cs.ROMar 19, 2026

Efficient and Versatile Quadrupedal Skating: Optimal Co-design via Reinforcement Learning and Bayesian Optimization

Authors: Hanwen Wang, Zhenlong Fang, Josiah Hanna, Xiaobin Xiong

Organizations: Hanwen Wang, Zhenlong Fang, and Josiah Hanna are with the University of Wisconsin - Madison. · Xiaobin Xiong was with the University of Wisconsin - Madison, and now with the Shanghai Innovation Institute (SII).

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Abstract

In this paper, we present a hardware-control co-design approach that enables efficient and versatile roller skating on quadrupedal robots equipped with passive wheels. Passive-wheel skating reduces leg inertia and improves energy efficiency, particularly at high speeds. However, the absence of direct wheel actuation tightly couples mechanical design and control. To unlock the full potential of this modality, we formulate a bilevel optimization framework: an upper-level Bayesian Optimization searches the mechanical design space, while a lower-level Reinforcement Learning trains a motor control policy for each candidate design. The resulting design-policy pairs not only outperform human-engineered baselines, but also exhibit versatile behaviors such as hockey stop (rapid braking by turning sideways to maximize friction) and self-aligning motion (automatic reorientation to improve energy efficiency in the direction of travel), offering the first system-level study of dynamic skating motion on quadrupedal robots.

Figures & tables

Fig. 1 : Quadrupedal skating setup with passive wheels. Each foot is equipped with a 3D-printed roller support that holds a passive wheel. In the default standing configuration, the wheel yaw installation angle ψ\psi is defined as the deviation of the wheel’s x-axis, xwheelx_{\text{wheel}} , from the x-axis of the robot’s sagittal plane. This angle is our key design parameter.
Fig. 2 : Bilevel co-design framework for quadrupedal skating. The outer loop uses BO to propose the next candidate design dnext\mathbf{d}_{\text{next}} . For each candidate design, the inner loop uses RL to optimize the policy πθ\pi_{\theta} . The resulting performance J(πθ∗,d,T)J(\pi_{\theta}^{*},\mathbf{d},\mathcal{T}) is fed back to BO, enabling efficient search over design-policy pairs that maximize skating performance.
Fig. 3 : Naive P configuration ( ψ=0∘\psi=0^{\circ} ) on Unitree Go1 [ 33 ] yields uncontrollable forward velocity vxv_{x} due to leg kinematic constraints. Similar limitations arise on many state-of-the-art quadrupedal robots.
NameExpression
Base VelocityBv{}^{\mathcal{B}}\mathbf{v} , Bω{}^{\mathcal{B}}\bm{\omega}
Commanded Base VelocityBvxyd,Bωzd{}^{\mathcal{B}}\mathbf{v}_{xy}^{d},{}^{\mathcal{B}}\omega_{z}^{d}
Projected GravityBug{}^{\mathcal{B}}\mathbf{u}_{g}
Joint Position and Velocityqj,q˙j\mathbf{q}_{j},\dot{\mathbf{q}}_{j}
Last Actionsaprev\mathbf{a}_{\text{prev}}
TABLE I : Observation space.
NameExpression
Base Linear Velocity xyrvxyr_{\mathbf{v}_{xy}} using ( 6a ) or ( 7a )
Base Angular Velocity zrωzr_{\omega_{z}} using ( 6b ) or ( 7 )
Base Height(z−zd)2(z-z^{d})^{2}
Base Orientation∥Bug,xy∥22\left\lVert{}^{\mathcal{B}}\mathbf{u}_{g,xy}\right\rVert_{2}^{2}
Base Linear Velocity zBvz2{}^{\mathcal{B}}v_{z}^{2}
Base Angular Velocity xy∥Bωxy∥22\left\lVert{}^{\mathcal{B}}\bm{\omega}_{xy}\right\rVert_{2}^{2}
TABLE II : Reward terms.
Fig. 4 : Illustration of World Frame Command angular velocity tracking reward scaling to prioritize linear velocity tracking.
Fig. 5 : Illustration of the human-engineered roller wheel angles and the resultant gait pattern.
Fig. 6 : A rapid stopping strategy known as the “hockey stop” emerged when using World Frame Command with the human-engineered design in Fig. 5 . When the robot is moving forward quickly and is suddenly commanded to stop, it rotates its body so that its lateral direction (which corresponds to the direction of maximal wheel friction) aligns with the velocity direction, thereby generating a large lateral friction force for deceleration.
Fig. 7 : An energy-efficient self-aligning behavior emerged when combining World Frame Command with the 2D design parameterization. For the optimal design in Fig. 9 , the backward direction is the most energy-efficient. As a result, the robot learns to consistently align its backward direction with the commanded velocity direction.
Fig. 8 : Directional CoT comparison. The left figure compares the CoT of walking and skating using human-engineered and 1D co-designed roller wheels. The right figure compares skating with 1D and 2D co-designed roller wheels.
Fig. 9 : 2D co-design optimal design.
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