Mosquito-inspired distributed swarming and pursuit for cooperative defense against fast intruders

Abstract

Inspired by the swarming behavior of male mosquitoes that aggregate to attract and subsequently pursue a female mosquito, we study how random swarming motion in autonomous vehicles affects the success of target capture. We consider the scenario in which multiple guardians with limited perceptual range and bounded acceleration are deployed to protect an area from an intruder. The main challenge for the guardian (male mosquito) is to quickly respond to a fast intruder (female) by matching its velocity. We focus on the motion strategy for the guardians before they perceive the intruder, which we call the swarming phase. In the parameter space consisting of the intruder’s speed and guardians’ ability (i.e., maximum acceleration and perceptual range) we identify necessary and sufficient conditions for target capture. We propose a swarming algorithm inspired by the behavior of male mosquitoes to improve the target-capture capability. The theoretical results are illustrated by experiments with an indoor quadrotor swarm.

Appendices

Appendix A: Calculation of $\beta ^*$

Figure 15 depicts the case where the damping term $b{\mathbf {v}}_{T/P}$ has to be saturated to give ${\mathbf {F}}_P^{\text {(pursuit)}} = u_{max}$. Let $n = \beta ^*\Vert b {\mathbf {v}}_{T/P}\Vert$, $m = (1-\beta ^*)\Vert b {\mathbf {v}}_{T/P}\Vert$, $A = n+m$, $B = \Vert c{\mathbf {r}}_{T/P}\Vert$, $C = \Vert c{\mathbf {r}}_{T/P}+b {\mathbf {v}}_{T/P}\Vert$, and $D = {\mathbf {F}}_P^{\text {(pursuit)}} = u_{max}$. Stewart’s theorem states that

$$\begin{aligned} B^2 m + C^2n = A(D^2+mn). \end{aligned}$$

(30)

Using (30) and $A = m+n$, we can solve for n to obtain

$$\begin{aligned} n = \frac{E\pm \sqrt{E^2 + F} }{2A}, \end{aligned}$$

(31)

where $E = A^2+B^2-C^2$ and $F = 4A^2(D^2-B^2)$. Noting that F is always positive, the solution (31) with $+$ is the only valid solution. The scaling factor is $\beta ^* = n/A$, i.e.,

$$\begin{aligned} \beta ^* = \frac{E+\sqrt{E^2 + F} }{2A^2}. \end{aligned}$$

(32)

Fig. 15

Computing the saturation factor $\beta$ to obtain the control law ${\mathbf {F}}_P^{\text {(pursuit)}}$

Fig. 16

Definitions of angles and speeds in the velocity space (Color figure online)

Appendix B: Required $N_P$ for guaranteed target capture using circling strategy

Consider a circling motion with radius $\rho _p$. Let $v_P$ denote the circling speed. Let $\theta _{T/P}=\cos ^{-1}\left( \frac{{\mathbf {v}}_T\cdot {\mathbf {v}}_P}{\Vert {\mathbf {v}}_T\Vert \Vert {\mathbf {v}}_P\Vert }\right)$ denote the difference between the direction of motion of the target and the pursuer. First, we seek to find the maximum angle $\theta ^*$ such that ${\mathbf {v}}_P \in B_{v_0}({\mathbf {v}}_T(t_0))$ (see Fig. 16 for the definitions of the relevant quantities). For a given guardian speed $v_P$, the angle $\theta ^*$ is the maximum allowable difference in the direction of motion to guarantee target capture. From Fig. 16 and the law of cosines, we have

$$\begin{aligned} \theta ^* = \cos ^{-1}\left( \frac{v_P^2 + v_T^2 - v_0^2}{2v_Pv_T} \right) . \end{aligned}$$

(33)

The angle $\theta ^*$ is maximized when the limiting ${\mathbf {v}}_P$ is tangent to the circle $B_{v_0}({\varvec{v}}_T)$, i.e., the blue dashed line in Fig. 16. This geometry is achieved when $v_P$ satisfies

$$\begin{aligned} v_P = v_P^{(1)} =\sqrt{ v_T^2 - v_0^2} = v_T \sqrt{1- {\varGamma }(1-\alpha )/2}. \end{aligned}$$

