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

• AcceptedAug 8, 2018
• PublishedNov 22, 2018
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### 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 $$V_2 = \frac{1}{2}(\lambda-\lambda^*)^2. \tag{28}$$

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

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*}

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• 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.

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