Distributed Static and Dynamic Circumnavigation Control with Arbitrary Spacings for a Heterogeneous Multi-robot System


Circumnavigation control algorithms enable multiple robots to rotate around a target while they still preserve a circular formation, which is useful in real world applications such as entrapping a hostile target. Specifically, four quantities are involved: the circumnavigation radius, the angular speed, the height and the phase differences among robots, which are termed spacings in this paper. Based on whether these quantities vary or not, the circumnavigation control problem is divided into two categories: the static one and the dynamic one. Corresponding to these two classes, distributed control algorithms are proposed for any number of mobile robots in random 3D positions to circumnavigate a target with arbitrarily given spacings or dynamic spacings. It should be noted that arbitrary spacings or dynamic spacings are useful for a heterogeneous multi-robot system in which robots may possess different kinematics capabilities; robots with higher movement speeds, for instance, can compensate for the insufficiency of those with lower movement speeds by decreasing the corresponding spacings. The robots can only perceive the positions of their two neighbouring robots, so the proposed control algorithms are distributed and scalable. Simulations along with real-robot experiments using soccer-playing robots are conducted to validate the theoretical results.

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  1. 1.

    Antsaklis, P., Michel, A.N.: Linear systems. Springer Science & Business Media (2006)

  2. 2.

    Chen, W.H.: Disturbance observer based control for nonlinear systems. IEEE/ASME Trans. Mechatron. 9(4), 706–710 (2004)

  3. 3.

    Dai, W., Yao, W., Luo, S., Ma, J., Wang, R., Hong, S., Zhou, Z., Li, X., Han, B., Xiao, J., et al.: Nubot team description paper (2018)

  4. 4.

    Fax, J.A., Murray, R.M.: Information flow and cooperative control of vehicle formations. IEEE Trans. Autom. Control 35(1), 115–120 (2002)

  5. 5.

    Franchi, A., Stegagno, P., Oriolo, G.: Decentralized multi-robot encirclement of a 3D target with guaranteed collision avoidance. Auton. Robot. 40(2), 245–265 (2016).

  6. 6.

    Gowers, T., Barrow-Green, J., Leader, I.: The Princeton Companion to Mathematics. Princeton University Press, Princeton (2010)

  7. 7.

    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (2012)

  8. 8.

    Lu, H., Yang, S., Zhang, H., Zheng, Z.: A robust omnidirectional vision sensor for soccer robots. Mechatronics 21(2), 373–389 (2011)

  9. 9.

    Marshall, B.J.A., Broucke, M.E., Francis, B.A.: Formation of vehicles in cyclic pursuit. IEEE Trans. Autom. Control 49(11), 1963–1974 (2015)

  10. 10.

    Mesbahi, M., Egerstedt, M.: Graph Theoretic Methods in Multiagent Networks. Princeton University Press, Princeton (2010)

  11. 11.

    Minc, H.: Nonnegative Matrices. Wiley, New York (1988)

  12. 12.

    Olfati-Saber, R., Murray, R.M.: Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans. Autom. Control 49(9), 1520–1533 (2004).

  13. 13.

    Pavone, M., Frazzoli, E.: Decentralized policies for geometric pattern formation and path coverage. J. Dyn. Syst. Meas. Control. 129(5), 633–643 (2007)

  14. 14.

    Ren, W., Beard, R.W.: Distributed Consensus in Multi-vehicle Cooperative Control. Springer, Berlin (2008)

  15. 15.

    Tang, S., Shinzaki, D., Lowe, G.C., Clark, C.M.: Multi-robot control for circumnavigation of particle distributions. Springer Tracts in Advanced Robotics 104, 149–152 (2014)

  16. 16.

    Wang, C., Xie, G., Cao, M.: Forming circle formations of anonymous mobile agents with order preservation. IEEE Trans. Autom. Control 58(12), 3248–3254 (2013)

  17. 17.

    Wang, C., Xie, G., Cao, M.: Controlling anonymous mobile agents with unidirectional locomotion to form formations on a circle. Automatica 50(4), 1100–1108 (2014)

  18. 18.

    Wang, X., Zeng, Z., Cong, Y.: Multi-agent distributed coordination control: developments and directions via graph viewpoint. Neurocomputing 199, 204–218 (2016).

  19. 19.

    Xiong, D., Xiao, J., Lu, H., Zeng, Z., Yu, Q., Huang, K., Yi, X., Zheng, Z.: The design of an intelligent soccer-playing robot. Ind. Robot. 43, 91–102 (2016).

