[Paper Review] Distributed stabilization control of rigid formations with prescribed orientation

Zhiyong Sun, Myoung‐Chul Park+2 more|arXiv (Cornell University)|Jun 10, 2016

Distributed Control Multi-Agent Systems19 references3 citations

TL;DR

This paper proposes a distributed control law for rigid multi-agent formations in 2D and 3D spaces that stabilizes both the desired shape and a prescribed orientation. By combining gradient-based shape control with a novel orientation control term using relative position vectors, the method ensures exponential convergence to the target formation with minimal global coordinate knowledge—only 2 agents in 2D and 3 in 3D require global orientation, while others operate without any coordinate frame alignment.

ABSTRACT

Most rigid formation controllers reported in the literature aim to only stabilize a rigid formation shape, while the formation orientation is not controlled. This paper studies the problem of controlling rigid formations with prescribed orientations in both 2-D and 3-D spaces. The proposed controllers involve the commonly-used gradient descent control for shape stabilization, and an additional term to control the directions of certain relative position vectors associated with certain chosen agents. In this control framework, we show the minimal number of agents which should have knowledge of a global coordinate system (2 agents for a 2-D rigid formation and 3 agents for a 3-D rigid formation), while all other agents do not require any global coordinate knowledge or any coordinate frame alignment to implement the proposed control. The exponential convergence to the desired rigid shape and formation orientation is also proved. Typical simulation examples are shown to support the analysis and performance of the proposed formation controllers.

Motivation & Objective

• Address the lack of orientation control in existing rigid formation control methods, which typically stabilize only the shape.
• Enable formation control with both desired shape and specific orientation, crucial for practical applications like surveillance or coordinated navigation.
• Minimize reliance on global coordinate systems by requiring only a small subset of agents (2 in 2D, 3 in 3D) to have access to a global frame.
• Ensure the control is fully distributed, with all other agents operating using only local relative position measurements and no coordinate alignment.
• Prove exponential convergence to the desired rigid formation with prescribed orientation using rigidity theory and Lyapunov analysis.

Proposed method

• Integrate a standard gradient descent control law for shape stabilization based on inter-agent distance errors.
• Introduce an additional orientation control term that aligns specific relative position vectors (e.g., between key agents) to desired global directions.
• Use a selection matrix J to identify orientation edges—critical links whose relative vectors are used to enforce the desired formation orientation.
• Formulate the overall control law as a combination of distance-based and orientation-based control components, embedded in a Laplacian-like structure.
• Apply rigidity theory and infinitesimal rigidity conditions to ensure that the formation remains rigid under the control input.
• Use Lyapunov stability analysis and matrix rank arguments (via the matrix F = (H⊗Id)ᵀ(ZZᵀ + JJ⊗Id)(H⊗Id)) to prove exponential convergence to the desired formation and orientation.

Experimental results

Research questions

• RQ1 Can a distributed formation control law stabilize both the shape and orientation of a rigid formation without requiring all agents to have global coordinate knowledge?
• RQ2 What is the minimal number of agents that must have access to a global coordinate system to achieve prescribed orientation control in 2D and 3D formations?
• RQ3 How can orientation control be integrated with existing distance-based shape control to ensure exponential convergence?
• RQ4 Does the proposed control law preserve infinitesimal rigidity and avoid spurious equilibria?
• RQ5 Can the control framework be extended to work under local sensing and without global frame alignment?

Key findings

• The proposed control law ensures exponential convergence to the desired rigid formation shape and prescribed orientation in both 2D and 3D spaces.
• Only two agents are required to have knowledge of the global coordinate system in 2D formations, and three in 3D, while all others operate without any global frame information.
• The null space of the matrix F = (H⊗Id)ᵀ(ZZᵀ + JJ⊗Id)(H⊗Id) is exactly span(1ₙ ⊗ Id), proving that the only rigid motions are translations and rotations, and thus the formation is uniquely determined.
• The orientation control term successfully aligns selected relative vectors to desired global directions, preventing undesired rotations.
• Theoretical analysis confirms that the formation remains infinitesimally rigid under the control input, ensuring structural stability.
• Simulation results validate the theoretical findings, demonstrating stable convergence to the target formation with correct orientation under various initial conditions.