# Synchronization Control with Dynamics Compensation for Three-Axis Parallel Motion Platform

^{1}

^{2}

^{*}

## Abstract

**:**

_{1}motor by 71.88% and 73.02%, respectively, demonstrating its performance advantages in trajectory tracking and synchronization.

## 1. Introduction

## 2. Kinematic Modeling of the TAPMP

#### 2.1. Structural Description

#### 2.2. Kinematic Modeling

_{1}, M

_{2}, M

_{3}) and solid lines (M

_{1}′, M

_{2}′, M

_{3}′) correspondingly denote the initial and final positions of the three motor-driven components. Endpoints A

_{1}and C

_{1}are linked to the left and right auxiliary blocks, respectively, while B

_{1}serves as the central point of the moving platform. The variables ${q}_{1}$, ${q}_{2}$, and ${q}_{3}$ represent the displacement of the three motor-driven components as they move from their initial to final positions. ${L}_{1}$ signifies the width of the moving platform. Coordinate systems O-XYZ and o-xyz are established, with O and o denoting the midpoints of the moving platform at its initial and final positions, respectively.

## 3. Dynamic Modeling of the TAPMP

#### 3.1. Moving Platform Dynamics

#### 3.2. Auxiliary Block Dynamics

#### 3.3. Overall Dynamics of the TAPMP

## 4. Synchronization Controller with Dynamics Compensation

#### 4.1. Definitions of Synchronization Error and Coupling Error

#### 4.2. Formulation of Synchronization Controller with Dynamics Compensation

#### 4.3. Stability Analysis

**Theorem**

**1.**

**Proof.**

^{T}(t) and subsequently inserting the result into Equation (29) leads to the following:

## 5. Simulation Analysis of the SC–DC

#### 5.1. Simulation Setup

#### 5.2. Simulation Results

^{T}, with a radius of 0.01 m and a frequency of 0.4 Hz, as well as sinusoidal trajectories with an amplitude of 5° and a frequency of 0.4 Hz, served as reference inputs for the XZ plane and the B-axis, respectively, as shown in Figure 4. The tracking error and synchronization error of each motor, along with the tracking error of each drive axis, are depicted in Figure 5, Figure 6 and Figure 7, respectively.

_{1}is reduced from 6.87 μm and 6.98 μm to 2.74 μm and 0.4 μm, respectively, representing reductions of 60.12% and 94.27% compared to the APD–DC method. Additionally, the MAE tracking error for the X- and Z-axes also decreases from 12.53 μm and 4.26 μm to 2.85 μm and 0.72 μm, respectively, constituting reductions of 77.25% and 83.1% compared to the APD–DC method.

## 6. Experimental Verification of the SC–DC

#### 6.1. Experimental Setup

#### 6.2. Experimental Results

^{T}and a radius of 0.01 m, as well as sinusoidal trajectories with an amplitude of 5° and a frequency of 0.18 Hz, were designated as the desired motion trajectories for the XZ plane and the B-axis, respectively. We utilized both the proposed controller and the APD–DC to achieve precise control of the motion trajectories.

_{1}is reduced from 4.16 μm and 1.89 μm to 1.17 μm and 0.51 μm, respectively, representing only 71.88% and 73.02% of the APD–DC method. Simultaneously, the MAE tracking error for the X- and Z-axes decreases from 5 μm and 1.63 μm to 1.43 μm and 0.32 μm, respectively, accounting for only 71.4% and 80.37% of the APD–DC method. Furthermore, by comparing the simulation and experimental results in Table 1, Table 2, Table 3, Table 4, Table 5 and Table 6, it can be observed that there are certain discrepancies in the extent of error reduction between them. These disparities primarily arise from the inconsistency between the error dynamics of the simulation model and those of the actual system, as well as the limitations imposed by the servo cycle. Nevertheless, the trend of error reduction remains consistent in both scenarios. This mutual validation further corroborates the feasibility and effectiveness of the proposed strategy.

## 7. Conclusions

_{1}motor by 71.88% and 73.02%, respectively. Additionally, the proposed controller decreases the MAE of the end-effector’s tracking error on the X- and Z-axes by 71.4% and 80.37%, respectively. This paper thus offers a novel strategy for the trajectory tracking and synchronization control of multi-axis parallel platforms. In future research, we aim to further refine the proposed synchronization control strategy by incorporating model-free decoupling control based on time-delay estimation for the application to a five-axis hybrid motion platform.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**Tracking error of each motor with APD–DC and the proposed controller. (

**a**) Motor q

_{1}, (

**b**) motor q

_{2}, (

**c**) motor q

_{3}.

**Figure 6.**Synchronization error of each motor with APD–DC and the proposed controller. (

**a**) Motor q

_{1}, (

**b**) motor q

_{2}, (

**c**) motor q

_{3}.

**Figure 7.**Tracking error of each drive axis with APD–DC and the proposed controller. (

**a**) X-axis, (

**b**) Z-axis, (

**c**) B-axis.

**Figure 10.**Experimental results of the tracking error for each motor with APD–DC and the proposed controller. (

**a**) Motor q

_{1}, (

**b**) motor q

_{2}, (

**c**) motor q

_{3}.