(34)

However, because of the centripetal acceleration, the achievable circling speed $v_P$ is bounded as

$$\begin{aligned} v_P \le v_P^{(2)} = \sqrt{\rho _p u_{max}} = v_T\sqrt{\frac{{\varGamma }\alpha }{2(1+\alpha )}}. \end{aligned}$$

We choose the circling speed $v_P$ to be

$$\begin{aligned} v_P = \min \left( v_P^{(1)},\;v_P^{(2)} \right) , \end{aligned}$$

(35)

i.e., use $v_P^{(1)}$ when it is achievable, otherwise, use maximum possible speed which is $v_P^{(2)}$. If the guardians are uniformly distributed on the circle, and if the number of guardians N satisfies

$$\begin{aligned} N > \frac{\pi }{\theta ^*}, \end{aligned}$$

(36)

there will be at least one guardian whose direction of motion satisfies $\theta _{T/P}<\theta ^*$. See Fig. 17 for the illustration of the case with $N_P=3$. When the target reaches the center, the velocity of the pursuer in the fan-shaped region satisfies $\theta _{T/P} < \theta ^*$. If the condition (36) is satisfied, then there is always at least one guardian in the fan-shaped region.

Fig. 17

Example of circling motion where $\theta ^*=\pi /3$. Guardians are uniformly spaced and there is always one guardian in the fan-shaped region. When the target reaches the center, the velocity of the pursuer in the fan-shaped region satisfies $\theta _{T/P} < \theta ^*$

Fig. 18

Sufficient number of guardians to guarantee target capture with circling motion

Figure 18 shows the required number of guardians obtained from conditions (33), (35) and (36). Close to the boundary $\partial _2$, the angle $\theta ^*\rightarrow 0$ and the sufficient number $N\rightarrow \infty$. Close to the boundary $\partial _3$, the angle $\theta ^*\rightarrow \pi$ and the sufficient number $N\rightarrow 2$.

Appendix C: Proof of proposition 3

For a given deflection angle $\phi$, the magnitude of normal acceleration exerted by the target increases as the time of execution ${\varDelta }t$ reduces. It is easy to see that the worst-case scenario for the guardian who is pursuing the target is when ${\varDelta }t$ approaches 0, i.e., the target makes a sudden instantaneous change in its direction of motion. This corresponds to the target applying a linear impulse with magnitude equal to $2 v_T \sin (\phi /2)$.

Consider the energy function used in the proof of Proposition 2. Let ${\varDelta }V$ denote the increase in the energy function due to the target maneuver. Then we have

$$\begin{aligned} 2c{\varDelta }V= & {} 2c V(t_2) - 2cV(t_1) \end{aligned}$$

(37a)

$$\begin{aligned}= & {} \Vert {\mathbf {v}}_{T/P} (t_2)\Vert ^2 - \Vert {\mathbf {v}}_{T/P} (t_1)\Vert ^2 \end{aligned}$$

(37b)

$$\begin{aligned}= & {} -{\mathbf {v}}_P \cdot \left( {\mathbf {v}}_T(t_2) - {\mathbf {v}}_T(t_1) \right) \end{aligned}$$

(37c)

$$\begin{aligned}\le & {} \Vert {\mathbf {v}}_P \Vert 2 v_T \sin (\phi /2) \end{aligned}$$

(37d)

$$\begin{aligned}\le & {} (v_T + v_0) 2 v_T \sin (\phi /2). \end{aligned}$$

(37e)

Target capture is guaranteed if the initial energy $V(t_0)$ is sufficiently small that the distance, $\Vert {\mathbf {r}} _{T/P}\Vert$, is bounded by $\rho _a$ even after the energy increase by ${\varDelta }V$, i.e.,

$$\begin{aligned} V(t_0) + {\varDelta }V\le & {} \frac{1}{2}\rho _a^2 \end{aligned}$$

(38a)

$$\begin{aligned} \Vert {\mathbf {v}}_{T/P}(t_0)\Vert\le & {} \sqrt{ c (\rho _a^2 -\rho _s^2)- 2c {\varDelta }V }, \end{aligned}$$

(38b)

which reduces to (10).

