# Repeated Collision of a Planar Robotic Arm with a Surface Using Generalized Active Forces

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### 2.1. Configuration of the Robotic Arm

#### 2.2. Equations of Motion

#### 2.3. Control Strategy

#### 2.4. Contact Force during Impact

#### 2.4.1. The Normal Impact Force

#### 2.4.2. The Friction Force

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Comparison of PID controller, Q controller, and PD controller: (

**a**) response of link 1

**(b**) response of link 2.

**Figure 4.**Change of link angles with time: (

**a**) angle ${q}_{1}\left(t\right)$ and (

**b**) angle ${q}_{2}\left(t\right)$.

**Figure 5.**Phase portraits: (

**a**) ${q}_{1},\phantom{\rule{3.33333pt}{0ex}}{q}_{2},\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}{\dot{q}}_{1}$, (

**b**) ${q}_{1},\phantom{\rule{3.33333pt}{0ex}}{q}_{2},\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}{\dot{q}}_{2}$.

**Figure 6.**Poincare maps (

**a**) ${q}_{1}$ and ${q}_{2}$ for ${\dot{q}}_{1}=0$ crossing, and (

**b**) ${q}_{1}$ and ${q}_{2}$ for ${\dot{q}}_{2}=0$ crossing.

**Figure 9.**Velocity of the end effector during the first impact: (

**a**) tangential velocity of impact (

**b**) normal velocity of impact.

**Figure 11.**Kinetic energy during impact: (

**a**) the kinetic energy of link 1; (

**b**) the kinetic energy of link 2; (

**c**) total kinetic energy.

Robotic Arm | Fixed Surface | ||
---|---|---|---|

$\rho $ | 7800 (kg/m${}^{3}$) | $\rho $ | 7800 (kg/m${}^{3}$) |

E | 210 (GPa) | E | 210 (GPa) |

$\nu $ | 0.29 | $\nu $ | 0.29 |

${S}_{Y}$ | 1.12 (GPa) | ${S}_{Y}$ | 1.12 (GPa) |

$\mu $ | 0.2 | $\mu $ | 0.2 |

L | 1 (m) | ||

R | 0.005 (m) | ||

m | 1 (kg) |

Control Gains | Initial Position | Target Position | |||
---|---|---|---|---|---|

${K}_{p1}$ | 600 (N m/rad) | ${q}_{10}$ | ${60}^{\xb0}$ | ${q}_{1f}$ | $\pm {80}^{\xb0}$ |

${K}_{p2}$ | 300 (N m/rad) | ${q}_{20}$ | ${45}^{\xb0}$ | ${q}_{2f}$ | $\pm {70}^{\xb0}$ |

${K}_{d1}$ | 100 (N m/rad) | ${\dot{q}}_{10}$ | 0 | ||

${K}_{d2}$ | 100 (N m/rad) | ${\dot{q}}_{20}$ | 0 |

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**MDPI and ACS Style**

Akhan, A.F.; Zhao, J.; Tarnita, D.; Marghitu, D.B.
Repeated Collision of a Planar Robotic Arm with a Surface Using Generalized Active Forces. *Machines* **2023**, *11*, 773.
https://doi.org/10.3390/machines11080773

**AMA Style**

Akhan AF, Zhao J, Tarnita D, Marghitu DB.
Repeated Collision of a Planar Robotic Arm with a Surface Using Generalized Active Forces. *Machines*. 2023; 11(8):773.
https://doi.org/10.3390/machines11080773

**Chicago/Turabian Style**

Akhan, Ahmet Faruk, Jing Zhao, Daniela Tarnita, and Dan B. Marghitu.
2023. "Repeated Collision of a Planar Robotic Arm with a Surface Using Generalized Active Forces" *Machines* 11, no. 8: 773.
https://doi.org/10.3390/machines11080773