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

Abstract

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.

Appendix

1.1 Proof of Theorem 4

Proof

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:

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

(66)

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

(67)

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 w_r = 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 φ_i 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. □