Learning Energy-Efficient Air--Ground Actuation for Hybrid Robots on Stair-Like Terrain

Jiaxing Li; Ishaan Bhimwal; Wen Tian; Xinhang Xu; Junbin Yuan; Yuxin Guo; Sebastian Scherer; Muqing Cao

cs.ROMar 13, 2026

Learning Energy-Efficient Air--Ground Actuation for Hybrid Robots on Stair-Like Terrain

Authors: Jiaxing Li, Ishaan Bhimwal, Wen Tian, Xinhang Xu, Junbin Yuan, Yuxin Guo, Sebastian Scherer, Muqing Cao

Organizations: Carnegie Mellon University, US · Technische Hochschule Würzburg-Schweinfurt, Germany · Nanyang Technological University, Singapore · Singapore University of Technology and Design, Singapore

Abstract

Hybrid aerial--ground robots can use thrust to cross obstacles that impede wheel-driven motion, but deciding how much thrust to apply during contact remains challenging. We present an energy-aware reinforcement learning framework that jointly commands wheels, tilt servos and propellers through a single continuous policy, without prescribing locomotion modes. Hardware-calibrated power models penalise estimated electrical energy, while a terrain curriculum and a terminal reward for upright, settled arrivals support learning of thrust-assisted climbing. In simulation, continuous thrust allocation improves single-step clearance over fixed-thrust and mode-switching baselines as steps become taller. At the wheel-radius step height, it draws approximately 27 percent less mean power than the best fixed allocation. An energy-weight ablation shows the accompanying trade-off between efficiency and reliability. On a physical DoubleBee prototype, the unchanged network with a deployment interface clears two 6 centimetre steps in 8 of 10 trials. Additional hardware tests show both the potential for transfer to different terrain geometries and the remaining limitations in heading control, contact robustness and recovery.

Figures & tables

Fig. 1: Thrust-assisted climbing on DoubleBee. Four exposures show the robot approaching and traversing two $6\text{\,}\mathrm{cm}$ steps spaced $40\text{\,}\mathrm{cm}$ apart. The wheels provide forward travel while thrust assists passage over each edge, and the robot covers $1.3\text{\,}\mathrm{m}$ in $2.9\text{\,}\mathrm{s}$ before settling upright on the top step.

Panel of Figure 2 — Fig. 2: DoubleBee and its actuators. The policy uses a shared propeller command, a shared servo command, and common and differential wheel commands. This four-dimensional action is mapped to the six low-level channels ( Section III-B ).

Fig. 3: Training terrain and the policy’s view of it. Each $10\times 10$ $\mathrm{m}$ tile is a procedurally generated inverted-pyramid staircase with a $0.40\text{\,}\mathrm{m}$ tread and step heights from $0.03\text{\,}\mathrm{m}0.08\text{\,}\mathrm{m}$ , instantiated at five difficulty levels with five tiles each. Episodes start on a flat patch at the base of the staircase, with the goal (red) sampled $1.5\text{\,}\mathrm{m}3.2\text{\,}\mathrm{m}$ ahead and $0.03\text{\,}\mathrm{m}0.20\text{\,}\mathrm{m}$ above it. Inset: the height scan (cyan), a $4\times 4$ grid of rays at $0.07\text{\,}\mathrm{m}$ spacing over a $0.21\text{\,}\mathrm{m}$ square centred on the base and aligned with its heading.

Signal	Dim.	Information
$\boldsymbol{\omega}_{w}$	2	Wheel angular velocities
$\boldsymbol{\sigma}$	2	Servo tilt angles
$\boldsymbol{\omega}_{p}$	2	Propeller spin rates
$\mathbf{v}^{B}$	3	Body-frame linear velocity
$\boldsymbol{\omega}^{B}$	3	Body-frame angular velocity
$\hat{\mathbf{g}}$	3	Projected gravity (attitude)

TABLE I: Policy observation. Paired actuator quantities contain one value per left/right unit.

