logo

SCIENCE CHINA Information Sciences, Volume 61, Issue 12: 120201(2018) https://doi.org/10.1007/s11432-018-9588-0

Cooperative transportation control of multiple mobile manipulators through distributed optimization

More info
  • ReceivedMay 15, 2018
  • AcceptedAug 8, 2018
  • PublishedNov 22, 2018

Abstract

This paper investigates the problem of distributed control of multiple redundant mobile manipulators to collectively transport an object tracking a desired trajectory with energy and manipulability optimized.To solve this optimization problem, formation control tasks are introduced as equality constraints with the variables being the velocities. In this paper, we propose a distributed proximal gradient algorithm searching for the optimal solution, with which the stability of the closed-loop system is proved. Simulations demonstrate the effectiveness of the proposed distributed optimization scheme and proximal algorithm.


Acknowledgment

This work was supported in part by National Natural Science Foundation of China (Grant Nos. 61621063, 61573062, 61603094), in part by Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT1208), in part by Beijing Education Committee Cooperation Building Foundation Project (Grant No. 2017CX02005), and in part by Beijing Advanced Innovation Center for Intelligent Robots and Systems (Beijing Institute of Technology), Key Laboratory of Biomimetic Robots and Systems (Beijing Institute of Technology), Ministry of Education, Beijing, China. The authors wished to thank Prof. Hao FANG, Dr. Xianlin ZENG, and Dr. Qingkai YANG for constructive comments and suggestions.


Supplement

Appendix

Convergence analysis

The algorithm proximal21, 17b can be rewritten as \begin{equation*}\begin{aligned} &\dot \omega + \omega= {\rm prox}_{g_1}[(1-C_{1})\omega+C_{2}J^{\rm T}(v-J\omega)+C_{3}\Upsilon] , \\ &\dot v + v= {\rm prox}_{g_{2}}[(1-C_{2})v+C_{2}J \omega-L^{\rm T}\lambda]. \end{aligned}\end{equation*}

It follows from 1 that \begin{align} &(1-C_{1})\omega+C_{2}J^{\rm T}(v-J\omega) +C_{3}\Upsilon-(\dot \omega + \omega) \in \partial g_{1}(\dot \omega + \omega), \tag{18} \\ &(1-C_{2})v+C_{2}J \omega-L^{\rm T}\lambda - (\dot v+ v) \in \partial {g_{2}}(\dot v+ v). \tag{19} \end{align} Let $(\omega^*,v^*,\lambda^*)$ be an equilibrium of algorithm 17a. The following can be obtained: \begin{align} &-C_{1}\omega^*+C_{2}J^{\rm T}(v^*-J\omega^*) + C_{3}\Upsilon \in \partial g_{1}( \omega^*), \tag{20} \\ &-C_{2}(v^*-J \omega^*)-L^{\rm T}\lambda^* \in \partial {g_{2}}( v^*). \tag{21} \end{align} Because $g_1(\cdot)$, $g_2(\cdot)$ are convex function, $\partial~g_1(\cdot)$ and $\partial~g_2(\cdot)$ are monotone. By combining 1821, it follows that \begin{align}&[-C_{1}\omega+C_{2}J^{\rm T}(v-J\omega) -\dot \omega +C_{1}\omega^* - C_{2}J^{\rm T}(v^*-J\omega^*)]^{\rm T}(\dot \omega +\omega-\omega^*) \ge 0, \tag{22} \\ &[-C_{2}(v-J \omega)-L^{\rm T}\lambda - \dot v +C_{2}(v^*-J \omega^*)+L^{\rm T}\lambda^* ]^{\rm T}(\dot v+v-v^*) \ge 0. \tag{23} \end{align} According to 22, the following is derived: \begin{align} &(C_{1}\omega^*-C_{1}\omega)^{\rm T}(\omega-\omega^*)+[C_{2}J^{\rm T}(v-J\omega)-C_{2}J^{\rm T}(v^*-J\omega^*)]^{\rm T}(\omega-\omega^*)-\dot\omega^{\rm T}\dot\omega \\ & \ge \dot\omega^{\rm T}(\omega-\omega^*)+(C_{1}\omega-C_{1}\omega^*)^{\rm T}\dot \omega+[C_{2}J^{\rm T}(v^*-J\omega^*)-C_{2}J^{\rm T}(v-J\omega)]^{\rm T}\dot \omega. \tag{24} \end{align} Then, 24 is rewritten as \begin{equation*}\begin{aligned} & -(\omega-\omega^*)^{\rm T}(\nabla f_1(\omega)-\nabla f_1(\omega^*))-(\omega-\omega^*)^{\rm T}(\nabla_\omega f_2(\omega,v)-\nabla_\omega f_2(\omega^*,v^*)) -\dot\omega^{\rm T}\dot\omega \\ & \ge \dot\omega^{\rm T}(\omega-\omega^*) + \dot \omega^{\rm T}(\nabla f_1(\omega)-\nabla f_1(\omega^*)) + \dot \omega^{\rm T}(\nabla_\omega f_2(\omega,v)-\nabla_\omega f_2(\omega^*,v^*)). \end{aligned}\end{equation*}

