Organizations: Center for X-Mechanics and Institute of Applied Mechanics, Zhejiang University, Hangzhou 310012, China · ZJU-Hangzhou Global Scientific and Technological Innovation Center, Hangzhou 311200, China · MirrorMe Robotics Co., Ltd., Hangzhou 311215, China
Achieving high-speed locomotion in quadrupedal robots remains highly challenging, as actuators operate near their physical limits and exhibit pronounced nonlinearities. However, many existing methods neglect actuator nonlinearities and physical constraints during training, leading to a significant sim-to-real gap under highly dynamic motions and limiting achievable performance. To address this issue, we propose a high-speed locomotion framework that reduces sim-to-real discrepancies and stabilizes learning over a wide command distribution. A refined actuator model explicitly captures high-speed voltage coupling and magnetic saturation, enabling a more accurate representation of the torque-speed envelope. In addition, a reinforcement learning framework incorporating a two-stage curriculum and adaptive command scheduling (ACS) ensures stable training. Experiments on the 36.5 kg quadruped BlackPanther2 (BP2) demonstrate speeds of up to 13.2 m/s on a treadmill and 11.65 m/s outdoors, establishing a new state-of-the-art and, to the best of our knowledge, a world record for quadrupedal robot locomotion. The results further highlight the importance of accurate actuator modeling in preventing non-physical policy exploitation, and show that ACS improves robustness without sacrificing performance.
Figures & tables
Fig. 1: First-quadrant operating envelopes of a representative joint actuator under different actuator modeling paradigms.
Fig. 2: Training pipeline for high-speed locomotion. In simulation, policy actions are mapped to joint torques and constrained by actuator limits, including torque–speed envelope constraints and magnetic saturation. During hardware deployment, the commanded torques are directly tracked by low-level motor controllers.
Reward Term
Equation ( ri )
Weight
Part A: Constant Weights
Lin. vel. track.
exp{−(vxycmd−vxy)2}
1.0
Ang. vel. track.
exp{−(ωzcmd−ωz)2}
0.5
Lin. vel. ( z )
vz2
−2.0
Joint power
∣τ⋅θ˙∣
−1.0×10−6
Action rate
(at−at−1)2
−0.01
TABLE I: Reward Function Elements and Multi-Stage Weight Schedules
Fig. 3: Comparison between the baseline without torque–speed envelope constraints and the proposed method in their respective training environments, where both policies operate at the limits of actuator performance.
Fig. 4: Visualization of locomotion patterns under diverse command scheduling modes enabled by the proposed ACS. The method generalizes across different command update paradigms while preserving acceleration profiles consistent with the robot dynamics.
Method
Max Velocity (m/s)
Relative Stability ( Jnorm )
Baseline
9.4
0.876
w/o torque–speed envelope
11.0
1.212
Simplified DC
11.7
1.070
Fixed acceleration scheduling
—
—
Uniform command sampling
11.0
1.452
Ours
12.4
1.000
TABLE II: Comparison of Velocity and Stability Metrics
Fig. 5: Experimental setup and representative snapshots from indoor locomotion experiments. (a)–(d) depict one complete gait cycle. (e) Snapshot at the instant when the robot reaches its maximum velocity.
Method
Max Sim Speed (m/s)
Max Outdoor Speed (m/s)
WildCat [ 12 ]
—
8.89
RLvRL [ 1 ]
5.7
3.9
HACL [ 30 ]
6.7
4.1
Hound [ 2 ]
6.5
5.9
Spot [ 22 ]
—
5.2
Ours
12.4
11.65
TABLE III: Comparison of Reported Maximum Speeds in Simulation and Outdoor Hardware
Fig. 6: Outdoor locomotion experiment results. (a) Time series of body velocity measured by the RTK system. (b) Corresponding torque–speed profiles of representative actuators, illustrating operation near actuator limits and close agreement with simulated behaviors.
Center for X-Mechanics, Zhejiang University, Hangzhou, China. · ZJU-Hangzhou Global Scientific and Technological Innovation Center, Hangzhou, China. · Mirrorme Technology Co., Ltd., Shanghai, China.
Department of Electrical and Computer Engineering at the University of Louisville. · Faculty of Electrical and Computer Engineering, University of Louisville, Louisville, KY, 40208, USA.
Department of Electrical and Computer Engineering, University of Louisville · Faculty of Electrical and Computer Engineering, University of Louisville, Louisville, KY, 40208, USA