# 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

- Subedi, D.; Tyapin, I.; Hovland, G. Review on Modeling and Control of Flexible Link Manipulators. Model. Identif. Control
**2020**, 41, 141–163. [Google Scholar] [CrossRef] - Theodore, R.J.; Ghosal, A. Comparison of the Assumed Modes and Finite Element Models for Flexible Multilink Manipulators. Int. J. Robot. Res.
**1995**, 14, 91–111. [Google Scholar] [CrossRef] [Green Version] - Krauss, R. An Improved Approach for Spatial Discretization of Transfer Matrix Models of Flexible Structures. In Proceedings of the 2019 American Control Conference (ACC), Philadelphia, PA, USA, 10–12 July 2019; pp. 3123–3128. [Google Scholar] [CrossRef]
- Tokhi, M.O.; Azad, A.K.M. Flexible Robot Manipulators: Modelling, Simulation and Control; IET: London, UK, 2008; Volume 68. [Google Scholar]
- De Luca, A.; Lanari, L.; Lucibello, P.; Panzieri, S.; Ulivi, G. Control experiments on a two-link robot with a flexible forearm. In Proceedings of the 29th IEEE Conference on Decision and Control, Honolulu, HI, USA, 5–7 December 1990; Volume 2, pp. 520–527. [Google Scholar] [CrossRef]
- Luca, A.D.; Siciliano, B. Closed-Form Dynamic Model of Planar Multilink Lightweight Robots. IEEE Trans. Syst. Man Cybern.
**1991**, 21, 826–839. [Google Scholar] [CrossRef] - Vera, F.G.D. Modeling and Sliding-Mode Control of Flexible-Link Robotic Structures for Vibration Suppression; Technische Universität Clausthal: Clausthal-Zellerfeld, Germany, 2016. [Google Scholar]
- Zhang, X.; Mills, J.K.; Cleghorn, W.L. Dynamic Modeling and Experimental Validation of a 3-PRR Parallel Manipulator with Flexible Intermediate Links. J. Intell. Robot. Syst.
**2007**, 50, 323–340. [Google Scholar] [CrossRef] - Book, W.J. Modeling, design, and control of flexible manipulator arms: A tutorial review. In Proceedings of the 29th IEEE Conference on Decision and Control, Honolulu, HI, USA, 5–7 December 1990; Volume 2, pp. 500–506. [Google Scholar] [CrossRef]
- Kurfess, T.R. Robotics and Automation Handbook; CRC Press: Boca Raton, FL, USA, 2018. [Google Scholar]
- Rahimi, H.N.; Nazemizadeh, M. Dynamic analysis and intelligent control techniques for flexible manipulators: A review. Adv. Robot.
**2014**, 28, 63–76. [Google Scholar] [CrossRef] - Lochan, K.; Roy, B.K.; Subudhi, B. A review on two-link flexible manipulators. Annu. Rev. Control
**2016**, 42, 346–367. [Google Scholar] [CrossRef] - Koutsovasilis, P.; Beitelschmidt, M. Comparison of model reduction techniques for large mechanical systems. Multibody Syst. Dyn.
**2008**, 20, 111–128. [Google Scholar] [CrossRef] - Vidoni, R.; Scalera, L.; Gasparetto, A.; Giovagnoni, M. Comparison of model order reduction techniques for flexible multibody dynamics using an equivalent rigid-link system approach. In Proceedings of the 8th ECCOMAS Thematic Conference on Multibody Dynamics, Prague, Czech Republic, 19–22 June 2017; pp. 269–280. [Google Scholar]
- Wu, L.; Tiso, P.; van Keulen, F. A modal derivatives enhanced Craig-Bampton method for geometrically nonlinear structural dynamics. In Proceedings of the ISMA, Leuven, Belgium, 19–21 September 2016; pp. 3615–3624. [Google Scholar]
- Tang, L.; Gouttefarde, M.; Sun, H.; Yin, L.; Zhou, C. Dynamic modelling and vibration suppression of a single-link flexible manipulator with two cables. Mech. Mach. Theory