According to 23, the following can be obtained: \begin{align} & -[C_{2}J^{\rm T}(v^*-J\omega^*)-C_{2}J^{\rm T}(v-J\omega)]^{\rm T}(v^*-v)+(\lambda^*-\lambda)^{\rm T}L[v+\dot v +(x-d)]-\dot v^{\rm T}\dot v \\ & \ge \dot v^{\rm T}(v-v^*) +[C_{2}J^{\rm T}(v-J\omega)-C_{2}J^{\rm T}(v^*-J\omega^*)]^{\rm T}\dot v. \tag{25} \end{align} Then, 25 can be rewritten as \begin{equation*}\begin{aligned} & -(v-v^*)^{\rm T}(\nabla_v f_2(\omega,v)-\nabla_u f_2(\omega^*,v^*))+(\lambda^*-\lambda)^{\rm T}L[v+\dot v +(x-d)]-\dot v^{\rm T}\dot v \\ & \ge \dot v^{\rm T}(v-v^*) + \dot v^{\rm T}(\nabla_v f_2(\omega,v)- \nabla_v f_2(\omega^*,v^*)). \end{aligned}\end{equation*}

The function $V_1(\omega,u)$ is constructed as \begin{align} V_1 =& \frac{1}{2}(\omega-\omega^*)^2+\frac{1}{2}(v-v^*)^2+[f_1(\omega)-f_1(\omega^*)-(\omega-\omega^*)^{\rm T}\nabla f_1(\omega^*)] \\ & +[f_2(\omega,v)-f_2(\omega^*,v^*)-(\omega-\omega^*)^{\rm T}\nabla_\omega f_2(\omega^*,v^*) -(v-v^*)^{\rm T}\nabla_v f_2(\omega^*,v^*)]. \tag{26} \end{align} Computing time derivative of $V_1$ in 26, the following can be derived: \begin{align} \dot V_1 =& \dot \omega^{\rm T}(\omega-\omega^*)+\dot \omega^{\rm T} (\nabla f_1(\omega)-\nabla f_1(\omega^*))+\dot \omega^{\rm T}(\nabla_\omega f_2(\omega,v)-\nabla_\omega f_2(\omega^*,v^*)) \\ &+\dot v^{\rm T}(v-v^*)+\dot v^{\rm T}(\nabla_v f_2(\omega,v)-\nabla_v f_2(\omega^*,v^*) ) \\ \le& -(\omega-\omega^*)^{\rm T}[\nabla f_1(\omega)-\nabla f_1(\omega^*)] \\ & -(\omega-\omega^*)^{\rm T}[\nabla_\omega f_2(\omega,v)-\nabla_\omega f_2(\omega^*,v^*)] \\ & -(v-v^*)^{\rm T}[\nabla_v f_2(\omega,v)-\nabla_v f_2(\omega^*,v^*)] \\ & +(\lambda^*-\lambda)^{\rm T}L[v+\dot v +(x-d)]-\dot\omega^{\rm T}\dot\omega-\dot v^{\rm T}\dot v. \tag{27} \end{align} Based on $\dot~V_1$ 27, another function $V_2(\lambda)$ is constructed as \begin{equation} V_2 = \frac{1}{2}(\lambda-\lambda^*)^2. \tag{28}\end{equation}

By calculating the time derivative of $V_2$, $\dot~V_2$ can be obtained \begin{equation} \dot V_2 = \dot \lambda^{\rm T}(\lambda-\lambda^*) =(\lambda-\lambda^*)^{\rm T} L[v+\dot v + (x-d)]. \tag{29}\end{equation}

