Experimental investigation of distributed navigation and collision avoidance for a robotic swarm

Abstract

In this study, we focus on the navigation of a robotic swarm and perform an experimental investigation. A decentralized control method using only local information has been proposed in many previous studies. However, because global coordinates are used and control input is calculated on a single PC in experiments, the system is regarded as partially centralized control. Thus, we first realize machines that are suitable for the distributed navigation control method. In conducting experiments using these machines, we encountered two problems. The first was loss of connectivity caused by the target’s coordinates being momentarily out of the sensing range. The second was collisions between robots caused by the target being in the blind spot of the camera’s field of view and loss of sight of the target. Note that each robot had a target and moved by following it. To solve these problems, we propose a method that removes unexpectedly output coordinates and adds rotational motion to each robot. The experimental results demonstrated that we achieved distributed navigation control with collision avoidance in an actual machine experiment.

Appendix

We briefly describe the control input (5). In the control input, we determine ${u}_{ir}(t)$ and ${u}_{i\theta }(t)$ as follows. We divide the control input into three cases: (i) the control input before the target is determined, (ii) the control input for maintaining connectivity during navigation, and (iii) the control input for collision avoidance.

• Case (i)  $t<t_i$:

  $$\begin{aligned} u_{ir}(t) = 0,\qquad u_{i\theta }(t) = 0. \end{aligned}$$

  (13)

• Case (ii)  $t \ge t_i \cap d_i(t):=\Vert {\varvec{x}}_k^i(t)\Vert > \rho ''_i$:

• (a) When $\rho ^{e-}_i(t) \le r_i(t):=\Vert {\varvec{x}}_j^i(t)\Vert \le \rho ''_i$,

  $$\begin{aligned} u_{ir}(t) = \tfrac{U_i(r_i(t) - \rho ''_i)}{\rho ''_i - \rho '''_i},\qquad u_{i\theta }(t) = 0. \end{aligned}$$

  (14)

• (b) When $\rho ''_i < r_i(t) \le (\rho ''_i + \rho '_i) / 2$,

  $$\begin{aligned} u_{ir}(t) = 0,\qquad u_{i\theta }(t) = \tfrac{2\sigma _i(t) U'_i(t)}{\rho '_i - \rho ''_i} (r_i(t) - \rho ''_i). \end{aligned}$$

  (15)

• (c) When $(\rho ''_i + \rho '_i) / 2< r_i(t) < \rho '_i$,

  $$\begin{aligned} u_{ir}(t) = 0,\qquad u_{i\theta }(t) = \tfrac{2\sigma _i(t) U'_i(t)}{\rho '_i - \rho ''_i} (\rho '_i - r_i(t)). \end{aligned}$$

  (16)

• (d) When $\rho '_i \le r_i(t) \le \rho ^c_i(t)$,

  $$\begin{aligned} u_{ir}(t) = \tfrac{U_i(r_i(t) - \rho '_i)}{\rho _i - \rho '_i},\qquad u_{i\theta }(t) = \sigma _i(t) u_{ir}(t). \end{aligned}$$

  (17)

• (e) When $\rho ^c_i(t) < r_i(t) \le \rho ^{e+}_i(t)$,

  $$\begin{aligned} u_{ir}(t) = \tfrac{U_i(r_i(t) - \rho '_i)}{\rho _i - \rho '_i},\qquad u_{i\theta }(t) = \sigma _i(t) (U'_i(t) - u_{ir}(t)), \end{aligned}$$

  (18)

where

$$\begin{aligned}&U'_i(t) := \underset{0 \le \tau \le t}{\mathrm {max}}\Vert u_{ir}(\tau )\Vert , \end{aligned}$$

(19)

$$\begin{aligned}&\left\{ \begin{array}{l} \rho _i^{e-}(t) := \rho ''_i - \tfrac{U'_i(t)}{U_i} (\rho ''_i - \rho '''_i), \quad \rho _i^{c}(t) := \rho '_i + \tfrac{U'_i(t)}{2 U_i} (\rho _i - \rho '_i), \quad \\ \rho _i^{e+}(t) := \rho '_i + \tfrac{U'_i(t)}{U_i} (\rho _i - \rho '_i) \end{array} \right. \end{aligned}$$

(20)

and $\sigma _i (t) \in [-1, 1]$ is a parameter used to change the shape of the swarm. Note that, when $r_i(t)\le \rho _i''$, $r_i(t)\ge \rho _i^{e-}$ always holds from Eqs. (14) and (19). Furthermore, when $r_i(t)\ge \rho _i'$, $r_i(t)\le \rho _i^{e+}$ always holds from Eqs. (17)–(19). Thus, it is sufficient to design the control input in the range $\rho _i^{e-}(t)< r_i(t) < \rho _i^{e+}(t)$.

• Case (iii)  $t \ge t_i \cap d_i(t) \le \rho ''_i$:

$$\begin{aligned} {\varvec{u}}_i(t) = {\varvec{u}}_i^j + \tfrac{t - t_i^{\mathrm {start}}}{T_i} ({\varvec{u}}_i^k(t) - {\varvec{u}}_i^j), \end{aligned}$$

(21)

where $T_i = h_i / (\zeta U_i)$, $h_i = \min \{\rho _i - \rho '_i , \rho ''_i - \rho '''_i\}$, $\zeta$ is a positive constant that satisfies $2\pi (1 - \mathrm {e}^{-1/\zeta }) /\zeta \le 1$ (we used $\zeta = 2.253$ in this paper). Furthermore, let $t^{\mathrm {start}}_i$ be the time at which $d_i(t) = \rho ''_i$, and agent k be the new target at $t = t^{\mathrm {start}}_i$. Note that if another agent comes into the area closer than $\rho _i''$, it becomes a new target. In addition, ${\varvec{u}}_i^j:={\varvec{u}}_i(t^{\mathrm {start}}_i)$. From the implementation point of view, it is difficult to switch targets instantly. Thus, the input is changed from the previous target reference to the new target reference over $T_i$.

We designed the control input $u_{ir}(t)$ to guarantee that followers maintain LSC. By contrast, we designed $u_{i\theta }(t)$ to widen the swarm shape and prevent it from becoming a chain structure. As a result, the follower behaves as follows in each case: In Case (II)-(a), the follower moves away from the target to avoid colliding with it. In Case (II)-(b) and (c), the follower moves away from the other agents to avoid approaching them. In Case (II)-(d) and (e), the follower moves away from the other agents while approaching the target to maintain connectivity. An outline of the control input in case (ii) is shown in Fig. 3.

For more details, see [7].