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 fully-actuated space robot), and the Quadrotor (an underactuated system). Results show that a symmetry-aware approach both accelerates training and reduces tracking error at convergence.

The use of bidirectional propellers provides quadrotors with greater maneuverability which is advantageous in constrained environments. This paper addresses the development of a trajectory planning algorithm for quadrotors with bidirectional motors. Previous work has shown that the property of differential flatness can be leveraged for efficient trajectory planning. However, planners that leverage flatness for quadrotors fail at points where the acceleration of the center of mass is equal to gravity, i.e., when the vehicle experiences free fall. The central contribution of this paper is a flatness-based trajectory planning method that allows quadrotors to use bidirectional propellers and pass through the so-called free-fall singularity. We model our system as a differentially flat hybrid system with the aid of coordinate charts derived from the Hopf fibration and develop an algorithm that computes forward and reverse thrusts for each propeller, resulting in smooth trajectories everywhere in SE(3). We demonstrate the planner's versatility by planning knife-edge maneuvers and trajectories passing through the free-fall singularity, while transitioning from forward to reverse thrust.

Mechanical systems naturally evolve on principal bundles describing their inherent symmetries. The ensuing factorization of the configuration manifold into a symmetry group and an internal shape space has provided deep insights into the locomotion of many robotic and biological systems. On the other hand, the property of differential flatness has enabled efficient, effective planning and control algorithms for various robotic systems. Yet, a practical means of finding a flat output for an arbitrary robotic system remains an open question. In this work, we demonstrate surprising new connections between these two domains, for the first time employing symmetry directly to construct a flat output. We provide sufficient conditions for the existence of a trivialization of the bundle in which the group variables themselves are a flat output. We call this a geometric flat output, since it is equivariant (i.e. it preserves the symmetry) and often global or almost global, properties not typically enjoyed by other flat outputs. In such a trivialization, the motion planning problem is easily solved, since a given trajectory for the group variables will fully determine the trajectory for the shape variables that exactly achieves this motion. We provide a partial catalog of robotic systems with geometric flat outputs and worked examples for the planar rocket, planar aerial manipulator, and quadrotor.

In this work, we derive a coordinate-free formulation of the coupled dynamics of a class of 6DOF aerial manipulators consisting of an underactuated quadrotor equipped with a 2DOF articulated manipulator, and demonstrate that the system is differentially flat with respect to the end effector pose. In particular, we require the center of mass of the entire system to be fixed in the end effector frame, suggesting a reasonable mechanical design criterion. We make use of an inertial decoupling transformation to demonstrate differential flatness, allowing us to plan dynamically feasible trajectories for the system in the space of the 6DOF pose of the end effector, which is ideal for achieving precise manipulator tasks. Simulation results validate the flatness-based planning methodology for our dynamic model, and its usefulness is demonstrated in a simulated aerial videography task.