# Cooperative Passivity-Based Control of Nonlinear Mechanical Systems

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## Abstract

**:**

## 1. Introduction

- A unified IDA-PBC scheme for cooperative stabilization of heterogeneous networks of fully actuated and underactuated agents that builds on existing single-agent IDA-PBC controllers. Our approach satisfies the matching conditions of each underactuated agent, independently of the cooperative input, by suitably defining the cooperative variables.
- A cooperative PBC scheme for fully actuated systems that is robust to time-varying delays and loss of communication packets, while the end-effector dynamics, by default, are transformed to behave as point-masses in the cooperative space, or follow desired dynamics as specified by the control designer.

#### Related Work

## 2. Problem Formulation

**Problem**

**1.**

**Problem**

**2.**

## 3. Review of Passivity and Single-Agent IDA-PBC

**Definition**

**1**

## 4. Cooperative IDA-PBC

#### 4.1. Guaranteed Local Matching

**Assumption**

**1.**

#### 4.2. Multi-Agent IDA-PBC

**Theorem**

**1.**

**Proof.**

#### 4.3. Concluding Remarks

## 5. Cooperative r-Passivity-Based Control

#### 5.1. Cooperative PBC with Communication Delays

**Theorem**

**2.**

#### 5.2. r-Passivity for Coordinate Synchronization

#### 5.3. Local Controller Design for r-Passivity

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

#### 5.4. Cooperative Synchronization

**Theorem**

**5.**

**Proof.**

**Remark**

**1.**

#### 5.5. Cooperative Kinetic Energy Shaping

**Theorem**

**6.**

**Proof.**

## 6. Simulation Results

## 7. Experimental Results

## 8. Discussion

## 9. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

PBC | Passivity-Based Control |

IDA-PBC | Interconnection-and-Damping Assignment Passivity-Based Control |

LFFC | Leader-Follower Formation Control |

ST | Scattering Transformation |

EL | Euler-Lagrange |

PDEs | Partial Differential Equations |

TDPA | Time Domain Passivity Approach |

PSPM | Passive Set Position Modulation |

WVM | Wave-Variable Modulation |

UAV | Unmanned Aerial Vehicle |

DOF | Degree-Of-Freedom |

## References

- Wurman, P.; D’Andrea, R.; Mountz, M. Coordinating Hundreds of Cooperative, Autonomous Vehicles in Warehouses. AI Mag.
**2008**, 29, 9–20. [Google Scholar] - Leitner, J. Multi-robot cooperation in space: A survey. In Advanced Technologies for Enhanced Quality of Life; IEEE: Piscataway, NJ, USA, 2009; pp. 144–151. [Google Scholar]
- Valk, L.; Keviczky, T. Unified Passivity-Based Distributed Control of Mechanical Systems. In Proceedings of the 37th Benelux Meeting on Systems and Control, Soesterberg, The Netherlands, 27–29 March 2018; p. 38. [Google Scholar]
- Ortega, R.; van der Schaft, A.; Maschke, B.; Escobar, G. Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica
**2002**, 38, 585–596. [Google Scholar] [CrossRef] - Niemeyer, G.; Slotine, J.J.E. Stable adaptive teleoperation. IEEE J. Ocean. Eng.
**1991**, 16, 152–162. [Google Scholar] [CrossRef] - Valk, L.; Keviczky, T. Distributed Control of Heterogeneous Underactuated Mechanical Systems. In Proceedings of the 7th IFAC Workshop on Distributed Estimation and Control in Networked Systems, Groningen, The Netherlands, 27–28 August 2018; Volume 51, pp. 325–330. [Google Scholar]
- De Groot, O.; Keviczky, T. Cooperative r-Passivity-Based Control for Mechanical Systems. IFAC Pap.
**2020**, 53, 3476–3481. [Google Scholar] [CrossRef] - Arcak, M. Passivity as a Design Tool for Group Coordination. IEEE Trans. Autom. Control.
**2007**, 52, 1380–1390. [Google Scholar] [CrossRef] - Franchi, A.; Secchi, C.; Son, H.I.; Bulthoff, H.H.; Giordano, P.R. Bilateral Teleoperation of Groups of Mobile Robots with Time-Varying Topology. IEEE Trans. Robot.
**2012**, 28, 1019–1033. [Google Scholar] [CrossRef] - Niemeyer, G.; Slotine, J.J.E. Telemanipulation with Time Delays. Int. J. Robot. Res.
**2004**, 23, 873–890. [Google Scholar] [CrossRef] - Yüksel, B.; Secchi, C.; Bülthoff, H.H.; Franchi, A. Aerial physical interaction via IDA-PBC. Int. J. Robot. Res.
**2019**, 38, 403–421. [Google Scholar] [CrossRef] - Brogliato, B. Dissipative Systems Analysis and Control: Theory and Applications, 2nd ed.; Springer: Berlin/Heidelberg, Germany, 2007. [Google Scholar]
- Mei, J.; Ren, W.; Ma, G. Distributed Coordinated Tracking with a Dynamic Leader for Multiple Euler-Lagrange Systems. IEEE Trans. Autom. Control
