Learning Flatness-Preserving Residuals for Pure-Feedback Systems

Problem Statement
Differential flatness is a powerful property that enables the transformation of nonlinear systems into linear chains of integrators, facilitating efficient trajectory planning and feedback control. However, flatness is typically established for idealized nominal models. In practice, unmodeled dynamics (such as aerodynamic drag in quadrotors) degrade control performance. While residual learning can approximate these unknown dynamics to improve model accuracy, generic residual parameterizations often destroy the flatness property of the augmented system. This loss of flatness renders standard flatness-based planning and control methods inapplicable. Furthermore, even if flatness is preserved, there is often no systematic method to construct the new flatness diffeomorphism (the mapping between flat outputs and state/control) for the augmented system.

Methodology
The authors propose a framework for learning residual dynamics that strictly preserves the differential flatness of systems whose nominal models admit a pure-feedback form.

1. Structural Restriction: The paper restricts attention to nominal dynamics $\bar{f}$ that can be expressed in a multi-input multi-output pure-feedback form (Assumption 1).
2. Lower-Triangular Residuals: The core theoretical insight is that if the learned residual dynamics $\hat{\Delta}$ possess a lower-triangular structure (Definition 2), the augmented system $\bar{f} + \hat{\Delta}$ remains differentially flat. Specifically, the $i$-th component of the residual depends only on the first $i$ sub-states, preserving the recursive solvability of the pure-feedback structure.
3. Constructive Diffeomorphism: The paper provides a constructive procedure (Theorem 3) to recover the flatness diffeomorphism of the augmented system using the known diffeomorphism of the nominal model and the learned residual. This allows the flat outputs to remain unchanged (e.g., position for a quadrotor) while the mapping to states and inputs is updated to account for the residual.
4. Learning Algorithm: The authors parameterize the lower-triangular residual components using smooth function approximators (specifically, feedforward neural networks with GeLU activations). The model is trained from trajectory data by minimizing the squared error between the true residual (estimated via finite differences) and the learned residual. Once trained, the flatness diffeomorphism is constructed offline using automatic differentiation to compute necessary Jacobians.

Key Contributions

• Theoretical Characterization: The authors identify a broad class of residual dynamics (lower-triangular) that preserve the differential flatness of pure-feedback systems (Theorem 2) and prove that the flat outputs of the nominal system remain valid for the augmented system.
• Constructive Procedure: They provide a systematic method to construct the flatness diffeomorphism for the augmented model directly from the nominal model's diffeomorphism and the learned residual (Theorem 3).
• Learning Framework: They develop a parameterization and learning algorithm that enforces the lower-triangular structure, enabling the direct learning of flatness-preserving residual models from trajectory data.
• Empirical Validation: The approach is validated on a 2D quadrotor subject to unmodeled aerodynamic effects (linear and parasitic drag).

Results
Experiments were conducted on a 2D quadrotor tracking circular and lemniscate trajectories under unmodeled drag:

• Open-Loop Performance: Augmenting the nominal model with the learned residual reduced the mean open-loop position tracking error by nearly an order of magnitude compared to the nominal flat model (e.g., from ~0.31m to ~0.026m on circular trajectories).
• Closed-Loop Performance: A flatness-based tracking controller using the learned model achieved a tracking error approximately 5 times lower than the controller using only the nominal flat model.
• Computational Efficiency: When compared to a Nonlinear Model Predictive Control (NMPC) baseline using the same learned residual, the flatness-based controller achieved similar tracking performance but was approximately 20 times faster in computation time (3.3 ms vs. 85.2 ms per control update).

Significance and Claims
The paper claims to be the first work to address the learning of flatness-preserving residual dynamics for differentially flat systems. The significance lies in bridging the gap between data-driven model improvement and the structural requirements of flatness-based control. By ensuring the augmented model remains flat and providing a method to construct the necessary coordinate transformations, the approach allows for the retention of the computational benefits of flatness-based methods (speed and simplicity) while eliminating the need for case-by-case controller redesign or computationally expensive alternatives like NMPC. The authors position this as a foundational step toward more principled methods for learning residual models in differentially flat systems, with future work planned to relax structural assumptions and deploy on physical hardware.