Drone flocking optimization using NSGA-II and principal component analysis

Abstract

Individual agents in natural systems like flocks of birds or schools of fish display a remarkable ability to coordinate and communicate in local groups and execute a variety of tasks efficiently. Emulating such natural systems into drone swarms to solve problems in defense, agriculture, industrial automation, and humanitarian relief is an emerging technology. However, flocking of aerial robots while maintaining multiple objectives, like collision avoidance, high speed etc., is still a challenge. This paper proposes optimized flocking of drones in a confined environment with multiple conflicting objectives. The considered objectives are collision avoidance (with each other and the wall), speed, correlation, and communication (connected and disconnected agents). Principal Component Analysis (PCA) is applied for dimensionality reduction and understanding of the collective dynamics of the swarm. The control model is characterized by 12 parameters which are then optimized using a multi-objective solver (NSGA-II). The obtained results are reported and compared with that of the CMA-ES algorithm. The study is particularly useful as the proposed optimizer outputs a Pareto Front representing different types of swarms that can be applied to different scenarios in the real world.

Appendices

Appendix A: Background

1.1 Principal component analysis

The covariance matrix K is formulated as follows: Let X be an n x m design matrix with n rows as the samples and m columns as objectives. A pre-processing step often carried out is the normalization of the design matrix which brings the mean of samples for each objective to 0.0 and the variance to 1.0 (Eq. A1). The covariance matrix is then calculated by taking the mean of all samples of the pairwise products for each objective (Eq. A2). In a vectorized format, this is equivalent to taking the matrix product of the design matrix X with its transpose (Eq. A3).

$$\begin{aligned} X_{ij}^{norm}&= \frac{X_{ij}-\mu _{j}}{\sigma _{j}} \end{aligned}$$

(A1)

$$\begin{aligned} K_{ij}&= \frac{1}{n} \sum ^{n}_{k=1} X_{ki}X_{kj} \end{aligned}$$

(A2)

$$\begin{aligned} K&= \frac{1}{n} (X^{norm})^{T} X^{norm} \end{aligned}$$

(A3)

• where,

• $X_{ij} \ \equiv$ Element of X at $i\hbox {th}$ row and $j\hbox {th}$ column

• $X_{ij}^{norm} \ \equiv X_{ij}$ normalized to 0.0 mean and 1.0 standard deviation

• $\mu _{j} \ \equiv$ Mean of all n samples of $j\hbox {th}$ objective

• $\sigma _{j} \ \equiv$ Standard deviation of all n samples of $j\hbox {th}$ objective

• $n \ \equiv$ Number of samples

• $m \ \equiv$ Number of objectives

• $K \equiv$ Covariance matrix

• $K_{ij} \ \equiv$ Element of K at $i\hbox {th}$ row and $j\hbox {th}$ column

1.2 Non-dominated sorting genetic algorithm

Let $P_{o}$ be an N sized initial random population. This population is sorted based on non-domination according to the following rules: An individual $X_{1}$ in the population is said to be dominated by individual $X_{2}$ if satisfies both of the following conditions:

• All fitnesses of $X_{1}$ must be less than or equal to that of $X_{2}$ particle.

• At least one fitness of $X_{1}$ must be strictly less than that of $X_{2 }$.

Mathematically, individual $X_{1}$ dominates $X_{2}$ if $d = 1$, and the individuals are non-dominated if $d=0$.

