Adaptive formation-switching of a multi-robot system in an unknown occluded environment using BAT algorithm

Abstract

This decade has witnessed a paradigm shift in human-labor based rescue-surveillance operations. Research propositions have explored the possibility of replacing manual intervention for the management of catastrophic situations by a team of robots. However, to implement the concept into practice, the robotics community has faced several challenges. The multi-robotic system has to be duly coordinated efficaciously by controllers to automate the operations thereby saving the lives of the rescuers. Subsequently, the controller/s should be able to ensemble the robots forming a particular shape depending on the varying environmental conditions. Moreover, it would allow the group to switch its current formation so that the system could maneuver towards the target while avoiding static/dynamic obstacles. To address these challenges, we have proposed a hierarchical control strategy so that the robots could maintain a strong inter-agent cohesiveness and simultaneously could switch their formation in the face of the changing situations and could pursue the goal of arriving towards the target. The formation control law has been designed based on the echolocation principle of the bio-inspired bat algorithm. The algorithm, corroborated by the simulation results and the real-time experiments is exceptionally useful for forming the desired pattern, changing the formation adaptively whenever obstructions show up in their trajectories.

Appendices

Appendix

A stability analysis of the Eq. 5

In order to analyze the stability of the proposed controllers, the following assumption is taken.

Assumption: All identical agents are navigating in an unknown environment and each agent can sense information about the position and velocity of its neighboring agents ($N_i$).

Definition 1

The error dynamics (Gazi and Passino 2002a) of the entire system is defined as $E=[E_1^T\quad E_2^T\quad \cdots \quad E_N^T]$; where $E_i=\begin{bmatrix} e_{pi}&e_{vi} \end{bmatrix}$.

From Eq. 5, the state equations of the error dynamics can be described as,

$$\begin{aligned} \begin{aligned} \dot{e}_{pi}=e_{vi} \\ \dot{e}_{vi}=\dot{v}_i-\dot{{\bar{v}}} \end{aligned} \end{aligned}$$

(15)

where $\dot{{\bar{v}}}=\frac{1}{N}\sum _{l=1}^{N}u_{l}$. As all the agents are identical in nature, $k_i^a=k^a$, $k_i^v=k^v$, $k_i^r=k^r$ and $r_i^s=r^s$ for all $i\in N$. Therefore, $\dot{e}_{vi}$ will be

$$\begin{aligned} \begin{aligned} \dot{e}_{vi}=u_i - \frac{1}{N}\sum _{l=1}^{N}u_l=-k^ae_{pi}-k^ve_{vi}+k^r\sum _{\begin{array}{c} j\in N_i \end{array}}\exp (-\frac{x^2_{ij}}{\delta \times {r^s}^2})\times (x_i-x_j) \\ +\frac{1}{N} \left[ k^a\sum _{l=1}^{N} e_{pl} + k^v\sum _{l=1}^{N}e_{vl}-k^r\sum _{l=1}^{N}\left[ \sum _{\begin{array}{c} j\in N_i \end{array}}\exp (-\frac{x^2_{lj}}{\delta \times {r^s}^2})\times (x_l-x_j)\right] \right] \end{aligned} \end{aligned}$$

(16)

The $3^{rd}$ and $4^{th}$ terms of the right-hand-side (RHS) of Eq. 16 are zero, as $\sum _{l=1}^{N}e_{pl}=\sum _{l=1}^{N}(x_{l}-{\bar{x}})=0$ and $\sum _{l=1}^{N}e_{vl}=\sum _{l=1}^{N}(v_{l}-{\bar{v}})=0$ respectively. Hence, the position and velocity errors are zero as expected for perfect control. Moreover, expanding the last term of the RHS of Eq. 16, we obtain that the result is equal to zero (derivation has been provided in Appendix B).

So, $\dot{e}_{vi}$ will be (expanding the exponential term and neglecting the higher orders);

$$\begin{aligned} \dot{e}_{vi}=-k^a e_{pi}-k^v e_{vi}+k^r \sum _{\begin{array}{c} j\in N_i \end{array}}[1 - (\frac{x^2_{ij}}{\delta \times {r^s}^2})]\times (x_i-x_j) \end{aligned}$$

(17)

Hence, the error dynamics will be (from Eq. 5 and Eq. 17),

$$\begin{aligned} \dot{E}_i^T=A_i\times E_i^T+B_i\left[ \phi _i^r-\phi _i^\delta \right] \end{aligned}$$

(18)

where $A_i=\begin{bmatrix} 0&{}1 \\ -k^a&{}-k^v \end{bmatrix}$, $B_i=\begin{bmatrix} 0\\ 1 \end{bmatrix}$, $\phi _i^r=k^r\sum _{\begin{array}{c} j\in N_i \end{array}}(x_i-x_j)$ and $\phi _i^\delta =k^r\sum _{\begin{array}{c} j\in N_i \end{array}}(\frac{x_{ij}^2}{\delta \times {r^s}^2})(x_i-x_j)$.

