Solving a Giant Rubik's Cube with a Quadruped Robot

Lukas Amiet; Gabriel Sánchez-Blanco Repgen; Mayank Mittal; Victor Klemm; Andrei Cramariuc; Marco Hutter; Cesar Cadena

cs.ROOct 17, 2026

Solving a Giant Rubik's Cube with a Quadruped Robot

Authors: Lukas Amiet, Gabriel Sánchez-Blanco Repgen, Mayank Mittal, Victor Klemm, Andrei Cramariuc, Marco Hutter, Cesar Cadena

Organizations: All authors are with ETH Zürich, Switzerland. · M. Mittal is also with NVIDIA.

Abstract

Pedipulating a freestanding articulated object requires controlling its pose and internal configuration while maintaining balance. We demonstrate this capability by autonomously solving a giant Rubik's Cube with a quadruped using only its feet. The task demands repeated precise interactions, since unintended layer motion can invalidate subsequent operations. We develop an onboard RGB-D perception pipeline that estimates the cube's metric pose and its full color state from partial views acquired through locomotion and flipping. A planner converts the computed solve into robot-executable skill sequences. Learned whole-body skills reorient the cube while preserving its layer configuration and turn its layers with one foot pinning the cube as the other pushes. We train these skills using reinforcement learning with privileged observations and distill them into policies that map onboard depth and proprioception to joint targets. The deployed policies tolerate variations in relative robot-cube placement and recover from external disturbances. The system completes three autonomous hardware solves, each about 30 steps over 8-9 minutes. These results show that combining onboard perception with coordinated pedipulation enables a quadruped to complete extended multi-step physical tasks. Additional material can be found at: https://leggedrobotics.github.io/rubik-skills/

Figures & tables

Fig. 2: System architecture. Between skills, cube perception provides the poses and cube net for planning and robot positioning. During a skill, the pedipulation policy acts only on proprioception and the front RealSense depth images.

Fig. 3: Cube perception. From one RGB-D frame (a), the pipeline segments the cube (b), fits a cube to the masked depth (c), and classifies the facelets (d). It outputs the geometric pose (c), the semantic pose (e), and the partial cube net (f). Black cells mark unseen facelets.

Fig. 4: Foot placement targets. Wireframe boxes $\mathcal{B}^{j}$ define target regions for the right front (RF, green) and left front (LF, blue) feet.

Fig. 5: Geometric pose error against labeled camera distance. Bars show the median (a) position and (b) rotation error in 0.5 m bins. The silhouette gate accepts a pose if the IoU between the rendered cube and the segmentation mask is at least 0.90.

Fig. 6: Facelet Color Confusion Matrix. Ground-truth versus predicted colors. Each cell shows indoor counts in the upper-left and outdoor counts in the lower-right.

Fig. 7: Cube scanning frames. Estimated pose, observed faces, and aggregated cube net for indoor (a, b) and outdoor (c–f) frames. Dashed outlines mark misclassified facelets: orange as yellow (c), red as orange (d), yellow as green (e). In (f) the top face does not pass the visibility gate, so the pipeline publishes no net for it.

Agent, Skill	Success	Layer error	Force (N) $\downarrow$	Drift
	(%) $\uparrow$	( ∘ ) $\downarrow$	RF / LF	(cm) $\downarrow$
Teacher, Flip Bwd.	99.8	0.34 [0.70]	47 [71] / –	1.3 [6.6]
Student, Flip Bwd.	99.7	0.35 [0.76]	46 [71] / –	1.5 [6.5]
Teacher, Flip Fwd.	99.8	0.35 [0.90]	27 [34] / –	1.5 [6.2]
Student, Flip Fwd.	99.7	0.36 [0.79]	27 [33] / –	1.5 [6.7]
Teacher, Rot. CCW	99.7	0.51 [1.58]	32 [53] / 55 [72]	1.0 [3.4]

TABLE II: Skill performance in simulation. Layer error is the maximum angular deviation of any layer during flipping, or the final angular error of the rotated layer. Force is the mean foot–cube contact force during contact for the robot’s feet. Drift is the cube displacement after the skill execution. Continuous metrics are reported across non-terminated episodes as median [p95].

Agent	Flip Bwd.	Flip Fwd.	Rotate CCW
Teacher (joint-space)	99.8	99.8	99.7
Hierarchical foot-target control	90.3	96.1	70.0
Student (both cameras)	99.7	99.7	99.0
w/o lower depth camera	99.1	99.4	96.2
w/o upper depth camera	99.4	99.8	98.0
w/o any depth cameras	77.8	87.5	76.8

TABLE III: Ablations and comparisons in simulation. Success rate (%). Ablated students are distilled from the same teacher. The hierarchical variant uses the teacher’s privileged observations.

Fig. 8: Student robustness to base placement. Axes give the position of the robot base relative to the cube center. The robot is spawned uniformly within each cell. Color encodes success.

Fig. 9: Ground-friction robustness of the deployed students. Ground friction is scaled while all other contacts remain nominal. Imprecise pedipulation achieves the commanded transformation but violates a success tolerance. Bars mark the training ranges of the cube-ground and foot-ground friction.

Fig. 10: Hardware robustness tests. Skills are tested under external disturbances, blocked cube motion, varying terrain and friction, and increased initial distance to the cube.

Fig. 11: Final 17 out of 25 steps of an autonomous solve. The robot scans the cube and executes the sequence without human intervention. Flip Forward/Backward rotate the cube $90^{\circ}$ away from/toward the robot. Rotate CCW turns the upper two layers $90^{\circ}$ counterclockwise (top view): the right front foot pins the bottom layer while the left pushes. Repositioning moves the robot around a $90^{\circ}$ arc. Together, flips and repositioning place any face at the bottom for turning. A video of the complete solve is available on the project website .

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Robotic manipulation studies how robots perceive, plan, and control physical interactions to grasp, move, assemble, and use objects.

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FINGR: Learning Dexterous Hand Control for Real-World Rubik's Cube Solving

Yutong Liang, Quanquan Peng, Matthew Kim, Xiaolong Wang

University of California San Diego

Robotic Control In-Hand Manipulation Contact-Rich Robotic Manipulation Robotic Manipulation

Open paper AskPDF arXiv

Manipulating a Rubik's Cube with a single dexterous hand is a challenging test of sustained, contact-rich control: the hand must execute successive layer turns while keeping the cube secure. Each turn requires some fingers to support the cube while others push a moving layer, release contact, and reset for the next move. To learn this coordination, we introduce FINGR (Future-supervised Interaction Network with Geometric Representations), a policy that combines finger-relative geometry with future interaction prediction. A shared point encoder expresses the cube relative to each fingertip and aggregates its points without depending on cubie indexing. Learned future tokens share the observation encoder and receive supervision for contact-force changes, layer-turn progress, and finger joint displacement at multiple time scales. The resulting representation conditions a flow policy that directly generates finger actions. On a real dexterous hand, our policy achieves 99.0% success over 300 turn attempts, compared with 79.7% for the base flow policy. Integrated with grasping and table-assisted regrasping, the policy solves all ten scrambled

2\times2\times2

cubes in a mean complete-system time of approximately 137 seconds. The project website is available at https://www.lyt0112.com/projects/FINGR

Abstract

Robotic Manipulation

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