Jake Welde

Fengjun Yang, Jake Welde, Nikolai Matni

We study distributed control for a network of nonlinear, differentially flat subsystems subject to dynamic coupling. Although differential flatness simplifies planning and control for isolated subsystems, the presence of coupling can destroy this property for the overall joint system. Focusing on subsystems in pure-feedback form, we identify a class of compatible lower-triangular dynamic couplings that preserve flatness and guarantee that the flat outputs of the subsystems remain the flat outputs of the coupled system. Further, we show that the joint flatness diffeomorphism can be constructed from those of the individual subsystems and, crucially, its sparsity structure reflects that of the coupling. Exploiting this structure, we synthesize a distributed tracking controller that computes control actions from local information only, thereby ensuring scalability. We validate our proposed framework on a simulated example of planar quadrotors dynamically coupled via aerodynamic downwash, and show that the distributed controller achieves accurate trajectory tracking.

Jake Welde, Pieter van Goor

Many nonlinear dynamical systems exhibit symmetry, affording substantial benefits for control design, observer architecture, and data-driven control. While the classical notion of group invariance enables a cascade decomposition of the system into highly structured subsystems, it demands very rigid structure in the original system. Conversely, much more general notions (e.g., partial symmetry) have been shown to be sufficient for obtaining less-structured decompositions. In this work, we propose a middle ground termed "weak invariance", studying diffeomorphisms (resp., vector fields) that are group invariant up to a diffeomorphism of (resp., vector field on) the symmetry group. Remarkably, we prove that weak invariance implies that this diffeomorphism of (resp., vector field on) the symmetry group must be an automorphism (resp., group linear). Additionally, we demonstrate that a vector field is weakly invariant if and only if its flow is weakly invariant, where the associated group linear vector field generates the associated automorphisms. Finally, we show that weakly invariant systems admit a cascade decomposition in which the dynamics are group affine along the orbits. Weak invariance thus generalizes both classical invariance and the important class of group affine dynamical systems on Lie groups, laying a foundation for new methods of symmetry-informed control and observer design.

Jake Welde, Nishanth Rao, Pratik Kunapuli et al.

Tracking controllers enable robotic systems to accurately follow planned reference trajectories. In particular, reinforcement learning (RL) has shown promise in the synthesis of controllers for systems with complex dynamics and modest online compute budgets. However, the poor sample efficiency of RL and the challenges of reward design make training slow and sometimes unstable, especially for high-dimensional systems. In this work, we leverage the inherent Lie group symmetries of robotic systems with a floating base to mitigate these challenges when learning tracking controllers. We model a general tracking problem as a Markov decision process (MDP) that captures the evolution of both the physical and reference states. Next, we prove that symmetry in the underlying dynamics and running costs leads to an MDP homomorphism, a mapping that allows a policy trained on a lower-dimensional "quotient" MDP to be lifted to an optimal tracking controller for the original system. We compare this symmetry-informed approach to an unstructured baseline, using Proximal Policy Optimization (PPO) to learn tracking controllers for three systems: the Particle (a forced point mass), the Astrobee (a fullyactuated space robot), and the Quadrotor (an underactuated system). Results show that a symmetry-aware approach both accelerates training and reduces tracking error at convergence.