Advertisement

Decentralized navigation method for a robotic swarm with nonhomogeneous abilities

  • 369 Accesses

  • 1 Citations

Abstract

This paper addresses the navigation of a robotic swarm with nonhomogeneous abilities, including sensing range, maximum velocity, and acceleration. With this method, the robotic swarm moves in a two-dimensional plane, and each follower distributedly constructs and maintains local directed connection using only local information to achieve maintenance of global connectivity. We also ensure the swarm is stable when the leader moves at a constant velocity. Validity and effectiveness of the proposed control strategy are shown by theoretical analysis, experiments with real robots, and numerical simulations.

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 99

This is the net price. Taxes to be calculated in checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22

References

  1. Alligood, K. T., Sauer, T. D., & Yorke, J. A. (1996). Chaos: An introduction to dynamical systems. New York: Springer.

  2. Ardito, C. F., Paola, D. D., & Gasparro, A. (2012). Decentraized estimation of the minimum strongly connected subdigraph for robotic networks with limited field of view. In Proceedings of 51st IEEE CDC (pp. 5304–5309).

  3. Brambilla, M., Ferrante, E., Birattari, M., & Dorigo, M. (2013). Swarm robotics: A review from the swarm engineering perspective. Swarm Intelligence, 7(1), 1–41.

  4. Bullo, F., Cortes, J., & Martinez, S. (2009). Distributed control of robotic networks: A mathematical approach to motion coordination algorithms. Princeton: Princeton Univ Press.

  5. Cezayirli, A., & Kerestecioglu, F. (2013). Navigation of non-communicating autonomous mobile robots with guaranteed connectivity. Robotica, 31(5), 767–776.

  6. Dimarogonas, D. V., & Johansson, K. H. (2010). Bounded control of network connectivity in multi-agent systems. IET Control Theory and Applications, 4(8), 1330–1338.

  7. Durham, J. W., Franchi, A., & Bullo, F. (2012). Distributed pursuit-evasion without mapping or global localization via local frontiers. Autonomous Robots, 32, 81–95.

  8. Fang, H., Wei, Y., Chen, J., & Xin, B. (2017). Flocking of second-order multiagent systems with connectivity preservation based on algebraic connectivity estimation. IEEE Transactions on Cybernetics, 47(4), 1067–1077.

  9. Feng, Y., Xu, S., Lewis, F. L., & Zhang, B. (2015). Consensus of heterogeneous first- and second-order multi-agent systems with directed communication topologies. International Journal of Robust Nonlinear Control, 25(3), 362–375.

  10. Feng, Z., Sun, C., & Hu, G. (2017). Robust connectivity preserving rendezvous of multi-robot systems under unknown dynamics and disturbances. IEEE Transactions on Control of Network Systems, 4(4), 725–735.

  11. Gasparri, A., Priolo, A., & Ulivi, G. (2012) A swarm aggregation algorithm for multi-robot systems based on local interaction. In Proceedings of 2012 IEEE multi-conference on systems and control (pp. 1497–1502).

  12. Howard, A., Parker, L. E., & Sukhatme, G. S. (2006). Experiments with a large heterogeneous mobile robot team: Exploration mapping, deployment and detection. The International Journal of Robotics Research, 25, 431–447.

  13. Ji, M., & Egerstedt, M. (2007). Distributed coordination control of multiagent systems while preserving connectedness. IEEE Transactions on Automatic Control, 23(4), 693–703.

  14. Kawakami, H., & Namerikawa, T. (2009). Cooperative target-capturing strategy for multi-vehicle systems with dynamic network topology. In Proceedings of 2009 American control conference (pp. 635–640).

  15. Li, Y., & Muldowney, J. S. (1993). On Bendixson’s criterion. Journal of Differential Equations, 106, 27–39.

  16. Mei, J., Ren, W., & Chen, J. (2016). Distributed consensus of second-order multi-agent systems with heterogeneous unknown inertias and control gains under a directed graph. IEEE Transactions on Automactic Control, 61(8), 2019–2034.

  17. Miyata, N., Ota, J., Arai, T., & Asama, H. (2002). Cooperative transport by multiple mobile robots in unknown static environments associated with real-time task assignment. IEEE Transactions on Robotics Automation, 18(5), 769–780.

  18. Navarro, I., & Matia, F. (2013). A survey of collective movement of mobile robots. International Journal of Advanced Robotic Systems, 10(73), 1–9.

  19. Olfati-Saber, R. (2006). Flocking for multi-agent dynamic systems: Algorithms and theory. IEEE Transactions on Automatic Control, 51(3), 401–420.

  20. Ota, J. (2006). Multi-agent robot systems as distributed autonomous systems. Advanved Engineering Informatics, 20(1), 59–70.

  21. Panagou, D., Stipanovic, D. M., & Voulgaris, P. G. (2016). Distributed coordination control for multi-robot networks using Lyapunov-like barrier functions. IEEE Transactions on Automatic Control, 61(3), 617–632.

  22. Paola, D. D., Asmundis, R. D., Gasparri, A., & Rizzo, A. (2012). Decentralized topology control for robotic network with limited field of view sensors. In Proceedings of 2012 American Control Conference (pp. 3167–3172).

  23. Qu, Z., Li, C., & Lewis, F. (2014). Cooperative control with distributed gain adaptation and connectivity estimation for directed networks. International Journal of Robust Nonlinear Control, 24, 450–476.

  24. Ren, W., & Beard, R. W. (2005). Consensus seeking in multiagent systems under dynamically changing interaction topologies. IEEE Transactions on Automatic Control, 50(5), 655–661.

  25. Sabattini, L., Secchi, C., & Chopra, N. (2015). Decentralized estimation and control for preserving the strong connectivity of directed graphs. IEEE Transactions on Cybernetics, 45(10), 2273–2286.

  26. Schuresko, M., & Cortes, J. (2012). Distributed tree rearrangement for reachability and robust connectivity. SIAM Journal on Control and Optimization, 50(5), 2588–2620.

  27. Shi, G., Hong, Y., & Johansson, K. H. (2012). Connectivity and set tracking of multi-agent systems guided by multiple moving leaders. IEEE Transactions on Automatic Control, 57(3), 663–676.

  28. Sontag, E. D. (2003). A remark on the converging-input converging-state property. IEEE Transactions on Automatic Control, 48(2), 313–314.

  29. Verginis, C. K., Bechlioulis, C. P., Dimarogonas, D. V., & Kyriakopoulos, K. J. (2015). Decentralized 2-D control of vehicular platoons under limited visual feedback. In Proceedings of 2015 IEEE/RSJ international conference on intelligent robots and systems (IROS) (pp. 3566–3571).

  30. Zavlanos, M. M., Egerstedt, M. B., & Pappas, G. J. (2011). Graph-theoretic connectivity control of mobile robot networks. Proceedings of the IEEE, 99(9), 1525–1540.

  31. Zheng, Y., Zhu, Y., & Wang, L. (2011). Consensus of heterogeneous multi-agent systems. IET Control Theory and Applications, 5(16), 1881–1888.

Download references

Author information

Correspondence to Takahiro Endo.

Additional information

This is one of several papers published in Autonomous Robots comprising the “Special Issue on Distributed Robotics: From Fundamentals to Applications”.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (mp4 6673 KB)

Supplementary material 1 (mp4 6673 KB)

Appendices

Appendix A: Proof of 2nd step in Theorem 2

Now, we show that \(\Vert \dot{\varvec{u}}_i(t) \Vert < A_i\) if \(\Vert \varvec{u}_{j}(t) \Vert \le U_{n+1}\). Here, we consider \(t \ge t_i\) because \(\dot{\varvec{u}}_i \equiv \varvec{0}\) at \(t < t_i\). Substituting (23) into the time-derivative of (8), the acceleration of agent i is computed as

$$\begin{aligned} \dot{\varvec{u}}_i = \left( \dot{u}_{ir} - u_{i\theta }\omega \right) \varvec{e}_r + \left( \dot{u}_{i\theta } + u_{ir}\omega \right) \varvec{e}_\theta , \end{aligned}$$
(31)

where \(\omega \) is defined in (22). From (31), the norm of the acceleration of agent i is calculated as

$$\begin{aligned} \Vert \dot{\varvec{u}}_i \Vert ^2 =\ (\dot{u}_{ir}^2 + \dot{u}_{i\theta }^2) + \omega ^2(u_{ir}^2 + u_{i\theta }^2) + 2\omega (u_{ir}\dot{u}_{i\theta } - u_{i\theta }\dot{u}_{ir}). \end{aligned}$$
(32)

