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.