Extraterrestrial multi-robot planner.

Multi-robot teams have been used for a variety of navigation and mapping applications. They have proven to provide better time efficacy and output than using a singular robot in doing the same. A place where such an implementation may have a lot of potential is in the exploration of planets and other similar terrestrial bodies, which is what this study aims to focus on. With space research thriving more than ever, a decentralized multi-robot system has the potential to deliver results many more times than a single ground robot would.

Theoretical Framework

Multi-Robot Coordination

For a team of $n$ robots, each robot $i$ has a state vector $x_i \in \mathbb{R}^d$ and control input $u_i \in \mathcal{U}_i$. The collective state space is:

$\mathcal{X} = \prod_{i=1}^n \mathbb{R}^d$

The decentralized control law for each robot is given by:

$\dot{x}i = f_i(x_i, u_i) + \sum{ {j \in \mathcal{N}i} } g{ij}(x_i,x_j)$

where:

  • $f_i$ represents the individual robot dynamics
  • $g_{ij}$ represents the interaction dynamics between robots $i$ and $j$
  • $\mathcal{N}_i$ is the set of neighboring robots

Space-Filling Optimization

The space-filling problem is formulated as a coverage optimization:

$\min_{X} \sum_{i=1}^n \int_{\mathcal{W}} \min_{j} |x - x_j|^2 dx$

where:

  • $X = {x_1, …, x_n}$ is the set of robot positions
  • $\mathcal{W}$ is the workspace
  • The objective minimizes the maximum distance from any point to its nearest robot

Collision Avoidance

To ensure no path overlap, we implement a potential field-based approach:

$U_{ij}(x_i, x_j) = \begin{cases} \frac{1}{2}\eta\left(\frac{1}{|x_i - x_j|} - \frac{1}{d_0}\right)^2 & \text{if } |x_i - x_j| \leq d_0
0 & \text{otherwise} \end{cases}$

where:

  • $\eta$ is the repulsion strength
  • $d_0$ is the minimum safe distance

The total potential field for robot $i$ is:

$U_i(x_i) = \sum_{j \in \mathcal{N}i} U{ij}(x_i, x_j)$

Decentralized Control Law

The final control law for each robot combines coverage optimization and collision avoidance:

$u_i = -\nabla_{x_i} \left( \int_{\mathcal{W}} \min_{j} |x - x_j|^2 dx + U_i(x_i) \right)$

This ensures:

  1. Optimal space coverage
  2. Collision-free navigation
  3. Decentralized operation

This paper introduces an optimized space-filling technique and tries to achieve autonomous multi-robot navigation without any path overlap.

Code repository and Report: