Solving the Right Problem with Multi-Robot Formations

TL;DR

This work tackles the mismatch between fixed formation shapes and the true, environment-dependent costs in multi-robot missions. It introduces a two-stage formation planner that first fits a surrogate cost via adaptive weights to approximate the true cost, and then optimizes the ideal formation with Adam, realized by a Lyapunov-based non-cooperative controller. The approach yields significant improvements, reducing a single true cost by up to 75% and achieving 20–40% reductions when multiple costs are considered, outperforming traditional shape-based methods. The combination of surrogate-cost fitting, adaptive weighting, and a principled control law offers a practical path to solving the right problem in dynamic, uncertain environments.

Abstract

Formation control simplifies minimizing multi-robot cost functions by encoding a cost function as a shape the robots maintain. However, by reducing complex cost functions to formations, discrepancies arise between maintaining the shape and minimizing the original cost function. For example, a Diamond or Box formation shape is often used for protecting all members of the formation. When more information about the surrounding environment becomes available, a static shape often no longer minimizes the original protection cost. We propose a formation planner to reduce mismatch between a formation and the cost function while still leveraging efficient formation controllers. Our formation planner is a two-step optimization problem that identifies desired relative robot positions. We first solve a constrained problem to estimate non-linear and non-differentiable costs with a weighted sum of surrogate cost functions. We theoretically analyze this problem and identify situations where weights do not need to be updated. The weighted, surrogate cost function is then minimized using relative positions between robots. The desired relative positions are realized using a non-cooperative formation controller derived from Lyapunov's direct approach. We then demonstrate the efficacy of this approach for military-like costs such as protection and obstacle avoidance. In simulations, we show a formation planner can reduce a single cost by over 75%. When minimizing a variety of cost functions simultaneously, using a formation planner with adaptive weights can reduce the cost by 20-40%. Formation planning provides better performance by minimizing a surrogate cost function that closely approximates the original cost function instead of relying on a shape abstraction.

Figures (6)

• Figure 1: Difference between a configuration and formation. \ref{['configuration']} Example configuration in $\mathbb{R}^2$. \ref{['formation']} Example formation in $\mathbb{R}^2$ with relative connections.
• Figure 2: Diagram showing the elements of the formation system.
• Figure 3: Four different simulated environments (3.2 m $\times$ 2.0 m) for testing. There are four different types of entities in each environment: a leader, agents, obstacles, and threats. Each environment is randomly generated. The black squares indicate the waypoints the leader will follow.
• Figure 4: Diagrams showing the agents' relative locations used for calculating costs. \ref{['protection_cost_fig']} Relative locations for protection cost \ref{['protection_cost']}. \ref{['obstacle_cost_fig']} Relative locations for obstacle cost \ref{['obstacle_cost']}.
• Figure 5: Costs for the FP-AW and FP methods in Environment 3. The FP-AW method was able to perform better than the FP method by shifting focus to costs in the surrogate cost that affect the true obstacle avoidance cost. We see this in effect in how the FP-AW is able to achieve a low true obstacle avoidance cost by ignoring the proximity cost.

Theorems & Definitions (2)

• Theorem 1
• proof