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.
Figures & tables
Fig. 1: Underactuated robot with one leg and two reaction wheels balancing on a table
Fig. 2: Robot platform from above. The 15-degree cant angle, θ , of the wheels allows for balancing and aerial reorientation while maintaining differential drive features.
Fig. 3: Isometric view of robot platform displayed in CAD
Fig. 4: Rack and pinion design for jumping locomotion
Fig. 5: Robot rolls on the ground in a figure-eight path, blue trajectory overlaid from camera tracking.
Fig. 6: Robot leaning and jumping, then reorientating itself in the air to point along the velocity vector. The data collected in this test is shown in Fig. 7 and Fig. 8 .
Fig. 7: Data traces from the lean jump depicted in figure 6 . Chassis orientation Euler angles, wheel velocity and torques, and leg pointing direction are displayed. The colored regions indicate the change in robot state: blue is balancing, green is jumping, and yellow is aerial.
Fig. 8: Leg position, velocity, and commanded torque from the leg motor. The first peak is the robot jumping by extending the rack and pinion. The second peak is the landing as the leg motor provides a counter torque to dampen the impact force.
Fig. 9: Test set-up for determining impulse rejection of the balance controller. Here the robot is hit by a mass at the end on a pendulum against the highest point of the robot.
Fig. 10: Euler angles and angular velocity of the robot in balance disturbance experiments. A pendulum struck the robot in the front-back direction at 22 and 32 s with impulse magnitudes of 0.013Nms and 0.009Nms respectively. The offsets from 0 in the Euler angles are due to the balance offset of the robot since the center of mass is not located exactly along the leg.
Fig. 11: Overlay of self-righting from rolling to balancing. Top: Robot rolls forward and stops suddenly to flip over. Middle: Robot extends leg to shift weight backwards and tip upright. Bottom: Robot lifts itself off the ground and begins balancing.
Fig. 12: Euler angles during self-righting shown in figure 11 . The red background denotes the system in rolling phase, and the blue background denotes the system in balancing phase. The discontinuity in pitch is due to the system passing the domain limits of ±π/2 .
Fig. 13: Robot Modes during the trial shown in figure 14 . The robot goes from rolling to balancing, then to jumping into the air, and then landing and rolling again.
Fig. 14: Overlay of combined behaviors: Rolling and self-righting into balancing mode, then leaning and jumping up. After jumping it reorients to point along the velocity vector, and finally lands softly, returning to rolling.
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
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.
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.