Complex-Valued GNNs for Distributed Basis-Invariant Control of Planar Systems

Abstract

Graph neural networks (GNNs) are a well-regarded tool for learned control of networked dynamical systems due to their ability to be deployed in a distributed manner. However, current distributed GNN architectures assume that all nodes in the network collect geometric observations in compatible bases, which limits the usefulness of such controllers in GPS-denied and compass-denied environments. This paper presents a GNN parametrization that is globally invariant to choice of local basis. 2D geometric features and transformations between bases are expressed in the complex domain. Inside each GNN layer, complex-valued linear layers with phase-equivariant activation functions are used. When viewed from a fixed global frame, all policies learned by this architecture are strictly invariant to choice of local frames. This architecture is shown to increase the data efficiency, tracking performance, and generalization of learned control when compared to a real-valued baseline on an imitation learning flocking task.

Figures (6)

• Figure C1: In the swarms considered in this paper, each robot has one or several geometric features (orange and magenta arrows), such as velocity or acceleration, expressed in its body frame (red and lime green axes). Robots share a communication edge (grey line) if they are within a communication radius, indicated by the dotted cyan circle in this diagram. An edge $\mathbf{e}_{ij}$ from robot $v_i$ to robot $v_j$ comes with an encoding in $SO(2)$ representing the rotation from the body frame of robot $i$, $\mathcal{B}_i$ to that of robot $j$.
• Figure E1: As seen in the left-hand plot, the smallest and largest equivariant models tested both track the nominal controller very well. The seven intermediate cases are excluded so not to crowd the plot, but they show similar results to those presented in this figure. The right-hand figure shows that the baseline GNN was not able to approximate the nominal controller as well as the basis-invariant model. Aside from not tracking the nominal controller well, the standard deviation of velocity variance is also much higher. This indicates that the controller learned by the baseline GNN is less consistent than that learned by the basis-invariant model.
• Figure E2: Shown above are the results of the extended 5 second rollouts of the basis-invariant GNN, nominal controller, and baseline GNN on an environment with 100 agents. Pink arrows indicate the velocity of each agent, and the faded red and green arrows the $x$ and $y$-axes of their body frames. The swarm visualizations for the extended-run rollouts demonstrates how similar the behavior of the basis-invariant GNN is to that of the nominal controller. The magnitudes of the velocities in the basis-invariant GNN and the nominal controller are similar to that of the baseline and the spacing is consistent. In the swarm controlled by the baseline GNN, the velocities of its agents are inconsistent and the agents are unevenly spaced. There are two contingents of agents pulling in different directions, which indicates that velocity information from far away neighbors is not being propagated or utilized well enough.
• Figure E3: The extended run velocity variance of the best-performing basis-invariant and baseline models indicate different levels of generalization to situations proximal to those seen in training data. The invariant model continues to faithfully follow the performance curve of the nominal controller for the duration of the experiment. The standard deviation of the baseline model's performance increases as it encounters unfamiliar states. A spike in variance can be observed at 2.2 seconds, though the log scale does exaggerate its magnitude.
• Figure E4: Lowering the communication radius increased the overall velocity variance of both the basis-invariant and baseline GNNs. However, the basis-invariant GNN is less affected, remaining within an order of magnitude of the nominal controller's velocity variance while the baseline GNN is nearly two orders of magnitude removed.

Theorems & Definitions (5)

• Remark C.1: Global invariance requires local equivariance
• Theorem C.1: Global Invariance to Local Frames
• Lemma C.1
• Theorem D.1: $SO(2)$ Equivariance of complex matrix multiplication
• proof