(case 1: If \(0 \le r < \rho ')\) Obviously \(\Vert \dot{\varvec{u}}_i \Vert = 0 < A_i\).

(case 2: If \(\rho ' \le r < r_c)\) From (10),

$$\begin{aligned} \dot{u}_{ir}&= a\dot{r} = a(u_{jr} - u_{ir}), \end{aligned}$$
(33)
$$\begin{aligned} \dot{u}_{i\theta }&= \sigma \dot{u}_{ir} = \sigma a (u_{jr} - u_{ir}). \end{aligned}$$
(34)

Substituting (10), (22), (33), (34) and \(\sigma ^2 \le 1\) into (32), we have

$$\begin{aligned} \Vert \dot{\varvec{u}}_i \Vert ^2 < 2a^2&\left\{ \left( u_{jr} - u_{ir}\right) ^2 + \left( u_{j\theta } - u_{i\theta }\right) ^2 \right\} \\ = 2a^2&\bigl \{ \left( u_{jr}^2 + u_{j\theta }^2\right) + \left( 1 +\sigma ^2 \right) u_{ir}^2 \\&-2u_{ir}\left( u_{jr} + \sigma u_{j\theta } \right) \bigr \} \\ \le 2a^2&\left\{ \left( u_{jr}^2+ u_{j\theta }^2\right) + 2u_{ir}^2 + 2u_{ir} \left( \vert u_{jr} \vert + \vert u_{j\theta } \vert \right) \right\} . \end{aligned}$$

This inequality yields

$$\begin{aligned} \Vert \dot{\varvec{u}}_i \Vert ^2 < (1 + \sqrt{2})^2 a^2 U_{n+1}^2, \end{aligned}$$
(35)

since \(u_{jr}^2 + u_{j\theta }^2 \le U_{n+1}^2\) , \(\vert u_{jr} \vert + \vert u_{j\theta } \vert \le \sqrt{2}U_{n+1}\) using the method of Lagrange multiplier, and \(u_{ir} < U' / 2\) from (20). Thus, substituting (7) into (35), we obtain

$$\begin{aligned} \Vert \dot{\varvec{u}}_i \Vert< & {} \left( 1 + \sqrt{2} \right) a U_{n+1} \le \frac{1 + \sqrt{2}}{2 + \sqrt{2}}a \min _{k \in \mathcal {F}}\frac{A_k}{a_k}\\\le & {} \frac{1 + \sqrt{2}}{2 + \sqrt{2}}a\frac{A_i}{a} < A_i. \end{aligned}$$

(case 3: If \(r_c \le r \le r_e)\) In this interval of r,

$$\begin{aligned} \dot{u}_{ir}&= a\dot{r} = a(u_{jr} - u_{ir}),\end{aligned}$$
(36)
$$\begin{aligned} \dot{u}_{i\theta }&= -\sigma a (u_{jr} - u_{ir}) \end{aligned}$$
(37)

from (11). Substituting (22), (36), (37) and \(\sigma ^2 \le 1\) into (32), we have

$$\begin{aligned} \Vert \dot{\varvec{u}}_i \Vert ^2 =&\ (1 + \sigma ^2)a^2 (u_{jr} - u_{ir})^2 \\&- \frac{2a}{r}(u_{jr} - u_{ir})(u_{j\theta } - u_{i\theta })(\sigma u_{ir} + u_{i\theta })\\&+ \frac{1}{r^2}(u_{j\theta } - u_{i\theta })^2 (u_{ir}^2 + u_{i\theta }^2). \end{aligned}$$

