# Modeling Dual-Drive Gantry Stages with Heavy-Load and Optimal Synchronous Controls with Force-Feed-Forward Decoupling

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Physical Modeling of Heavy-Load Dual-Drive Gantry System

#### 2.1.1. Equivalent Dynamic Model of the System

**Remark**

**1.**

#### 2.1.2. Full State-Space Equation of the System

#### 2.1.3. Validity of the Established Model

#### 2.2. Virtual-Centroid-Based GSLQR Optimal Control and Force-FF Decoupling Control Algorithm Design

#### 2.2.1. GSLQR Optimal Control Algorithm Design

#### 2.2.2. Virtual-Centroid-Based Force-FF Decoupling Control Algorithm Design

## 3. Simulation Experiments

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Bode diagram of model (open loop), heavy-load close ${Y}_{1}$ motor side: (1)—${G}_{11}\left(s\right)$; (2)—${G}_{22}\left(s\right)$; (3)—${G}_{12}\left(s\right)$.

**Figure 4.**Bode diagram of actual system (closed loop), heavy-load close ${Y}_{1}$ motor side: (1)—${Y}_{1}$ motor; (2)—${Y}_{2}$ motor.

**Figure 5.**Bode diagram of model (open loop), heavy-load close ${Y}_{2}$ motor side: (1)—${G}_{11}\left(s\right)$; (2)—${G}_{22}\left(s\right)$.

**Figure 6.**Bode diagram of actual system (closed loop), heavy-load close ${Y}_{2}$ motor side: (1)—${Y}_{1}$ motor; (2)—${Y}_{2}$ motor.

**Figure 7.**Bode diagram of model (open loop), heavy-load centered. (1)—${G}_{11}\left(s\right)$; (2)—${G}_{22}\left(s\right)$.

**Figure 8.**Bode diagram of actual system (closed loop), heavy-load centered. (1)—${Y}_{1}$ motor; (2)—${Y}_{2}$ motor.

**Figure 9.**Block diagram of the virtual-centroid-based GSLQR optimal control and force-FF decoupling control algorithm. AMP referred to an amplifier.

**Figure 10.**Simulation result of GSLQR optimal control algorithm: (

**a**) the load ${M}_{l}$ is close to ${Y}_{1}$ motor side, ${l}_{1}=0.2$ m, ${l}_{2}=0.6$ m; (

**b**) the load ${M}_{l}$ is centered, ${l}_{1}={l}_{2}=0.4$ m; (

**c**) the load ${M}_{l}$ close to ${Y}_{2}$ motor side, ${l}_{1}=0.5$ m, ${l}_{2}=0.3$ m. (1)—speed of ${Y}_{1}$ motor; (2)—speed of ${Y}_{2}$ motor.

**Figure 11.**The synchronization error curve of each algorithm: (1)—GSLQR optimal control algorithm; (2)—thrust allocation algorithm [1]; (3)—proposed algorithm. Where heavy-load ${M}_{l}$ close to the ${Y}_{1}$ motor side, ${l}_{1}=0.2\phantom{\rule{3.33333pt}{0ex}}\mathrm{m},{l}_{2}=0.6\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$.

**Figure 12.**The synchronization error curve of each algorithm: (1)—GSLQR optimal control algorithm; (2)—thrust allocation algorithm [1]; (3)—proposed algorithm. Where the heavy-load ${M}_{l}$ close to the ${Y}_{2}$ motor side, ${l}_{1}=0.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m},{l}_{2}=0.3\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$.

Name | Symbol | Value |
---|---|---|

midrule mass of crossbeam (including load ${M}_{l}$) | M | 25 kg |

mass of load | ${M}_{l}$ | 10 kg |

length of crossbeam | L | 0.8 m |

damping of ${Y}_{1}$ rotor | ${b}_{1}$ | 5 N·m·s |

damping of ${Y}_{2}$ rotor | ${b}_{2}$ | 5 N·m·s |

stiffness of joint between crossbeam and rails | ${K}_{\alpha}$ | 52,520 N/m |

thrust constant of ${Y}_{1}$ Motor | ${k}_{t1}$ | 61.0 N/A |

thrust constant of ${Y}_{2}$ Motor | ${k}_{t2}$ | 61.0 N/A |

back EMF constant of ${Y}_{1}$ motor | ${k}_{e1}$ | 49.6 V/M/S |

back EMF constant of ${Y}_{2}$ motor | ${k}_{e2}$ | 49.6 V/M/S |

inductance of ${Y}_{1}$ Motor | ${L}_{a1}$ | $5.07\times {10}^{-3}$ H |

inductance of ${Y}_{2}$ Motor | ${L}_{a2}$ | $5.07\times {10}^{-3}$ H |

resistance of ${Y}_{1}$ Motor | ${R}_{a1}$ | 8.4 $\Omega $ |

resistance of ${Y}_{2}$ Motor | ${R}_{a2}$ | 8.4 $\Omega $ |

**Table 2.**The heavy-load ${M}_{l}$ close to the ${Y}_{1}$ motor side, the maximum value of the synchronization error of each algorithm: (1)—GSLQR optimal control algorithm; (2)—thrust allocation algorithm [1]; (3)—proposed algorithm.

Index | Algo (1) | Algo (2) | Algo (3) |
---|---|---|---|

$max\left(\left|{y}_{1}-{y}_{2}\right|\right)$, mm | 0.76 | 0.54 | 0.22 |

**Table 3.**The heavy-load ${M}_{l}$ close to the ${Y}_{2}$ motor side, the maximum value of the synchronization error of each algorithm: (1)—GSLQR optimal control algorithm; (2)—thrust allocation algorithm [1]; (3)—proposed algorithm.

Index | Algo (1) | Algo (2) | Algo (3) |
---|---|---|---|

$max\left(\left|{y}_{1}-{y}_{2}\right|\right)$, mm | 0.39 | 0.26 | 0.11 |

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**MDPI and ACS Style**

Xie, H.; Wang, Q.
Modeling Dual-Drive Gantry Stages with Heavy-Load and Optimal Synchronous Controls with Force-Feed-Forward Decoupling. *Entropy* **2022**, *24*, 1153.
https://doi.org/10.3390/e24081153

**AMA Style**

Xie H, Wang Q.
Modeling Dual-Drive Gantry Stages with Heavy-Load and Optimal Synchronous Controls with Force-Feed-Forward Decoupling. *Entropy*. 2022; 24(8):1153.
https://doi.org/10.3390/e24081153

**Chicago/Turabian Style**

Xie, Hanjun, and Qinruo Wang.
2022. "Modeling Dual-Drive Gantry Stages with Heavy-Load and Optimal Synchronous Controls with Force-Feed-Forward Decoupling" *Entropy* 24, no. 8: 1153.
https://doi.org/10.3390/e24081153