Graph-Based Dynamics and Network Control of a Single Articulated Robotic System
Jonathan Lane, Nak-seung Patrick Hyun
TL;DR
This work reframes a single articulated robotic system as a physically constrained network over an arborescence graph, linking link dynamics to graph operators. By over-parameterizing the configuration with node coordinates $Q$ and edge coordinates $Q_e$, the authors derive a second-order, consensus-like dynamics $\ddot{Q} = -M^{-1}\Gamma + (M^{-1}F - G)$ with constraint forces $\Gamma = L_w(\mathcal{G}) Q$ and $L_w(\mathcal{G}) = D(\mathcal{G})\Lambda D(\mathcal{G})^\top$, and an edge-centric form $\ddot{Q}_e = -L_e(\mathcal{G})\Lambda Q_e + D(\mathcal{G})^\top M^{-1}F$. A decentralized control strategy is developed to decouple constraint forces from the control inputs under a structural condition, enabling leader-follower regulation of relative configurations. The paper proves how to compute Lagrange multipliers $\lambda$ algebraically from the graph structure and edge coordinates, and demonstrates the approach on two- and five-link SARs, showing effective edge-trajectory tracking and scalability. This framework provides a scalable, distributed method for controlling physically coupled robots while preserving the underlying graph topology in the dynamics.
Abstract
Extensive research on graph-based dynamics and control of multi-agent systems has successfully demonstrated control of robotic swarms, where each robot is perceived as an independent agent virtually connected by a network topology. The strong advantage of the network control structure lies in the decentralized nature of the control action, which only requires the knowledge of virtually connected agents. In this paper, we seek to expand the ideas of virtual network constraints to physical constraints on a class of tree-structured robots which we denote as single articulated robotic (SAR) systems. In our proposed framework, each link can be viewed as an agent, and each holonomic constraint connecting links serves as an edge. By following the first principles of Lagrangian dynamics, we derive a consensus-like matrix-differential equation with weighted graph and edge Laplacians for the dynamics of a SAR system. The sufficient condition for the holonomic constraint forces becoming independent to the control inputs is derived. This condition leads to a decentralized leader-follower network control framework for regulating the relative configuration of the robot. Simulation results demonstrate the effectiveness of the proposed control method.