# Study on the Sensitivity of a Gyroscope System Homing a Quadcopter onto a Moving Ground Target under the Action of External Disturbance

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## Abstract

**:**

## 1. Introduction

## 2. Determining Optimal Parameters for a Controlled Gyroscope System

- ${\mathit{x}}_{g}={\left[\begin{array}{cccc}{\vartheta}_{g}& {\dot{\vartheta}}_{g}& {\psi}_{g}& {\dot{\psi}}_{g}\end{array}\right]}^{T}$—state vector, ${\mathit{u}}_{g}={\left[\begin{array}{cc}{M}_{b}& {M}_{c}\end{array}\right]}^{T}$—control vector,
- ${\mathit{A}}_{g}=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& -{b}_{b}& 0& -1\\ 0& 0& 0& 1\\ 0& 1& 0& -{b}_{c}\end{array}\right]$—state matrix, ${\mathit{B}}_{g}=\left[\begin{array}{cc}0& 0\\ {c}_{b}& 0\\ 0& 0\\ 0& {c}_{c}\end{array}\right]$—control matrix,
- ${b}_{b}=\frac{{\eta}_{b}}{{J}_{gk}\Omega}$, ${b}_{c}=\frac{{\eta}_{c}}{{J}_{gk}\Omega}$, ${c}_{b}={c}_{c}=\frac{1}{{J}_{gk}{\Omega}^{2}}$, $\Omega =\frac{{J}_{go}{n}_{g}}{{J}_{gk}}$,
- ${\vartheta}_{g},{\psi}_{g}$—angles defining the position of the GS axis in space,
- ${M}_{b},{M}_{c}$—control moments,
- ${\eta}_{b},{\mathsf{\eta}}_{c}$—damping coefficients in GS frame suspension bearings,
- ${J}_{go}$—moment of inertia of a GS rotor relative to the longitudinal axis,
- ${J}_{gk}$—moment of inertia of a GS rotor relative to the transverse axis,
- ${n}_{g}$—rotary speed of the GS rotor.

## 3. Test Results

#### 3.1. Test Results Regarding the Sensitivity of a Gyroscope System during Tracking and Laser Illumination of a Ground Target

_{b}coefficient, which was ultimately selected so that the control moments did not exceed the permissible absolute values of 0.5 Nm on one hand, and the total error between the set and implemented gyroscope axis position was below 0.5 degrees (0.0087 rad) on the other. Other coefficients were determined based on the relationships (7)–(15). Numerous tests involving gyroscope system sensitivity indicated that with an optimally selected k

_{b}coefficients, other coefficients, namely ${k}_{c}$ and ${h}_{g}$, varying within 30% of the optimum values, did not cause significant errors in maintaining the gyroscope axis in accordance with the target line of sight. Errors exceeding permissible values, i.e., axis deviation from the set value higher than 0.0087 rad appeared after leaving the aforementioned change range of the coefficients. At the same time, the control moments reached unacceptable values.

#### 3.2. Simulation Studies Involving the Control over an Optimum Gyroscope System for Homing onto a Ground Target from Onboard a Quadcopter

- IAE (Integral Absolute Error) quality indicator:$$\mathrm{IAE}=\underset{0}{\overset{\infty}{{\displaystyle \int}}}\left|{e}_{c}\right|dt$$

- 2.
- ISSC (Integral Square State and Control) quality indicator:$$\mathrm{ISSC}=\underset{0}{\overset{\infty}{{\displaystyle \int}}}\left({x}^{T}x\right)dt+\underset{0}{\overset{\infty}{{\displaystyle \int}}}\left({u}^{T}u\right)dt$$$$x=\left[\begin{array}{cccc}{\vartheta}_{g}& {\psi}_{g}& {\dot{\vartheta}}_{g}& {\dot{\psi}}_{g}\end{array}\right],u=\left[\begin{array}{cc}{M}_{b}& {M}_{c}\end{array}\right].$$

**Figure 20.**Changes of realizing and pre-set deflection and inclination angles of GS as a time function.

**Figure 25.**Changes of realizing and pre-set deflection and inclination angles of GS as a time function.

**Figure 30.**Changes of realizing and pre-set deflection and inclination angles of GS as a time function.

**Figure 35.**Changes of realizing and preset deflection and inclination angles of GS as a time function.

**Figure 40.**Changes of realizing and pre-set deflection and inclination angles of GS as a time function.

**Figure 45.**Changes of realizing and preset deflection and inclination angles of GS as a time function.

## 4. Conclusions

_{b}was determined for the controlled gyroscope system in question. The GS is very sensitive to changes in this coefficient. Other coefficients, ${k}_{c}$ and ${h}_{g}$, are functions of ${k}_{b}$ and the GS is not really sensitive to changes in their values, since they can vary by up to 30% of optimum values without a significant impact on the precision of ground target tracking and laser illumination.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Graph of optimal inter-relations between the regulator damping coefficient ${h}_{g}$, angular velocity ${n}_{g}$ and the gain coefficient ${k}_{b}$.

**Figure 2.**Graph of optimal inter-relations between the regulator gain coefficient ${k}_{c}$, angular velocity ${n}_{g}$ and the gain coefficient ${k}_{b}$.

**Figure 3.**Real and set angle of gyroscope system (GS) deflection and inclination as a function of time.

Variant | Regulator Parameters | ISSC | IAE |
---|---|---|---|

1 | ${k}_{b}=10,{k}_{c}=10,{h}_{g}=10$ | 9.2403 × 10^{8} | 5.3726 × 10^{3} |

2 | ${k}_{b}=10,{k}_{c}=100,{h}_{g}=100$ | 9.6861 × 10^{8} | 7.2331 × 10^{3} |

3 | ${k}_{b}=10,{k}_{c},{h}_{g}$ − optimum | 1.1070 × 10^{9} | 4.3352 × 10^{3} |

4 | ${k}_{b}=348,{k}_{c},{h}_{g}$ − optimum | 1.6691 × 10^{9} | 1.7290 × 10^{3} |

5 | ${k}_{b}=348,{k}_{c},{h}_{g}=0.7\ast \mathrm{optimum}$ | 1.6845 × 10^{9} | 1.8137 × 10^{3} |

6 | ${k}_{b}=348,{k}_{c},{h}_{g}=1.3\ast \mathrm{optimum}$ | 1.6263 × 10^{9} | 1.9237 × 10^{3} |

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**MDPI and ACS Style**

Krzysztofik, I.; Koruba, Z.
Study on the Sensitivity of a Gyroscope System Homing a Quadcopter onto a Moving Ground Target under the Action of External Disturbance. *Energies* **2021**, *14*, 1696.
https://doi.org/10.3390/en14061696

**AMA Style**

Krzysztofik I, Koruba Z.
Study on the Sensitivity of a Gyroscope System Homing a Quadcopter onto a Moving Ground Target under the Action of External Disturbance. *Energies*. 2021; 14(6):1696.
https://doi.org/10.3390/en14061696

**Chicago/Turabian Style**

Krzysztofik, Izabela, and Zbigniew Koruba.
2021. "Study on the Sensitivity of a Gyroscope System Homing a Quadcopter onto a Moving Ground Target under the Action of External Disturbance" *Energies* 14, no. 6: 1696.
https://doi.org/10.3390/en14061696