# Dynamic Modeling of Planar Multi-Link Flexible Manipulators

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modeling

#### 2.1. Kinematics

- Each link of the manipulator can undergo bending deformations (transversal deflection) in the plane of motion.
- The torsional effects and shear deformations are neglected.
- All joints are rigid and revolute. This assumption is considered because of higher joint stiffness compared to link stiffness.
- Link deflections are small.

#### 2.2. Assumed Modes Method

#### 2.3. Equations of Motion

## 3. Explicit Dynamic Model of a Three-Link Flexible Manipulator

## 4. Simulation Results

#### 4.1. Effect of Payload on Mode Shapes and Eigenfrequencies

#### 4.2. Effect of Arm Configuration on Mode Shapes and Eigenfrequencies

#### 4.3. Time-Domain Simulation

#### 4.4. Frequency-Domain Analysis

## 5. Conclusions and Discussions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Mode shapes for link 1, 2, and 3 with no payload (${m}_{p}$ = 0 kg), ${q}_{r2}={0}^{\circ}$, and ${q}_{r3}={0}^{\circ}$, (

**b**) Mode shapes for link 1, 2, and 3 with nominal payload (${m}_{p}$ = 2 kg), ${q}_{r2}={0}^{\circ}$, and ${q}_{r3}={0}^{\circ}$.

**Figure 3.**(

**a**) ${J}_{D1}$ with nominal payload, (

**b**) ${M}_{D1}$ with nominal payload, (

**c**) ${M}_{D2}$ with nominal payload.

**Figure 4.**(

**a**) Mode shapes for link 1 with nominal payload and ${q}_{r3}={0}^{\circ}$, (

**b**) Mode shapes for link 1 with nominal payload and ${q}_{r2}={0}^{\circ}$, (

**c**) Mode shapes for link 2 with nominal payload.

**Figure 5.**(

**a**) Mode shapes for link 1 (red), 2 (green), and 3 (blue) with nominal payload, ${q}_{r1}={0}^{\circ}$, ${q}_{r2}={0}^{\circ}$, and ${q}_{r3}={0}^{\circ}$ (I), ${90}^{\circ}$ (II), ${180}^{\circ}$ (III) (

**b**) Mode shapes for link 1 (red), 2 (green), and 3 (blue) with nominal payload, ${q}_{r1}={0}^{\circ}$, ${q}_{r3}={0}^{\circ}$, and ${q}_{r2}={0}^{\circ}$ (I), ${90}^{\circ}$ (II), ${180}^{\circ}$ (III).

**Figure 6.**(

**a**) Eigenfrequencies for link 1 and 2 with nominal payload and ${q}_{r2}={0}^{\circ}$, (

**b**) Eigenfrequencies for link 1 and 2 with nominal payload and ${q}_{r3}={0}^{\circ}$, (

**c**) Eigenfrequencies for link 1 and 2 with nominal payload and ${q}_{r2}={90}^{\circ}$, (

**d**) Eigenfrequencies for link 1 and 2 with nominal payload and ${q}_{r3}={90}^{\circ}$, (

**e**) Eigenfrequencies for link 1 and 2 with nominal payload and ${q}_{r2}={180}^{\circ}$, (

**f**) Eigenfrequencies for link 1 and 2 with nominal payload and ${q}_{r3}={180}^{\circ}$.

**Figure 7.**(

**a**) Eigenfrequency ${f}_{11}$ for link 1 with nominal payload, (

**b**) Eigenfrequency ${f}_{12}$ for link 1 with nominal payload, (

**c**) Eigenfrequency ${f}_{21}$ for link 2 with nominal payload, (

**d**) Eigenfrequencies ${f}_{22}$ for link 2 with nominal payload.

**Figure 8.**Free vibration response without damping under gravity starting with initial deformation (${q}_{f31}\left(0\right)$ = 0.1 m and ${q}_{f32}\left(0\right)$ = 0.002 m): (

**a**) Joint Position, (

**b**) Deflections of link 1, (

**c**) Deflections of link 2, (

**d**) Deflections of link 3, (

**e**) Manipulator tip position.

**Figure 9.**Free vibration response with damping under gravity starting with initial deformation (${q}_{f31}\left(0\right)$ = 0.1 m and ${q}_{f32}\left(0\right)$ = 0.002 m): (

**a**) Joint Position, (

**b**) Deflections of link 1, (

**c**) Deflections of link 2, (

**d**) Deflections of link 3, (

**e**) Manipulator tip position.

**Figure 10.**Energy of the manipulator system under free vibration without damping under gravity starting with initial deformation (${q}_{f31}\left(0\right)$ = 0.1 m and ${q}_{f32}\left(0\right)$ = 0.002 m): (

**a**) Elastic energy, (

**b**) Potential energy due to gravity, kinetic and total energy.

**Figure 11.**Energy of the manipulator system under free vibration with damping under gravity starting with initial deformation (${q}_{f31}\left(0\right)$ = 0.1 m and ${q}_{f32}\left(0\right)$ = 0.002 m): (

