# Two-Degree-Of-Freedom Dynamic Model-Based Terminal Sliding Mode Control with Observer for Dual-Driving Feed Stage

^{1}

^{2}

^{3}

^{4}

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## Abstract

**:**

## 1. Introduction

## 2. Modelling and Analysis of 2-DOF Dual-Driving Feed Stage

#### 2.1. 2-DOF Dual-Driving Dynamic Model

#### 2.2. Comparison between Experimental and Model Simulated Results

## 3. Two-Degree-Of-Freedom Dynamic Model-Based Sliding Mode Control

#### 3.1. Terminal Sliding Mode Control Design

#### 3.2. Disturbance Observer and State Observer Design

_{1}and P

_{2}are the parameters that need to be designed. The derivative of the Lyapunov function can be given as [29]

## 4. Experimental Validations

#### 4.1. Experimental Set-Up

#### 4.2. Experimental Results

^{2}. In the configuration of the experiment system, the initial imbalance between the dual-drive axes at each feed direction is zero. The mechanical limit for the desynchronization between dual-axes in each feed direction is $\pm 2$ mm. The desynchronization limits of position and torque have been designed by program, and the system will come to a stop to avoid damage.

_{1}and y

_{2}axis of the proposed dynamic model-based TSMC with observer are shown in Figure 16a, and the tracking errors of the ${x}_{1}$ and ${x}_{2}$ axis of the proposed dynamic model-based TSMC with observer are shown in Figure 16b. The synchronous errors between the ${y}_{1}$ and ${y}_{2}$ axis of the proposed dynamic model-based TSMC with observer are shown in Figure 16c, and the synchronous errors between the ${x}_{1}$ and ${x}_{2}$ axis of the proposed dynamic model-based TSMC with observer are shown in Figure 16d.

## 5. Conclusions

- (1)
- The 2-DOF dynamic model considering the mechanical coupling and torsion errors of dual-axes is developed based on the Lagrange method. The geometric relationship and unbalanced forces of the 2-DOF dual-driving stage are analyzed by the translation from basic coordinates (${X}_{1}$, ${X}_{2}$, ${Y}_{1}$, ${Y}_{2}$) to the equivalent coordinates ($X$, ${\theta}_{1}$, $Y$, ${\theta}_{2}$). Furthermore, the parameters of the dynamic model have been identified by the acceleration signal test platform, and the dynamic model has been validated by the modal test.
- (2)
- The 2-DOF dynamic model-based TSMC with observer is designed in detail, considering the mechanical coupling and diversity of mechanism characteristics between dual-axes. In order to improve the robustness against the mechanical coupling between dual-axes in 2-DOF, the disturbance observer is developed. The state observer is applied to estimate the unmeasurable state variable. Moreover, the stability of the proposed control scheme has been verified by using Lyapunov criterion.
- (3)
- The performance of the proposed dynamic model-based TSMC with observer is validated experimentally on a 2-DOF dual-driving feed stage. In comparison to the cross-coupled PID control and cross-coupled normal TSMC, the proposed control scheme leads to a significant improvement of the tracking and synchronization accuracy. Particularly, the mean square errors indicates that the vibration of synchronous error was effectively suppressed.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${m}_{1}$ | mass of the lower layer stage | ${F}_{d}$ | actual disturbances of dual-driving system |

${m}_{2}$ | mass of the upper layer stage | ${\widehat{F}}_{d}$ | estimation of the disturbances |

${x}_{i}$ | actual displacement of the $i\mathrm{th}$ axis in X direction | ${\tilde{F}}_{d}$ | estimation error of the disturbances |

${y}_{i}$ | actual displacement of the $i\mathrm{th}$ axis in Y direction | $\widehat{X}$ | estimation of the states |

${\theta}_{i}$ | rotation angle of the lower layer and upper layer stage | $\tilde{X}$ | estimation error of the states |

${x}_{ref}$ | reference position command in X direction | ${u}_{TSMC}$ | output of TSMC synchronous control |

${y}_{ref}$ | reference position command in Y direction | $S$ | sliding surface |

${e}_{xi}$ | tracking errors of ith axis in X direction | $\widehat{S}$ | estimation of sliding surface |

${e}_{yi}$ | tracking errors of ith axis in Y direction | $\tilde{S}$ | estimation error of sliding surface |

${\epsilon}_{x}$ | synchronous errors between dual axes in X direction | $\lambda $ | sliding mode control gain |

