Gait Parameters in Biological and Robotic Locomotion
In robotic quadrupeds, duty factor and other gait parameters have likewise been studied in relation to static stability and energetic efficiency. Collectively, this literature demonstrates that locomotion performance depends on a range of interacting gait parameters, including speed, frequency, step length, stance width, and duty factor.
Hybrid Systems
For a nominal trajectory consisting of continuous motion and a sequence of hybrid transitions, perturbations to the trajectory evolve according to the continuous-time variational dynamics within each mode and are modified by the saltation matrix at each event. This provides a local linear approximation of how perturbations propagate through the complete hybrid trajectory.
Methods
We present three investigations that provide evidence that duty factor is a key predictor of quadrupedal locomotion robustness, particularly in challenging narrow environments.
Investigation 1: Whole-Body Trajectory Optimization with LQR Feedback
In Investigation 1, we use whole-body direct collocation trajectory optimization with LQR feedback to systematically study how gait parameters affect local error convergence and energetic efficiency. This approach provides direct control over the contact sequence and timing of each gait, allowing us to independently vary duty factor, forward speed, and stance width while maintaining a specified nominal gait type.
For each optimized trajectory, we generate an LQR controller using fixed state and input weighting matrices. The resulting feedback controller stabilizes the nominal trajectory and enables us to evaluate local error convergence under closed-loop dynamics. Given an optimal trajectory and LQR feedback controller, the convergence metric measures local error convergence under the resulting closed-loop dynamics.
While this choice of state and input weighting matrices does affect the value of the convergence metric that quantifies local error convergence, we find that the trends in Investigation 1 hold across reasonable choices of these weighting matrices.
Investigation 3: Centroidal MPC
In the final investigation, we evaluate whether the relationship between duty factor and locomotion robustness persists under a separate planning and control architecture and on physical hardware. We use the open-source Quad-SDK package, which employs a centroidal model-predictive controller (MPC) to plan ground-reaction forces (GRFs) that are subsequently converted into joint commands. Unlike the approach in Investigation 1, this approach plans locomotion using a reduced-order centroidal dynamics model and tracks the resulting motion with an MPC controller.
The structure of Quad-SDK allows gait parameters and simulation environments to be modified without changing the underlying planning and control architecture. It also provides an existing local footstep planner and centroidal MPC implementation, enabling us to evaluate the effects of gait parameters using a controller that is independently developed from the whole-body optimization framework in Investigation 1.
Quad-SDK consists of three modules: a global body planner, a local planner, and a robot driver. For these experiments, we directly command body velocities through twist inputs and therefore bypass the global body planner. The local planner determines valid foothold locations using a predetermined traversability map and provides these footholds to the centroidal MPC, which computes the ground-reaction forces required to track the desired center-of-mass trajectory. The robot driver then converts the planned GRFs into joint-level commands using inverse kinematics and interfaces with the robot’s motor controllers.
We evaluate gait parameters using Quad-SDK on a Unitree Go2. By varying gait type, duty factor, forward velocity, and the width of the available foothold region, we assess whether the trends observed in Investigation 1 persist under a distinct locomotion architecture and in physical deployment. In particular, these experiments test whether higher duty factors enable more robust locomotion over narrow terrain.
Investigation 2
In this investigation, we conducted two experiments within the NVIDIA Isaac Sim simulator to evaluate: (1) the performance of the learned locomotion policy over commanded duty factors and stance widths; and (2) the relationship between stance width and the duty factor chosen by the selector network.
We evaluated walking and trotting at widths of 0.05–0.30 m in 0.05 m increments. Duty factor commands ranged from 0.50–0.75 for trot and 0.75–0.90 for walk in 0.05 increments.
To assess robustness as the robot traversed across the 3 m beam terrain, we applied randomized forces to the robot body of up to 25 N and torques of up to 3 N m. Disturbances were held for 0.10–0.40 s before resampling, with their magnitude envelope increasing over the first 10 s. Results were evaluated at a commanded speed of 0.3 m/s. For each condition, 64 trials were conducted for gait periods of 0.40, 0.48, and 0.54 s, giving 192 trials per duty-factor/stance-width combination for each gait type.
Increasing commanded duty factor generally improved robustness at constrained stance width. The effect is strongest at the stance width of 0.15 m, where the trot gait with a duty factor of 0.5 was successful in just 31.8% of trials. On the other hand, the success rate of the walk gait was over 70% at duty factors of 0.8, 0.85, and 0.9. At the 0.05 m stance width, none of the trials produced any successes.