  20. 20.

    Yao, W., Dai, W., Xiao, J., Lu, H., Zheng, Z.: A simulation system based on ros and gazebo for robocup middle size league. In: 2015 IEEE International Conference on Robotics and Biomimetics (ROBIO), pp 54–59 (2015),

  21. 21.

    Yao, W., Luo, S., Lu, H., Xiao, J.: Distributed circumnavigation control with dynamic spacing for a heterogeneous multi-robot system. In: 2018 RoboCup Symposium. Springer, Berlin (2018)

  22. 22.

    Yao, W., Zeng, Z., Wang, X., Lu, H., Zheng, Z.: Distributed encirclement control with arbitrary spacing for multiple anonymous mobile robots. In: 2017 36th Chinese Control Conference (CCC), pp 8800–8805 (2017),

  23. 23.

    Yu, X., Liu, L.: Distributed circular formation control of ring-networked nonholonomic vehicles. Automatica 68, 92–99 (2016).

  24. 24.

    Zeng, Z.: Multi-agent Coordination Control with Nonlinear Dynamics, Quantized Communication and Structure-constraint. Ph.D. thesis, National University of Defense Technology, Changsha (2016)

  25. 25.

    Zheng, R., Lin, Z., Fu, M., Sun, D.: Distributed circumnavigation by unicycles with cyclic repelling strategies. In: Control Conference (ASCC), 2013 9th Asian, pp 1–6. IEEE (2013)

  26. 26.

    Zheng, R., Liu, Y., Sun, D.: Enclosing a target by nonholonomic mobile robots with bearing-only measurements. Automatica 53, 400–407 (2015).

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Our work is supported by National Science Foundation of China (NO.61773393 and NO. 61503401), China Postdoctoral Science Foundation (NO. 2014M562648), and graduate school of National University of Defense Technology.

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Correspondence to Huimin Lu.

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Proof of Theorem 4


The proof is similar to that of Theorem 4 except for some minor changes. First, according to the classical control theory, \(\rho _{i} \) and \(z_{i} \) will converge exponentially to \(\rho ^{*} \) and 0 respectively. Since \(\mu _{i}, ~i = 1,\dots ,n\), is piecewise constant, it is obvious that \(\underset {t\to \infty }{\lim } f_{i}\) exists. As before, we define \(\widetilde {\varphi }=[\widetilde {\varphi }_{1} ... \widetilde {\varphi }_{n} ]^{T} \) and \(\varphi =[\varphi _{1} ... \varphi _{n} ]^{T} \), so Eqs. 56 and 57 can be written into compact forms as \( \dot {\varphi }=\omega ^{*} \boldsymbol {1}+k_{\varphi } (\widetilde {\varphi }-\varphi ), \) and \( \widetilde {\varphi }=\hat {A}\varphi + \hat {b}, \) where \( \hat {A}\in M_{n}\) is shown as Eq. 66, and \( \hat {b}\in R^{n} \) is as Eq. 67:

$$ \hat{A}=\left[ \begin{array}{cccc} {0} & {\frac{\mu_{n} + \mu_{1} }{\mu_{2} + 2 \mu_{1} + \mu_{n}} } & {{\ldots} } & {\frac{\mu_{1} + \mu_{2} }{\mu_{2} + 2 \mu_{1} + \mu_{n} } } \\ {\frac{\mu_{2} + \mu_{3} }{\mu_{3} + 2 \mu_{2} + \mu_{1}} } & {0} & {{\ldots} } & {0} \\ {{\vdots} } & {{\vdots} } & {{\vdots} } & {{\vdots} } \\ {\frac{\mu_{n-1} + \mu_{n} }{\mu_{1} + 2 \mu_{n} + \mu_{n-1} } } & {0} & {{\ldots} } & {0} \end{array} \right]. $$
$$ \hat{b} = 2\pi \left[ \begin{array}{ccccc} {\frac{-(\mu_{1} + \mu_{2}) }{\mu_{2} + 2 \mu_{1} + \mu_{n} } } & {0} & {{\ldots} } & {0} & {\frac{\mu_{n-1} + \mu_{n} }{\mu_{1} + 2 \mu_{n} + \mu_{n-1} } } \end{array} \right]^{T} $$

