs ESO dynamics:

Parameter	Role	Recommended Range
$\varepsilon$	Observer bandwidth: lower is faster	$0.1$–$0.5$
$\alpha$	High-gain scaling	$0.1$–$1.0$
$w$	High-gain ceiling	$0.3$–$1.0$
$d$	Low-gain floor, noise suppression	$3$–$10$

Tuning proceeds as follows:

1. Initialize $\varepsilon$ (e.g., $0.5$).
2. Simulate worst-case disturbance; decrease $\varepsilon$ until $\hat\Delta$ tracks $\Delta$ within $\delta_f$.
3. Adjust $\alpha$ for bandwidth matching.
4. Fine-tune $w, d$ to balance noise rejection and rise-time.

6. Fusion of ESO Estimates into Prescribed Performance Control

Within PreGME, two ESOs operate in cascade:

• Position Loop: The position ESO estimates $\Delta_v$ for thrust computation,

$$T\,n = (m_B + m_R)\left[ g\,n + \hat\Delta_v - \ddot p_d - \ddot\beta_p + \Lambda_p\dot z_p + K_p\,s_p \right] \tag{6}$$

where $z_p = (p - p_d) - \beta_p(t)$ and $s_p = \dot z_p + \Lambda_p z_p$.

• Attitude Loop: The attitude ESO estimates $\Delta_\omega$ for body torque,

$$\tau = 2\,I\,Q^{-1}\left[ -f_q(\tilde q_v,\dot{\tilde q}_v) -\frac{1}{2} Q \hat\Delta_\omega + \ddot\beta_q - \Lambda_q \dot z_q - K_q s_q \right] \tag{7}$$

with $z_q = \tilde q_v - \beta_q(t)$ and $s_q = \dot z_q + \Lambda_q z_q$.

ESO estimates are injected as feed-forward compensation terms, effectively canceling unmodeled base-arm coupling and stabilizing closed-loop error trajectories within prescribed envelopes.

7. Quantitative Performance: Simulation and Experiment

Empirical and simulated benchmarks illustrate variable-gain ESO utility:

• Fast-Swing Arm Simulation: Arm tip speed up to $0.76~\mathrm{m/s}$; ESO ($$\varepsilon_v = 0.5, \alpha_v = 0.1, w = 0.5, d = 5$$) converges in $\approx 0.2~\mathrm{s}$ with RMS error $\approx 0.02~\mathrm{m/s}^2$. Position error held within $0.05~\mathrm{m}$ prescribed envelope, steady $\approx 0.03~\mathrm{m}$ RMS error.
• Aerial Staff-Twirling Experiment: End-effector speed $0.51~\mathrm{m/s}$, staff spin $5.6~\mathrm{rad/s}$; torque tracking up to $0.8~\mathrm{Nm}$, peak error $< 0.08~\mathrm{Nm}$, position error mean ± std $$(4.24~\mathrm{cm} \pm 4.86~\mathrm{cm}, 3.97~\mathrm{cm} \pm 4.86~\mathrm{cm}, 0.79~\mathrm{cm} \pm 0.95~\mathrm{cm})$$, outperforming baseline (> $10~\mathrm{cm}$ mean error).
• Aerial Mixology Experiment: Arm shake yields $5.10~\mathrm{m/s}^2$ acceleration, ESO convergence $0.15~\mathrm{s}$, RMSE $0.04~\mathrm{m/s}^2$, closed-loop position RMS $0.04~\mathrm{m}$ vs baseline PX4 ($0.11~\mathrm{m}$) and ESO off ($0.09~\mathrm{m}$).
• Aerial Cart-Pulling: Cart mass $14.6~\mathrm{kg}$, external force $\sim 15~\mathrm{N}$, ESO errors $< 1~\mathrm{N} / < 0.1~\mathrm{Nm}$ in $0.2~\mathrm{s}$, position error $< 0.07~\mathrm{m}$ (vs $>0.2~\mathrm{m}$ without ESO).

This suggests variable-gain ESOs enable aerial manipulators to maintain high tracking accuracy under highly dynamic and coupled conditions.

8. Contextual Implications and Research Significance

Variable-gain ESOs as implemented in PreGME (Ji et al., 28 Dec 2025) achieve bounded disturbance estimation with convergence rates and error envelopes seldom attained by fixed-gain or adaptive observers in severe nonlinear regimes. By scheduling gain as a nonlinear function of estimation error, the ESO addresses the trade-off between noise robustness and responsiveness to abrupt disturbance changes. A plausible implication is broader applicability in any domain marked by severe unmodeled dynamics, e.g., aerial load transport, legged robotics, or mobile manipulation, where prescribed performance is critical. The empirical evidence substantiates the technical viability of variable-gain ESO frameworks for time-critical applications requiring quantifiable, tight error bounds.