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.
Figures & tables
Fig. 1 : Overview of the reaction-wheel based bipedal hopper operating on procedurally generated low-gravity terrain.
Fig. 2 : Planar dynamic model of the reaction-wheel hopper illustrating the torso-mounted reaction wheel (RW), wheel rotation Φw , hip and knee joint angles ( qhL,qhR,qkL,qkR ), and the ground reference frame used for system modeling.
Simulation Settings
Gravity g
1.625 m/s 2 (Moon)
Timestep Δt
0.001 s
Control frequency
1 kHz
Body Parameters
Torso mass
4.0 kg
Reaction wheel mass
3.0 kg
TABLE I : Simulation and Physical Parameters
Fig. 3 : Representative time instants of a single hopping cycle illustrating touchdown, propulsion, mid-air stabilization, and landing phases.
Fig. 4 : Continuous spatial trajectories across successive hops. The changing vertical baseline indicates successful adaptation to the uneven lunar heightmap, while forward progression remains consistent.
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
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.
Hopping robots often lose balance on slopes because the tilted ground creates unwanted rotation at landing. This work analyzes that effect using a simple spring mass model and identifies how slope induced impulses destabilize the robot. To address this, we introduce two straightforward fixes, adjusting the bodys touchdown angle based on the slope and applying a small corrective torque before takeoff. Together, these steps effectively cancel the unwanted rotation caused by inclined terrain, allowing the robot to land smoothly and maintain stable hopping even on steep slopes. Moreover, the proposed method remains simple enough to implement on low cost robotic platforms without requiring complex sensing or computation. By combining this analytical model with minimal control actions, this approach provides a practical path toward reliable hopping on uneven terrain. The results from simulation confirm that even small slope aware adjustments can dramatically improve landing stability, making the technique suitable for future autonomous field robots that must navigate natural environments such as hills, rubble, and irregular outdoor landscapes.
Heston Roberts, Pronoy Sarker, Sm Ashikul Islam +1
Department of Electrical and Computer Engineering, Georgia Southern University, Statesboro, GA, USA