${\epsilon}_{y}$ | synchronous errors between dual axes in Y direction |

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**Figure 1.**(

**a**) Machine tool with 2-DOF dual-driving frame (Mori Seiki CO., LTD.); (

**b**) configuration of 2-DOF dual-driving stage.

**Figure 5.**Friction identification and fitting of X-axis. (

**a**) the identification results of axis ${x}_{1}$; (

**b**) the identification results of axis ${x}_{2}$.

**Figure 6.**Time-frequency characteristics of 2-DOF feed stage. (

**a**) the time domain signal; (

**b**) the frequency domain signal.

**Figure 14.**Experimental results of the cross-coupled PID control. (

**a**) The tracking errors of the ${y}_{1}$ and ${y}_{2}$ axis; (

**b**) the tracking errors of the ${x}_{1}$ and ${x}_{2}$ axis; (

**c**) the synchronous errors between the ${y}_{1}$ and ${y}_{2}$ axis; (

**d**) the synchronous errors between the ${x}_{1}$ and ${x}_{2}$ axis.

**Figure 15.**Experimental results of the cross-coupled normal TSMC. (

**a**) The tracking errors of the ${y}_{1}$ and ${y}_{2}$ axis; (

**b**) the tracking errors of the ${x}_{1}$ and ${x}_{2}$ axis; (

**c**) the synchronous errors between the ${y}_{1}$ and ${y}_{2}$ axis; (

**d**) the synchronous errors between the ${x}_{1}$ and ${x}_{2}$ axis.

**Figure 16.**Experimental results of the proposed dynamic model-based TSMC with observer. (

**a**) The tracking errors of the ${y}_{1}$ and ${y}_{2}$ axis; (

**b**) the tracking errors of the ${x}_{1}$ and ${x}_{2}$ axis; (

**c**) the synchronous errors between the ${y}_{1}$ and ${y}_{2}$ axis; (

**d**) the synchronous errors between the ${x}_{1}$ and ${x}_{2}$ axis.

Axis | ${\mathit{F}}_{\mathit{C}}\text{}\left(\mathbf{N}\right)$ | ${\mathit{F}}_{\mathit{S}}\text{}\left(\mathbf{N}\right)$ | ${\mathit{v}}_{\mathit{\sigma}}\text{}(\mathbf{mm}/\mathbf{min})$ | Fv (Ns/m) |
---|---|---|---|---|

x_{1} | 15.1008 | 58.3671 | 73.2533 | 0.0027 |

x_{2} | 13.4760 | 51.0637 | 109.3258 | 0.0030 |

y_{1} | 10.0224 | 46.8655 | 57.1537 | 0.0019 |

y_{2} | 9.0956 | 39.7762 | 48.3761 | 0.0013 |

Name | Value | Description |
---|---|---|

${m}_{1}$ | 16.77 kg | Mass of the lower layer stage |

${m}_{2}$ | 10.53 kg | Mass of the upper layer stage |

${I}_{1}$ | 1.18 kg$\cdot $m^{2} | Moment of inertia of the lower layer stage |

${I}_{22}$ | 5.48 kg$\cdot $m^{2} | Moment of inertia of the upper layer stage |

${k}_{ex}$ | 1631.8 N$\cdot $m/rad | Equivalent stiffness of X-axis |

${k}_{ey}$ | 1347.6 N$\cdot $m/rad | Equivalent stiffness of Y-axis |

${k}_{bx}$ | 3.0146e6 N$\cdot $m/rad | Lateral stiffness of slider in X-axis |

${k}_{by}$ | 2.8734e6 N$\cdot $m/rad | Lateral stiffness of slider in Y-axis |

${l}_{x}$ | 0.8 m | Distance between dual screws in X-axis |

${l}_{y}$ | 0.6 m | Distance between dual screws in Y-axis |

a_{x} | 0.45 m | Axial distance of the sliders in X-axis |

a_{y} | 0.4 m | Axial distance of the sliders in Y-axis |

${r}_{g}$ | 5 mm | Screw lead |

${F}_{s1}$ | 117.7352 N | Maximum static friction of ${y}_{1}$ |

${F}_{s2}$ | 102.1365 N | Maximum static friction of ${y}_{2}$ |

${F}_{c1}$ | 30.2017 N | Coulumb friction of axis ${y}_{1}$ |

${F}_{c2}$ | 26.8520 N | Coulumb friction of axis ${y}_{2}$ |

${F}_{v1}$ | 0.0052 Ns/m | Viscosity coefficient of axis ${y}_{1}$ |

${F}_{v2}$ | 0.0060 Ns/m | Viscosity coefficient of axis ${y}_{2}$ |

${v}_{\sigma 1}$ | 73.2533 mm/min | Stribeck velocity of axis ${y}_{1}$ |

${v}_{\sigma 2}$ | 109.3258 mm/min | Stribeck velocity of axis ${y}_{2}$ |

Eigenvalue | Natural Frequency | Vector | Mode of Vibration |
---|---|---|---|

${w}_{1}$ | 0 Hz | [0, 0, 0, 0]^{T} | Axial |

${w}_{2}$ | 36.95 Hz | [0.55, −0.55, 1, −1]^{T} | Low–order torsion |

${w}_{3}$ | 121.7 Hz | [0.35, 0.15, 1, −0.15]^{T} | Axial and torsion |

${w}_{4}$ | 390.1 Hz | [0.75, −0.75, 1, −1]^{T} | High-order torsion |

${w}_{5}$ | 448.6 Hz | [0.5, 0.35, 1, 0.3] | High-order Axial and torsion |

Mode of Vibration | Prediction Natural Frequency | Experiment Natural Frequency | Relative Error |
---|---|---|---|

