Locomotion under reduced gravity is commonly realized through jumping, yet continuous pronking in lunar gravity remains challenging due to prolonged flight phases and sparse ground contact. The extended aerial duration increases landing impact sensitivity and makes stable attitude regulation over rough planetary terrain difficult. Existing approaches primarily address single jumps on flat surfaces and lack both continuous-terrain solutions and realistic hardware validation. This work presents a Dual-Horizon Hybrid Internal Model for continuous quadrupedal jumping under lunar gravity using proprioceptive sensing only. Two temporal encoders capture complementary time scales: a short-horizon branch models rapid vertical dynamics with explicit vertical velocity estimation, while a long-horizon branch models horizontal motion trends and center-of-mass height evolution across the jump cycle. The fused representation enables stable and continuous jumping under extended aerial phases characteristic of lunar gravity. To provide hardware-in-the-loop validation, we develop the MATRIX (Mixed-reality Adaptive Testbed for Robotic Integrated eXploration) platform, a digital-twin-driven system that offloads gravity through a pulley-counterweight mechanism and maps Unreal Engine lunar terrain to a motion platform and treadmill in real time. Using MATRIX, we demonstrate continuous jumping of a quadruped robot under lunar-gravity emulation across cratered lunar-like terrain.
Figures & tables
Fig. 1: Left: Training the quadruped robot in the Isaac Gym simulator under lunar gravity. Right: Deploying the learned policy on our MATRIX system while simultaneously emulating low gravity and lunar terrain.
Fig. 2: Overview of the proposed Dual-Horizon Hybrid Internal Model. Two temporal encoders process proprioceptive history at complementary time scales. The short-horizon encoder fs operates on the recent window Hs (6 steps, ≈ 0.12 s) to capture rapid vertical dynamics, producing an explicit vertical-velocity estimate v^z for phase awareness and a latent vector hs . The long-horizon encoder fl operates on a subsampled window Hl ( ≈ 0.9 s) to track slow trends across the full jump cycle, producing horizontal-velocity estimates v^x,v^y , CoM-height estimate h^ , and a latent vector hl . The fused representation zt=[v^x,v^y,v^z,h^,hs,hl] augments the current observation ot as input to the policy network. Training combines ground-truth regression supervision ( Lreg ) for the explicit estimates with InfoNCE-style contrastive objectives for the latent vectors, where each branch is pulled toward a target-encoder embedding of a future observation.
Fig. 3: Nine terrain types used for training. The top row includes Pure Flat, Flat, and Perlin Flat. The middle row includes Smooth Slope, Rough Slope, and Perlin Slope. The bottom row includes Discrete Obstacles, Crater, and Perlin Crater. The Crater and Perlin Crater terrains are defined by an exponential height function to approximate impact crater structures commonly observed on the lunar surface.
Fig. 4: The MATRIX hardware-in-the-loop simulation platform. Left – Physical World: the Low-Gravity Simulation Subsystem uses an overhead 2:1 pulley block with a counterweight ( mcw=125mrobot ) to reduce the robot’s effective weight to lunar gravity; the Terrain and Motion Emulation Subsystem comprises a 6-DoF Stewart platform and an integrated treadmill controlled by the Stewart Platform Controller and Treadmill Controller respectively. Right – Digital Twin: the Digital Twin and Control Subsystem runs in Unreal Engine; four ray-casts query the Landscape Mesh Collider to obtain raycast heights, from which tilt angles (ϕ,θ) are computed and sent as a Tilt Command to the Stewart Platform Controller, while the virtual robot avatar’s moving speed is sent as a Speed Command v to the Treadmill Controller.
Fig. 5: Representative lunar terrains configured in Unreal Engine. (a) relatively flat mare plains characterized by low surface roughness; (b) irregular uneven ground with multi-scale surface perturbations; (c) undulating hilly terrain with moderate inclination; and (d) crater-like depressions with smooth radial curvature.
Fig. 6: Comparison of state estimation across different temporal modeling strategies during a representative jump cycle.
Method
vz MSE
vx MSE
vy MSE
h MSE
Survival (s)
Landing SR (%)
Height Err. (m)
Short-only
0.041
0.013
0.013
0.028
13.6
71.2
0.054
Long-only
0.067
0.012
0.011
0.014
14.8
74.5
0.041
w/o Phase Reward
0.036
0.014
0.013
0.016
9.7
58.4
0.073
Ours
0.031
0.010
0.009
0.012
18.9
86.7
0.026
TABLE I: Ablation study in simulation.