**Figure 11.**Experimental results of the synchronization error for each motor with APD–DC and the proposed controller. (

**a**) Motor q

_{1}, (

**b**) motor q

_{2}, (

**c**) motor q

_{3}.

**Figure 12.**Experimental results of the tracking error for each drive axis with APD–DC and the proposed controller. (

**a**) X-axis, (

**b**) Z-axis, (

**c**) B-axis.

Control Strategy | Tracking Error (μm) | |||||
---|---|---|---|---|---|---|

STD | MAE | |||||

q_{1} | q_{2} | q_{3} | q_{1} | q_{2} | q_{3} | |

APD–DC | 7.83 | 14.21 | 6.69 | 6.87 | 12.53 | 5.87 |

Proposed | 3.16 | 3.28 | 3.29 | 2.74 | 2.85 | 2.89 |

Reduction (%) (proposed compared with APD–DC) | 59.64 | 76.92 | 50.82 | 60.12 | 77.25 | 50.77 |

**Table 2.**Comparison of the synchronization error for each motor with APD–DC and the proposed controller.

Control Strategy | Synchronization Error (μm) | |||||
---|---|---|---|---|---|---|

STD | MAE | |||||

q_{1} | q_{2} | q_{3} | q_{1} | q_{2} | q_{3} | |

APD–DC | 7.89 | 8.20 | 5.49 | 6.98 | 7.26 | 4.92 |

Proposed | 0.48 | 0.52 | 0.95 | 0.40 | 0.46 | 0.83 |

Reduction (%) (proposed compared with APD–DC) | 93.92 | 93.66 | 82.7 | 94.27 | 93.66 | 83.13 |

**Table 3.**Comparison of the tracking error for each drive axis with APD–DC and the proposed controller.

Control Strategy | Tracking Error | |||||
---|---|---|---|---|---|---|

STD | MAE | |||||

X (μm) | Z (μm) | B (μrad) | X (μm) | Z (μm) | B (μrad) | |

APD–DC | 14.21 | 4.75 | 75.95 | 12.53 | 4.26 | 67.16 |

Proposed | 3.28 | 0.82 | 3.18 | 2.85 | 0.72 | 1.62 |

Reduction (%) (proposed compared with APD–DC) | 76.92 | 82.74 | 95.81 | 77.25 | 83.1 | 97.59 |

**Table 4.**Experimental comparison of the tracking error for each motor with APD–DC and the proposed controller.

Control Strategy | Tracking Error (μm) | |||||
---|---|---|---|---|---|---|

STD | MAE | |||||

q_{1} | q_{2} | q_{3} | q_{1} | q_{2} | q_{3} | |

APD–DC | 4.93 | 5.98 | 4.27 | 4.16 | 5.00 | 3.62 |

Proposed | 1.59 | 2.02 | 1.54 | 1.17 | 1.43 | 1.21 |

Reduction (%) (proposed compared with APD–DC) | 67.75 | 66.22 | 63.93 | 71.88 | 71.4 | 66.57 |

**Table 5.**Experimental comparison of the synchronization error for each motor with APD–DC and the proposed controller.

Control Strategy | Synchronization Error (μm) | |||||
---|---|---|---|---|---|---|

STD | MAE | |||||

q_{1} | q_{2} | q_{3} | q_{1} | q_{2} | q_{3} | |

APD–DC | 2.17 | 1.86 | 2.16 | 1.89 | 1.47 | 1.89 |

Proposed | 0.65 | 0.88 | 0.46 | 0.51 | 0.62 | 0.37 |

Reduction (%) (proposed compared with APD–DC) | 70.01 | 52.69 | 78.7 | 73.02 | 57.82 | 80.42 |

**Table 6.**Experimental comparison of the tracking error for each axis with APD–DC and the proposed controller.

Control Strategy | Tracking Error | |||||
---|---|---|---|---|---|---|

STD | MAE | |||||

X (μm) | Z (μm) | B (μrad) | X (μm) | Z (μm) | B (μrad) | |

APD–DC | 5.98 | 1.87 | 33.01 | 5.00 | 1.63 | 26.55 |

Proposed | 2.02 | 0.39 | 14.21 | 1.43 | 0.32 | 10.17 |

Reduction (%) (proposed compared with APD–DC) | 66.22 | 79.14 | 56.95 | 71.4 | 80.37 | 61.69 |

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## Share and Cite

**MDPI and ACS Style**

Zhou, Z.; Gao, J.; Zhang, L.
Synchronization Control with Dynamics Compensation for Three-Axis Parallel Motion Platform. *Actuators* **2024**, *13*, 166.
https://doi.org/10.3390/act13050166

**AMA Style**

Zhou Z, Gao J, Zhang L.
Synchronization Control with Dynamics Compensation for Three-Axis Parallel Motion Platform. *Actuators*. 2024; 13(5):166.
https://doi.org/10.3390/act13050166

**Chicago/Turabian Style**

Zhou, Zhiwei, Jian Gao, and Lanyu Zhang.
2024. "Synchronization Control with Dynamics Compensation for Three-Axis Parallel Motion Platform" *Actuators* 13, no. 5: 166.
https://doi.org/10.3390/act13050166