[Paper Review] Controlling a triangular flexible formation of autonomous agents

Héctor García de Marina, Zhiyong Sun+2 more|arXiv (Cornell University)|Apr 3, 2017

Distributed Control Multi-Agent Systems1 references4 citations

TL;DR

This paper proposes a novel distributed control algorithm for forming a triangular shape using only two controlled agent pairs, combining benefits of distance- and position-based control. By introducing artificial biases in range sensors, the method achieves stable, orientation-free triangular formations through collective motion induced by sensing errors, enabling both stationary and moving configurations with precise shape control.

ABSTRACT

In formation control, triangular formations consisting of three autonomous agents serve as a class of benchmarks that can be used to test and compare the performances of different controllers. We present an algorithm that combines the advantages of both position- and distance-based gradient descent control laws. For example, only two pairs of neighboring agents need to be controlled, agents can work in their own local frame of coordinates and the orientation of the formation with respect to a global frame of coordinates is not prescribed. We first present a novel technique based on adding artificial biases to neighboring agents' range sensors such that their eventual positions correspond to a collinear configuration. Right after, a small modification in the bias terms by introducing a prescribed rotation matrix will allow the control of the bearing of the neighboring agents.

Motivation & Objective

• Address the trade-off between position-based and distance-based formation control by merging their advantages.
• Enable formation control with fewer controlled agent pairs than minimally rigid distance-based methods.
• Achieve orientation-free triangular formations using local coordinate frames and sensing-error-induced motion.
• Control both shape and collective motion (stationary or constant velocity) via artificial sensor biases.
• Provide a foundation for extending to arbitrary non-rigid formations with more than three agents.

Proposed method

• Introduce artificial biases in range measurements between two agent pairs to induce controlled collective motion.
• Use a modified distance-based gradient descent control law with bias terms to steer agents toward collinear configurations.
• Apply a rotation matrix to bias terms to control the angle between relative vectors, enabling triangular formation control.
• Formulate an augmented error system with a small gain parameter $ c $ to ensure local exponential stability.
• Leverage eigenvalue analysis of the Jacobian matrix to prove stability of the desired equilibrium set.
• Utilize sensing-error-induced dynamics to achieve shape control without requiring global orientation or common reference frames.

Experimental results

Research questions

• RQ1 Can a triangular formation be stabilized using only two controlled agent pairs while maintaining orientation freedom?
• RQ2 How can sensing errors or artificial biases be exploited to induce controlled collective motion in non-rigid formations?
• RQ3 What conditions ensure local exponential stability of a desired triangular shape under biased distance-based control?
• RQ4 Can the angle between agents be controlled independently to form a specific triangle shape without global orientation?
• RQ5 How does the number of controlled agent pairs compare to minimally rigid distance-based control in achieving the same formation?

Key findings

• With symmetric biases $ \mu_1 = -\mu_2 $, agents converge to a collinear configuration with zero distance errors, forming a stationary triangular shape.
• With antisymmetric biases $ \mu_2 = -\mu_1 $, agents achieve a steady-state motion with constant velocity, and distance errors converge to $ \frac{2}{3} $, independent of desired distances.
• The desired triangular formation with a prescribed angle (e.g., 60°) is locally exponentially stable for sufficiently small gain $ c > 0 $, as confirmed by Hurwitz analysis of the Jacobian.
• The equilibrium set $ \mathcal{U}_d $ is locally exponentially stable for all $ \theta \in (-\pi, \pi) $, ensuring robust shape control regardless of initial orientation.
• The method allows formation control without requiring a common global frame or prescribed orientation, enabling free rotational motion.
• The system outperforms standard distance-based control in that collinearity is not invariant, allowing escape from degenerate configurations.