# A Comparative Study between NMPC and Baseline Feedback Controllers for UAV Trajectory Tracking

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## Abstract

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## 1. Introduction

#### 1.1. Motivation

#### 1.2. Related Work

#### 1.2.1. Non-Predictive Controllers

#### 1.2.2. Predictive Controllers

#### 1.3. Contribution of This Work

- Differential Kinematics and Dynamic Formulation: this work develops the differential kinematics and dynamics formulation considering maneuverability velocities as the control input of the system because many commercial aerial vehicles are produced under this structure where the user does not have access to attitude control schemes. Thus, this formulation can be straightforwardly applied to another commercial platform with desired velocities as inputs of the system. To obtain the dynamic model of the system, this work used the formulation of Euler–Lagrange; additionally, to modify this model into the velocities space, it is necessary to introduce the concept of low-level PID in the mathematical formulation.
- Nonlinear Feedback Formulations: are generated by inversion of differential kinematics and dynamic compensation techniques. The design of these control structures was developed using the theory of nonlinear control and linear algebra, guaranteeing minimal computational time with asymptotically and robust behavior.
- NMPC Formulation: This work formulates the NMPC considering system and input constraints such as system dynamics, limits in control action, rate of change of control inputs, and a candidate Lyapunov function that guarantees the stability of the control structure. This work uses CasADI to solve the nonlinear optimization problem.
- Comparative Study: The experiments are carried out in simulations and real-world environments. The simulation experiments were performed under the commercial simulator of the DJI brand. On the other hand, real-world experiments were conducted using the DJI Matrice 100 platform, which was equipped with all the necessary hardware to run the controllers on the onboard computer. The agile trajectory selected to compare the performances of the controllers is the Lissajous; additionally, to guarantee a good comparison, this work considers low and high velocities through the reference signal where the aerodynamics effects, latency, and uncertainties are inevitable. Finally, with the comparative results is possible to show the benefits of each controller considering their predictive and non-predictive structures; the comparative consider the following metrics: tracking accuracy and average computational time.

## 2. Preliminary Materials

#### 2.1. Kinematics Formulation

#### 2.2. Quadrotor Dynamics

#### 2.3. Simplified Dynamic Model Based on Maneuvering Velocities

#### 2.4. Identification and Validation

## 3. Controllers Design

#### 3.1. Kinematic Controller

#### Kinematic Controller Stability Analysis

#### 3.2. Dynamic Controller

#### Dynamic Controller Stability Analysis

#### 3.3. NMPC Controller

## 4. Results

#### 4.1. Simulation Experiments Results

#### 4.1.1. Inverse Differential Kinematics

#### 4.1.2. Inverse Dynamic Compensation

#### 4.1.3. Nonlinear Model Predictive Control

#### 4.2. Comparative of Simulations Experiments

#### 4.3. Real-World Experiments Results

#### 4.3.1. Inverse Differential Kinematics

#### 4.3.2. Inverse Dynamic Compensation

#### 4.3.3. Nonlinear Model Predictive Control

#### 4.4. Comparative of Real-World Experiments

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Identification and validation, (

**a**) shows the identification process under step signals, and (

**b**) presents the validation of the proposed dynamic model considering smooth signals.

**Figure 6.**Simulation tracking results considering a Lissajous trajectory using the inverse differential kinematic controller; where (

**a**) shows the evolution of the system in the plane $\left\{{\mathbf{I}}_{x},{\mathbf{I}}_{y}\right\}$ and (

**b**) considering the plane $\left\{{\mathbf{I}}_{x},{\mathbf{I}}_{z}\right\}$; finally (

**c**,

**d**) illustrate the control error and actions, respectively.

**Figure 7.**Simulation tracking results considering a Lissajous trajectory using the inverse dynamic compensation controller; where (

**a**) shows the evolution of the system in the plane $\left\{{\mathbf{I}}_{x},{\mathbf{I}}_{y}\right\}$ and (

**b**) over the plane $\left\{{\mathbf{I}}_{x},{\mathbf{I}}_{z}\right\}$; finally (

**c**,

**d**) represent the control error and actions, respectively.

**Figure 8.**Simulation tracking results considering a Lissajous trajectory using the NMPC controller; where (

**a**) shows the evolution of the system in the plane $\left\{{\mathbf{I}}_{x},{\mathbf{I}}_{y}\right\}$ and (

**b**) over the plane $\left\{{\mathbf{I}}_{x},{\mathbf{I}}_{z}\right\}$; finally (

**c**,

**d**) represent the control error and actions, respectively.

**Figure 9.**Comparative results of the proposed controllers during the simulation experiments; where (

**a**) illustrates the norm of the control error during the simulation of each controller (

**b**) shows the RMSE of each controller along the simulation.

**Figure 11.**Real-world tracking results considering the inverse differential kinematic controller; where (

**a**,

**b**) show the evolution of the system; finally (

**c**,

**d**) represent the control error and actions, respectively.

**Figure 12.**Real-world tracking results considering the inverse dynamic compensation; where (

**a**,

**b**) show the evolution of the system; finally (

**c**,

**d**) represent the control error and actions.

**Figure 13.**Real-world tracking results considering the NMPC controller; where (

**a**,

**b**) show the evolution of the system; finally (

**c**,

**d**) represent the control error and actions, respectively.

**Figure 15.**Comparative results of the proposed controllers during the real-world experiments; where (

**a**) illustrates the Euclidean norm of the control error during the execution of each controller (

**b**) shows the RMSE of each controller.

Parameters | |||
---|---|---|---|

${\pi}_{1}=2.11$ | ${\pi}_{2}=-0.005$ | ${\pi}_{3}=1.8$ | ${\pi}_{4}=3.17$ |

${\pi}_{5}=1.78$ | ${\pi}_{6}=0.39$ | ${\pi}_{7}=-0.003$ | ${\pi}_{8}=-0.03$ |

${\pi}_{9}=0.006$ | ${\pi}_{10}=0.02$ | ${\pi}_{11}=0.002$ | ${\pi}_{12}=0.06$ |

${\pi}_{13}=0.70$ | ${\pi}_{14}=0.02$ | ${\pi}_{15}=-0.05$ | ${\pi}_{16}=-0.01$ |