**2011**, 56, 1415–1421. [Google Scholar] [CrossRef] - Ren, W. Distributed leaderless consensus algorithms for networked Euler–Lagrange systems. Int. J. Control.
**2009**, 82, 2137–2149. [Google Scholar] [CrossRef] - Avila-Becerril, S.; Espinosa-Perez, G.; Panteley, E.; Ortega, R. Consensus control of flexible joint robots. In Proceedings of the 52nd IEEE CDC, Florence, Italy, 10–13 December 2013; pp. 2288–2293. [Google Scholar]
- Nuño, E.; Valle, D.; Sarras, I.; Basañez, L. Leader–follower and leaderless consensus in networks of flexible-joint manipulators. Eur. J. Control.
**2014**, 20, 249–258. [Google Scholar] [CrossRef] - Acosta, J.A.; Ortega, R.; Astolfi, A.; Mahindrakar, A.D. Interconnection and damping assignment passivity-based control of mechanical systems with underactuation degree one. IEEE Trans. Autom. Control.
**2005**, 50, 1936–1955. [Google Scholar] [CrossRef] - Ryalat, M.; Laila, D.S. A Robust IDA-PBC Approach for Handling Uncertainties in Underactuated Mechanical Systems. IEEE Trans. Autom. Control.
**2018**, 63, 3495–3502. [Google Scholar] [CrossRef] - Nuño, E.; Ortega, R. Achieving Consensus of Euler–Lagrange Agents with Interconnecting Delays and without Velocity Measurements via Passivity-Based Control. IEEE Trans. Control. Syst. Technol.
**2018**, 26, 222–232. [Google Scholar] [CrossRef] - Aldana, C.I.; Romero, E.; Nuño, E.; Basañez, L. Pose consensus in networks of heterogeneous robots with variable time delays. Int. J. Robust Nonlinear Control.
**2015**, 25, 2279–2298. [Google Scholar] [CrossRef] - Aldana, C.I.; Tabarez, L.; Nuño, E.; Cruz-Zavala, E. Task Space Consensus of Heterogeneous Robots with Time-Delays and without Velocity Measurements. IEEE Control. Syst. Lett.
**2021**, 5, 1525–1530. [Google Scholar] [CrossRef] - Hannaford, B.; Ryu, J.H. Time-domain passivity control of haptic interfaces. IEEE Trans. Robot. Autom.
**2002**, 18, 1–10. [Google Scholar] [CrossRef] - Lee, D.; Huang, K. Passive-Set-Position-Modulation Framework for Interactive Robotic Systems. IEEE Trans. Robot.
**2010**, 26, 354–369. [Google Scholar] [CrossRef] - Secchi, C.; Ferraguti, F.; Fantuzzi, C. Catching the wave: A transparency oriented wave based teleoperation architecture. In Proceedings of the 2016 IEEE International Conference on Robotics and Automation (ICRA), Stockholm, Sweden, 16–21 May 2006; pp. 2422–2427. [Google Scholar] [CrossRef]
- Liu, Y.; Puah, S. Passivity-based control for networked robotic system over unreliable communication. In Proceedings of the IEEE ICRA, Hong Kong, China, 31 May–7 June 2014; pp. 1769–1774. [Google Scholar]
- Chopra, N.; Spong, M.W. Output Synchronization of Nonlinear Systems with Time Delay in Communication. In Proceedings of the 45th IEEE CDC, San Diego, CA, USA, 13–15 December 2006; pp. 4986–4992. [Google Scholar]
- Chopra, N.; Spong, M.W. On Synchronization of Networked Passive Systems with Time Delays and Application to Bilateral Teleoperation. In Proceedings of the SCIE Annual Conference, Okayama Japan, 8–10 August 2005; pp. 3424–3429. [Google Scholar]
- Liu, Y.; Chopra, N. Controlled Synchronization of Heterogeneous Robotic Manipulators in the Task Space. IEEE Trans. Robot.
**2012**, 28, 268–275. [Google Scholar] [CrossRef] - Valk, L. Distributed Control of Underactuated and Heterogeneous Mechanical Systems. Master’s Thesis, Delft University of Technology, Delft, The Netherlands, 2018. [Google Scholar]
- Lozano, R.; Chopra, N.; Spong, M. Passivation of Force Reflecting Bilateral Teleoperators With Time Varying Delay. In Proceedings of the 8. Mechatronics Forum; University of Twente: Enschede, The Netherlands, 2002; pp. 24–26. [Google Scholar]
- Berestesky, P.; Chopra, N.; Spong, M.W. Discrete time passivity in bilateral teleoperation over the Internet. In Proceedings of the IEEE ICRA, New Orleans, LA, USA, 26 April–1 May 2004; Volume 5, pp. 4557–4564. [Google Scholar]
- Hsu, P.; Hauser, J.; Sastry, S. Dynamic control of redundant manipulators. In Proceedings of the IEEE International Conference on Robotics and Automation Proceedings, Philadelphia, PA, USA, 24–29 April 1988; Volume 1, pp. 183–187. [Google Scholar]
- Wang, Y.; Wang, D.; Zhu, S. A New Navigation Function Based Decentralized Control of Multi-Vehicle Systems in Unknown Environments. J. Intell. Robot. Syst.
**2017**, 87, 363–377. [Google Scholar] [CrossRef] - Buss, S.R. Introduction to Inverse Kinematics with Jacobian Transpose, Pseudoinverse and Damped Least Squares Methods; Technical Report; Department of Mathematics, University of California: San Diego, CA, USA, 2004. [Google Scholar]
- Emika, F. Available online: https://www.franka.de/technology (accessed on 28 September 2023).
- GCtronic. Available online: https://www.gctronic.com/doc/index.php/Elisa-3 (accessed on 28 September 2023).
- Lawton, J.; Beard, R.; Young, B. A decentralized approach to formation maneuvers. IEEE Trans. Robot. Autom.
**2003**, 19, 933–941. [Google Scholar] [CrossRef]