Since \(r = \rho ' + u_{ir}/a > u_{ir}/a\) from (11), \(u_{ir}\ge u_{i\theta }\), and \(\sigma ^2 \le 1\),

$$\begin{aligned} \Vert \dot{\varvec{u}}_i \Vert ^2 <&\ 2a^2 \{ (u_{jr} - u_{ir})^2 \\&+2\vert u_{jr} - u_{ir} \vert \vert u_{j\theta } - u_{i\theta } \vert + (u_{j\theta } - u_{i\theta })^2 \} \\ =&\ 2a^2(\vert u_{jr} - u_{ir} \vert + \vert u_{j\theta } - u_{i\theta } \vert )^2 \\ \le&\ 2a^2(\vert u_{ir} \vert + \vert u_{i\theta } \vert + \vert u_{jr} \vert + \vert u_{j\theta } \vert )^2. \end{aligned}$$

Substituting \(\vert u_{ir} \vert + \vert u_{i\theta } \vert \le U' < U_{n+1}\) from (11), and \(\vert u_{jr} \vert + \vert u_{j\theta } \vert \le \sqrt{2}U_{n+1}\) as the second case to (32), \(\Vert \dot{\varvec{u}}_i \Vert ^2 < (2 + \sqrt{2})^2 a^2 U_{n+1}^2\). Thus, substituting (7) into (35), we obtain

$$\begin{aligned} \Vert \dot{\varvec{u}}_i \Vert< & {} (2 + \sqrt{2})aU_{n+1} \le \frac{2+\sqrt{2}}{2+\sqrt{2}}a \min _{k \in \mathcal {F}}\frac{A_k}{a_k}\\\le & {} \frac{2+\sqrt{2}}{2+\sqrt{2}}a \frac{A_i}{a} = A_i. \end{aligned}$$

Thus, \(\Vert \dot{\varvec{u}}_i \Vert \) is smaller than \(A_i\).

Appendix B: Proof of Lemma 1 (existence of the equilibrium)

(case 1: If \(\rho '< r^\mathrm {eq} < r_c)\)The equilibrium satisfies

$$\begin{aligned}&U^* \cos \theta ^\mathrm {eq} - a(r^\mathrm {eq} - \rho ') = 0,\end{aligned}$$
(38)
$$\begin{aligned}&\sigma a(r^\mathrm {eq} - \rho ') - U^* \sin \theta ^\mathrm {eq} = 0 \end{aligned}$$
(39)

from (10), (24) and (25). Since \(U^* > 0\) and \(r^\mathrm {eq} > \rho '\), \(\sin \theta ^\mathrm {eq} \ge 0\) and \(\cos \theta ^\mathrm {eq} \ge 0\) must hold, and then \(0 \le \theta ^\mathrm {eq} \le \pi / 2\). From (38) and (39), we obtain \(\sin \theta ^\mathrm {eq} - \sigma \cos \theta ^\mathrm {eq} = 0\), and \(\theta ^\mathrm {eq} = \arctan \sigma \).

Considering \(0 \le \theta ^\mathrm {eq} \le \pi / 2\),

$$\begin{aligned} \sin \theta ^\mathrm {eq} = \frac{\sigma }{\sqrt{1 + \sigma ^2}},~~ \cos \theta ^\mathrm {eq} = \frac{1}{\sqrt{1 + \sigma ^2}}. \end{aligned}$$
(40)

Substituting (40) into (38), \(r^\mathrm {eq} = \rho ' + \frac{U^*}{\sqrt{1 + \sigma ^2}a}\). Therefore, for \(U^* \in \left( 0, \sqrt{1+\sigma ^2}U'/2\right) \), the equilibrium \((r^\mathrm {eq}, \theta ^\mathrm {eq}) = \left( \rho ' + U^*/(\sqrt{1 + \sigma ^2}a) , \arctan \sigma \right) \) continuously exists in \(\rho '< r < r_c\), and \(r^\mathrm {eq}\) is monotonically increasing for \(U^*\).

