# Hybrid Multi-Domain Analytical and Data-Driven Modeling for Feed Systems in Machine Tools

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## Abstract

**:**

## 1. Introduction

## 2. Multi-Domain Analytical Model of the Ball Screw Feed System

#### 2.1. Multi-Domain Modeling Method for the Ball Screw Feed System

#### 2.2. Modeling and Analysis of the Mechanical Subsystem

#### 2.3. Modeling of the Servo Drive System

#### 2.4. Modeling of the Permanent Magnet Synchronous Motor

- (1)
- The conductivity of permanent magnet material is zero;
- (2)
- There is no damping winding on the rotor;
- (3)
- Stator winding current produces only sine distribution of magnetic potential in the air gap, ignoring the high-order harmonic of magnetic field;
- (4)
- During the steady-state operation, the waveform of induction electromotive force in phase winding is sinusoidal.

## 3. The Learned Data-Driven Model

## 4. Hybrid Model of the Ball Screw Feed System

## 5. Simulation and Experiments

#### 5.1. Single Axis Ball Screw Feed System Hybrid Model

#### 5.2. Double Axis Ball Screw Feed System Hybrid Model

## 6. Conclusions

- A hybrid multi-domain analytical and data-driven modeling method was proposed in this paper and a hybrid model of a ball screw feed drive system was established accurately using the hybrid modeling method.
- In contrast to the traditional causal modeling method based on signal flow, the multi-domain integrated analytical model of the ball screw feed system was established using the non-causal modeling method based on energy flow. The analytical model of a feed system developed in this paper realized seamless integrated modeling of a complicated multi-domain system.
- A data-driven error model based on a BP neural network was established and the model was trained using experimental data. Then the learned data-driven error model was coupled with the analytical model of the ball screw feed system and the hybrid model was obtained.
- The hybrid model was validated using experimental data at different speeds, and the results show that, whether for the tracking error of a single-axis feeding system or the contour error of a double-axis feeding system, the prediction effect of the hybrid model is better than that of a pure analytical model, and the prediction accuracy of the hybrid model reaches a higher level.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Module division of the ball screw feed system driven by a permanent magnet synchronous motor (PMSM).

**Figure 6.**Illustration for the modeling of space vector pulse width modulation (SVPWM) and inverter.

**Figure 7.**Equivalent circuit of PMSM. (

**a**) Equivalent circuit of d axis; (

**b**) Equivalent circuit of q axis.

**Figure 8.**Training data for the back propagation neural network (BPNN). (

**a**) Input data of position command; (

**b**) Input data of analytical simulated position; (

**c**) Output data of deviation.

**Figure 14.**Comparison of the pure analytical model simulation results and the hybrid model simulation results at different feed speed.

**Figure 15.**Comparisons between simulation results and experimental results of circular trajectory tracking at different velocity.

**Figure 16.**Comparison of contour error predicted by pure analytical model and hybrid model at different velocities.

Mechanical Connectors | Electrical Connectors | Control Connectors | |||
---|---|---|---|---|---|

Icon | ⬤ | Icon | ⬤ | Icon | |

Flow variable | Torque τ | Flow variable | Current i | Input variable u | ▶ |

Potential variable | Angle φ | Potential variable | Voltage v | Output variable y | ▷ |

Name of Parameters | Value |
---|---|

Nominal torque of the motor | 18 N m |

Nominal speed of the motor | 2000 rpm |

Nominal current of the motor | 12.5 A |

Nominal power of the power | 3.6 Kw |

Inertia of the motor rotor | 5.3 $\times {10}^{-3}\mathrm{Kg}\cdot {\mathrm{m}}^{2}$ |

Pole-pair number of the motor | 4 |

Equivalent resistance of phase winding | 0.184$\text{}\mathsf{\Omega}$ |

$\mathrm{d}$ axis inductance of the motor ${L}_{d}$ | 2.3 $\times {10}^{-3}$ H |

$\mathrm{q}$ axis inductance of the motor ${L}_{q}$ | 2.3 $\times {10}^{-3\text{}}$ H |

Control cycle of position loop | 0.001 s |

Position loop gain | 120 Hz |

Control cycle of velocity loop | 0.125 ms |

Velocity loop gain | 800 Hz |

Integral time constant of velocity loop | 40 ms |

Control cycle of current loop | 31.25 us |

Current loop gain | 5000 Hz |

Integral time constant of current loop | 9.8 ms |

DC-side voltage of inverter | 560 V |

Pitch of the ball screw | 16 mm |

backlash | 17 um |

Feed Rate (mm/min) | F1000 | F2000 | F3000 | F4000 | F5000 | F6000 | |
---|---|---|---|---|---|---|---|

Hybrid model | (MAE)/$\mathsf{\mu}\mathrm{m}$ | 5.22 | 5.16 | 5.98 | 5.27 | 4.55 | 6.79 |

(RMSE)/$\mathsf{\mu}\mathrm{m}$ | 1.87 | 2.33 | 1.72 | 1.41 | 1.64 | 2.63 | |

(RE) | 2.7% | 2.0% | 1.1% | 0.7% | 0.5% | 0.6% | |

Analytical model | (MAE)/$\mathsf{\mu}\mathrm{m}$ | 7.99 | 7.52 | 7.70 | 8.51 | 10.2 | 12.38 |

(RMSE)/$\mathsf{\mu}\mathrm{m}$ | 2.91 | 2.83 | 3.15 | 3.84 | 4.89 | 6.21 | |

(RE) | 4.2% | 4.0% | 1.4% | 1.1% | 1.1% | 1.2% |

**Table 4.**Comparison of maximum contour error predicted by analytical model and hybrid model at different velocity.

$\mathsf{\omega}\text{}(\mathbf{rad}/\mathbf{min})$ | 60 | 80 | 100 | 120 |
---|---|---|---|---|

Max Error (Analytical model predicted)/$\mathsf{\mu}\mathrm{m}$ | 7.19 | 8.62 | 10.26 | 12.54 |

Max Error (Hybrid model predicted)/$\mathsf{\mu}\mathrm{m}$ | 6.57 | 5.34 | 4.10 | 6.95 |

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**MDPI and ACS Style**

Mei, Z.; Ding, J.; Chen, L.; Pi, T.; Mei, Z.
Hybrid Multi-Domain Analytical and Data-Driven Modeling for Feed Systems in Machine Tools. *Symmetry* **2019**, *11*, 1156.
https://doi.org/10.3390/sym11091156

**AMA Style**

Mei Z, Ding J, Chen L, Pi T, Mei Z.
Hybrid Multi-Domain Analytical and Data-Driven Modeling for Feed Systems in Machine Tools. *Symmetry*. 2019; 11(9):1156.
https://doi.org/10.3390/sym11091156

**Chicago/Turabian Style**

Mei, Zaiwu, Jianwan Ding, Liping Chen, Ting Pi, and Zaidao Mei.
2019. "Hybrid Multi-Domain Analytical and Data-Driven Modeling for Feed Systems in Machine Tools" *Symmetry* 11, no. 9: 1156.
https://doi.org/10.3390/sym11091156