Axial | - | - | - |

Low-order torsion | 35.95 Hz | 33.82 Hz | 6.3 % |

Axial and torsion | 121.7 Hz | 112.4 Hz | 8.3 % |

High-order torsion | 390.1 Hz | 356.3 Hz | 9.5 % |

High-order Axial and torsion | 448.6 Hz | 407.1 Hz | 10.2 % |

Description | Name | Motor x_{1} | Motor x_{2} | $\mathbf{Motor}\text{}{\mathit{y}}_{1}$ | $\mathbf{Motor}\text{}{\mathit{y}}_{2}$ |
---|---|---|---|---|---|

Inertia | $J$ kg$\cdot $m^{2} | 0.000177 | 0.000176 | 0.000167 | 0.000165 |

Damping | $B$ kg$\cdot $m^{2}/sec | 0.00025 | 0.00022 | 0.00023 | 0.00019 |

Torque coefficient | ${k}_{\mathrm{t}}$ NmA | 1.37 | 1.37 | 1.37 | 1.37 |

Lead screw | ${r}_{g}$ mm/rad | 5 | 5 | 5 | 5 |

Amplifier | ${k}_{a}$ A/V | 8.8462 | 7.9741 | 8.4355 | 7.7686 |

Errors Control Schemes | Tracking Errors | Synchronous Errors | |||||
---|---|---|---|---|---|---|---|

Cross-Coupled PID Control | Cross-Coupled Normal TSMC | Cross-Coupled Dynamic Model-Based TSMC with Observer | Cross-Coupled PID Control | Cross-Coupled Normal TSMC | Cross-Coupled Dynamic Model-Based TSMC with Observer | ||

Max (μm) | Axis ${y}_{1}$ | 122.4 | 115.6 | 81.5 | 7.6 | 7.0 | 5.2 |

Axis ${y}_{2}$ | 129.7 | 123.4 | 86.1 | ||||

Axis ${x}_{1}$ | 235.6 | 133.2 | 108.7 | 10.4 | 7.8 | 6.3 | |

Axis ${x}_{2}$ | 247.1 | 141.3 | 115.2 | ||||

Average (μm) | Axis ${y}_{1}$ | 33.7 | 26.6 | 24.9 | 6.1 | 5.7 | 3.8 |

Axis ${y}_{2}$ | 27.5 | 32.7 | 28.1 | ||||

Axis ${x}_{1}$ | 59.3 | 36.3 | 30.3 | 8.3 | 6.6 | 4.6 | |

Axis ${x}_{2}$ | 51.6 | 43.5 | 35.1 | ||||

Mean Square (μm) | Axis ${y}_{1}$ | 41.7 | 37.1 | 19.4 | 6.7 | 5.5 | 2.7 |

Axis ${y}_{2}$ | 48.9 | 37.9 | 21.5 | ||||

Axis ${x}_{1}$ | 57.1 | 38.4 | 26.7 | 7.1 | 5.3 | 1.9 | |

Axis ${x}_{2}$ | 62.8 | 41.6 | 25.1 |

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## Share and Cite

**MDPI and ACS Style**

Fan, W.; Lu, H.; Zhang, X.; Zhang, Y.; Zeng, R.; Liu, Q.
Two-Degree-Of-Freedom Dynamic Model-Based Terminal Sliding Mode Control with Observer for Dual-Driving Feed Stage. *Symmetry* **2018**, *10*, 488.
https://doi.org/10.3390/sym10100488

**AMA Style**

Fan W, Lu H, Zhang X, Zhang Y, Zeng R, Liu Q.
Two-Degree-Of-Freedom Dynamic Model-Based Terminal Sliding Mode Control with Observer for Dual-Driving Feed Stage. *Symmetry*. 2018; 10(10):488.
https://doi.org/10.3390/sym10100488

**Chicago/Turabian Style**

Fan, Wei, Hong Lu, Xinbao Zhang, Yongquan Zhang, Rong Zeng, and Qi Liu.
2018. "Two-Degree-Of-Freedom Dynamic Model-Based Terminal Sliding Mode Control with Observer for Dual-Driving Feed Stage" *Symmetry* 10, no. 10: 488.
https://doi.org/10.3390/sym10100488