# Design and Analysis of a Variable Inertia Spatial Robotic Tail for Dynamic Stabilization

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Mechanical and Mechatronic Design

#### 2.1. Mechanical Design

#### 2.2. Mechatronics Design

## 3. System Modeling

#### 3.1. Kinematic Modeling

#### 3.2. Dynamics Modeling

## 4. Validation of Dynamic Model and Simulation Results

## 5. Robot Stabilization Using the Robotic Tail

#### 5.1. Biped Robot-Tail System

#### 5.2. Trajectory Planner

#### 5.3. Virtual Torque Estimator

#### 5.4. Model Based Controller

#### 5.5. Controller Performance

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Alexander, R.M.; Vernon, A. The mechanics of hopping by kangaroos (Macropodidae). J. Zool.
**2009**, 177, 265–303. [Google Scholar] [CrossRef] - HICKMAN, G.C. The mammalian tail: A review of functions. Mammal Rev.
**1979**, 9, 143–157. [Google Scholar] [CrossRef] - Yang, L.; Su, Y.; Xiao, Q. Numerical study of propulsion mechanism for oscillating rigid and flexible tuna-tails. J. Bionic Eng.
**2011**, 8, 406–417. [Google Scholar] [CrossRef] - Jusufi, A.; Goldman, D.I.; Revzen, S.; Full, R.J. Active tails enhance arboreal acrobatics in geckos. Proc. Natl. Acad. Sci. USA
**2008**, 105, 4215–4219. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Libby, T.; Moore, T.Y.; Chang-Siu, E.; Li, D.; Cohen, D.J.; Jusufi, A.; Full, R.J. Tail-assisted pitch control in lizards, robots and dinosaurs. Nature
**2012**, 481, 181–184. [Google Scholar] [CrossRef] [PubMed] - Jusufi, A.; Kawano, D.T.; Libby, T.; Full, R.J. Righting and turning in mid-air using appendage inertia: Reptile tails, analytical models and bio-inspired robots. Bioinspir. Biomim.
**2010**, 5, 045001. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bartholomew, G.A.; Caswell, H.H. Locomotion in Kangaroo Rats and Its Adaptive Significance. J. Mammal.
**1951**, 32, 155. [Google Scholar] [CrossRef] - Liu, G.H.; Lin, H.Y.; Lin, H.Y.; Chen, S.T.; Lin, P.C. A Bio-Inspired Hopping Kangaroo Robot with an Active Tail. J. Bionic Eng.
**2014**, 11, 541–555. [Google Scholar] [CrossRef] - Patel, A.; Braae, M. Rapid turning at high-speed: Inspirations from the cheetah’s tail. In Proceedings of the 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems, Tokyo, Japan, 3–7 November 2013; IEEE: Piscataway, NJ, USA, 2013; pp. 5506–5511. [Google Scholar] [CrossRef]
- Patel, A.; Boje, E. On the Conical Motion of a Two-Degree-of-Freedom Tail Inspired by the Cheetah. IEEE Trans. Robot.
**2015**, 31, 1555–1560. [Google Scholar] [CrossRef] - Dharmawan, A.G.; Koh, D.C.Y.; Soh, G.S.; Foong, S.; Bouffanais, R.; Wood, K.L. Tail Design of A Miniature Two-Wheg Climbing Robot for External Transitioning. In Mechanisms and Machine Science; Springer: Dordrecht, The Netherlands, 2019; Volume 73, pp. 2139–2148. [Google Scholar] [CrossRef]
- Suarez, A.; Grau, P.; Heredia, G.; Ollero, A. Winged Aerial Manipulation Robot with Dual Arm and Tail. Appl. Sci.
