Around the World: Unified Learned Locomotion on a 270 g Continuous-Rotation Quadruped

Arturo Flores Alvarez; Nathan Lintu; Dennis Hong

cs.ROOct 2, 2026

Around the World: Unified Learned Locomotion on a 270 g Continuous-Rotation Quadruped

Authors: Arturo Flores Alvarez, Nathan Lintu, Dennis Hong

Organizations: Robotics and Mechanisms Laboratory (RoMeLa), University of California, Los Angeles (UCLA)

Abstract

Closed-loop learned locomotion is established on commercial quadrupeds but remains uncommon at the sub-kilogram scale. Continuous-rotation legs give MiNI-Q, a 270 g quadruped, access to supporting configurations on either side of the body. We exploit this range with a single posture-conditioned reinforcement-learning policy that runs entirely onboard. A continuous joint-space reference on the torus $T^8$ and its gravity-conditioned transformation connect upright walking, inverted walking, and landing recovery without state machines or phase switching. Coordinated posture and release curricula train this behavior family; identified actuation, cross-engine validation, and embedded execution support hardware transfer. The same sub-100k-parameter network tracks forward velocity with RMSE of 0.037 m/s upright and 0.050 m/s inverted, resumes walking in 24 of 30 release trials, and operates with four interchangeable foot geometries across four indoor surfaces. Hardware experiments and simulation ablations connect these capabilities to the representation, conditioning, and training choices that exploit the platform's motion range. Demonstration videos and supplementary material are available on the project website: https://submissionreview.github.io/around-the-world/.

Figures & tables

Panel of Figure 1 — Fig. 2: Posture references and curriculum geometry. Left: three stance configurations (top) and four attitude references around ANYmal (bottom), all on MiNI-Q. Right: pairwise toroidal distances (rad). Lean-left–Q8bot is the only pair beyond the stage-2 cutoff of $4.0\text{\,}\mathrm{rad}$ . Training samples both anchors and their interpolations.

Fig. 3: Onboard data path at $50\text{\,}\mathrm{Hz}$ . The S3 constructs the reference and observation and maps policy outputs to position targets; the C3 services the IMU and actuator bus.

Iterations	Release conditions	Posture sampling
0–5k	Upright, small perturbations	Anchor + jitter
5k–10k	Lateral drops, $\leq$ $0.2\text{\,}\mathrm{m}$	Pairs $d_{T^{8}}\leq 4.0$
10k–20k	$\mathrm{SO}(3)$ , $\leq$ $0.5\text{\,}\mathrm{m}$	Any anchor pair

TABLE I: Joint release/posture curriculum; reference resampling every

6\text{\,}\mathrm{s}

Fig. 4: Velocity tracking with the same policy, upright (left) and inverted (right), on the real robot, Isaac Lab, and MuJoCo. Forward velocity and yaw rate follow a ramp-and-hold command. Dashed lines denote commands, solid lines measured motion, and bands $\pm 1\sigma$ . Hardware trials use $N=5$ per orientation.

Fig. 5: Hardware posture changes under a continuous velocity command ( $25\text{\,}\mathrm{s}$ , $N=5$ ). Plots report $v_{x}$ , $\omega_{z}$ , and toroidal distance $d_{T^{8}}$ from the current joint state to the fixed ANYmal anchor; the images show the commanded posture sequence.

Term	Coeff.	Formulation
Tracking and task (positive)
tracking_lin_vel_x	$+3.0$	$\exp(-e_{v}^{2}/\sigma)$
tracking_ang_vel	$+1.5$	$\exp(-e_{\omega}^{2}/\sigma)$
feet_air_time	$+1.2$	$\sum_{k}\max(t_{k}^{\text{air}}-0.5,0)$
orientation	$+0.5$	$\\|\hat{g}_{xy}-\hat{g}_{xy}^{\text{ref}}\\|^{2}<\theta_{c}$
survival	$+0.2$	$\mathbf{1}[\text{contact}]$

TABLE II: Reward configuration.

e_{v}=v_{x}^{\mathrm{cmd}}-v_{x}

e_{\omega}=\omega_{z}^{\mathrm{cmd}}-s\omega_{z}

s=-\mathrm{sign}(g_{z})

\sigma=0.005

;

\hat{g}_{xy}^{\mathrm{ref}}

is the FK-derived attitude,

\theta_{c}

the orientation cutoff, and

\mathbf{1}[\cdot]

the indicator.

Fig. 6: Foot-geometry transfer on hardware. Left: interchangeable printed variants. Right: forward-velocity RMSE (m/s), $N=3$ per cell. Rounded is represented in training; Hex, Square, and Spike are held-out physical shapes. Bold marks the lowest error on each surface.

Panel of Figure 8 — Fig. 7: Actuator identification. Left: single-leg fixture. Right: hip and knee trajectories for the command (dashed), measured motion (red), nominal-gain PD (orange), and fitted model (blue).

