Aggressiveness-Aware Learning-based Control of Quadrotor UAVs with Safety Guarantees

Abstract: This paper presents an aggressiveness-aware control framework for quadrotor UAVs that integrates learning-based oracles to mitigate the effects of unknown disturbances. Starting from a nominal tracking controller on $\mathrm{SE}(3)$, unmodeled generalized forces and moments are estimated using a learning-based oracle and compensated in the control inputs. An aggressiveness-aware gain scheduling mechanism adapts the feedback gains based on probabilistic model-error bounds, enabling reduced feedback-induced aggressiveness while guaranteeing a prescribed practical exponential tracking performance. The proposed approach makes explicit the trade-off between model accuracy, robustness, and control aggressiveness, and provides a principled way to exploit learning for safer and less aggressive quadrotor maneuvers.

• The paper develops an SE(3)-based controller that learns disturbance forces and moments with Gaussian processes, then schedules feedback gains according to the model-error bound to preserve practical exponential tracking guarantees.
• The paper proves that, with probability at least δ, sufficiently accurate disturbance estimates enable lower feedback gains while maintaining a prescribed tracking tolerance, with local guarantees under matched disturbances and finite dataset updates.
• The paper’s simulations show GP compensation reducing final position error to 0.028 m under severe disturbances—versus 0.190 m for fixed-high gains—while keeping thrust and torque variation near lower-aggressiveness levels, although hardware validation remains open.

This paper presents a control framework for quadrotor UAVs that explicitly quantifies and minimizes feedback-induced aggressiveness while maintaining probabilistic tracking guarantees. Building on geometric tracking control on $SE(3)$, the authors augment a nominal controller with a learned model of unknown generalized disturbance forces and moments, and derive a gain-scheduling rule in which the required feedback gains—and hence the aggressiveness—scale with the high-probability model-error bound of a Gaussian process (GP) oracle. The central claim is that improved disturbance compensation permits smaller feedback gains for the same prescribed practical exponential tracking tolerance.

Problem formulation and aggressiveness measure

The quadrotor is modeled on $SE(3)$ with unknown disturbance forces $f_{\text{trans}}$ and moments $f_{\text{rot}}$ stacked into $f(x)\in\mathbb{R}^6$, lifted into a 12-dimensional control-affine local representation. A nominal geometric controller $u = h(x,x_d) + h(x)He$ is assumed to render the error locally exponentially stable (Assumptions 1–2 give quadratic Lyapunov bounds with constants $c_1, c_2$). The paper's key conceptual contribution is the aggressiveness metric

$s(H,x) = \left\| h(x) H \right\|,$

i.e., the induced gain of the feedback map with respect to tracking-error variations, inspired by stiffness-alteration viewpoints in soft robotics. Because the feedback term is linear in the error, the objective $\sup_x s(H,x)$ is convex, so the gain-tuning problem—minimize aggressiveness subject to a practical exponential bound $$\|e(t)\| \le \gamma_1 e^{-\gamma_2 t}\|e(t_0)\| + \varepsilon$$—admits a minimizer when $SE(3)$0 is compact. This formalizes the trade-off that high-gain feedback, though robust, produces large thrust/torque variations, actuator stress, and unsafe behavior near obstacles.

Stability guarantees

The main theorem connects the GP oracle's uniform high-probability error bound $SE(3)$1 (Assumption 3, valid under a finite-RKHS-norm kernel assumption) to both tracking performance and aggressiveness. If the oracle bound satisfies

$SE(3)$2

then, with probability at least $SE(3)$3, the tracking error obeys the practical exponential bound with $SE(3)$4 and $SE(3)$5, and the aggressiveness satisfies $SE(3)$6 when the gains are selected affinely in the model error. A corollary shows that if $SE(3)$7, the aggressiveness $SE(3)$8 converges to $SE(3)$9, i.e., asymptotically the feedback need not be more aggressive than the residual floor. A block-diagonal scheduling proposition further separates translational and rotational gains using channel-specific error bounds. The proof is a standard Lyapunov perturbation argument with forward invariance of the $f_{\text{trans}}$0-ball on the high-probability event; the guarantees are local (valid in a chart of $f_{\text{trans}}$1), inheriting only the almost-global character of the underlying geometric controller. The framework assumes matched additive disturbances, inputs remaining within $f_{\text{trans}}$2, and a finite number of dataset updates (Assumption 4)—the latter excluding indefinitely evolving online learning from the formal guarantees.

Numerical results

Simulations use a 1 kg quadrotor tracking a 20 s trajectory with tolerance $f_{\text{trans}}$3 m under moderate ($f_{\text{trans}}$4) and severe ($f_{\text{trans}}$5) disturbances, comparing fixed-low, fixed-high, and sweep-based aggressiveness-aware gains. Under moderate disturbance, the scheduler selects $f_{\text{trans}}$6, achieving $f_{\text{trans}}$7 m (within tolerance) with lower transient aggressiveness than fixed-high ($f_{\text{trans}}$8 of 8.121 vs. 9.925 N/s). Under severe disturbance, no tested gain meets tolerance without learning (fixed-low: 0.351 m; fixed-high: 0.190 m), and the GP-compensated controller restores feasibility:

Metric	fixed-low	fixed-high	GP-comp (aware)
$f_{\text{trans}}$9 [m]	0.351	0.190	0.028
$f_{\text{rot}}$0 [N/s]	7.887	10.411	8.066
$f_{\text{rot}}$1 [Nm/s]	5.734	8.154	6.077
$f_{\text{rot}}$2	17.866	26.961	17.866

The GP oracle uses six independent RBF-ARD regressors over a 20-dimensional feature vector, trained on 1162 samples, with a sigmoid uncertainty gate (steady-state mean 0.930) suppressing compensation when epistemic uncertainty is high. An online experiment adding a gated residual GP on top of the offline prior yields a marginal improvement from 0.028 m to 0.025 m with essentially unchanged aggressiveness, indicating that most of the benefit comes from the offline model. These results support the paper's central mechanism: learning reduces the residual perturbation, restoring tolerance feasibility without raising gains. Notably, the aggressiveness-aware selection in simulation is implemented as an empirical gain sweep rather than via the certified model-error bound of the theory, so the simulation feasibility is not formally tied to the probabilistic certificate.

Limitations and open questions

The paper is explicit that validation is simulation-only; hardware experiments with realistic aerodynamics, sensing noise, input saturation, discrete-time implementation, and onboard computation remain unaddressed. The guarantees are local in $f_{\text{rot}}$3 coordinates and require compactness of $f_{\text{rot}}$4, finite dataset updates, and a computable $f_{\text{rot}}$5; how to calibrate the uncertainty bounds and safety gates online from flight data so that prescribed tolerances hold in practice is left open. The gain condition $f_{\text{rot}}$6 depends on conservative Lyapunov constants, and the paper does not address how tight these are in practice. Whether the convexity of the aggressiveness objective survives input saturation constraints is not analyzed.

Conclusion

The paper contributes a principled formulation of feedback-induced aggressiveness for $f_{\text{rot}}$7 quadrotor tracking, a Lyapunov-based proof that GP-based disturbance compensation reduces the gains needed for a prescribed practical exponential tolerance with probability $f_{\text{rot}}$8, and simulation evidence that GP compensation restores tracking feasibility under severe mismatch (0.028 m vs. 0.190 m for fixed-high gains) at comparable input-variation levels. The main gap between theory and results is that the numerical scheduler emulates, rather than certifies, the model-error-based feasibility condition, and experimental validation on hardware is outstanding.