Stability Proof: In order to prove the stability of the $i^{th}$ agent’s error dynamics (as shown in Eq. 18), considering the following (generalized) Lyapunov candidate (Yuan et al. 2018) as;

$$\begin{aligned} V_i(E_i)=E_i^TP_iE_i \end{aligned}$$

(19)

The time-derivative of Eq. 19 will be (Yuan et al. 2018);

$$\begin{aligned} \dot{V}_i(E_i)=-E_i^TQ_iE_i+2E_i^TP_iB_i(\phi _i^r-\phi _i^\delta ) \end{aligned}$$

(20)

where $Q_i=-(P_iA_i+A_i^TP_i)$. Now for the entire system, the Lyapunov candidate will be;

$$\begin{aligned} \dot{V}(E)=\sum _{i=1}^N\dot{V}_i(E_i)=\sum _{i=1}^N[-E_i^TQ_iE_i+2E_i^TP_iB_i(\phi _i^r-\phi _i^\delta )] \end{aligned}$$

(21)

In order to prove the stability in the sense of Lyapunov, $\dot{V}(E)$ should be negative semi-definite; hence the rightmost term of Eq. 21 needs to be less than or equal to zero. The stated terminology is only possible if $\phi _i^r \le \phi _i^\delta$. Then from Eq. 18, the above inequality can be represented as:

$$\begin{aligned} k^r\sum _{\begin{array}{c} j\in N_i \end{array}}(x_i-x_j) \le k^r\sum _{\begin{array}{c} j\in N_i \end{array}}(\frac{x_{ij}^2}{\delta \times {r^s}^2})(x_i-x_j) \end{aligned}$$

(22)

By solving Eq. 22, we obtain the following condition;

$$\begin{aligned} x_{ij}\ge \sqrt{\delta }\times r^s \end{aligned}$$

(23)

In a simplistic scenario, it should be noted that in order to make the assumption 1 to be satisfied, the value of $\sqrt{\delta }$ would be 2 which makes an equivalent adjustment between the $i^{th}$ agent with its neighbors in terms of repulsion region. Hence, $x_{ij} \ge 2\times r^s$. For this specific condition, Lyapunov stability (Yuan et al. 2018) holds true.

B Proof of Eq. 16

Let us assume that the last term ($6^{th}$ term) of Eq. 16 is equal to G. Therefore,

$$\begin{aligned} G = -\frac{k^r}{N}\sum _{l=1}^{N}\left[ \sum _{j\in N_i} \exp \left( -\frac{x^2_{lj}}{\delta \times r^{s^2}}\right) \times \left( x_l-x_j\right) \right] \end{aligned}$$

(24)

The generalized form of Eq. 24 can be written as (assuming all agents are neighbors, i.e. $N_i=\{1, 2, \ldots , N\}$):

$$\begin{aligned} G = -\frac{k^r}{N}\sum _{l=1}^{N}\left[ \sum _{j=1}^{N} \exp \left( -\frac{||x_l-x_j||^2}{\delta \times r^{s^2}}\right) \times \left( x_l-x_j\right) \right] \end{aligned}$$

(25)

After expanding Eq. 25, we obtain

$$\begin{aligned} G&= -\frac{k^r}{N}[[\exp (-\frac{||x_1-x_2||^2}{\delta \times r^{s^2}})(x_1-x_2) + \exp (-\frac{||x_1-x_3||^2}{\delta \times r^{s^2}})(x_1-x_3)\nonumber \\&\quad + \cdots + \exp (-\frac{||x_1-x_N||^2}{\delta \times r^{s^2}})(x_1-x_N)]\nonumber \\&\quad + [\exp (-\frac{||x_2-x_1||^2}{\delta \times r^{s^2}})(x_2-x_1) + \exp (-\frac{||x_2-x_3||^2}{\delta \times r^{s^2}})(x_2-x_3)\nonumber \\&\quad + \cdots + \exp (-\frac{||x_2-x_N||^2}{\delta \times r^{s^2}})(x_2-x_N)] \nonumber \\&\quad + \cdots + \nonumber \\&\quad + [\exp (-\frac{||x_N-x_1||^2}{\delta \times r^{s^2}})(x_N-x_1) + \exp (-\frac{||x_N-x_2||^2}{\delta \times r^{s^2}})(x_N-x_2) \nonumber \\&\quad + \cdots + \exp (-\frac{||x_N-x_{N-1}||^2}{\delta \times r^{s^2}})(x_N-x_{N-1})]] \end{aligned}$$

(26)

From Eq. 26, it is observed that for each agent there will be an equal and opposit force from it’s neighbors to stabilize the entire system. Therefore, the final result of G (Eq. 26) is zero which concludes that the multi-robotics system is stable for this specific condition.