It follows from 27 and 29 that \begin{equation*}\begin{aligned} \dot V =& \dot V_1+\dot V_2 \\ \le& -(\omega-\omega^*)^{\rm T}[\nabla f_1(\omega)-\nabla f_1(\omega^*)] \\ & -(\omega-\omega^*)^{\rm T}[\nabla_\omega f_2(\omega,v)-\nabla_\omega f_2(\omega^*,v^*)] \\ & -(v-v^*)^{\rm T}[\nabla_v f_2(\omega,v)-\nabla_v f_2(\omega^*,v^*)] \\ & -\dot\omega^{\rm T}\dot\omega-\dot v^{\rm T}\dot v \\ \le & -\dot\omega^{\rm T}\dot\omega-\dot v^{\rm T}\dot v \\ \le & 0. \end{aligned}\end{equation*}

As a result, $\{(\omega,v,\lambda):\dot~V=0\}~\subset\{(\omega,v,\lambda):~\dot~\omega=0,~\dot~v=0~\}$. In addition, because $f_1$ and $f_2$ are convex functions, then \begin{align*}&f_1(\omega)-f_1(\omega^*)-(\omega-\omega^*)^{\rm T}\nabla f_1(\omega^*) \ge 0, \\ &f_2(\omega,v)-f_2(\omega^*,v^*)-(\omega-\omega^*)^{\rm T}\nabla_\omega f_2(\omega^*,v^*) -(v-v^*)^{\rm T}\nabla_v f_2(\omega^*,v^*) \ge 0. \end{align*} By combining 26 and 28, it follows that \begin{equation*}\begin{aligned} V=& V_1+V_2 \\ = & \frac{1}{2}(\omega-\omega^*)^2+\frac{1}{2}(v-v^*)^2 +[f_1(\omega)-f_1(\omega^*)-(\omega-\omega^*)^{\rm T}\nabla f_1(\omega^*)] \\ &+[f_2(\omega,v)-f_2(\omega^*,v^*)-(\omega-\omega^*)^{\rm T}\nabla_\omega f_2(\omega^*,v^*)-(v-v^*)^{\rm T}\nabla_v f_2(\omega^*,v^*)] +\frac{1}{2}(\lambda-\lambda^*)^2 \\ \ge& \frac{1}{2}(\omega-\omega^*)^2+\frac{1}{2}(v-v^*)^2 + \frac{1}{2}(\lambda-\lambda^*)^2 \\ \ge & 0. \end{aligned}\end{equation*}


References

[1] Huntsberger T L, Trebi-Ollennu A, Aghazarian H. Distributed Control of Multi-Robot Systems Engaged in Tightly Coupled Tasks. Autonomous Robots, 2004, 17: 79-92 CrossRef Google Scholar

[2] Farivarnejad H, Wilson S, Berman S. Decentralized sliding mode control for autonomous collective transport by multi-robot systems. In: Proceedings of IEEE Conference on Decision and Control, Las Vegas, 2016. 1826--1833. Google Scholar

[3] Yoshikawa T. Foundations of Robotics: Analysis and Control. Cambridge: The MIT Press, 1990. Google Scholar

[4] Li Z J, Ge S S. Fundamentals in Modeling and Control of Mobile Manipulators. Boca Raton: CRC Press, 2013. Google Scholar

[5] Abeygunawardhana P K W, Murakami T. Vibration Suppression of Two-Wheel Mobile Manipulator Using Resonance-Ratio-Control-Based Null-Space Control. IEEE Trans Ind Electron, 2010, 57: 4137-4146 CrossRef Google Scholar

[6] Wei W. Neuro-fuzzy and model-based motion control for mobile manipulator among dynamic obstacles. Sci China Ser F, 2003, 46: 14-30 CrossRef Google Scholar

[7] Alonso-Mora J, Knepper R, Siegwart R, et al. Local motion planning for collaborative multi-robot manipulation of deformable objects. In: Proceedings of IEEE International Conference on Robotics and Automation, Seattle, 2015. 5495--5502. Google Scholar

[8] Li Z, Ge S S, Adams M. Robust adaptive control of uncertain force/motion constrained nonholonomic mobile manipulators. Automatica, 2008, 44: 776-784 CrossRef Google Scholar

[9] Tinos R, Terra M H, Ishihara J Y. Motion and force control of cooperative robotic manipulators with passive joints. IEEE Trans Contr Syst Technol, 2006, 14: 725-734 CrossRef Google Scholar

[10] Khatib, O, Yokoi K, Chang K. Coordination and decentralized cooperation of multiple mobile manipulators. J Robotic Syst, 1996, 13: 755-764 CrossRef Google Scholar