Component	Weight
Task progress
Goal progress, height gain, forward progress, settled climb	2 each
Per-step reward within $60^{\circ}$ of upright	0.5
Propeller engagement
Vertical thrust while leaning, increasing with lean	1.5
Keeping thrust vertical, upward thrust at a step	2 each

TABLE II: Principal reward components and weights. Group names correspond to ( 6 ). Terminal weights multiply a base value of 100 for the goal reward and

-10

for each failure.

Quantity	Perturbation or range
Body mass / CoM location	$\pm 5\%$ / $\pm$ 1\text{,}\mathrm{cm}$$
Per-propeller thrust	$\pm 20\%$
Static / dynamic friction	0.8–1.2 / 0.7–1.0
Wheel/servo stiffness, damping	$\pm 20\%$
Initial roll and pitch	$\pm$ 0.2\text{,}\mathrm{rad}$$
Push: linear / angular	$\pm$ 0.05\text{,}\mathrm{m}\text{,}{\mathrm{s}}^{-1} $/ $\pm$0.25\text{\,}\mathrm{rad}\text{\,}{\mathrm{s}}^{-1}$

TABLE III: Randomisation and delay settings. Physical parameters are resampled at every reset, except actuator stiffness and damping, which are sampled once at startup.

Fig. 4: Fixed-thrust comparison. Top: single-step clearance against step height for the learned policy and four fixed allocations. Bottom: clearance of a

6\text{\,}\mathrm{cm}

step against mean power. Wheels and servos remain learned in all variants. Each point is 2000 episodes and the bars denote one standard error across ten checkpoints. The line joins the fixed-allocation points.

Fig. 5: Thrust and posture through one simulated two-step climb, where shaded bands mark the step crossings. Forward speed ( a ) falls as each wheel loads against a step face and recovers between them. Height ( b ) rises in two stages. The body leans back to

-31\text{\,}\mathrm{\SIUnitSymbolDegree}

on the approach and forward to

33\text{\,}\mathrm{\SIUnitSymbolDegree}

through the crossings ( c ). Thrust ( d ) settles near

7\text{\,}\mathrm{N}

while driving, rises towards the simulated ceiling as each step is crossed, and falls away afterwards.

	Learned	Switch 0.31	Switch 0.46	Fixed 0.55
3 cm	$67.3\pm 5.6$	$\mathbf{70.7}\pm 4.7$	$62.6\pm 6.5$	$60.9\pm 7.3$
4 cm	$60.1\pm 6.1$	$\mathbf{60.4}\pm 7.0$	$54.1\pm 6.9$	$54.9\pm 7.7$
5 cm	$\mathbf{52.6}\pm 4.8$	$39.8\pm 5.6$	$42.6\pm 3.7$	$37.7\pm 6.7$
6 cm	$\mathbf{39.8}\pm 2.5$	$28.8\pm 5.0$	$25.6\pm 3.7$	$22.5\pm 6.2$
7 cm	$\mathbf{28.1}\pm 2.4$	$15.9\pm 3.8$	$13.1\pm 2.5$	$8.9\pm 5.9$
Power (W)	286	264	321	391

TABLE IV: Learned thrust against mode switching. Clearance is in percent, power is measured at

6\text{\,}\mathrm{cm}

, and values after

\pm

are one standard error across checkpoints.

Metric	Measurement
Both steps cleared	8/10 trials
Distance travelled	$1.26\text{\,}\mathrm{m}1.40\text{\,}\mathrm{m}$
Duration	$2.95\pm 64\text{\,}\mathrm{s}$
Mean electrical power	$275\pm 21\text{\,}\mathrm{W}$
Integrated electrical energy	$823\pm 229\text{\,}\mathrm{J}$

TABLE V: Primary hardware experiment. Distance, duration, power and energy refer to the eight completed climbs, and values after

\pm

are standard deviations across trials.

Fig. 6: Additional hardware terrains. Top: representative traversals on ( $\alpha$ ) closely spaced unequal steps, ( $\beta$ ) a ramp crest and ( $\gamma$ ) taller steps with a foam upper surface. Bottom: asymmetric wheel contact, departure from the ramp edge and chassis jamming. Geometry and trial outcomes in Table VI .

