[Paper Review] Weak Rigidity Theory and its Application to Multi-agent Formation Stabilization

Gangshan Jing, Guofeng Zhang+2 more|arXiv (Cornell University)|Apr 9, 2018

Structural Analysis and Optimization1 references3 citations

TL;DR

This paper introduces weak rigidity theory, which stabilizes multi-agent formations using pairwise inner products of inter-agent displacements rather than distances, enabling formation control with fewer constraints. It proposes a gradient-based and non-gradient-based control law that ensures local exponential stability under minimal sensing requirements, even when the sensing graph is not rigid—offering a communication-free, distributed solution for GPS-denied environments.

ABSTRACT

This paper introduces the notion of weak rigidity to characterize a framework by pairwise inner products of inter-agent displacements. Compared to distance-based rigidity, weak rigidity requires fewer constrained edges in the graph to determine a geometric shape in an arbitrarily dimensional space. A necessary and sufficient graphical condition for infinitesimal weak rigidity of planar frameworks is derived. As an application of the proposed weak rigidity theory, a gradient based control law and a non-gradient based control law are designed for a group of single-integrator modeled agents to stabilize a desired formation shape, respectively. Using the gradient control law, we prove that an infinitesimally weakly rigid formation is locally exponentially stable. In particular, if the number of agents is one greater than the dimension of the space, a minimally infinitesimally weakly rigid formation is almost globally asymptotically stable. In the literature of rigid formation, the sensing graph is always required to be rigid. Using the non-gradient control law based on weak rigidity theory, the sensing graph is unnecessary to be rigid for local exponential stability of the formation. A numerical simulation is performed for illustrating effectiveness of our main results.

Motivation & Objective

• Address the challenge of stabilizing multi-agent formations in GPS-denied environments where global positioning and inter-agent communication are unavailable.
• Overcome limitations of distance-based rigidity, which require rigid sensing graphs and global coordinate systems.
• Develop a distributed formation control strategy that relies only on local relative displacement measurements in each agent’s local coordinate frame.
• Establish a graphical condition for infinitesimal weak rigidity in the plane to enable efficient verification of formation stability.
• Design control laws that ensure local exponential stability even when the sensing graph is not rigid, reducing network information flow and system cost.

Proposed method

• Propose weak rigidity as a framework characterization using pairwise inner products of inter-agent displacement vectors, which captures geometric shape with fewer constraints than distance-based rigidity.
• Derive a necessary and sufficient graphical condition for infinitesimal weak rigidity in the plane using triangle-based edge sets and rank analysis of the weak rigidity matrix.
• Design a gradient-based control law using a cost function based on inner product deviations, with a gain matrix selected to ensure local exponential stability.
• Propose a non-gradient control law that stabilizes the formation under weakly rigid sensing graphs, avoiding the need for a rigid graph and reducing communication overhead.
• Use matrix completion and rank analysis to verify infinitesimal weak rigidity, particularly focusing on the rank of the matrix $ ar{R}_w^* $, which captures the weak rigidity condition.
• Apply numerical simulation with a 6-agent system forming a regular hexagon, using a path graph as the sensing graph, to validate the theoretical results.

Experimental results

Research questions

• RQ1 Can a formation be stabilized using only relative displacement measurements in local coordinate frames without requiring a global coordinate system or inter-agent communication?
• RQ2 What is the minimal set of constraints (in terms of inner products of displacements) needed to ensure a framework is infinitesimally weakly rigid in the plane?
• RQ3 How does weak rigidity compare to distance-based rigidity in terms of required graph connectivity and number of constraints?
• RQ4 Can local exponential stability of a desired formation be achieved even when the sensing graph is not rigid, using a non-gradient control law?
• RQ5 What conditions on the gain matrix ensure exponential stability in the non-gradient control scheme under weak rigidity?

Key findings

• A necessary and sufficient graphical condition for infinitesimal weak rigidity in the plane is derived using triangle-based edge sets, enabling efficient verification without rank computation of the rigidity matrix.
• For the gradient-based control law, an infinitesimally weakly rigid formation is locally exponentially stable, with convergence confirmed via eigenvalue analysis of the Jacobian matrix.
• When the number of agents is one more than the dimension of the space (e.g., 3 agents in 2D), a minimally infinitesimally weakly rigid formation achieves almost global asymptotic stability.
• The non-gradient control law achieves local exponential stability even when the sensing graph is not rigid, demonstrating that weak rigidity is sufficient and less stringent than distance-based rigidity.
• In the simulation, with a non-symmetric gain matrix $ K $, the eigenvalues of the Jacobian $ J^* $ were all in the open left half-plane except for three zero eigenvalues, confirming local exponential stability.
• Numerical results show that the cost function $ V $, based on inner product deviations, decays to zero exponentially, and edge lengths converge to the desired value of 2, validating the control law performance.