[11] Wang Z, Schwager M. Force-Amplifying N-robot Transport System (Force-ANTS) for cooperative planar manipulation without communication. Int J Robotics Res, 2016, 35: 1564-1586 CrossRef Google Scholar

[12] Pereira G A S, Campos M F M, Kumar V. Decentralized Algorithms for Multi-Robot Manipulation via Caging. Int J Robotics Res, 2004, 23: 783-795 CrossRef Google Scholar

[13] Fink J, Hsieh M A, Kumar V. Multi-robot manipulation via caging in environments with obstacles. In: Proceedings of IEEE International Conference on Robotics and Automation, Pasadena, 2008. 1471--1476. Google Scholar

[14] Wang L J, Meng B. Characteristic model-based consensus of networked heterogeneous robotic manipulators with dynamic uncertainties. Sci China Technol Sci, 2016, 59: 63-71 CrossRef Google Scholar

[15] Dong Y, Chen J, Huang J. A self-tuning adaptive distributed observer approach to the cooperative output regulation problem for networked multi-agent systems. Int J Control, 2017. doi: 10.1080/00207179.2017.1411610. Google Scholar

[16] Liu X Y, Dou L H, Sun J. Consensus for networked multi-agent systems with unknown communication delays. J Franklin I, 2016, 53: 4176-4190. Google Scholar

[17] Dong Y, Chen J, Huang J. Cooperative Robust Output Regulation for Second-Order Nonlinear Multiagent Systems With an Unknown Exosystem. IEEE Trans Autom Control, 2018, 63: 3418-3425 CrossRef Google Scholar

[18] Huang J. The Consensus for Discrete-Time Linear Multi-Agent Systems Under Directed Switching Networks. IEEE Trans Autom Control, 2017, 62: 4086-4092 CrossRef Google Scholar

[19] Yang Q, Fang H, Chen J. Distributed Global Output-Feedback Control for a Class of Euler-Lagrange Systems. IEEE Trans Autom Control, 2017, 62: 4855-4861 CrossRef Google Scholar

[20] Li S, Zhang Y, Jin L. Kinematic Control of Redundant Manipulators Using Neural Networks.. IEEE Trans Neural Netw Learning Syst, 2017, 28: 2243-2254 CrossRef PubMed Google Scholar

[21] Cai B, Zhang Y. Different-Level Redundancy-Resolution and Its Equivalent Relationship Analysis for Robot Manipulators Using Gradient-Descent and Zhang 's Neural-Dynamic Methods. IEEE Trans Ind Electron, 2012, 59: 3146-3155 CrossRef Google Scholar

[22] Jin L, Li S, La H M. Manipulability Optimization of Redundant Manipulators Using Dynamic Neural Networks. IEEE Trans Ind Electron, 2017, 64: 4710-4720 CrossRef Google Scholar

[23] Li S, He J, Li Y. Distributed Recurrent Neural Networks for Cooperative Control of Manipulators: A Game-Theoretic Perspective.. IEEE Trans Neural Netw Learning Syst, 2017, 28: 415-426 CrossRef PubMed Google Scholar

[24] Liang S, Zeng X, Hong Y. Distributed Nonsmooth Optimization With Coupled Inequality Constraints via Modified Lagrangian Function. IEEE Trans Autom Control, 2018, 63: 1753-1759 CrossRef Google Scholar

[25] Liu Q, Wang J. A Second-Order Multi-Agent Network for Bound-Constrained Distributed Optimization. IEEE Trans Autom Control, 2015, 60: 3310-3315 CrossRef Google Scholar

[26] Zeng X, Yi P, Hong Y. Distributed Continuous-Time Algorithm for Constrained Convex Optimizations via Nonsmooth Analysis Approach. IEEE Trans Autom Control, 2017, 62: 5227-5233 CrossRef Google Scholar

[27] Yi P, Hong Y, Liu F. Distributed gradient algorithm for constrained optimization with application to load sharing in power systems. Syst Control Lett, 2015, 83: 45-52 CrossRef Google Scholar

[28] Tang C, Li X, Wang Z. Cooperation and distributed optimization for the unreliable wireless game with indirect reciprocity. Sci China Inf Sci, 2017, 60: 110205 CrossRef Google Scholar

[29] Chen J, Gan M G, Huang J, et al. Formation control of multiple Euler-Lagrange systems via null-space-based behavioral control. Sci China Inf Sci, 2016, 59: 010202. Google Scholar

