Duty Factor Predicts Robust Constrained Quadrupedal Locomotion Across Gait Types
Authors: James Zhu, David Ologan, George Ortiz, Thomas Chun Fai Lee, Selvin Garcia Gonzalez, Ardalan Tajbakhsh, Pinhas Ben-Tzvi, Aaron M. Johnson
Organizations: Department of Electrical and Computer Engineering, University of Miami, Coral Gables, FL, USA · Department of Mechanical Engineering, 3 Robotics Institute, Carnegie Mellon University, Pittsburgh, PA, USA
Quadrupedal robots are increasingly deployed in environments where locomotion must remain robust to disturbances and constrained terrain. Gait type, such as walking or trotting, is commonly used to characterize quadrupedal locomotion. However, gait type does not uniquely define locomotion, as parameters such as duty factor, speed, and stance width can vary within a single gait type. In this work, we investigate the relationship between these gait parameters using three distinct quadrupedal locomotion control approaches. First, using whole body trajectory optimization with LQR feedback, we show that duty factor is a stronger predictor of local error convergence than nominal gait type. Second, we investigate duty factor selection with a learned locomotion controller, suggesting how duty factor may serve as a low-dimensional parameter for adapting locomotion robustness in narrow-terrain environments. Finally, we show that these trends persist under a centroidal model predictive control framework and validate them through narrow-terrain experiments on a physical quadruped. These results show that duty factor provides a simple and effective basis for understanding and selecting robust quadrupedal locomotion across gait types and control architectures.
Figures & tables
Fig. 1 : In this work, we demonstrate that the robustness of quadruped locomotion improves with an increase in duty factor. This finding is validated with the first demonstration of a quadruped robot walking across a narrow beam without the aid of external mechanisms like a reaction wheel.
Investigation
Planning/Control Architecture
Robot Model
Evaluation Environment
1
Whole-Body Optimization w/ LQR Feedback
Ghost Spirit 40
Pinocchio
2
Gait-Conditioned RL
Unitree Go2
NVIDIA Isaac Sim
3
Centroidal MPC
Unitree Go2
Hardware
TABLE I : Across three control architectures, two quadruped robot models, and three different evaluation environments in both simulation and hardware, we find consistent evidence that duty factor, as opposed to gait type, is a primary parameter that determines robustness of quadrupedal locomotion.
Predictor
Estimate β
Standard Error
p -value
Gait (Walk)
0.3867
0.221
8.82 ⋅ 10 -2
Speed
0.646
0.111
1.21 ⋅ 10 -6
Stance Width
-0.312
0.112
8.42 ⋅ 10 -3
TABLE II : Linear regression predicting convergence for walking and trotting gaits at matched DF=0.75.
Fig. 2 : Convergence across walking and trotting gaits over a range of speeds and duty factors. Walking and trotting exhibit a continuous transition at the matched duty factor of 0.75 . Overall, convergence is strongly associated with duty factor.
Predictor
Estimate β
Standard Error
p -value
Duty Factor
-0.677
0.0863
1.42 ⋅ 10 -10
Speed
0.262
0.0853
3.29 ⋅ 10 -3
Stance Width
-0.0621
0.0791
4.35 ⋅ 10 -1
TABLE III : Linear regression predicting convergence for walking and trotting gaits over a range of duty factors, speeds, and stance widths.
Fig. 3 : Across both trot and walk gaits, success rate of simulated trials increases with duty factor.
Fig. 4 : The duty factor selector network shows a trend of selecting higher duty factors as stance width narrows. Plot shows mean values and shaded regions represent minimum and maximum chosen values at each stance width.
Gait Type
Duty Factor
Speed (m/s)
Success Rate
Trot
0.5
0.3
4/5
0.4
4/5
0.5
4/5
Trot
0.75
0.3
5/5
0.4
5/5
0.5
5/5
TABLE IV : At the duty factor of 0.75, the trot and walk both achieved 100% success on the hardware experiment. The lower duty factor trot experienced several failures.
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
Technical University of Munich. · University of Tübingen. · Hertie Institute for Clinical Brain Research & Center for Integrative Neuroscience. +1
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.
Lingxue Lyu, Zihui Liu
School of Engineering and Applied Science, University of Pennsylvania, Philadelphia, PA 19104, USA · Department of Aeronautics & Astronautics, Stanford University, Stanford, CA 94305, USA
Quadruped robots often switch gaits to traverse mixed terrain safely. Many practical supervisors enforce hardware constraints (e.g., dwell or guard intervals) using a transition handler. However, the switch decision itself is often driven by near-field thresholds, which can delay commitment when rough terrain is only detected shortly before contact in the local height map. We propose a lightweight height-map-only gait supervisor that unifies look-ahead and locomotion constraints in a single decision policy. A robot-centered height map is partitioned into near-, mid-, and far-field bands, each discretized into a terrain class, and band evolution is modeled as a Markov process. Gait selection is cast as a look-ahead Markov Decision Process whose transition model and cost explicitly encode platform-specific switching constraints. Indoor and outdoor experiments on a Unitree Go2 platform show reduced late switching, lower time spent using ineffective gaits on rough terrain, and improved roll stability compared to tuned reactive baselines. Overall, the MDP formulation yields anticipative constraint-aware gait switching that improves robustness without course-specific retuning.