(case 2: If \(r_c \le r^\mathrm {eq} \le r_e)\)  The equilibrium satisfies

$$\begin{aligned}&U^* \cos \theta ^\mathrm {eq} - a(r^\mathrm {eq} - \rho ') = 0,\end{aligned}$$
(41)
$$\begin{aligned}&\sigma \left( U' - a(r^\mathrm {eq} - \rho ') \right) - U^* \sin \theta ^\mathrm {eq} = 0 \end{aligned}$$
(42)

from (11), (24) and (25). Since \(U^* > 0\) and \(r^\mathrm {eq} > \rho '\), \(\sin \theta ^\mathrm {eq} \ge 0\) and \(\cos \theta ^\mathrm {eq} > 0\) must hold, and then \(0 \le \theta ^\mathrm {eq} < \pi / 2\). From (41) and (42), we obtain

$$\begin{aligned} \sin \theta ^\mathrm {eq} + \sigma \cos \theta ^\mathrm {eq} = \frac{\sigma U'}{U^*} \end{aligned}$$
(43)

and a condition of existence of \(\theta ^\mathrm {eq}\) is

$$\begin{aligned} U^* \ge \frac{\sigma U'}{\sqrt{1+\sigma ^2}}. \end{aligned}$$
(44)

Since \(\frac{\sigma U'}{\sqrt{1+\sigma ^2}} \le \frac{\sqrt{1+\sigma ^2}}{2}U'\) holds for any \(\sigma \in [0, 1]\) and \(U' \ge 0\), the equilibrium exists continuously in \(r_c \le r \le r_e\) for \(U^* \in \left[ \sqrt{1+\sigma ^2}U'/2, U' \right] \), which satisfies (44). Since \(\theta ^\mathrm {eq}\) (\(\ge 0\)) is monotonically decreasing for \(U^*\) from (43) and \(\theta ^\mathrm {eq} = \arctan \sigma \) at \(U^* = \sqrt{1+\sigma ^2}U'/2\), we have \(\theta ^\mathrm {eq} \le \arctan \sigma \) for \(U^* \in \left[ \sqrt{1+\sigma ^2}U'/2, U' \right] \). Moreover, \(r^\mathrm {eq}\) is monotonically increasing for \(U^*\) in the same way as in the first case.

At \(U^* = \sqrt{1+\sigma ^2}U'/2\), \((r^\mathrm {eq}, \theta ^\mathrm {eq})\) is continuous from equations (38), (39), (41) and (42). Thus, this lemma is proved.

Appendix C: Proof of Lemma 1 (stability of the equilibrium)

Consider fixed \(U^* \in (0, U']\) and the corresponding equilibrium \((r^\mathrm {eq}, \theta ^\mathrm {eq})\). Jacobi matrix J at \((r^\mathrm {eq}, \theta ^\mathrm {eq})\) is computed as follows:

$$\begin{aligned} J = \left[ \begin{array}{c c} - \, a &{}\quad - \, U^* \sin \theta ^\mathrm {eq} \\ J_{21} &{}\quad -\,\frac{U^*}{r^\mathrm {eq}}\cos \theta ^\mathrm {eq} \end{array} \right] , \end{aligned}$$

where

$$\begin{aligned} J_{21} = {\left\{ \begin{array}{ll} \frac{a\sigma }{r^\mathrm {eq}} ~~~~~ (\rho '< r^\mathrm {eq} < r_c), \\ -\frac{a\sigma }{r^\mathrm {eq}} ~~~ (r_c \le r^\mathrm {eq} \le r_e). \end{array}\right. } \end{aligned}$$