Fig. 8: Full-policy training diagnostics, mean $\pm 1\sigma$ over four seeds. Panels report total, tracking, and orientation rewards and gravity-sign crossings. Boundaries mark curriculum transitions at 5k and 10k; the first 15k of 20k iterations are shown.

Condition	Successes / trials	Success rate
Upright, pitch	4/6	67%
Inverted, pitch	4/6	67%
Upright, roll	5/6	83%
Inverted, roll	6/6	100%
Throw	5/6	83%
Overall	24/30	80%

TABLE III: Hardware release recovery: stable walking within

5\text{\,}\mathrm{s}

of first contact. All failures count in the outcome.

Symbol	Parameter	Value (SI units)
$k_{p}$	Position stiffness	$5.95\times 10^{-1}$
$k_{d}$	Position damping	$2.74\times 10^{-2}$
$b$	Viscous damping	$1.01\times 10^{-4}$
$I_{a}$	Rotor armature	$6.57\times 10^{-4}$
$\tau_{f}$	Friction loss	$2.86\times 10^{-2}$
$\tau_{\max}$	Simulated torque bound	$2.28\times 10^{-1}$

TABLE IV: Effective actuator-model settings used for training and evaluation.

\tau_{\max}

specifies the simulated torque bound.

Hidden layers	RAM (KB)	Flash (KB)	Inference ( $\mu$ s)
$128\times 64$	115.8	812.4	400.74
$\mathbf{128\times 128}$	150.3	846.7	594.33
$256\times 128$	247.3	943.2	1154.55

TABLE V: Implementation memory footprint and forward-pass latency for candidate policy sizes. Bold marks the deployed configuration.

Design	$v_{x}^{u}$	$v_{x}^{i}$	$\omega_{z}$	Recovery
Unified Baseline	0.98	1.00	0.79	6.9
No toroidal encoding	0.78	0.21	0.61	9.7
No $\mathbb{Z}_{2}$ mirror	0.59	0.02	0.27	10.0
No throw curriculum	0.76	0.80	0.77	9.7
Discrete morphology	0.69	0.00	0.39	10.0
No foot DR	0.96	0.94	0.76	8.4

TABLE VI: Simulation ablations. Tracking columns report achieved / commanded velocity (

u

: upright;

i

: inverted); recovery time is in seconds.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Engine	Step	Drive	Role
Isaac Gym	2 ms	Position	Training
MuJoCo	1 ms	Explicit effort	Identification, evaluation
Isaac Lab	2 ms	Position	Evaluation

Appendix

TABLE VII: Physics integration and drive interfaces. All engines use the identified actuator model and a

50\text{\,}\mathrm{Hz}

actor.

Fig. 9: Periodic joint representation. Top: the front-right hip at nine positions covering $\pm 4\pi/3$ ; poses at $\pm 180^{\circ}$ and $\pm 240^{\circ}$ ( $\equiv\mp 120^{\circ}$ ) coincide physically but differ by $2\pi$ in raw $\theta$ . Bottom: raw $\theta$ wrapped to $(-\pi,\pi]$ jumps at every odd multiple of $\pi$ ; the $(\sin\theta,\cos\theta)$ encoding used by the policy remains smooth.

Hyperparameter	Value
Parallel environments	4096
Rollout horizon	24 steps
PPO mini-epochs	20 per rollout
KL target (adaptive LR)	$\approx 0.01$
Entropy coefficient	$-0.015$
Discount $\gamma$	0.99

Appendix

TABLE VIII: PPO and training hyperparameters. The entropy term enters an objective that is minimized, so the negative coefficient corresponds to a conventional entropy bonus.

Condition	Trials	Success
Upright, pitch	$\bullet\,\circ\,\circ\,\bullet\,\bullet\,\bullet$	4/6
Inverted, pitch	$\circ\,\bullet\,\circ\,\bullet\,\bullet\,\bullet$	4/6
Upright, roll	$\bullet\,\bullet\,\bullet\,\circ\,\bullet\,\bullet$	5/6
Inverted, roll	$\bullet\,\bullet\,\bullet\,\bullet\,\bullet\,\bullet$	6/6
Throw	$\bullet\,\bullet\,\bullet\,\bullet\,\circ\,\bullet$	5/6
Overall		24/30

Appendix

TABLE IX: Per-trial hardware recovery (same hardware trials as the main paper).

\bullet

= resumed commanded locomotion,

\circ

= did not (

N=6

per condition).

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Quadruped locomotion provides a natural setting for understanding when model-free learning can outperform model-based control design, by exploiting data patterns to bypass the difficulty of optimizing over discrete contacts and the combinatorial explosion of mode changes. We give a principled analysis of why imitation learning with quadrupeds can be inherently effective in a small data regime, based on the structure of its limit cycles, Poincaré return maps, and local numerical properties of neural networks. The understanding motivates a new imitation learning method that regulates the alignment between variations in a latent space and those over the output actions. Hardware experiments confirm that a few seconds of demonstration is sufficient to train various locomotion policies from scratch entirely offline with reasonable robustness.

Abstract

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