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.
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This is one of several papers published in Autonomous Robots comprising the “Special Issue on Distributed Robotics: From Fundamentals to Applications”.
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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
where \(\omega \) is defined in (22). From (31), the norm of the acceleration of agent i is calculated as
(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),
Substituting (10), (22), (33), (34) and \(\sigma ^2 \le 1\) into (32), we have
This inequality yields
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
(case 3: If \(r_c \le r \le r_e)\) In this interval of r,
from (11). Substituting (22), (36), (37) and \(\sigma ^2 \le 1\) into (32), we have
Since \(r = \rho ' + u_{ir}/a > u_{ir}/a\) from (11), \(u_{ir}\ge u_{i\theta }\), and \(\sigma ^2 \le 1\),
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
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
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 \(0 \le \theta ^\mathrm {eq} \le \pi / 2\),
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
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
and a condition of existence of \(\theta ^\mathrm {eq}\) is
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:
where
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
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
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.
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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
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DOI: https://doi.org/10.1007/s10514-018-9774-x