Where, $d = \{ \forall m \ F(X_{1})^m \le F(X_{2})^m \} \cap \{ \exists m \ F(X_{1})^m < F(X_{2})^m \}$

This method divides the population into dominating and non-dominating solutions, which is a heuristic used to guide the population towards better solutions through the generations. Each solution in this population is also ranked based on the number of other members it is dominated by, and accordingly, it is assigned a front rank. Next, an offspring population Q is created from the sorted population by applying tournament selection, recombination, and mutation operators. A new 2N sized population is made using $P \cup Q$ and is again sorted and ranked to retain the best solution across generations (elitism). To make the next population $P_{t+1}$ from this combined set, solutions are taken in order of their front ranking. In case the number of solutions belonging to a front exceeds the amount that can be accommodated into the new N sized population, the remaining solutions in that front are ranked based on a crowding operator as follows:

Let $\mathcal {F}^{k}$ be the set of solutions on the $k\hbox {th}$ ranked Pareto front. The crowding distance ($c^{m}_{i}$) for the $m\hbox {th}$ objective for $i\hbox {th}$ solution on this front is defined as the normalized distance between the two nearest solutions i.e. $(i+1)\hbox {th}$ and $(i-1)\hbox {th}$ (Eq. A4). The overall crowding distance ($c_{i}$) is the sum taken for each objective (Eq. A5).

$$\begin{aligned} \forall X_{i} \in \mathcal {F}^{k}: c^{m}_{i}&= \frac{F^{m}(X_{i+1}) -F^{m}(X_{i-1})}{F^{m}_{max}-F^{m}_{min}} \end{aligned}$$

(A4)

$$\begin{aligned} c_{i}&= \sum ^{M}_{m=1} c^{m}_{i} \end{aligned}$$

(A5)

This crowding operator ensures that the Pareto Front is uniformly distributed, and the range of each objective value is minimized as the search progresses. The remaining solutions are ranked according to $c_{i}$ and the new population $P_{t+1}$ moves forward to the next generation. NSGA-II is faster than NSGA-I and has a worst-case complexity of $O(MN^{2})$.

Appendix B: Decision variables

1.1 Separation

The following two equations (B7) and (B8) are used to find a repulsion vector for agent i after scaling it according to the relative distances in ${\textbf {r}}_{i}^{\mathrm {mag}}$ and a gain $p^{\mathrm {rep}}$.

$$\begin{aligned} {\textbf {r}}^{\mathrm {mag}}&= \Vert R^{rel}\Vert ^{\bot r_{0}^{\mathrm {rep}} } \end{aligned}$$

(B6)

$$\begin{aligned} V^{\mathrm {rep}}&= p^{\mathrm {rep}}. ({\textbf {r}}^{\mathrm {mag}} - r_{0}^{\mathrm {rep}}).\frac{R^{rel}}{{\textbf {r}}^{\mathrm {mag}}} \end{aligned}$$

(B7)

$$\begin{aligned} {\textbf {v}}^{\mathrm {rep}}_{i}&=\sum _{j=1}^{\mathcal {N}_{o}} V^{\mathrm {rep}}_{j} \end{aligned}$$

(B8)

• where,

• $a_{\top b} = max(a,b)$ i.e. a is at least b

• $a^{\bot c} = min(a,c)$ i.e. a is at most c

• ${\textbf {r}}^{\mathrm {mag}} \equiv \mathcal {N}_{o} \ \text {x} \ 1$ sized vector containing inter-agent distances

• $r_{0}^{\mathrm {rep}} \ \equiv$ Repulsion cutoff distance (user-dependent parameter)

• $p^{\mathrm {rep}} \ \equiv$ Repulsion gain (user-dependent parameter)

• $V^{\mathrm {rep}} \ \equiv \mathcal {N}_{o} \ \text {x} \ 2$ sized matrix of scaled repulsion velocities for all neighbors

• $V^{\mathrm {rep}}_{j} \ \equiv$ Repulsion velocity of $j\hbox {th}$ neighbor i.e. $j\hbox {th}$ row of $V^{\mathrm {rep}}$

• ${\textbf {v}}^{\mathrm {rep}}_{i}\ \equiv$ Desired collective repulsion vector

Note that the upper bound of ${\textbf {r}}^{\mathrm {mag}}$ is the parameter $r_{0}^{\mathrm {rep}}$ to enable short-range effects. The matrix norm in Eq. (B6) is only taken along the row axis, i.e. for each neighbor. An Nx2 matrix $(R^{rel})$ of position vectors divided by the distance vector $(r^{\mathrm {mag}})$ yields unit position vectors. $V^{\mathrm {rep}}$ contains all the corresponding scaled repulsion velocities, and the division and multiplication in Eq. (B7) is done element-wise.

