# A Near-Hover Adaptive Attitude Control Strategy of a Ducted Fan Micro Aerial Vehicle with Actuator Dynamics

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Attitude Control Problem Statement of the Prototype Ducted Fan MAV

## 3. The Adaptive Attitude Control Strategy of the MAV with Actuator Dynamics

#### 3.1. The Online Parameter Estimation Method for the MAV

#### 3.2. The Adaptive Gain Scheduling Algorithm

## 4. Numerical and Experimental Tests

#### 4.1. Numerical Simulation

**Figure 3.**Numerical simulation results of the proposed strategy. (

**a**) ${\mathrm{\theta}}_{c}$ and $\mathrm{\theta}$; (

**b**) ${q}_{m}$ and $q$; (

**c**) $e$; (

**d**) $\varphi $; (

**e**) $p$; (

**f**), (

**g**) and (

**h**) Identification parameters.

#### 4.2. Near-hover Flight Tests

**Figure 5.**Test results of the proposed strategy. (

**a**) $\mathrm{\theta}$ and $\varphi $; (

**b**) ${q}_{m}$ and $q$; (

**c**) $e$; (

**d**); (

**e**) and (

**f**) Identification parameters; (

**g**) The ground track view of trajectory.

## 5. Conclusions

**Figure 6.**Test results of the conventional controller. (

**a**) $\mathrm{\theta}$; (

**b**) $q$; (

**c**) $\varphi $; (

**d**) $p$; (

**e**) The ground track view of trajectory.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

${a}_{q},{a}_{\dot{q}},{b}_{u},{b}_{w},{b}_{p},{b}_{{\mathrm{\delta}}_{pit}},{b}_{{\mathrm{\delta}}_{rol}},{b}_{{\mathrm{\delta}}_{rot}},{b}_{e}$ | identification parameters |

${a}_{q}^{\ast},{a}_{\dot{q}}^{\ast},{b}_{{\mathrm{\delta}}_{pit}}^{\ast}$ | desired model parameters |

$A,B$ | system matrix and control matrix |

$e$ | tracking error |

${e}_{\mathrm{\theta}}$ | attitude tracking error |

$E$ | unknown equivalent trim error vector |

$f$ | nonlinear kinematic function |

${G}_{a}$ | actuator transfer function |

${G}_{f},{G}_{f1},{G}_{f2}$ | filter transfer functions |

${k}_{p},{k}_{d}$ | proportional and differential coefficients |

${k}_{{q}_{0}}$ | predetermined feedback gain |

$K$ | identification parameter vector |

${M}_{u}^{\dot{q}},{M}_{v}^{\dot{q}},{M}_{w}^{\dot{q}},{M}_{\mathrm{\theta}}^{\dot{q}},{M}_{\varphi}^{\dot{q}},{M}_{\psi}^{\dot{q}},{M}_{q}^{\dot{q}},{M}_{p}^{\dot{q}},{M}_{r}^{\dot{q}},{M}_{{\mathrm{\delta}}_{pit}}^{\dot{q}},{M}_{{\mathrm{\delta}}_{rol}}^{\dot{q}},{M}_{{\mathrm{\delta}}_{yaw}}^{\dot{q}},{M}_{{\mathrm{\delta}}_{rot}}^{\dot{q}}$ | aerodynamic parameters |

$p,q,r$ | linearized roll, pitch and yaw rates, deg/s |

$\stackrel{\u2322}{p},\stackrel{\u2322}{q},\stackrel{\u2322}{r}$ | roll, pitch and yaw rates, deg/s |

${q}_{m}$ | identification model output |

${R}_{f}$ | filtered state and input vector |

${T}_{a},{T}_{f1},{T}_{f2}$ | time constants of ${G}_{a}$, ${G}_{f1}$ and ${G}_{f2}$, respectively |

$\stackrel{\u2322}{U},U$ | input vector and its linearized version |

${U}_{e},{\tilde{U}}_{e}$ | trim input vector and its estimate error |

${U}_{0}$ | nominal trim input vector |

$u,v,w$ | linearized forward, lateral and vertical velocities, m/s |

$\stackrel{\u2322}{u},\stackrel{\u2322}{v},\stackrel{\u2322}{w}$ | forward, lateral and vertical velocities, m/s |

$V$ | Lyapunov function candidate |

$\stackrel{\u2322}{X},X$ | state vector and its linearized version |

${X}_{0}$ | nominal trim state vector |

$\mathrm{\beta}$ | optional positive number |

$\Delta {a}_{q},\Delta {a}_{\dot{q}},\Delta {b}_{{\mathrm{\delta}}_{pit}},\Delta {b}_{u},\Delta {b}_{w},\Delta {b}_{p},\Delta {b}_{{\mathrm{\delta}}_{rol}},\Delta {b}_{{\mathrm{\delta}}_{rot}},\Delta {b}_{e}$ | identification parameter errors |

