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Closing the Loop on the Poppy Humanoid: Bipedal Locomotion with Linear-Quadratic Control and Learned Cost Functions
Xulin Chen, Borui He, Ruipeng Liu, Naveed Tahir, Zhenyu Gan, Garrett E. Katz
TL;DR
Reliable, unassisted bipedal locomotion on standard Poppy hardware lacked an established published methodology despite the platform’s research and education value. The paper learns dynamics and a quadratic cost from open-loop runs for a closed-loop LQR controller, which significantly improves walking performance over open-loop playback. The authors validate the controller on hardware while identifying remaining gait-optimization and modeling limits.
Problem
Published Poppy studies had required expert assistance or non-standard hardware modifications, leaving reliable autonomous walking on the standard design unaddressed.
Method
The method learns linearized dynamics and a fall-sensitive quadratic cost from open-loop hardware runs, then uses closed-loop LQR to track a nominal walking trajectory.
Results
Statistically significant improvements in walking performance were demonstrated against open-loop playback, including higher success rates and more successful steps in office and lab environments.
Takeaways & Limitations
The methodology is a first step toward reliable and versatile bipedal locomotion on standard off-the-shelf Poppy hardware.
Takeaways & Limitations
The controller takes steps and avoids falls but has not optimized forward velocity, turning radius, or cost of transport.
Abstract
from arXiv · showhide
The Poppy Humanoid is an open-source, low-cost robot suitable for research and education in artificial intelligence. However, we are unaware of any published methodology that achieves reliable, unassisted bipedal locomotion on the standard Poppy hardware. This paper contributes a functional closed-loop walking controller for Poppy, based on the linear-quadratic regulator (LQR) framework for trajectory tracking. Starting with data collected from open-loop playback of a nominal walking trajectory, our proposed method learns a quadratic cost function for an LQR controller that substantially improves the reliability of the motion. The closed-loop controller is validated empirically, demonstrating statistically significant improvements in walking performance compared to open-loop trajectory playback.
1 Introduction
The standard Poppy Humanoid is an accessible but difficult platform for reliable autonomous bipedal walking. This paper addresses that gap with a learned-cost, closed-loop LQR controller validated on hardware.
- The Poppy Humanoid is an open-source, child-sized, lightweight, low-cost platform for research and education.Its limbs are 3D-printed, supporting an extensible hardware design.
- Position-only motors, low control rates, small foot contacts, limited ankle rotation, and partial observability constrain reliable locomotion.The standard hardware lacks foot pressure sensors and inertial measurement units, making robot-state estimation difficult.
- Published Poppy walking studies have required continuous expert assistance or non-standard hardware modifications, leaving standard-hardware autonomous walking unresolved.The standard design remains attractive for software-focused researchers and educators because of its low cost and off-the-shelf availability.
- The proposed controller uses closed-loop LQR to drive the robot toward a nominal open-loop trajectory while learning linearized dynamics and a fall-sensitive quadratic cost from prior runs.The learning procedure uses a convex optimization formulation.
- Hardware validation found statistically significant improvement in walking performance, with reproducible data and deployment code released openly.The release is intended to support reproduction of the results and controller deployment.
2 Background
The background describes Poppy’s constrained sensing and actuation, the instability and hybrid dynamics of bipedal locomotion, and established modeling, learning, LQR, and system-identification approaches.
- 2.1 Details of the Poppy Humanoid Platform: Poppy weighs roughly 3.5kg, stands 83cm tall, and has 25 position-controlled Dynamixel joints, an RGB camera, and an embedded Odroid UX4 computer.PyPot provides access to motor commands and sensor readings such as temperature, voltage, and joint angles.
- 2.1 Details of the Poppy Humanoid Platform: On the authors’ hardware, target-position commands became choppy and unreliable above 5Hz, although joint measurements could be read near 100Hz.This permits about 20 joint observations before each target-position command.
- 2.2 Bipedal Locomotion Methods: Bipedal locomotion is inherently unstable relative to quadrupedal locomotion and can become underactuated when foot contact or ankle degrees of freedom are limited.Walking also involves single- and double-stance phases with discontinuous contact-event dynamics.
- 2.2 Bipedal Locomotion Methods: Prior locomotion methods include physics-based ZMP pattern generation, finite-state representations of stance phases, and deep-learning systems for terrain-aware or versatile humanoid gaits.These approaches span simplified dynamics models, geometric phase transitions, and learned terrain or proprioceptive representations.
- 2.3 Linear Quadratic Regulators: LQR stabilizes linear systems by minimizing a quadratic cost balancing performance and control effort, and has been extended to nonlinear robotic systems.Related work includes iterative LQR, LQR-Trees, and time-varying LQR frameworks for humanoid walking.
