Breaking speed scaling in quadrupedal robots via Huygens' coupled-pendulum dynamics

Yucheng Tao; Yongbin Jin; Shaowen Cheng; Xianwei Liu; Yanyan Yuan; Yanhong Liang; Chengkai Su; Chaojie Fu

cs.ROSep 9, 2026

Breaking speed scaling in quadrupedal robots via Huygens' coupled-pendulum dynamics

Authors: Yucheng Tao, Yongbin Jin, Shaowen Cheng, Xianwei Liu, Yanyan Yuan, Yanhong Liang, Chengkai Su, Chaojie Fu, +3 more

Organizations: Center for X-Mechanics, Zhejiang University, Hangzhou, China. · ZJU-Hangzhou Global Scientific and Technological Innovation Center, Hangzhou, China. · Mirrorme Technology Co., Ltd., Shanghai, China.

Abstract

Achieving biological-level running speeds has largely been pursued through advances in control algorithms, which improve the utilization of existing hardware. However, the ultimate speed limits remain governed by the underlying force and torque requirements of rapid locomotion, which are typically addressed through increased actuator capacity. Inspired by Huygens' coupled pendulums, we demonstrate that superior locomotion can emerge from principled exploitation of intrinsic dynamics rather than brute-force hardware scaling. Inter-limb inertial coupling redistributes energy across the gait cycle and reduces peak joint torque required for rapid periodic motion, thereby expanding the achievable speed without proportional increases in actuator capability. Incorporating hardware parameters as additional design variables further extends this analysis into a co-optimization framework, enabling the systematic utilization of inertial coupling in robot design. Guided by this framework, a quadruped robot achieves a running speed of 10.74 m/s (Froude number 21.4) and completes a 100-meter sprint in 12.2 seconds, representing the first legged robot to surpass 10 m/s. These results establish inertial coupling as an underlying mechanism governing high-speed legged locomotion and highlight its role in reducing force requirements, offering new insights into the design of agile robotic systems.

Figures & tables

Figure 1: Statistics of the maximum speed and body mass of mammals and quadrupedal robots in logarithmic scales. There are total 171 mammalian species of mammals with size ranging from an adult mouse to an elephant. Data for mammalian species are obtained from Hirt et al. [ 17 ] , in which body mass and running speed adhere to a general scaling law: $v=25.5m^{0.26}(1-e^{-22m^{-0.6}})$ . The lower-left inset summarizes peak locomotion performance and actuator capability for representative quadrupedal robots. Detailed data are provided in Table S1 .

Figure 2: Mechanical structure diagram of BP2. BP2 has a standing height of 55 cm, a body length of 80 cm, a width of 44 cm, and a total mass of 36 kg with battery. The shank structure is enlarged to clarify the assembly of the spring components and foot.

Figure 3: Analysis of the coupled dynamic model. ( A ) Experimental setup for Huygens’ coupled pendulums. ( B ) The simplified coupled dynamic model. ( C ) Dimensionless per-pendulum energies $\tilde{E}_{i}=E_{i}/(mgl)$ vs. normalized time $t/T_{0}$ (with $T_{0}=2\pi\sqrt{l/g}$ ). Light traces are instantaneous, bold traces are trailing time-averages $\langle\tilde{E}_{1,2}\rangle_{T}$ . ( D ) Poincaré section analysis of the coupled dynamic system. ( E ) Active torque requirements under specific kinematic conditions ( $R=1,\,A=1\,\text{rad},\,\omega=5\,\text{Hz},\,\phi=\tfrac{1}{4}\pi$ ).

Figure 4: Force decomposition and contribution analysis of the coupled dynamic model. ( A-C ) Stacked force profiles at 10 m/s for key degrees of freedom. ( D ) Optimized robot trajectory at 10 m/s, with the red arrow indicating the direction of forward motion. ( E-H ) Mean magnitude of each force/torque component in different running speeds. ( I-L ) Relative contribution of each dynamic term in different running speeds.

Figure 5: Experimental results of high-speed locomotion. ( A ) Representative keyframes of three consecutive gait cycles. Intermediate frames are faded and spatially shifted to illustrate the temporal evolution of leg configurations. ( B ) Bird’s-eye view of the experimental environment. ( C ) GPS-recorded global velocity profile. ( D ) Poincaré section of body angular-velocity dynamics during steady high-speed running. ( E ) Motor torque-speed operating points over three cycles during steady high-speed running.

Figure 6: Hundred-Meter Sprint Benchmark. ( A ) Field snapshot of the race between BP2 and Lyles [ 39 ] . ( B ) Speed–distance profiles for BP2 and world-class athletes. The kinematic data for Noah Lyles and Usain Bolt are derived from the official 10 m split times reported for the 2024 Paris Olympics [ 42 ] and the biomechanical report of the 2009 Berlin World Championships [ 14 ] , respectively, and the corresponding continuous curves are obtained through curve fitting. ( C ) Speed–distance profiles for BP2 and trained athletes. ( D ) Speed–distance profiles for BP2 and untrained adult participants.