**a**) Elastic energy, (

**b**) Potential energy due to gravity, kinetic and total energy.

**Figure 12.**Forced vibration response without damping without gravity starting with undeformed configuration: (

**a**) Joint Position, (

**b**) Deflections of link 1, (

**c**) Deflections of link 2, (

**d**) Deflections of link 3, (

**e**) Manipulator tip position, (

**f**) Energy of the manipulator system.

**Figure 13.**Forced vibration response with damping without gravity starting with undeformed configuration: (

**a**) Joint Position, (

**b**) Deflections of link 1, (

**c**) Deflections of link 2, (

**d**) Deflections of link 3, (

**e**) Manipulator tip position, (

**f**) Energy of the manipulator system.

**Figure 14.**Time-domain and frequency-domain representation of tip deflection of the links with damping under gravity starting with initial deformation (${q}_{f31}\left(0\right)$ = 0.1 m and ${q}_{f32}\left(0\right)$ = 0.002 m): (

**a**) Tip deflection of link 1, (

**b**) Frequency response of the tip deflection of link 1, (

**c**) Tip deflection of link 2, (

**d**) Frequency response of the tip deflection of link 2, (

**e**) Tip deflection of link 3, (

**f**) Frequency response of the tip deflection of link 3.

Length (m) | Width (m) | Height (m) | Thickness (m) | |
---|---|---|---|---|

Link 1 | $1.5$ | $50\times {10}^{-3}$ | $50\times {10}^{-3}$ | $4\times {10}^{-3}$ |

Link 2 | $1.5$ | $40\times {10}^{-3}$ | $40\times {10}^{-3}$ | $3\times {10}^{-3}$ |

Link 3 | $1.5$ | $30\times {10}^{-3}$ | $30\times {10}^{-3}$ | $2.5\times {10}^{-3}$ |

Parameters | Values |
---|---|

${\ell}_{1}$ | 1.5 m |

${\ell}_{2}$ | 1.5 m |

${\ell}_{3}$ | 1.5 m |

${\rho}_{1}$ | 1.9872 kgm${}^{-1}$ |

${\rho}_{2}$ | 1.1988 kgm${}^{-1}$ |

${\rho}_{3}$ | 0.7425 kgm${}^{-1}$ |

${m}_{\ell 1}$ | 2.9808 kg |

${m}_{\ell 2}$ | 1.7982 kg |

${m}_{\ell 3}$ | 1.1138 kg |

${\left(EI\right)}_{1}$ | 1.8045 × 10${}^{4}$ Nm${}^{2}$ |

${\left(EI\right)}_{2}$ | 7.0361 × 10${}^{3}$ Nm${}^{2}$ |

${\left(EI\right)}_{3}$ | 2.4114 × 10${}^{3}$ Nm${}^{2}$ |

${J}_{\ell 1}$ | 0.5589 kgm${}^{2}$ |

${J}_{\ell 2}$ | 0.3372 kgm${}^{2}$ |

${J}_{\ell 3}$ | 0.2088 kgm${}^{2}$ |

${J}_{h1}$ | 0.0022 kgm${}^{2}$ |

${J}_{h2}$ | 6.631 × 10${}^{-4}$ kgm${}^{2}$ |

${J}_{h3}$ | 7.0100 × 10${}^{-5}$ kgm${}^{2}$ |

${J}_{p}$ | 3.2 × 10${}^{-4}$ kgm${}^{2}$ |

${\mathit{g}}_{v}$ | $\left[\begin{array}{cc}0& -9.81\end{array}\right]$${}^{T}$ ms${}^{-2}$ |

Eigenfrequencies (Hz) ${\mathit{f}}_{\mathit{i}}=\left[\begin{array}{cc}{\mathit{f}}_{\mathit{i}1}& {\mathit{f}}_{\mathit{i}2}\end{array}\right]$ | ||
---|---|---|

${\mathit{m}}_{\mathit{p}}$ = 0 kg | ${\mathit{m}}_{\mathit{p}}$ = 2 kg | |

Link 1 | $\left[\begin{array}{cc}2.56& 17.38\end{array}\right]$ | $\left[\begin{array}{cc}2.02& 15.03\end{array}\right]$ |

Link 2 | $\left[\begin{array}{cc}4.34& 28.74\end{array}\right]$ | $\left[\begin{array}{cc}2.67& 18.82\end{array}\right]$ |

Link 3 | $\left[\begin{array}{cc}14.17& 88.82\end{array}\right]$ | $\left[\begin{array}{cc}4.90& 63.97\end{array}\right]$ |

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**MDPI and ACS Style**

Subedi, D.; Tyapin, I.; Hovland, G. Dynamic Modeling of Planar Multi-Link Flexible Manipulators. *Robotics* **2021**, *10*, 70.
https://doi.org/10.3390/robotics10020070

**AMA Style**

Subedi D, Tyapin I, Hovland G. Dynamic Modeling of Planar Multi-Link Flexible Manipulators. *Robotics*. 2021; 10(2):70.
https://doi.org/10.3390/robotics10020070

**Chicago/Turabian Style**

Subedi, Dipendra, Ilya Tyapin, and Geir Hovland. 2021. "Dynamic Modeling of Planar Multi-Link Flexible Manipulators" *Robotics* 10, no. 2: 70.
https://doi.org/10.3390/robotics10020070