The characteristic equation of J is \(\lambda ^2 - (\mathrm {tr}J)\lambda + \mathrm {det}J = 0\), where \(\lambda \) is an eigenvalue of J. Since \(0 \le \theta ^\mathrm {eq} \le \arctan \sigma \le \pi / 4\) from Lemma 1, \(\mathrm {tr}J = -a -\frac{U^*}{r^\mathrm {eq}}\cos \theta ^\mathrm {eq} < 0\), and

$$\begin{aligned} \mathrm {det}J= & {} \frac{aU^*}{r^\mathrm {eq}}\cos \theta ^\mathrm {eq} \pm \frac{aU^*\sigma }{r^\mathrm {eq}}\sin \theta ^\mathrm {eq}\nonumber \\\ge & {} \frac{aU^*}{r^\mathrm {eq}}(\cos \theta ^\mathrm {eq} - \sigma \sin \theta ^\mathrm {eq}) \ge 0 \end{aligned}$$
(45)

are obtained. Here, the last equality of (45) is attained if and only if \(\sigma = 1\) and \(U^* = U' / \sqrt{2}\). Otherwise, we have \(\mathrm {Re}(\lambda ) < 0\) from the theorem of Hurwitz. Moreover, since \((\mathrm {tr}J)^2 - 4\mathrm {det}J >0\), \(\lambda \) is a real number, and \(\lambda \le 0\).

If \(\lambda < 0\), the equilibrium is stable. Otherwise, one of the eigenvalues equals 0. One of the eigenvectors corresponding to \(\lambda = 0\) is \([U^* / \sqrt{2}a, -1]^{\mathrm {T}}\). The equilibrium is \((r_0, \theta _0) = (r_c, \pi / 4)\), where \(u_{i\theta }\) is not differentiable. \(\lambda = 0\) is adopted to the positive direction of r, while \(\lambda < 0\) to the negative direction from (45). Thus, we consider the direction of the vector \([\varDelta r, \varDelta \theta ]^{\mathrm {T}} = \epsilon [U^* / \sqrt{2}a, -1]^{\mathrm {T}}\), where \(\epsilon > 0\). By the Taylor expansion of \(\dot{r}\) and \(\dot{\theta }\) around \((r^\mathrm {eq}, \theta ^\mathrm {eq})\), we have

$$\begin{aligned}&\dot{r}(r_0 + \varDelta r, \theta _0 + \varDelta \theta ) = -a\varDelta r - U^* \sin \theta _0 \varDelta \theta \nonumber \\&\qquad \qquad \qquad \qquad \qquad - \frac{U^*\cos \theta _0}{2}(\varDelta \theta )^2 + \dots ,\end{aligned}$$
(46)
$$\begin{aligned}&\dot{\theta }(r_0 + \varDelta r, \theta _0 + \varDelta \theta ) = -\frac{\sigma a}{r_0}\varDelta r\nonumber \\&\qquad \qquad - \frac{U^*\cos \theta _0}{r_0}\varDelta \theta + \frac{\sigma a}{r_0^2}(\varDelta r)^2 \nonumber \\&\qquad \qquad + \frac{U^*\sin \theta _0}{2r_0} (\varDelta \theta )^2 + \frac{U^*\cos \theta _0}{r_0^2}\varDelta r \varDelta \theta + \dots . \end{aligned}$$
(47)

Substituting \([\varDelta r, \varDelta \theta ]^{\mathrm {T}} = \epsilon [U^* / \sqrt{2}a, -1]^{\mathrm {T}}\) into (46) and (46), we have \(\dot{r}(r_0 + \varDelta r, \theta _0 + \varDelta \theta ) = -\frac{\epsilon a}{2\sqrt{2}}\varDelta r\), and \(\dot{\theta }(r_0 + \varDelta r, \theta _0 + \varDelta \theta ) = -\frac{U^*\epsilon }{2\sqrt{2}r_0}\varDelta \theta \), which show the equilibrium attracts points in the direction of \(\epsilon [U^* / \sqrt{2}a, -1]^{\mathrm {T}}\). Therefore, \((r^\mathrm {eq}, \theta ^\mathrm {eq})\) is stable.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yoshimoto, M., Endo, T., Maeda, R. et al. Decentralized navigation method for a robotic swarm with nonhomogeneous abilities. Auton Robot 42, 1583–1599 (2018). https://doi.org/10.1007/s10514-018-9774-x

Download citation

Keywords

  • Swarm robotics
  • Connectivity maintenance
  • Spanning tree
  • Nonhomogeneous