Distributed Newton and Quasi-Newton methods for formation control of autonomous vehicles

ABSTRACT

This paper presents a distributed approach based on a Newton-type method for solving the fast formation problems of a group of autonomous underwater vehicles (AUVs) in the plane. Each AUV of this group aims at minimising its own local alignment error function that formulates the difference between the desired relative formation of the AUV and its neighbours and their current positions. In addition, the group jointly minimises the total cost composed by local alignment error functions. The presented approach utilises a modified Jacobi algorithm based on local position data to approximate the Newton direction. It is shown that when combining the descent direction with distributed line search algorithms, the approach exhibits the performance of super-linear convergence within the neighbourhood of the desired position. Two numerical examples are considered here to illustrate that using the proposed approach, all AUVs rapidly achieve the desired formation from any initial configuration and initial position in both static and dynamic formation scenarios.

KEYWORDS:

• Autonomous underwater vehicles
• distributed newton method
• formation control
• leader-follower structure

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes on contributors

Mansour Karkoub received the BS degree in mechanical engineering (with highest distinction) in 1984, the MSME degree in 1990, the PhD degree in 1994, all from the University of Minnesota, Minneapolis, MN, USA, and the HDR (Habilitation Diriger des Recherches) degree from the University of Versailles, Versailles, France, in 2003. Previously he held a faculty positions at Kuwait University, the Petroleum Institute, and the French National Institute for Research in Informatics and Automation, Rocquencourt, France. He is currently a Professor of mechanical engineering with Texas A&M University at Qatar, Doha, Qatar. His research interests include robust control, robotics, autonomous systems, mechatronics, and vibration engineering. Dr Karkoub is a Fellow of ASME and a Fellow of the Institute of Engineering Technology.

Huiwei Wang received the BS degree in information and computing science and the ME degree in computer application from Chongqing Jiao Tong University, Chongqing, China, in 2008 and 2011, respectively, and the PhD degree in computer science from Chongqing University, Chongqing, in 2014. He is currently a Lecturer with the College of Electronic and Information Engineering, Southwest University, Chongqing. He was a Postdoctoral Research Associate from 2014 to 2016 and a Program Aide from 2012 to 2013 with Texas A&M University at Qatar, Doha, Qatar. His research interests include neural networks, multi-agent networks, wireless sensor networks, cyber physical system, and smart grids.

Tzu-Sung Wu received the BS degree in electrical engineering from Lunghwa University of Science and Technology, Taoyuan, Taiwan, ROC, in 2003, and the MS and PhD degrees in electrical engineering from Tatung University, Taipei, Taiwan, in 2005 and 2011, respectively. He is currently an Engineer with the Automotive Research & Testing Center, Changhua, Taiwan. Before joining the Automotive Research & Testing Center, he was a Research Scientist of mechanical engineering at Texas A&M University at Qatar. His major research interests include robust control, robotics, fuzzy control, and adaptive control.

Notes

1 Throughout this paper, we denote by $\parallel \mathbf{x}{\parallel }_A^2={\mathbf{x}}^{\mathrm{\top }}A\mathbf{x}$ the weighted norm of vector $\mathbf{x}$ for a matrix A with proper dimensions.

2 Theorem 3.1.4. [4] Let $\parallel \cdot \parallel$ be any norm on ${\mathbb{R}}^{N\times N}$ that obeys $\parallel AB\parallel \le \parallel A\parallel \parallel B\parallel$ and let $E\in {\mathbb{R}}^{N\times N}$. If $\parallel E\parallel <1$, then $(I-E{)}^{-1}$ exists and $\parallel (I-E{)}^{-1}\parallel \le 1/[1-\parallel E\parallel ]$. If A is nonsingular and $\parallel A^{-1}(B-A)\parallel <1$, then B is nonsingular and $\parallel B^{-1}\parallel \le \parallel A^{-1}\parallel /[1-\parallel A^{-1}(B-A)\parallel ]$.