For feasibility of the condition, we also require that the right-hand-side of (10) is positive, i.e.,

$$\begin{aligned} \chi ^2 - 2a \chi -2a > 0, \end{aligned}$$

(39)

which reduces to (11).

Appendix D: Proof of Remark 1

For notational simplicity, let ${\mathbf {r}} \triangleq {\mathbf {r}}_{T/P}(t)$ and ${\mathbf {v}} \triangleq {\mathbf {v}}_{T/P}(t)$. Consider a Lyapunov function

$$\begin{aligned} V = 2bc\Vert {\mathbf {r}}\Vert ^2 + b\Vert {\mathbf {v}}\Vert ^2 + 2 c{\mathbf {r}} \cdot {\mathbf {v}}, \end{aligned}$$

which is positive definite if

$$\begin{aligned} 2b^2>c . \end{aligned}$$

(40)

Assuming that the control is never saturated, i.e., $\beta = 1$ (sufficient condition for this assumption is given later), the time derivative is

$$\begin{aligned} 0.5 \dot{V}= & {} -c^2 \Vert {\mathbf {r}}\Vert ^2 - (b^2-c) \Vert {\mathbf {v}}\Vert ^2 + b {\mathbf {v}} \cdot {\mathbf {a}}_T + c {\mathbf {r}} \cdot {\mathbf {a}}_T \end{aligned}$$

(41)

$$\begin{aligned}\le & {} -c^2 \Vert {\mathbf {r}}\Vert ^2 {-} (b^2{-}c) \Vert {\mathbf {v}}\Vert ^2 {+} b \left( \frac{b}{2}\Vert {\mathbf {v}}\Vert ^2 {+}\frac{1}{2b} \Vert {\mathbf {a}}_T\Vert ^2\right) \nonumber \\&+\, b \left( \frac{b}{2}\Vert {\mathbf {v}}\Vert ^2 +\frac{1}{2b} \Vert {\mathbf {a}}_T\Vert ^2\right) \end{aligned}$$

(42)

$$\begin{aligned}= & {} -\frac{c^2}{2} \Vert {\mathbf {r}}\Vert ^2 - \left( \frac{b^2}{2}-c\right) \Vert {\mathbf {v}}\Vert ^2 + \Vert {\mathbf {a}}_T\Vert ^2 \end{aligned}$$

(43)

$$\begin{aligned}\le & {} -\frac{c^2}{2} \Vert {\mathbf {r}}\Vert ^2 - \left( \frac{b^2}{2}-c\right) \Vert {\mathbf {v}}\Vert ^2 + u_T^2 \end{aligned}$$

(44)

Let $\sigma _1 = \frac{c^2}{2}$, $\sigma _2 = \frac{b^2}{2}-c$, and $D = u_T^2$. For $\sigma _2$ to be positive, we require

$$\begin{aligned} b^2>2c, \end{aligned}$$

(45)

which is stronger than (40). Also let ${\mathbf {z}} = [\Vert {\mathbf {r}} \Vert ,\; \Vert {\mathbf {v}}\Vert ]^T = [r,\;v]^T$. Then we have $\dot{V}\le 0$ for ${\mathbf {z}} \notin B_e = \{[r,v]\in {\mathbb {R}}^2\; |\; \sigma _1^2r^2+\sigma _2^2v^2\le D\}$, where $B_e$ is an ellipsoid centered at ${\mathbf {z}} = 0$, with axis length $\rho _1 = \sqrt{D/\sigma _1}$ and $\rho _2 = \sqrt{D/\sigma _2}$. Figure 19 depicts $B_e$ with other relevant regions.

Fig. 19

Definition of the regions for the proof of ultimate boundedness

If a compact set ${\varOmega }$ is such that $V\le \omega$ for ${\mathbf {z}} \in {\varOmega }$, and also $B_e \in {\varOmega }$, then by ultimate boundedness (Khalil and Grizzle 2002), we know that there exists $T>0$ such that ${\mathbf {z}} \in {\varOmega }$ for all $t > T$ (see Lemma 1 in Shishika et al. 2016).

