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A flatness-based saturated controller design for a quadcopter with experimental validation

Huu-Thinh Do, Franco Blanchini, Ionela Prodan

TL;DR

The paper addresses enforcing input constraints for quadcopter control within a differential flatness framework by developing a flatness-based saturated controller that operates in the linear flat-output space while keeping the original inputs feasible. Stability and constraint satisfaction are ensured via Lyapunov analysis and an ellipsoidal invariant set, supported by an explicit, low-complexity saturation rule derived from a finite candidate set. The method is validated through simulations and Crazyflie 2.1 experiments, achieving tracking accuracy under 10 cm RMS and fast computation times suitable for embedded deployment. Overall, the approach offers a computationally efficient, provably correct control strategy for constrained, differentially flat systems with practical applicability to quadcopters and other platforms.

Abstract

Using the properties of differential flatness, a controllable system, such as a quadcoper model, may be transformed into a linear equivalent system via a coordinate change and an input mapping. This is a straightforward advantage for the quadcopter's controller design and its real-time implementation. However, one significant hindrance is that, while the dynamics become linear in the new coordinates (the flat output space), the input constraints become convoluted. This paper addresses an explicit pre-stabilization based control scheme which handles the input constraints for the quadcopter in the flat output space with a saturation component. The system's stability is shown to hold by Lyapunov-stability arguments. Moreover, the practical viability of the proposed method is validated both in simulation and experiments over a nano-drone platform. Hence, the flatness-based saturated controller not only ensures stability and constraints satisfaction, but also requires very low computational effort, allowing for embedded implementations.

A flatness-based saturated controller design for a quadcopter with experimental validation

TL;DR

The paper addresses enforcing input constraints for quadcopter control within a differential flatness framework by developing a flatness-based saturated controller that operates in the linear flat-output space while keeping the original inputs feasible. Stability and constraint satisfaction are ensured via Lyapunov analysis and an ellipsoidal invariant set, supported by an explicit, low-complexity saturation rule derived from a finite candidate set. The method is validated through simulations and Crazyflie 2.1 experiments, achieving tracking accuracy under 10 cm RMS and fast computation times suitable for embedded deployment. Overall, the approach offers a computationally efficient, provably correct control strategy for constrained, differentially flat systems with practical applicability to quadcopters and other platforms.

Abstract

Using the properties of differential flatness, a controllable system, such as a quadcoper model, may be transformed into a linear equivalent system via a coordinate change and an input mapping. This is a straightforward advantage for the quadcopter's controller design and its real-time implementation. However, one significant hindrance is that, while the dynamics become linear in the new coordinates (the flat output space), the input constraints become convoluted. This paper addresses an explicit pre-stabilization based control scheme which handles the input constraints for the quadcopter in the flat output space with a saturation component. The system's stability is shown to hold by Lyapunov-stability arguments. Moreover, the practical viability of the proposed method is validated both in simulation and experiments over a nano-drone platform. Hence, the flatness-based saturated controller not only ensures stability and constraints satisfaction, but also requires very low computational effort, allowing for embedded implementations.
Paper Structure (13 sections, 5 theorems, 48 equations, 16 figures, 3 tables, 1 algorithm)

This paper contains 13 sections, 5 theorems, 48 equations, 16 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model representation and prerequisites
  3. Quadcopter's model and its characterization in the flat output space
  4. Input constraints description in the flat output space
  5. Invariance and Lyapunov-based tools
  6. Explicit saturated control design
  7. Controller characterization
  8. Invariant set characterization
  9. Implementation algorithm for the saturation function and simulation study
  10. Experimental validation
  11. Experimental setup and scenarios
  12. Experimental results and discussion
  13. Conclusion

Key Result

Proposition 1

(Nagumo's invariance principle blanchini2008set) Consider a dynamical system described as: with $\boldsymbol{\xi}\in\mathbb{R}^n,\boldsymbol{v}\in\mathbb{R}^m$ representing the state and the input vector, respectively. Then, the ellipsoid set $\mathcal{E}=\{\boldsymbol{\xi}\in\mathbb{R}^n:\boldsymbol{\xi}^\top P\boldsymbol{\xi} \leq \varepsilon\}$ with $P\succ 0, \varepsilon>0$ is an inva where

Figures (16)

  • Figure 1: Flatness-based control scheme for the quadcopter.
  • Figure 2: Quadcopter's input constraints in the flat output space with $\psi=0\;(rad)$, $g={T}_{max}/1.45= 9.81\; (m/s^2)$, ${\phi}_{max}={\theta}_{max}=\pi/18 \;(rad)$).
  • Figure 3: Numerical illustration for the solution of \ref{['eq:ball_in_Vc_rho']} - a ball maximally inscribed in $\mathcal{V}_c$.
  • Figure 4: The projection of $\mathcal{B}_P(\varepsilon)$ into the subspace $(x,\dot x)$ with different choices of $\alpha$.
  • Figure 5: The quadcopter's trajectory with different initial states (black points) implemented with different values of $\gamma$ (projected into $(x,\dot x)$ subspace).
  • ...and 11 more figures

Theorems & Definitions (7)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Remark 1
  • Remark 2
  • Proposition 5