**2020**, 10, 4783. [Google Scholar] [CrossRef] - Rone, W.S.; Ben-Tzvi, P. Continuum Robotic Tail Loading Analysis for Mobile Robot Stabilization and Maneuvering. In Volume 5A: 38th Mechanisms and Robotics Conference; American Society of Mechanical Engineers: New York, NY, USA, 2014; Volume 5A. [Google Scholar] [CrossRef]
- Saab, W.; Rone, W.S.; Ben-Tzvi, P. Discrete modular serpentine robotic tail: Design, analysis and experimentation. Robotica
**2018**, 36, 994–1018. [Google Scholar] [CrossRef] [Green Version] - Rone, W.S.; Ben-Tzvi, P. Mechanics Modeling of Multisegment Rod-Driven Continuum Robots. J. Mech. Robot.
**2014**, 6. [Google Scholar] [CrossRef] [Green Version] - Singh, A.; Libby, T.; Fuller, S.B. Rapid Inertial Reorientation of an Aerial Insect-sized Robot Using a Piezo-actuated Tail. In Proceedings of the 2019 International Conference on Robotics and Automation (ICRA), Montreal, QC, Canada, 20–24 May 2019; IEEE: Piscataway, NJ, USA, 2019; pp. 4154–4160. [Google Scholar] [CrossRef]
- Berenguer, F.; Monasterio-Huelin, F. Zappa, a Quasi-Passive Biped Walking Robot with a Tail: Modeling, Behavior, and Kinematic Estimation Using Accelerometers. IEEE Trans. Ind. Electron.
**2008**, 55, 3281–3289. [Google Scholar] [CrossRef] - Briggs, R.; Lee, J.; Haberland, M.; Kim, S. Tails in biomimetic design: Analysis, simulation, and experiment. In Proceedings of the 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems, Vilamoura, Portugal, 7–12 October 2012; IEEE: Piscataway, NJ, USA, 2012; pp. 1473–1480. [Google Scholar] [CrossRef]
- Liu, Y.; Wang, J.; Ben-Tzvi, P. A Cable Length Invariant Robotic Tail Using a Circular Shape Universal Joint Mechanism. J. Mech. Robot.
**2019**, 11. [Google Scholar] [CrossRef] - Transeth, A.A.; Pettersen, K.Y.; Liljebäck, P. A survey on snake robot modeling and locomotion. Robotica
**2009**, 27, 999–1015. [Google Scholar] [CrossRef] [Green Version] - Jazar, R.N. Theory of Applied Robotics; Springer: Boston, MA, USA, 2007. [Google Scholar] [CrossRef]
- Saab, W.; Rone, W.S.; Ben-Tzvi, P. Robotic Modular Leg: Design, Analysis, and Experimentation. J. Mech. Robot.
**2017**, 9. [Google Scholar] [CrossRef] - Fernandes, C.; Gurvits, L.; Li, Z. Attitude Control of a Space Platform/Manipulator System Using Internal Motion. Int. J. Robot. Res.
**1994**, 13, 289–304. [Google Scholar] [CrossRef] - Chang-Siu, E.; Libby, T.; Tomizuka, M.; Full, R.J. A lizard-inspired active tail enables rapid maneuvers and dynamic stabilization in a terrestrial robot. In Proceedings of the 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems, San Francisco, CA, USA, 25–30 September 2011; IEEE: Piscataway, NJ, USA, 2011; pp. 1887–1894. [Google Scholar] [CrossRef]
- Keo, L.; Yamakita, M. Controlling balancer and steering for bicycle stabilization. In Proceedings of the 2009 IEEE/RSJ International Conference on Intelligent Robots and Systems, St. Louis, MO, USA, 10–15 October 2009; IEEE: Piscataway, NJ, USA, 2009; pp. 4541–4546. [Google Scholar] [CrossRef]
- Alcaraz-Jiménez, J.; Herrero-Pérez, D.; Martínez-Barberá, H. Robust feedback control of ZMP-based gait for the humanoid robot Nao. Int. J. Robot. Res.
**2013**, 32, 1074–1088. [Google Scholar] [CrossRef] - Ugurlu, B. Bipedal Motion Planning Based on Composite Rigid Body Angular Momentum Resolution. Ph.D. Thesis, Yokohama National University, Yokohama, Japan, 2010. [Google Scholar]