Since it is not easy to visualize ${\varOmega }$, we introduce two compact sets ${\varOmega }_{min}$ and ${\varOmega }_{max}$ with the property ${\varOmega }_{min}\in {\varOmega }\in {\varOmega }_{max}$. Noting that $V = {\mathbf {z}}^T P {\mathbf {z}}$ where

$$\begin{aligned} P = \left[ \begin{array}{c c} 2bc &{} c \\ c &{} b \end{array} \right] , \end{aligned}$$

(46)

we obtain ${\varOmega }_{min}$ and ${\varOmega }_{max}$ to be discs with radii $\rho _{min} = \sqrt{\omega /\lambda _{max}\{P\} }$ and $\rho _{max} = \sqrt{\omega / \lambda _{min}\{P\}}$, where $\lambda _{min}\{P\}$ and $\lambda _{max}\{P\}$ are the smallest and largest eigenvalue of P.

Conditions $B_e\in {\varOmega }_{min}$ and ${\mathbf {z}}(t_0) \in {\varOmega }_{max}$ guarantee that ${\mathbf {z}}\in {\varOmega }_{max}$ for all time $t>t_0$. If $\rho _{max} = \rho _a$, then ${\mathbf {z}}\in {\varOmega }_{max}$ guarantees target capture, i.e., $\Vert {\mathbf {r}}\Vert \le \rho _a$. The latter condition determines the value of $\omega$, which defines ${\varOmega }$, as follows:

$$\begin{aligned} \omega = \rho _a^2 \lambda _{min}\{P\}. \end{aligned}$$

For $B_e\in {\varOmega }_{min}$, it is sufficient if $\max \{\rho _1, \rho _2\} \le \rho _{min}$, which gives

$$\begin{aligned} u_T \le \rho _a \sqrt{\frac{\lambda _{min}\{P\}}{\lambda _{max}\{P\}}} \big / \sqrt{2}\max \left\{ \frac{1}{c}, \frac{1}{\sqrt{b^2-2c}}\right\} . \end{aligned}$$

(47)

Equation (47) corresponds to condition (i), sufficiently small $u_T$, in Remark 1. Note that (40) guarantees that the right-hand side is positive.

Next, ${\mathbf {z}}(t_0)\in {\varOmega }_{max}$ is true if the initial condition satisfies $V(t_0)\le \omega$, which is equivalent to

$$\begin{aligned}&2bc\rho _p^2 + b\Vert {\mathbf {v}}(t_0)\Vert ^2 + 2c{\mathbf {r}}(t_0) \cdot {\mathbf {v}}(t_0)\le \omega \nonumber \\&b\Vert {\mathbf {v}}_{}(t_0)\Vert ^2 + 2c{\mathbf {r}}_{}(t_0) \cdot {\mathbf {v}}_{}(t_0) \le \rho _a^2 ( \lambda _{min}\{P\} - 2\alpha bc) \end{aligned}$$

(48)

Since ${\mathbf {r}}(t_0) \cdot {\mathbf {v}}(t_0) \le 0$ for the close encounter to occur, a conservative version of the above condition is

$$\begin{aligned} \Vert {\mathbf {v}}_{}(t_0)\Vert ^2 \le \frac{\rho _a^2}{b} ( \lambda _{min}\{P\} - 2\alpha bc) \end{aligned}$$

(49)

Equation (49) corresponds to condition (iv), sufficiently small $\Vert {\mathbf {v}}_{T/P}(t_0) \Vert$, in Remark 1. By explicitly calculating $\lambda _{min}\{P\}$, one can prove that the right-hand side of (49) is positive if

$$\begin{aligned} c \le \frac{1}{2\alpha }\left( 1 - \frac{1}{4(1-\alpha )}\right) . \end{aligned}$$

(50)

Conditions for the gain selection, which corresponds to (ii) in Remark 1, are thus (45) and (50).

For this proof, we assume that the pursuer control is never saturated, which is true if

$$\begin{aligned} u_{max}\ge \rho _{max}\max \{b,c\} = \rho _a \max \{ b,c\}, \end{aligned}$$

(51)

which corresponds to (iii), sufficiently large $u_{max}$, in Remark 1. $\square$