$\Delta $ | identification parameter error vector |

${\mathrm{\delta}}_{e}$ | equivalent input disturbance |

${\stackrel{\u2322}{\mathrm{\delta}}}_{l},{\stackrel{\u2322}{\mathrm{\delta}}}_{r},{\stackrel{\u2322}{\mathrm{\delta}}}_{f},{\stackrel{\u2322}{\mathrm{\delta}}}_{b}$ | movements of the left, right, front and back servo motors, respectively, deg |

${\mathrm{\delta}}_{pit},{\mathrm{\delta}}_{rol},{\mathrm{\delta}}_{yaw},{\mathrm{\delta}}_{rot}$ | linearized manipulated input signals of pitch axis, roll axis, yaw axis and rotational speed of the ducted fan, respectively, deg |

${\stackrel{\u2322}{\mathrm{\delta}}}_{pit},{\stackrel{\u2322}{\mathrm{\delta}}}_{rol},{\stackrel{\u2322}{\mathrm{\delta}}}_{yaw},{\stackrel{\u2322}{\mathrm{\delta}}}_{rot}$ | manipulated input signals of pitch axis, roll axis, yaw axis and rotational speed of the ducted fan, respectively, deg |

${\mathrm{\delta}}_{piti}$ | input signal of actuator |

${\mathrm{\delta}}_{pitc}$ | attitude control signal |

$\Gamma $ | ${\Lambda}^{-1}$ |

${\mathrm{\epsilon}}_{f}$ | optional component part of learning laws |

$\Lambda $ | optional positive definite diagonal matrix |

${\mathrm{\lambda}}_{\dot{q}},{\mathrm{\lambda}}_{q},{\mathrm{\lambda}}_{{\mathrm{\delta}}_{pit}},{\mathrm{\lambda}}_{u},{\mathrm{\lambda}}_{w},{\mathrm{\lambda}}_{p},{\mathrm{\lambda}}_{{\mathrm{\delta}}_{rol}},{\mathrm{\lambda}}_{{\mathrm{\delta}}_{rot}},{\mathrm{\lambda}}_{e}$ | diagonal elements of $\Lambda $ |

$\mathrm{\Theta}$ | unknown aerodynamic parameter set |

${X}_{e},{\tilde{X}}_{e}$ | trim state vector and its estimate error |

$\Upsilon $ | optional positive number |

$\mathrm{\theta},\varphi ,\psi $ | linearized pitch, roll and yaw angles, deg |

$\stackrel{\u2322}{\mathrm{\theta}},\stackrel{\u2322}{\varphi},\stackrel{\u2322}{\psi}$ | pitch, roll and yaw angles, deg |

${\mathrm{\theta}}_{c}$ | pitch command, deg |

${\mathrm{\rho}}_{\dot{q}},{\mathrm{\rho}}_{q},{\mathrm{\rho}}_{{\mathrm{\delta}}_{pit}},{\mathrm{\rho}}_{u},{\mathrm{\rho}}_{w},{\mathrm{\rho}}_{p},{\mathrm{\rho}}_{{\mathrm{\delta}}_{rol}},{\mathrm{\rho}}_{{\mathrm{\delta}}_{rot}},{\mathrm{\rho}}_{e}$ | diagonal elements of $\Gamma $ |

${\mathrm{\tau}}_{e}^{\dot{q}}$ | element of $E$ |

${\Vert \Vert}_{\infty}$ | infinite-norm |

${p}_{d},{u}_{d},{w}_{d},{\mathrm{\delta}}_{rold},{\mathrm{\delta}}_{rotd}$ | outputs of ${G}_{a}^{-1}$ in response to $p,u,w$,${\mathrm{\delta}}_{rol},{\mathrm{\delta}}_{rot}$, respectively |

${e}_{f},{p}_{df},{q}_{f},{u}_{df},{w}_{df},{\mathrm{\delta}}_{pitif},{\mathrm{\delta}}_{roldf},{\mathrm{\delta}}_{rotdf}$ | outputs of ${G}_{{}_{f}}$ in response to $e,{p}_{d},q,{u}_{d}$,${w}_{d},{\mathrm{\delta}}_{piti},{\mathrm{\delta}}_{rold},{\mathrm{\delta}}_{rotd}$, respectively |

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**MDPI and ACS Style**

Sheng, S.; Sun, C. A Near-Hover Adaptive Attitude Control Strategy of a Ducted Fan Micro Aerial Vehicle with Actuator Dynamics. *Appl. Sci.* **2015**, *5*, 666-681.
https://doi.org/10.3390/app5040666

**AMA Style**

Sheng S, Sun C. A Near-Hover Adaptive Attitude Control Strategy of a Ducted Fan Micro Aerial Vehicle with Actuator Dynamics. *Applied Sciences*. 2015; 5(4):666-681.
https://doi.org/10.3390/app5040666

**Chicago/Turabian Style**

Sheng, Shouzhao, and Chenwu Sun. 2015. "A Near-Hover Adaptive Attitude Control Strategy of a Ducted Fan Micro Aerial Vehicle with Actuator Dynamics" *Applied Sciences* 5, no. 4: 666-681.
https://doi.org/10.3390/app5040666