Dual-Horizon Hybrid Internal Model for Low-Gravity Quadrupedal Jumping with Hardware-in-the-Loop Validation

This paper introduces a Dual-Horizon Hybrid Internal Model that enables stable, continuous quadrupedal jumping under lunar gravity using only proprioceptive sensing, validated through the MATRIX hardware-in-the-loop testbed which emulates reduced gravity and lunar terrain in real time.

The Gist

Imagine you are trying to teach a robot dog how to hop around on the Moon. It sounds like a fun game, but the Moon is a tricky playground. Because gravity is so weak there (about 1/6th of Earth's), when the robot jumps, it doesn't just pop up and down quickly like a kangaroo on Earth. Instead, it floats in the air for a long time—like a slow-motion balloon drifting.

This "slow-motion float" creates two big problems:

1. The "Where am I?" Problem: While the robot is floating, its feet aren't touching the ground. It can't feel the terrain, so it has to guess where it is and how fast it's falling. If it guesses wrong, it might land face-first.
2. The "Bumpy Road" Problem: The Moon isn't flat; it's covered in craters, rocks, and slopes. Hopping continuously over this uneven ground without falling over is incredibly hard.

This paper introduces a clever solution to teach a robot dog to hop continuously across the Moon's surface, and they built a special "Moon simulator" in a lab to prove it works.

The Brain: The "Dual-Horizon" Thinker

Most robot brains are like people looking at a car's speedometer: they only care about what's happening right now (the last split second). This works fine for walking on Earth. But on the Moon, a jump lasts so long that "right now" isn't enough information.

The researchers gave the robot a Dual-Horizon Brain. Think of it like a driver who has two different ways of looking at the road:

• The "Fast Glance" (Short Horizon): This part of the brain looks at the last 0.1 seconds. It's like checking the speedometer and the brake pedal. It tells the robot, "I just pushed off the ground hard!" or "I'm about to hit the ground!" This helps with the quick, sharp movements of jumping up and landing.
• The "Long Gaze" (Long Horizon): This part looks back over the last 0.9 seconds (a long time in robot terms). It's like looking out the windshield to see the curve of the road ahead. It helps the robot understand, "I've been in the air for a while, I'm drifting forward, and my center of gravity is slowly changing."

By combining these two views, the robot knows exactly what to do at every stage of the jump: when to push off, how to tuck its legs in mid-air, and how to land softly.

The Reward System: The "Traffic Light" Coach

To teach the robot, they didn't just say "jump." They created a Phase-Adaptive Reward System. Imagine a coach standing on the sidelines with a traffic light:

• Green Light (Takeoff): When the robot pushes off, the coach yells, "Go high! Push hard!"
• Yellow Light (Flight): While the robot is floating, the coach says, "Stay straight! Don't spin around!"
• Red Light (Landing): As it gets close to the ground, the coach screams, "Slow down your legs! Get ready to touch down gently!"

The robot learns to listen to these changing instructions automatically, without needing a human to tell it which phase it's in.

The Lab: The "MATRIX" Moon Simulator

You can't just send a robot to the Moon to test this first; it's too expensive and risky. So, the team built a machine called MATRIX.

Think of MATRIX as a high-tech trampoline gym:

• The Gravity Trick: The robot is tied to the ceiling with a rope and a heavy weight (like a counterweight on an elevator). This weight pulls up on the robot, canceling out most of Earth's gravity. It makes the robot feel like it weighs only 1/6th of its normal weight, just like on the Moon.
• The Moving Floor: The robot stands on a treadmill. But this isn't a normal treadmill. It's connected to a giant motion platform (like the ones used in movie theaters for 3D movies).
• The Digital Twin: A computer program (Unreal Engine) creates a virtual Moon with craters and rocks. As the robot hops in the real world, the computer instantly tilts the treadmill and the motion platform to match the virtual terrain. If the robot jumps toward a virtual crater, the floor tilts down to match it.

This setup lets them test the robot's "Moon hopping" skills in a real lab with real physics, but with a virtual landscape.

The Results

They trained the robot in a computer simulation first, then tested it on the MATRIX platform.

• The Winner: The robot with the "Dual-Horizon" brain and the "Traffic Light" coach could hop continuously for much longer than the other robots. It didn't fall over, even when the treadmill was moving fast or the virtual ground was bumpy.
• The Lesson: The experiment proved that to hop on the Moon, a robot needs to look at both the immediate moment and the longer trend of its jump.

In a Nutshell

This paper is about teaching a robot dog to play "hopscotch" on the Moon. They realized that because Moon jumps are slow and floaty, the robot needs a special brain that looks at both the split-second and the whole jump. They built a fancy lab with a rope-pulley system and a tilting treadmill to practice this on Earth, and the robot learned to hop across craters without falling. It's a major step toward sending robots to explore the Moon's craters and lava tubes in the future.

Technical Summary

Here is a detailed technical summary of the paper "Dual-Horizon Hybrid Internal Model for Low-Gravity Quadrupedal Jumping with Hardware-in-the-Loop Validation."

1. Problem Statement

The paper addresses the challenge of enabling continuous quadrupedal jumping (pronking) on the lunar surface. While jumping is energy-efficient under reduced gravity, the lunar environment presents unique difficulties:

• Prolonged Flight Phases: Lunar gravity ($\approx 1.62 \, m/s^2$) results in flight times more than twice as long as on Earth for the same jump height.
• Sparse Contact: The extended aerial duration means ground contact events are infrequent, making state estimation difficult.
• Sensitivity: Long flight phases increase sensitivity to landing impacts and make attitude regulation over rough, cratered terrain challenging.
• Limitations of Existing Methods: Current approaches often rely on single-jump scenarios, flat surfaces, or short observation windows that fail to capture the full state evolution of a lunar jump cycle. Furthermore, there is a lack of realistic hardware validation systems that can simultaneously emulate low gravity and complex 3D terrain.