1.2 Alignment

The equations are similar to separation with one major difference: the upper bound for the velocity magnitude (${\textbf {v}}^{frictmax}$) is now calculated dynamically with decay function D in Eq. (B9), which is dependent on the inter-agent distance (Vásárhelyi et al. 2018). Eqs. (B10) - (B12) describe the process of finding out the agent’s combined alignment vector.

$$\begin{aligned} {\textbf {v}}^{frictmax}&= D({\textbf {r}}_{i}^{\mathrm {mag}}- r_{0}^{\mathrm {frict}} - r_{0}^{\mathrm {rep}}, a^{\mathrm {frict}}, p^{\mathrm {frict}}) _{\top {v}^{\mathrm {frict}}} \end{aligned}$$

(B9)

$$\begin{aligned} {\textbf {v}}^{\mathrm {mag}}&= \Vert V^{rel}\Vert _{\top {\textbf {v}}^{frictmax}} \end{aligned}$$

(B10)

$$\begin{aligned} V^{\mathrm {frict}}&= c^{\mathrm {frict}}.({\textbf {v}}^{\mathrm {mag}}-{\textbf {v}}^{frictmax}).\frac{V^{rel}}{{\textbf {v}}_{mag}} \end{aligned}$$

(B11)

$$\begin{aligned} {\textbf {v}}^{\mathrm {frict}}_{i}&=\sum _{j=1}^{\mathcal {N}_{o}} V^{\mathrm {frict}}_{j} \end{aligned}$$

(B12)

• where,

• D is the velocity decay function from the Vásárhelyi et al. model and takes a vector as the first argument

• $p^{\mathrm {frict}} \equiv$ Slope for the linear part of the decay curve (user-dependent parameter)

• $a^{\mathrm {frict}} \equiv$ Acceleration for the non-linear part of the decay curve (user-dependent parameter)

• $c^{\mathrm {frict}} \equiv$ Overall Gain for alignment (user-dependent parameter)

• $v^{\mathrm {frict}} \equiv$ Velocity slack for alignment (user-dependent parameter)

• $r_{0}^{\mathrm {frict}} \ \equiv$ Alignment cutoff distance for maximum alignment (user-dependent parameter)

• $V^{\mathrm {frict}} \ \equiv \mathcal {N}_{o} \ \text {x} \ 2$ sized matrix of scaled alignment velocities for all neighbors

• $V^{\mathrm {frict}}_{j} \ \equiv$ Alignment velocity of $j\hbox {th}$ neighbor i.e. $j\hbox {th}$ row of $V^{\mathrm {frict}}$

• ${\textbf {v}}^{\mathrm {frict}}_{i}\ \equiv$ Desired collective alignment vector

Eq. (B9) gives a vector composed of each neighbor’s maximum allowable velocity differences. The maximum is proportional to the inter-agent distance. This ensures that the alignment for two agents in close proximity is larger and vice-versa. Also, the maximum allowable difference is lower bound by an optimization parameter $v^{\mathrm {frict}}$ so the agents do not strive for perfect alignment, and there is some slack. Eq. (B11) compensates for the velocity difference for each neighbor, and Eq. (B12) sums the alignment velocities for each neighbor.

Appendix C: Software

See Figure 7.

Fig. 7

Class diagram for the Multi-Objective flocking simulator. The Env class has a 1 to n relationship with the CoBot as each environment can contain multiple CoBots. Similarly, the Optimizer class can run multiple environments on multiple cores/processes. The complete simulator is available on GitHub (Supplementary material S1, 2022)