**Figure 1.**Multi-agent PBC scheme of [26], depicted on a bidirectional edge.

**Figure 4.**Simulated end-effector synchronization of two robotic manipulators. (

**a**) IDA-PBC trajectories. (

**b**) r-PBC trajectories. (

**c**) End-effector coordinates over time for IDA-PBC (top) and r-PBC (bottom). (

**d**) The case with communication delays for IDA-PBC (top) and r-PBC (bottom).

**Figure 5.**Simulations of end-effector synchronization with an underactuated system (IDA-PBC) and subtask optimization (r-PBC), without communication delay. (

**a**) IDA-PBC trajectories with UAV and manipulator. (

**b**) r-PBC trajectories with subtasks. (

**c**) Generalized end-effector coordinates over time for IDA-PBC (top) and r-PBC (bottom). (

**d**) Joint-coordinates of the UAV in (

**a**) (top) and the blue manipulator in (

**b**) (bottom).

**Figure 7.**Experimental results of consensus control between heterogeneous systems, using the proposed cooperative IDA-PBC and r-PBC methods. (

**a**) Leader at $(-0.25,0)$, no communication delays. (

**b**) Leaderless, no communication delays. (

**c**) Leaderless, no communication delays.

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**MDPI and ACS Style**

de Groot, O.; Valk, L.; Keviczky, T.
Cooperative Passivity-Based Control of Nonlinear Mechanical Systems. *Robotics* **2023**, *12*, 142.
https://doi.org/10.3390/robotics12050142

**AMA Style**

de Groot O, Valk L, Keviczky T.
Cooperative Passivity-Based Control of Nonlinear Mechanical Systems. *Robotics*. 2023; 12(5):142.
https://doi.org/10.3390/robotics12050142

**Chicago/Turabian Style**

de Groot, Oscar, Laurens Valk, and Tamas Keviczky.
2023. "Cooperative Passivity-Based Control of Nonlinear Mechanical Systems" *Robotics* 12, no. 5: 142.
https://doi.org/10.3390/robotics12050142