**2021**, 162, 104347. [Google Scholar] [CrossRef] - Vidoni, R.; Scalera, L.; Gasparetto, A. 3-D ERLS based dynamic formulation for flexible-link robots: Theoretical and numerical comparison between the finite element method and the component mode synthesis approaches. Int. J. Mech. Control
**2018**, 19, 39–50. [Google Scholar] - Korayem, M.H.; Dehkordi, S.F. Dynamic modeling of flexible cooperative mobile manipulator with revolute-prismatic joints for the purpose of moving common object with closed kinematic chain using the recursive Gibbs-Appell formulation. Mech. Mach. Theory
**2019**, 137, 254–279. [Google Scholar] [CrossRef] - Zhang, C.; Yang, T.; Sun, N.; Zhang, J. A Simple Control Method of Single-Link Flexible Manipulators. In Proceedings of the 3rd International Symposium on Autonomous Systems, ISAS 2019, Shanghai, China, 29–31 May 2019; pp. 300–304. [Google Scholar] [CrossRef]
- Liu, Z.; Liu, J.; He, W. Dynamic modeling and vibration control for a nonlinear 3-dimensional flexible manipulator. Int. J. Robust Nonlinear Control
**2018**, 28, 3927–3945. [Google Scholar] [CrossRef] - Meng, Q.X.; Lai, X.Z.; Wang, Y.W.; Wu, M. A fast stable control strategy based on system energy for a planar single-link flexible manipulator. Nonlinear Dyn.
**2018**, 94, 615–626. [Google Scholar] [CrossRef] - He, W.; He, X.; Zou, M.; Li, H. PDE Model-Based Boundary Control Design for a Flexible Robotic Manipulator with Input Backlash. IEEE Trans. Control Syst. Technol.
**2018**, 27, 790–797. [Google Scholar] [CrossRef] - Sun, C.; Gao, H.; He, W.; Yu, Y. Fuzzy Neural Network Control of a Flexible Robotic Manipulator Using Assumed Mode Method. IEEE Trans. Neural Netw. Learn. Syst.
**2018**, 29, 5214–5227. [Google Scholar] [CrossRef] [PubMed] - Reddy, M.P.P.; Jacob, J. Vibration control of flexible link manipulator using SDRE controller and Kalman filtering. Stud. Inform. Control
**2017**, 26, 143–150. [Google Scholar] [CrossRef] [Green Version] - Ghasemi, A.H. Slewing and vibration control of a single-link flexible manipulator using filtered feedback linearization. J. Intell. Mater. Syst. Struct.
**2017**, 28, 2887–2895. [Google Scholar] [CrossRef] - Ouyang, Y.; He, W.; Li, X.; Liu, J.K.; Li, G. Vibration Control Based on Reinforcement Learning for a Single-link Flexible Robotic Manipulator. IFAC-PapersOnLine
**2017**, 50, 3476–3481. [Google Scholar] [CrossRef] - Ouyang, Y.; He, W.; Li, X. Reinforcement learning control of a singlelink flexible robotic manipulator. IET Control Theory Appl.
**2017**, 11, 1426–1433. [Google Scholar] [CrossRef] - Lochan, K.; Roy, B.K. Second-order SMC for tip trajectory tracking and tip deflection suppression of an AMM modelled nonlinear TLFM. Int. J. Dyn. Control
**2018**, 6, 1310–1318. [Google Scholar] [CrossRef] - Singla, A.; Singh, A. Dynamic Modeling of Flexible Robotic Manipulators. In Harmony Search and Nature Inspired Optimization Algorithms; Yadav, N., Yadav, A., Bansal, J.C., Deep, K., Kim, J.H., Eds.; Springer Singapore: Singapore, 2019; pp. 819–834. [Google Scholar]
- Qiu, Z.c.; Li, C.; min Zhang, X. Experimental study on active vibration control for a kind of two-link flexible manipulator. Mech. Syst. Signal Process.
**2019**, 118, 623–644. [Google Scholar] [CrossRef] - Gao, H.; He, W.; Zhou, C.; Sun, C. Neural Network Control of a Two-Link Flexible Robotic Manipulator Using Assumed Mode Method. IEEE Trans. Ind. Inform.
**2018**, 15, 755–765. [Google Scholar] [CrossRef] - Pradhan, S.K.; Subudhi, B. Position control of a flexible manipulator using a new nonlinear self-Tuning PID controller. IEEE/CAA J. Autom. Sin.
**2020**, 7, 136–149. [Google Scholar] [CrossRef] - Giorgio, I.; Del Vescovo, D.D. Non-linear lumped-parameter modeling of planar multi-link manipulators with highly flexible arms. Robotics
**2018**, 7, 60. [Google Scholar] [CrossRef] [Green Version] - Subedi, D.; Tyapin, I.; Hovland, G. Modeling and Analysis of Flexible Bodies Using Lumped Parameter Method. In Proceedings of the 2020 IEEE 11th International Conference on Mechanical and Intelligent Manufacturing Technologies (ICMIMT), Cape town, South Africa, 20–22 January 2020; pp. 161–166. [Google Scholar] [CrossRef]

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

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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