- 2.4 System Identification: System identification estimates a mathematical dynamical model from observed input-output behavior, complementing LQR when the dynamics model is not known.Prior work has estimated physical, friction, and motor parameters, while reinforcement-learning studies use system identification to improve simulation fidelity and sim-to-real transfer.
3 Methodology
The method builds a nominal walking trajectory, learns local dynamics and a fall-sensitive quadratic cost from open-loop hardware runs, and uses LQR for closed-loop waypoint tracking. The approach is validated on Poppy hardware, with failure-related deviations informing the learned controller and an explicit scope limitation in fall labeling.
- 3.1 Nominal Trajectory Design: The nominal gait combines kinematic analysis, trajectory optimization, and manual tuning, but simulation-optimized trajectories transferred poorly to physical hardware.Manual tuning ultimately produced a reasonably effective open-loop trajectory using leg motors and selected torso and arm motors.
- 3.2 Trajectory Playback: Each open-loop run executed six footsteps, recording the number completed before a fall; 125 hardware runs supplied the training data.The dataset was collected from March to May 2025 across three locations, while a fourth location was reserved for later generalization assessment.
- 3.3 Open-Loop Data Collection: Open-loop errors accumulated over time, and failure runs showed greater joint-observation distance from central successful-run or initial-stance references.This motivated using observed deviations from the nominal trajectory as inputs for closed-loop correction.
- 3.5 Linear System Identification: The controller interpolates variable-length joint measurements into fixed-width states and uses learned linearized dynamics to predict the next state.Dynamics matrices are fit by least squares from pooled residuals, excluding data during and after failed footsteps because human intervention can make those measurements unrepresentative.
- 3.4 Residual Closed-Loop Trajectory Tracking with LQR: LQR continually adjusts waypoint targets by minimizing a quadratic cost that penalizes deviations associated with falls.The learned cost is fit by convex optimization, enforcing higher average cost for fall trajectories than successful trajectories while maintaining positive-semidefinite cost structure.
- 3.6 Learned Cost Function: The learned-cost approach assumes each fall’s point of no return occurs no later than the first waypoint of the falling footstep.Footstep-level labels cannot identify the precise fall time, and after a fall begins, joint observations no longer determine the full system state.
4 Empirical Validation
The controller was validated through stability analysis and controlled hardware experiments comparing closed-loop LQR control with open-loop playback. Negative values of ϵ enabled stability, and closed-loop control significantly improved walking performance in both tested locations.
- 4.1 LQR Stability Analysis: ϵ = 0 produced an unstable controller, whereas ϵ < 0 achieved stability in the analyzed controller variants.The stability criterion considers maximum eigenvalue magnitudes and cumulative products over one gait period.
- 4.1 LQR Stability Analysis: Stricter cost-separation constraints near ϵ = 0 made more Cn matrices singular, creating unpenalized directions that could translate into unstable LQR directions.Singular cost matrices behave like parabolic valleys rather than bowls because cost does not increase along zero-eigenvalue directions.
- 4.1 LQR Stability Analysis: The ϵ = −2 controller was selected for hardware validation because stable control was prioritized despite limited overlap between failure and success costs.The remaining overlap was confined to approximately ±ϵ around a decision boundary near d ≈5.
- 4.2 Closed-Loop Hardware Performance: The hardware experiment compared successful footsteps before falling under closed-loop LQR control and open-loop trajectory playback across office and lab environments.The experiment used 100 runs per condition in the office and 80 runs per condition in the lab, with batches alternated between conditions.
- 4.2 Closed-Loop Hardware Performance: 78% versus 62% office success rates and 5.13 versus 4.18 average steps before falling favored closed-loop control, with p value 0.00225.The comparison reports closed-loop versus open-loop performance in the office environment.
- 4.2 Closed-Loop Hardware Performance: 42.5% versus 30% lab success rates and 4.8 versus 3.75 average successful steps likewise favored closed-loop control, with p value 0.00014.The lab location was not used for training-data collection, providing a separate evaluation setting.
5 Conclusion
The paper presents a first step toward reliable and versatile bipedal locomotion on standard off-the-shelf Poppy hardware. Future work could improve system identification, exploit additional sensors, and optimize gait properties beyond fall avoidance.
- Conclusion: The methodology combines data-learned quadratic costs with LQR control and achieves statistically significant improvement in walking performance on standard Poppy hardware.The conclusion characterizes the work as a first step toward reliable and versatile bipedal locomotion.
- Future Research: The system identification currently uses a simple linear fit, while more sophisticated methods and additional sensor measurements remain potential improvements.Candidate additional measurements include joint temperatures, loads, and egocentric RGB video.
- Future Research: Forward velocity, turning radius, and cost of transport have not yet been optimized, although they could be measured or estimated in future work.The stated difficulty is partly due to measuring these gait properties with the current experimental tooling.