	Terrain	Observed outcome
$\alpha$ Narrow	Steps of 6, 7, 8 cm with 20 cm spacing	Third step reached 4/5 Upright on top 3/5
$\beta$ Ramp	Two $12.9^{\circ}$ ramps meeting at a crest	Crest crossed and far side descended 4/5
$\gamma$ Foam	Steps of 8 and 9 cm with second of foam	First step cleared 5/5 Upright on second 2/5

TABLE VI: Additional terrain tests, five trials each. Outcome definitions differ by configuration and are stated explicitly.

Robot Locomotion

Robot locomotion studies how robots move efficiently and reliably across varied terrains, environments, and movement modes.

See all 233 papers

Dynamics-Informed Reinforcement Learning for Agile and Energy-Efficient Locomotion of a Monopedal Hopping Quadcopter
Sep 14, 2026Ruigang Chen, Qi Zhang, Zhicheng Zhong +3Energy-Efficient Robot Locomotion Robot Locomotion
Department of Mechanical Engineering and Robotics, Guangdong Technion - Israel Institute of Technology, Shantou, 515063, Guangdong, China. · Department of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa, 3200003, Israel.
Energy-Efficient Gait Adaptation via Hierarchical Reinforcement Learning for Quadrupedal Locomotion Across Diverse Terrains
Oct 7, 2026Ammar Issa, Anubhav Singh, Anton Tsaritsin +1Legged Robotics Energy-Efficient Robot Locomotion
Faculty of Control Systems and Robotics, ITMO University, and the Biomechatronics and Energy-Efficient Robotics Lab (BE2R), St. Petersburg, Russia
LoComposition: Terrain-Adaptive Energy-Efficient Quadruped Locomotion without Gait Priors
Jun 14, 2026Loukas Kordos, Leonard T. Franz, Simon Rappenecker +4Legged Robotics RL for Robotics
Technical University of Munich. · University of Tübingen. · Hertie Institute for Clinical Brain Research & Center for Integrative Neuroscience. +1

Sep 14, 2026J/K move · Enter open · S save

Dynamics-Informed Reinforcement Learning for Agile and Energy-Efficient Locomotion of a Monopedal Hopping Quadcopter

Ruigang Chen, Qi Zhang, Zhicheng Zhong, Zhuorui Yun+2

Department of Mechanical Engineering and Robotics, Guangdong Technion - Israel Institute of Technology, Shantou, 515063, Guangdong, China. · Department of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa, 3200003, Israel.

Energy-Efficient Robot Locomotion Robot Locomotion RL for Legged Locomotion Quadrotor Control Physics-Informed RL

Open paper AskPDF arXiv

Although aerial-legged robots offer combined agility and efficiency, controlling high-speed hopping under complex hybrid dynamics is challenging. Reinforcement Learning (RL) is promising but prone to energy-inefficient "reward hacking". We propose a Dynamics-Informed RL framework for a monopedal hopping quadcopter. By embedding a target Specific Energy into the reward, we constrain the optimization to a physically viable energy manifold, ensuring stable hopping behaviour. By rewarding the phase-consistent behavior, it can encourage bio-inspired stance-phase impulse. Furthermore, penalizing the electro-mechanical power waste induces the motors generate an efficient impulse. This enables the policy to inject energy strictly during spring restitution without heuristic state machines. MuJoCo simulations validate robust height regulation and forward velocity tracking up to 2.0 m/s despite severe attitude-contact coupling. Ultimately, our approach yields a highly agile hopping gait, reducing energy consumption by 82% and 73% compared to hovering baselines and inefficiency baseline, respectively.

Abstract

Robot Locomotion

Explore similar work

Dynamics-Informed Reinforcement Learning for Agile and Energy-Efficient Locomotion of a Monopedal Hopping Quadcopter

Energy-Efficient Gait Adaptation via Hierarchical Reinforcement Learning for Quadrupedal Locomotion Across Diverse Terrains

LoComposition: Terrain-Adaptive Energy-Efficient Quadruped Locomotion without Gait Priors