During each time period where \(\mu _{i}\) is constant, \(\hat {A}\) and \(\hat {b}\) are constant matrix and vector respectively. Similarly, \(\mathcal {G}(\hat {A})\) is strongly connected and the error signal is \( e_{\varphi } = -\hat {L}_{p} \varphi + \hat {b}\), where \(\hat {L}_{p} =I_{n} - \hat {A}\), which is the Laplacian matrix of \(\mathcal {G}(\hat {A})\). Since \(\hat {L}_{p}\) is constant at each time period, the derivative of \(e_{\varphi } \) is \(\dot {e}_{\varphi } =-\hat {L}_{p} \dot {\varphi }\). Then \(e_{\varphi } (t)=\exp (-k_{\varphi } \hat {L}_{p} t)e_{\varphi }(0)\) and \( \underset {t\to \infty }{\lim } e_{\varphi } (t) = w_{r} (-{w_{l}^{T}} \hat {L}_{p} \varphi +{w_{l}^{T}} \hat {b}) ={w_{l}^{T}} \hat {b} w_{r}\). Let wr = 1 and \( w_{l} =\frac {w_{L}}{{\sum }_{w_{L}}}\), where the i th entry of \(w_{L} \) is

$$\left[ w_{L_{i}} =(\mu_{i^{+}} + 2 \mu_{i} + \mu_{i^{-}})\prod\limits_{j = 1,j\ne i,i^{-} }^{n} (\mu_{j} + \mu_{j^{+}}) \right], $$

and \({\sum }_{w_{L}}={\sum }_{i = 1}^{n}w_{L_{i}}\). It can be easily verified that \({w_{l}^{T}} \)and \(w_{r} \) are the left and right eigenvector of the Laplacian matrix \(L_{p} \) associated with the zero eigenvalue respectively, and \({w_{l}^{T}} w_{r} = 1\). Therefore, Eq. 25 becomes \(\underset {t\to \infty }{\lim } e_{\varphi } (t)=\boldsymbol {0}\), or \( \underset {t\to \infty }{\lim } \varphi (t)=\underset {t\to \infty }{\lim } \widetilde {\varphi }(t). \) According to \(\dot {\varphi }=\omega ^{*} \boldsymbol {1}+k_{\varphi } (\widetilde {\varphi }-\varphi )\), the circumnavigation speed of each robot converges to the desired angular speed \(\omega ^{*} \). In addition, under this condition, \(\widetilde {\varphi }_{i} \) is replaced by φi in Eq. 57 and therefore, for robots with indices i = 2,...,n, the equation \(\varphi _{i} =\varphi _{i^{-}} +\frac {\mu _{i^{-}}+\mu _{i}}{\mu _{i^{+}}+ 2\mu _{i}+\mu _{i^{-}}} ({\Delta }_{i} +{\Delta }_{i^{-}}) \) further becomes \( \frac {{\Delta }_{i} }{{\Delta }_{i^{-} } } =\frac {\mu _{i} + \mu _{i^{+}} }{\mu _{i} + \mu _{i^{-}}}. \) This means a sequence of equations \(\frac {{\Delta }_{n} }{{\Delta }_{n-1} } =\frac {\mu _{n} + \mu _{1} }{\mu _{n-1} + \mu _{n} } ,...,\frac {{\Delta }_{2} }{{\Delta }_{1} } =\frac {\mu _{2} + \mu _{3} }{\mu _{1} + \mu _{2} } \) . Assuming Δ1 = k(μ1 + μ2),k≠ 0, we have \({\Delta }_{i} =k (\mu _{i} + \mu _{i^{+}}), i = 2,...,n\). According to Eq. 8, it follows that \(2 k {\sum }_{i = 1}^{n} \mu _{i} = 2 \pi \), and hence \(k=\pi / {\sum }_{i = 1}^{n} \mu _{i}\). Therefore, \({\Delta }_{i} =(\mu _{i} + \mu _{i^{+}}) \pi / {\sum }_{i = 1}^{n} \mu _{i} = f_{i}(t, \mu _{1},\dots ,\mu _{n}) , i = 1,...,n\). So the expected spacings expressed by Eqs. 49 and 53 can be achieved. □

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Yao, W., Lu, H., Zeng, Z. et al. Distributed Static and Dynamic Circumnavigation Control with Arbitrary Spacings for a Heterogeneous Multi-robot System. J Intell Robot Syst 94, 883–905 (2019).

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  • Static circumnavigation
  • Dynamic circumnavigation
  • Distributed control
  • Multi-robot system