[30] Li C H, Zhang E C, Jiu L, et al. Optimal control on special Euclidean group via natural gradient algorithm. Sci China Inf Sci, 2016, 59: 112203. Google Scholar

[31] Fang H, Shang C S, Chen J. An optimization-based shared control framework with applications in multi-robot systems. Sci China Inf Sci, 2018, 61: 014201. Google Scholar

[32] Taghirad H D, Bedoustani Y B. An Analytic-Iterative Redundancy Resolution Scheme for Cable-Driven Redundant Parallel Manipulators. IEEE Trans Robot, 2011, 27: 1137-1143 CrossRef Google Scholar

[33] Patchaikani P K, Behera L, Prasad G. A Single Network Adaptive Critic-Based Redundancy Resolution Scheme for Robot Manipulators. IEEE Trans Ind Electron, 2012, 59: 3241-3253 CrossRef Google Scholar

[34] Li S, Chen S, Liu B. Decentralized kinematic control of a class of collaborative redundant manipulators via recurrent neural networks. Neurocomputing, 2012, 91: 1-10 CrossRef Google Scholar

[35] Zhang Y, Ge S S, Lee T H. A Unified Quadratic-Programming-Based Dynamical System Approach to Joint Torque Optimization of Physically Constrained Redundant Manipulators. IEEE Trans Syst Man Cybern B, 2004, 34: 2126-2132 CrossRef Google Scholar

[36] Tang W S, Wang J. A recurrent neural network for minimum infinity-norm kinematic control of redundant manipulators with an improved problem formulation and reduced architecture complexity. IEEE Trans Syst Man Cybern B, 2001, 31: 98-105 CrossRef PubMed Google Scholar

[37] Gueaieb W, Karray F, Al-Sharhan S. A Robust Hybrid Intelligent Position/Force Control Scheme for Cooperative Manipulators. IEEE/ASME Trans Mechatron, 2007, 12: 109-125 CrossRef Google Scholar

[38] Parikh N, Boyd S. Proximal algorithms. Found & Trend in Opt, 2013, 1: 127-239. Google Scholar

[39] Boyd S, Vandenberghe L. Convex Optimization. Cambridge: Cambridge University Press, 2004. Google Scholar

[40] Haddad W M, Chellaboina V. Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach. Princeton: Princeton University Press, 2008. Google Scholar

[41] Yoshikawa T. Manipulability of robotic mechanisms. Int J Robot Res, 1985, 4: 3-9. Google Scholar

[42] Sverdrup-Thygeson J, Moe S, Pettersen K Y, et al. Kinematic singularity avoidance for robot manipulators using set-based manipulability tasks. In: Proceedings of IEEE Conference on Control Technology and Applications, Kohala Coast, 2017. 142--149. Google Scholar

  • Figure 1

    (Color online) Transportation task of multiple mobile manipulators.

  • Figure 2

    (Color online) Simulation results on object-tracking error using the proposed scheme 6afor transporting the object tracking a circular path. (a) Object-tracking error of the $X$-axis; (b) object-tracking error of the $Y$-axis.

  • Figure 3

    (Color online) Simulation results on position errors and velocity errors of end-effectors using the proposed scheme 6afor transporting the object tracking a circular path. (a) Position error of end-effectors of the $X$-axis; (b) position error of end-effectors of the $Y$-axis; (c) velocity error of end-effectors of the $X$-axis; (d) velocity error of end-effectors of the $Y$-axis.

  • Figure 4

    (Color online) Simulation results on manipulability of four mobile manipulators using the proposed scheme 6afor transporting the object tracking a circular path.

  • Figure 5

    (Color online) Simulation results on velocity limits using the proposed scheme 6afor transporting the object tracking a circular path. (a) End-effector velocities of the $X$-axis; (b) end-effector velocities of the $Y$-axis; (c) angular velocities of joint $1$; (d) angular velocities of joint $2$.

  • Figure 6

    (Color online) Simulation results on manipulability change with different coefficients. (a) Manipulability change without manipulability optimization in the first simulation; (b) manipulability change with manipulability optimization with constant $c_{i3}$ in the second simulation; (c) manipulability change with manipulability optimization with adaptive $c_{i3}$ in the third simulation.

Copyright 2019 Science China Press Co., Ltd. 《中国科学》杂志社有限责任公司 版权所有

京ICP备18024590号-1