Learned quadruped locomotion policies commonly map observations directly to joint-level actions, leaving the periodic structure of locomotion implicit in the policy. We investigate an alternative representation in which each joint trajectory is expressed as a command-conditioned Fourier series and modified online using feedback from the robot state. A context network generates the Fourier coefficients and the weights of a per-step feedback network, whose outputs adjust joint offsets, harmonic gains, frequency, and phase during execution. In simulation, we examine this explicit frequency structure alongside the hidden activations of an MLP policy that directly outputs joint targets. The harmonic waveforms change frequency and shape with commanded speed. Dynamic mode decomposition of selected MLP rollouts reveals dominant activation modes near the foot-height oscillation frequency and its second harmonic, showing periodic structure without an explicit Fourier generator. On a Unitree Go2, the simulation-trained harmonic controller records a provisional onboard-estimated peak speed of 3.67 meter per second and carries added loads up to 5.883 kilogram in separate trials.
LIPM is everywhere in legged-locomotion control, but almost always as a modeling choice rather than as something the controller's cost actually prefers. This note tries to make that link more explicit. Working from a small centroidal OCP that penalizes the rate of angular momentum, we look at what its optimum tends to look like. Three things come out. With full-rank stance, the optimum drifts toward a pendular force pattern at a rate determined by the SVD of the moment Jacobian; the constant is set by foot-span geometry and matches the experiments to within 16%. With N=2 stance, as in trot, the friction cone introduces a lower bound on ∥H˙G∥ that no amount of weight tuning fixes; we also see a non-smooth feasibility kink at a critical horizontal acceleration that we can write in closed form. Adding a task term that asks for a nonzero H˙G moves the optimum off the pendular set in a predictable way. None of this is far from the classical ZMP/DCM picture. We test these claims on a point-mass quadruped and on the Unitree Go1 in MuJoCo (open-loop QP and a torque-level closed-loop controller), and we note where the asymptotic story stops being a good description of what the closed loop actually does.
Learning-based quadrupedal locomotion typically relies on complex reward formulations that entangle task specification, operational limits, gait preference, and terrain adaptation within a single optimization objective. We instead treat these functions through distinct mechanisms: rewards for task specification, constraints for operational limits, energy minimization for gait preference, and exteroceptive perception for adapting energy use to terrain difficulty. We show that these components jointly enable efficient, terrain-adaptive locomotion, and that removing each component exposes a distinct failure mode. Our formulation removes explicit gait priors (including air-time, contact-count, and foot-clearance targets) in favor of emergent behavior. Compared to a conventional complex-reward baseline, our formulation achieves comparable terrain traversal while reducing cost of transport by 56% and operational-limit violations by 96%. The resulting policies transfer zero-shot to a physical Unitree Go2 using LiDAR-based elevation mapping. Project website with videos: https://tinyurl.com/locomposition.
Loukas Kordos, Leonard T. Franz, Simon Rappenecker +4
This work focuses on sampling strategies of configuration variations for generating robust universal locomotion policies for quadrupedal robots. We investigate the effects of sampling physical robot parameters and joint proportional-derivative gains to enable training a single reinforcement learning policy that generalizes to multiple parameter configurations. Three fundamental joint gain sampling strategies are compared: parameter sampling with (1) linear and polynomial function mappings of mass-to-gains, (2) performance-based adaptive filtering, and (3) uniform random sampling. We improve the robustness of the policy by biasing the configurations using nominal priors and reference models. All training was conducted using the RaiSim simulation environment, tested in simulation on a range of diverse quadrupeds, and zero-shot deployed onto hardware using the ANYmal quadruped robot. Compared to multiple baseline implementations, our results demonstrate the need for significant joint controller gains randomization for robust closing of the sim-to-real gap.