**Figure 1.**(

**A**) The prototype of the robotic tail design. (

**B**) Link assignments and details of the robotic tail mechanical components.

**Figure 4.**The designed trajectory for the end effector mass. (

**A**) Position, velocity, and acceleration of end effector in X, Y, and Z components. (

**B**) Positions of the end effector in space.

**Figure 5.**The torques generated at the base when the end effector is at the (

**A**) lowest position (

**B**) highest position along Link 3.

**Figure 7.**ZMP illustration with simplified mathematical model. (

**A**) an illusion of the tail robot attached to a biped robot with a tilt angle $\alpha $. (

**B**) a simplified mathmatical model of the Biped Robot-Tail System with a tilt angle $\alpha $ and actuated tail angle $\beta $.

**Figure 8.**(

**A**) $\alpha $, $\beta $ and $\dot{\beta}$ trajectories for $c=10$ and $d=\pi /2$ and ${\dot{\alpha}}_{0}=0.587$ rad/s. (

**B**) $\alpha $, $\beta $ and $\dot{\beta}$ trajectories for $a=0.44$, $b=1.35$, $d=\pi /2$ and ${\dot{\alpha}}_{0}=0.587$ rad/s with different c.

**Figure 9.**(

**A**) Desired trajectory of $\alpha $. (

**B**) Expected trajectory of $\alpha $ obtained from inverse dynamic model. (

**C**,

**D**) Expected trajectory of $\beta $ and $\dot{\beta}$ obtained from inverse dynamic model.

**Figure 11.**Robot stabilization using tail dynamics: (

**A**) without virtual torque estimator. (

**B**) With virtual torque estimator in trajectory generation (using end effector linear motion) when subjected to an impulse of 2.5 Nm-s.

i | a | $\mathit{\alpha}$ | $\mathit{\theta}$ | d |
---|---|---|---|---|

0 | 0 | 0 | 0 | ${\delta}_{B}$ |

1 | 0 | 90${}^{\circ}$ | ${\theta}_{1}$ | 0 |

2 | 0 | 90${}^{\circ}$ | ${\theta}_{2}$ | 0 |

3 | 0 | 90${}^{\circ}$ | ${\theta}_{3}$ | 0 |

4 | 0 | 0 | 0 | $\delta $ |

Parameter | Length (m) | Mass (kg) |
---|---|---|

Link0 | $0.094$ | $0.76$ |

Link1 | $0.073$ | $0.33$ |

Link2 | $0.139$ | $0.1$ |

Link3 | $0.416$ | $0.33$ |

Total Mass of Tail | − | $1.69$ |

Mass of End Effector | − | $0.17$ |

Parameter | Unit |
---|---|

Height of Biped Robot | 0.482 (m) |

${l}_{CoM}$ | 0.382 (m) |

Mass of Biped Robot | 15.88 (Kg) |

Mass of End Effector | 0.83 (Kg) |

Total Mass of Biped Robot –Tail system | 18.3 (Kg) |

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

© 2020 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, X.; Ren, H.; Kumar, A.; Ben-Tzvi, P.
Design and Analysis of a Variable Inertia Spatial Robotic Tail for Dynamic Stabilization. *Biomimetics* **2020**, *5*, 55.
https://doi.org/10.3390/biomimetics5040055

**AMA Style**

Wang X, Ren H, Kumar A, Ben-Tzvi P.
Design and Analysis of a Variable Inertia Spatial Robotic Tail for Dynamic Stabilization. *Biomimetics*. 2020; 5(4):55.
https://doi.org/10.3390/biomimetics5040055

**Chicago/Turabian Style**

Wang, Xinran, Hailin Ren, Anil Kumar, and Pinhas Ben-Tzvi.
2020. "Design and Analysis of a Variable Inertia Spatial Robotic Tail for Dynamic Stabilization" *Biomimetics* 5, no. 4: 55.
https://doi.org/10.3390/biomimetics5040055