[Paper] Exploiting Differential Flatness for Efficient Learning-based Model Predictive Control of Constrained Multi-Input Control Affine Systems

Source: arXiv - 2604.24706v1

Overview

This paper presents a new learning‑based Model Predictive Control (MPC) scheme that leverages differential flatness—a structural property common in many robots—to dramatically cut the computational cost of online control. By marrying flatness with a probabilistic learning model, the authors deliver an MPC that respects input limits and state constraints while remaining fast enough for real‑time deployment on multi‑input, nonlinear, control‑affine systems.

Key Contributions

• Flatness‑aware learning MPC: Introduces a controller that explicitly exploits differential flatness to simplify the optimization problem.
• General multi‑input support: Extends previous flatness‑based approaches (which were limited to single‑input systems) to arbitrary‑dimensional input vectors.
• Constraint handling: Incorporates both hard input bounds and half‑space constraints on flat states, something earlier methods often ignored.
• Probabilistic Lyapunov guarantee: Provides a theoretical guarantee of expected Lyapunov decrease using only two sequential convex programs per control step.
• Computational efficiency: Demonstrates a multiple‑fold speed‑up over a standard Gaussian‑process (GP) MPC while achieving comparable tracking performance.
• Real‑world validation: Validates the approach on both high‑fidelity simulations and physical hardware experiments, showing competitive tracking accuracy.

Methodology

1. System Extension & Flat Output Selection
   • The original control‑affine dynamics are augmented with auxiliary states so that a flat output (a set of outputs whose trajectories uniquely define the full state and inputs) exists.
2. Learning the Uncertain Dynamics
   • A Gaussian Process (GP) models the residual dynamics that are not captured by the known nominal model. The GP provides mean predictions and uncertainty estimates used for safety.
3. Flat‑Space MPC Formulation
   • By expressing the control problem in the flat output space, the nonlinear dynamics become linear in the flat coordinates, turning the MPC into a quadratic program (QP) with a block‑diagonal cost matrix.
4. Sequential Convex Optimizations
   • At each time step, two convex programs are solved:
     a. A certainty‑equivalent QP that computes a nominal trajectory ignoring uncertainty.
     b. A robustification QP that tightens constraints based on GP variance to ensure probabilistic safety.
5. Constraint Enforcement
   • Input limits are directly imposed on the control variables. Flat‑state half‑space constraints (e.g., staying within a corridor) are enforced via linear inequalities in the flat space.
6. Lyapunov‑Based Safety Check
   • The authors prove that, under the GP uncertainty model, the expected value of a chosen Lyapunov function decreases, guaranteeing stability in a probabilistic sense.

Results & Findings

• Tracking performance: The flat‑MPC tracks reference trajectories almost as accurately as the full GP‑MPC.
• Computation time: Solving the two small QPs takes only a few milliseconds on an embedded processor, compared to tens of milliseconds for the full GP‑MPC.
• Safety: Both input saturation and flat‑state constraints are respected throughout the experiments, confirming the probabilistic Lyapunov guarantee in practice.

Practical Implications

• Real‑time deployment on resource‑constrained platforms (e.g., drones, mobile manipulators) becomes feasible without sacrificing safety or performance.
• Simplified controller design: Engineers can reuse existing flatness analyses of their robots and plug in the learning‑based MPC with minimal retuning.
• Scalable to high‑DOF systems: Because the optimization scales linearly with the number of flat outputs, even complex manipulators can benefit from fast MPC updates.
• Hybrid model‑learning pipelines: The approach shows a concrete pathway to combine physics‑based models (the nominal part) with data‑driven residuals, reducing the amount of data needed for accurate control.
• Potential for autonomous fleets: Fast, constraint‑aware MPC can be embedded in fleet‑level motion planners where each vehicle must react quickly to dynamic environments while respecting safety envelopes.

Limitations & Future Work

• Flatness requirement: The method hinges on the existence (or construction) of a flat output; systems lacking this property cannot directly benefit.
• GP scalability: Although the control problem is cheap, the GP regression still scales cubically with the number of training points, which may become a bottleneck for long‑term learning. Sparse GP or neural‑network surrogates are suggested as remedies.
• Half‑space constraints only: The current formulation handles linear (half‑space) constraints on flat states; extending to arbitrary nonlinear state constraints remains an open challenge.
• Robustness to model mismatch: The theoretical guarantees assume the GP accurately captures the residual dynamics; large unmodeled disturbances could degrade stability. Future work aims to integrate robust tube‑based MPC or adaptive uncertainty bounds.

Authors

• Tobias A. Farger
• Adam W. Hall
• Angela P. Schoellig

Paper Information

• arXiv ID: 2604.24706v1
• Categories: eess.SY, cs.LG, cs.RO
• Published: April 27, 2026
• PDF: Download PDF