# Position Tracking of Multiple Robotic Manipulator Systems Associated with Communication Strength Dynamics

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

- (i)
- The dynamical model of an RMS is described using a vector differential equation with the second derivative term and interconnected term due to Newton’s law of motion; the model is more general.
- (ii)
- A vector differential equation is used to model the dynamics of CSS, and there are few studies considering the variation of the strength of communication between multiple robotic manipulators in the existing literature.
- (iii)
- The position tracking for an MRMS is achieved by employing the position tracking control protocol for the RMS and the coupling matrix function for the CSS designed in this paper, which has been rarely reported in existing studies.

## 2. Model Description and Control Design

**Remark**

**1.**

**Remark**

**2.**

**Definition**

**1**

**Remark**

**3.**

**Assumption**

**1.**

**Assumption**

**2.**

**Remark**

**4.**

**Assumption**

**3.**

**Remark**

**5.**

## 3. The Design of Tracking Control Protocol and Coupling Matrix Function

**Remark**

**6.**

**Theorem**

**1.**

**Proof**

**of Theorem.**

**Remark**

**7.**

**Remark**

**8.**

## 4. Illustrative Example

^{2}, $\sigma =5.7$, and $\beta =5\phantom{\rule{3.33333pt}{0ex}}randn\left(1\right)$. Finally, the model parameters and the matrices obtained from the above steps are substituted into the position tracking control protocols ${\tau}_{i}$ (6) and ${\eta}_{i}$ (7) and the coupling matrix function ${\Theta}_{i}\left({z}_{i}\right)$ (8) designed in this paper. In addition, in order to demonstrate the advantages of the position tracking control scheme synthesized in this paper, we introduce an experiment comparing the position tracking proposed in [33,34,35] with ours. For simplicity, let $\u2225e\left(t\right)\u2225=\sqrt{{\displaystyle \sum _{i=1}^{N}}{\u2225{e}_{i}\left(t\right)\u2225}^{2}}$ be the total position tracking error of the RMS in the MRMS. The simulation results are shown in Figure 3, Figure 4, Figure 5 and Figure 6.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MRMS | multiple robotic manipulator system |

RMS | robotic manipulator subsystem |

CSS | communication strength subsystem |

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**Figure 1.**The schematic diagram of ${p}_{ij}\left(t\right)$ influenced by ${p}_{ik}\left(t\right)$, ${z}_{i\rho}\left(t\right)$ and ${\dot{z}}_{i\rho}\left(t\right)$ in Equation (2).

**Figure 3.**(

**a**) The state curves of position for RMS under the control scheme from [33]; (

**b**) the state curves of position for RMS under the control scheme from [34]; (

**c**) the state curves of position for RMS under the control scheme from [35]; (

**d**) the state curves of position for RMS under the control scheme from this paper.

**Figure 4.**(

**a**) The position error curves of RMS under the control scheme from [33]; (

**b**) the position error curves of RMS under the control scheme from [34]; (

**c**) the position error curves of RMS under the control scheme from [35]; (

**d**) the position error curves of RMS under the control scheme from this paper.

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## Share and Cite

**MDPI and ACS Style**

Zhao, J.; Wang, Y.; Gao, P.; Li, S.; Chen, H.
Position Tracking of Multiple Robotic Manipulator Systems Associated with Communication Strength Dynamics. *Sensors* **2023**, *23*, 9275.
https://doi.org/10.3390/s23229275

**AMA Style**

Zhao J, Wang Y, Gao P, Li S, Chen H.
Position Tracking of Multiple Robotic Manipulator Systems Associated with Communication Strength Dynamics. *Sensors*. 2023; 23(22):9275.
https://doi.org/10.3390/s23229275

**Chicago/Turabian Style**

Zhao, Juanxia, Yinhe Wang, Peitao Gao, Shengping Li, and Haoguang Chen.
2023. "Position Tracking of Multiple Robotic Manipulator Systems Associated with Communication Strength Dynamics" *Sensors* 23, no. 22: 9275.
https://doi.org/10.3390/s23229275