Investigation 3
To evaluate the effect of speed, duty factor, and stance width on the locomotion performance of a physical robot, we built a custom narrow beam terrain with a width of 0.15 m. A Unitree Go2 quadruped robot was tasked with traversing these beam terrains with planning and control as described in the centroidal MPC method.
Nine sets of trials were run at three different gaits—trot at 0.5 duty factor, trot at 0.75 duty factor, and walk at 0.75 duty factor—and three commanded speeds: 0.3, 0.4, and 0.5 m/s. Each of these gait and speed combinations was evaluated on the 0.15 m width beam. Each set of trials was executed five times, giving 45 total trials. Each trial was evaluated on whether the robot successfully traversed the entire beam length without falling.
For both of the higher duty factor gaits, the robot successfully traversed the beam in 100% of the trials. On the other hand, the trot gait with a duty factor of 0.5 failed at each speed, completing a total of 12 of 15 trials. In each of these three failures, the robot lost balance in the middle of the beam and fell off.
While this set of hardware experiments suggests a difference in performance between duty factors, the results do not show a vast gap in performance. Since the robot is highly dependent on accurate state estimation and sensing, the robot leveraged motion capture to estimate its position relative to the world. Prior to execution of the trials, the motion capture system was carefully calibrated to ensure accuracy. This precise calibration and estimation ability may have amplified the robot’s ability to successfully traverse the beam, even at lower duty factors. In future work, we would like to explore how noisier vision-based estimation methods may be more sensitive to duty factor in similar experiments.
Discussion and Limitations
The three investigations presented in this work provide converging evidence that duty factor is an important predictor of quadrupedal locomotion robustness, particularly when the robot must operate with constrained stance width. Nevertheless, the scope of these conclusions should be interpreted in light of several limitations.
First, isolating the effects of individual gait parameters requires holding other aspects of locomotion approximately constant, such as gait period, swing-leg timing and trajectory, and body height. These parameters may also affect locomotion performance and may interact with duty factor. We therefore do not claim that duty factor is the sole determinant of locomotion robustness. Rather, it consistently provided substantially more explanatory power than gait type. A more complete characterization of these interactions is an important direction for future work.
Second, our analysis focuses on walking and trotting, which are among the most commonly used gait families in contemporary quadrupedal locomotion. We do not evaluate other gaits, such as bounding, pronking, or cantering. The relationship identified here should therefore not be interpreted as a universal statement that gait type is irrelevant to all forms of quadrupedal locomotion. Instead, our results demonstrate that even two gait types traditionally treated as fundamentally distinct do not necessarily yield distinct performance.
Despite these limitations, the consistency of the results across the three investigations provides strong evidence that gait type is not a good predictor of locomotion performance and duty factor is much more highly correlated. The practical implication is that gait selection need not begin with a discrete choice between “walk” and “trot”. Instead, duty factor can serve as a continuous and interpretable control parameter for adapting locomotion to task and environmental constraints. This perspective can simplify gait selection and provide a direct mechanism for increasing robustness when environmental constraints demand it.
Conclusion
This work investigated the role of duty factor in quadrupedal locomotion across different gait types, control architectures, and operating conditions. Across three investigations, we found consistent evidence that duty factor provides a more informative description of robustness than nominal gait type for the walking and trotting gaits studied.
These results suggest that gait selection should not necessarily be treated as a discrete choice between predefined gait types. Instead, continuous gait parameters such as duty factor may provide a basis for selecting locomotion behavior according to environmental constraints and desired performance. Future work will investigate whether this relationship extends to additional gait parameters, locomotion regimes, and sources of uncertainty, as well as whether duty factor can be automatically adapted online to balance competing objectives like robustness and energetic efficiency during real-world locomotion.
Acknowledgments
ChatGPT was used for text editing while preparing this manuscript.
Frequently Asked Questions
What is duty factor in quadrupedal locomotion? Duty factor is a gait parameter examined as a continuous control variable for adapting quadrupedal locomotion.
How did higher duty factors affect locomotion robustness? Higher commanded duty factors generally improved robustness at constrained stance widths, particularly in narrow terrain experiments.
Which robots and control systems were evaluated? The experiments used a Unitree Go2, whole-body trajectory optimization with LQR feedback, Isaac Sim, and the Quad-SDK centroidal MPC architecture.
What were the main limitations of the study? The analysis focused on walking and trotting while holding other gait parameters approximately constant and did not evaluate bounding, pronking, or cantering.