2. Methodology

The proposed solution consists of a novel control framework and a specialized hardware-in-the-loop (HIL) validation platform.

A. Dual-Horizon Hybrid Internal Model (Control Framework)

The core algorithmic contribution is a reinforcement learning-based controller (using PPO) that relies solely on proprioceptive sensing (no external terrain sensors). It introduces a Dual-Horizon Hybrid Internal Model to handle the multi-timescale dynamics of lunar jumping:

1. Two Temporal Encoders:
   • Short-Horizon Branch ($H_s \approx 0.12s$): Captures rapid vertical dynamics. It explicitly estimates vertical velocity ($\hat{v}_z$) to provide immediate phase awareness (takeoff/landing transitions).
   • Long-Horizon Branch ($H_l \approx 0.9s$): Captures slow trends across the entire jump cycle. It estimates horizontal velocities ($\hat{v}_x, \hat{v}_y$) and Center of Mass (CoM) height evolution ($\hat{h}$).
   • Fusion: The outputs are fused to augment the policy input, allowing the robot to infer terrain and state information implicitly without external sensors.
2. Phase-Adaptive Gated Reward:
   • Instead of a fixed reward function, the system infers the jumping stage (Takeoff, Flight, Landing) based on CoM height and vertical velocity.
   • Specific rewards are activated for each phase: encouraging sufficient takeoff velocity, mid-air attitude stability, and coordinated touchdown.
3. Sim-to-Real Gap Mitigation:
   • Gravity Domain Randomization: Randomizes gravity during training to simulate cable tension fluctuations.
   • Phase-Triggered Disturbance: Injects impulse disturbances during takeoff/landing to simulate cable slack events common in suspension systems.

B. The MATRIX Platform (Hardware Validation)

To validate the policy physically, the authors developed MATRIX (Mixed-reality Adaptive Testbed for Robotic Integrated eXploration), a digital-twin-driven HIL system:

• Low-Gravity Emulation: Uses an overhead pulley-counterweight mechanism (2:1 mechanical advantage) to offload 5/6 of the robot's weight, simulating lunar gravity ($1/6$ Earth gravity).
• Terrain Emulation: A 6-DoF Stewart motion platform combined with a motorized treadmill.
• Digital Twin Loop: An Unreal Engine simulation runs in real-time. It casts rays from the robot's footprint to the virtual terrain to calculate surface normals. These normals drive the Stewart platform's tilt (roll/pitch) and the treadmill speed, creating a continuous, moving terrain experience for the physical robot.

3. Key Contributions

1. Dual-Horizon Control Framework: A novel internal model using complementary time scales (short for vertical dynamics, long for trajectory trends) to enable stable continuous jumping under prolonged aerial phases.
2. MATRIX Platform: The first hardware-in-the-loop system capable of simultaneously emulating reduced gravity and dynamic 3D lunar terrain (craters, slopes) for legged robots.
3. Experimental Validation: Successful demonstration of continuous quadrupedal jumping on a Unitree A1 robot across cratered, uneven, and sloped lunar-like terrain under emulated low gravity.

4. Results

Simulation Results (Ablation Studies)

• State Estimation: The Dual-Horizon model achieved the lowest Mean Squared Error (MSE) for vertical velocity and CoM height compared to "Short-only" or "Long-only" models.
• Performance: The full method achieved a 18.9s survival time (vs. 13.6s for Short-only) and an 86.7% landing success rate (vs. 71.2% for Short-only).
• Reward Design: Removing the Phase-Adaptive Gated Reward caused a significant drop in performance (survival time dropped to 9.7s), proving the necessity of stage-specific regulation.

Real-World Experiments (MATRIX)

• Robustness: The proposed method (combining Gravity Randomization and Phase-Triggered Disturbance) achieved the longest survival times across all treadmill speeds (0.3, 0.5, 0.7 m/s).
  • At 0.3 m/s: 20.0s survival time.
  • At 0.7 m/s: 15.1s survival time.
• Terrain Adaptability: The robot successfully maintained continuous jumping on four terrain types: Mare Plains, Uneven Ground, Hilly Terrain, and Crater Terrain. Performance degraded slightly on Crater Terrain due to steep exponential slopes, but the robot remained stable at lower speeds.

5. Significance

This work bridges a critical gap between simulation and reality in planetary robotics.

• Scientific Impact: It demonstrates that continuous, adaptive jumping is feasible on rough lunar terrain, offering a potential alternative to wheeled rovers for exploring craters and lava tubes.
• Technical Innovation: The Dual-Horizon model solves the specific problem of state estimation during long flight phases, a limitation of previous Earth-gravity-centric algorithms.
• Validation Standard: The MATRIX platform sets a new standard for hardware validation, moving beyond static flat-ground tests to dynamic, 3D terrain emulation under reduced gravity, which is essential for reliable deployment in space exploration missions.

Limitations: The current suspension system constrains lateral movement to a limited area, and cable tension variations introduce disturbances not fully modeled. Additionally, the platform does not yet simulate the complex contact mechanics of lunar regolith (slip/sinkage).