Method
0.3 m/s
0.5 m/s
0.7 m/s
None
8.4
5.2
2.1
DR
12.7
9.3
5.4
PD
11.5
8.7
4.6
Ours
20.0
18.3
15.1
TABLE II: Average survival time (s) under different treadmill speeds.
Terrain
0.3 m/s
0.5 m/s
0.7 m/s
Mare Plains
20.0
18.3
15.1
Uneven Ground
18.6
15.9
11.8
Hilly Terrain
17.4
14.2
10.1
Crater Terrain
16.8
13.5
9.4
TABLE III: Average survival time (s) of the proposed method under different lunar terrain scenarios and treadmill speeds.
Planetary bodies characterized by low gravitational acceleration, such as the Moon and near-Earth asteroids, impose unique locomotion constraints due to diminished contact forces and extended airborne intervals. Among traversal strategies, hopping locomotion offers high energy efficiency but is prone to mid-flight attitude instability caused by asymmetric thrust generation and uneven terrain interactions. This paper presents an underactuated bipedal hopping robot that employs an internal reaction wheel to regulate body posture during the ballistic flight phase. The system is modeled as a gyrostat, enabling analysis of the dynamic coupling between torso rotation and reaction wheel momentum. The locomotion cycle comprises three phases: a leg-driven propulsive jump, mid-air attitude stabilization via an active momentum exchange controller, and a shock-absorbing landing. A reduced-order model is developed to capture the critical coupling between torso rotation and reaction wheel dynamics. The proposed framework is evaluated in MuJoCo-based simulations under lunar gravity conditions (g = 1.625 m/s^2). Results demonstrate that activation of the reaction wheel controller reduces peak mid-air angular deviation by more than 65% and constrains landing attitude error to within 3.5 degrees at touchdown. Additionally, actuator saturation per hop cycle is reduced, ensuring sufficient control authority. Overall, the approach significantly mitigates in-flight attitude excursions and enables consistent upright landings, providing a practical and control-efficient solution for locomotion on irregular extraterrestrial terrains.
Shriram Hari, M Venkata Sai Nikhil, R Prasanth Kumar
Department of Mechanical and Aerospace Engineering, Indian Institute of Technology Hyderabad, Sangareddy, India
We present a rolling and jumping underactuated monopedal robot designed to explore multimodal locomotion on low-gravity bodies. It uses only two reaction wheels to control its spatial orientation with two controllers: a balancing controller which can aim the robot's jump direction on the ground, and an aerial reorientation controller which can aim the robot's leg for landing after flight. We demonstrate rolling, targeted jumping and landing, and self-righting using only three actuators total, keeping system size to 0.33m and 1.25kg. Simple switching between locomotion modes enables the system to deal with differing landscapes and environmental conditions.
Neil R. Wagner, Justin K. Yim
Department of Mechanical Science and Engineering, University of Illinois Urbana-Champaign
This paper presents reinforcement learning (RL) policies for dynamic quadrupedal locomotion in planetary exploration scenarios. Building on a taskoptimized quadruped with a 5-bar leg design, we develop RL policies for walking, vertical jumping, forward jumping, and in-flight attitude control, explicitly tailored to the reduced gravity on Mars. These policies jointly enable such robots to overcome obstacles larger than themselves through coordinated jumping and precise in-flight reorientation for safe landings. We demonstrate Sim2Real transfer of the attitude control policy on the Olympus quadruped through single-axis reorientation tests, while all locomotion policies are validated in simulation. A complete Mars exploration mission scenario demonstrates coordinated policy deployment across challenging terrain. Experimental results show 90° attitude reorientation in 2.6 seconds, with simulations demonstrating 3.1 meter vertical jumps and 3.9 meter forward jumps under Martian gravity conditions. - Supplementary video: https://www.youtube.com/watch?v=qlSJ3P87A4A
Jørgen Anker Olsen, Kostas Alexis
The authors are with the Autonomous Robots Lab, NTNU, O.S. Bragstads Plass 2D, 7034, Trondheim, Norway.