${\pi}_{17}=-0.005$ | ${\pi}_{18}=-0.01$ |

Initial Positions [m]-[rad] | Initial Velocities [m/s]-[rad/s] | Reference Trajectory [m]-[rad] |
---|---|---|

${\eta}_{{x}_{o}}=0\phantom{\rule{3.33333pt}{0ex}}$ | ${\mu}_{{l}_{o}}=0$ | ${\eta}_{{x}_{d}}=3sin\left(0.4t\right)+3$ |

${\eta}_{{y}_{o}}=0$ | ${\mu}_{{m}_{o}}=0$ | ${\eta}_{{y}_{d}}=3sin\left(0.8t\right)$ |

${\eta}_{{z}_{o}}=1$ | ${\mu}_{{n}_{o}}=0$ | ${\eta}_{{z}_{d}}=1.5sin\left(0.2t\right)+8$ |

${\eta}_{{\psi}_{o}}=0$ | ${\omega}_{o}=0$ | ${\eta}_{{\psi}_{d}}=0$ |

Parameters | Values | Parameters | Values |
---|---|---|---|

${\mathbf{K}}_{1}$ | $diag\left(1\right)\in {\mathbb{R}}^{4\times 4}$ | ${\mathbf{K}}_{2}$ | $diag\left(2\phantom{\rule{3.33333pt}{0ex}}2\phantom{\rule{3.33333pt}{0ex}}2\phantom{\rule{3.33333pt}{0ex}}10\right)\in {\mathbb{R}}^{4\times 4}$ |

Parameters | Values | Parameters | Values |
---|---|---|---|

${\mathbf{K}}_{3}$ | $diag\left(1\right)\in {\mathbb{R}}^{4\times 4}$ | ${\mathbf{K}}_{4}$ | $diag\left(1\right)\in {\mathbb{R}}^{4\times 4}$ |

Parameters | Values | Parameters | Values |
---|---|---|---|

$\mathbf{Q}$ | $diag\left(0.9\right)\in {\mathbb{R}}^{4\times 4}$ | $\mathbf{R}$ | $diag\left(0.1\right)\in {\mathbb{R}}^{4\times 4}$ |

${\mathbf{\mu}}_{{\mathbf{ref}}_{min}}$ | $-[2.4,2.4,2.4,11.5]$ | ${\mathbf{\mu}}_{{\mathbf{ref}}_{max}}$ | $[2.4,2.4,2.4,11.5]$ |

$T\left[s\right]$ | $1\phantom{\rule{3.33333pt}{0ex}}$ |

Kinematic | Dynamic | NMPC | |
---|---|---|---|

Avg. dt [ms] | $0.054$ | $0.141$ | $17.54$ |

Initial Positions [m]-[rad] | Initial Velocities [m/s]-[rad/s] | Reference Trajectory [m]-[rad] |
---|---|---|

${\eta}_{{x}_{o}}=2\phantom{\rule{3.33333pt}{0ex}}$ | ${\mu}_{{l}_{o}}=0$ | ${\eta}_{{x}_{d}}=3sin\left(0.16t\right)+3$ |

${\eta}_{{y}_{o}}=0$ | ${\mu}_{{m}_{o}}=0$ | ${\eta}_{{y}_{d}}=3sin\left(0.32t\right)$ |

${\eta}_{{z}_{o}}=1$ | ${\mu}_{{n}_{o}}=0$ | ${\eta}_{{z}_{d}}=1.5sin\left(0.2t\right)+8$ |

${\eta}_{{\psi}_{o}}=0$ | ${\omega}_{o}=0$ | ${\eta}_{{\psi}_{d}}=0$ |

Kinematic | Dynamic | NMPC | |
---|---|---|---|

Avg. dt [ms] | $0.075$ | $0.26$ | $25.36$ |

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## Share and Cite

**MDPI and ACS Style**

Guevara, B.S.; Recalde, L.F.; Varela-Aldás, J.; Andaluz, V.H.; Gandolfo, D.C.; Toibero, J.M.
A Comparative Study between NMPC and Baseline Feedback Controllers for UAV Trajectory Tracking. *Drones* **2023**, *7*, 144.
https://doi.org/10.3390/drones7020144

**AMA Style**

Guevara BS, Recalde LF, Varela-Aldás J, Andaluz VH, Gandolfo DC, Toibero JM.
A Comparative Study between NMPC and Baseline Feedback Controllers for UAV Trajectory Tracking. *Drones*. 2023; 7(2):144.
https://doi.org/10.3390/drones7020144

**Chicago/Turabian Style**

Guevara, Bryan S., Luis F. Recalde, José Varela-Aldás, Victor H. Andaluz, Daniel C. Gandolfo, and Juan M. Toibero.
2023. "A Comparative Study between NMPC and Baseline Feedback Controllers for UAV Trajectory Tracking" *Drones* 7, no. 2: 144.
https://doi.org/10.3390/drones7020144