Figure 7: Peak Speed Endurance Test. ( A ) Experimental setup, snapshot of the robot at peak velocity. ( B ) Thermal images captured at the initial pose and during peak speed operation. ( C ) Torque-speed profile of the key joint motor during the endurance test. ( D ) Temperature curve of the key actuator during the endurance test.

Figure S1 : Hybrid dynamic model of the spring-legged robot with a complete motor inertia model. Schematic of the sagittal-plane spring-legged model used for deriving the hybrid dynamics. The model is used for the forelimbs and adapted for the hindlimbs by setting $L_{0}$ and $l_{0}$ to negative values. In the derived equations of motion, the motor- and transmission-related parameters enter only through the rotor inertia $I_{m}$ and the gear ratio $r$ .

Figure S2 : Co-optimization results for the sagittal-plane model at 10 m/s. Throughout, blue and red curves represent the left and right legs, respectively. ( A ) Robot trajectory, with the red arrow indicating forward motion. ( B-E ) Velocity profiles for horizontal center-of-mass (CoM) motion (B), vertical CoM motion (C), hip rotation (D), and knee rotation (E). ( F, G ) Ground reaction forces. ( H, I ) Hip (H) and knee (I) joint torques. Gray dashed lines indicate the upper and lower torque limits of the hind leg motor at the corresponding joint velocities.

Robot Model	Body Mass	Motor Peak Torque	Motor Max Speed	Peak Speed
	$m$ (kg)	$\tau^{max}$ (Nm)	$\omega_{max}$ (rad/s)	$v$ (m/s)
MIT Mini Cheetah [ 26 ]	9	18	40	3.9
BlackPanther [ 25 ]	10	18	42	5
BlackPanther2	36	120	21.4	10.74
StarlETH [ 20 ]	23	24	10.5	0.75
ANYmal [ 22 ]	30	40	12	1.6

Table S1 : Performance specifications of representative legged robots. The table summarizes the body mass, motor peak torque, maximum motor speed, and achievable peak locomotion speed of different robotic platforms.

Category	Level	Event	Qualifying Time	Units
Male	Level-1	100 m	10.93	s
Male	Level-2	100 m	11.74	s
Male	Level-3	100 m	12.64	s
Female	Level-1	100 m	12.33	s
Female	Level-2	100 m	13.04	s
Female	Level-3	100 m	14.04	s

Table S2 : Performance standards for China National Level athletes in the 100 m sprint. The table summarizes the qualifying times for male and female athletes under the official athletics grading system.

Symbols	Physical Meaning	Typical Values	Units
$m_{0}$	Mass of torso	6.15	kg
$m_{1}$	Mass of thigh	2.2	kg
$m_{2}$	Mass of shank	0.1	kg
$m_{3}$	Mass of toe	0.5	kg
$I_{0}$	Inertia of torso	0.10	kg $\cdot$ m 2
$I_{1}$	Inertia of thigh	0.014	kg $\cdot$ m 2

Table S3 : Typical physical parameters of the sagittal-plane model. Values correspond to the nominal configuration.

Reward term	Formulation	Base coefficient
Body height	$r_{h}=c_{h}\exp[-80(z-z^{ref})^{2}]$	$0.05/0$
Body attitude	$r_{R}=c_{R}\exp[-80\\|\mathbf{e}_{z}-\mathbf{e}^{ref}_{z}\\|^{2}]$	$0.05/0$
Joint position imitation	$r_{q}=0.2c_{m}\exp[-2\\|\mathbf{q}^{ref}-\mathbf{q}\\|^{2}]$	$\bar{c}_{J}=0.6/0.1$
Joint velocity imitation	$r_{\dot{q}}=0.7c_{m}\exp[-0.02\\|\dot{\mathbf{q}}^{ref}-\dot{\mathbf{q}}\\|^{2}]$
Velocity tracking	$r_{v}=c_{v}[w\exp(-2\\|\mathbf{v}-\mathbf{v}^{cmd}\\|^{2})+(1-w)\exp(-2\\|\boldsymbol{\omega}-\boldsymbol{\omega}^{cmd}\\|^{2})]$	$\bar{c}_{V}=0.1/0.4$
Torque regularization	$r_{\tau}=c_{\tau}[0.9\exp(-0.5\\|\boldsymbol{\tau}_{n}\\|^{2})+0.1\exp(-0.1\\|\boldsymbol{\tau}_{n}-\boldsymbol{\tau}_{n,last}\\|^{2}/\Delta t)]$	$0.1/0.2$

Table S4 : Reward terms used for reinforcement learning controller training. The imitation stage emphasizes reference-motion guidance, whereas the relaxed stage continues training from the imitation policy with reweighted objectives for high-speed locomotion. Base coefficients are given as imitation/relaxed. Terms below the separator are added for the higher-speed BP2 training.

Symbol	Description
$\mathbf{q}^{ref}$ , $\dot{\mathbf{q}}^{ref}$	Reference joint position and velocity generated by the kinematic gait generator.
$\mathbf{q}$ , $\dot{\mathbf{q}}$	Measured joint position and velocity of the robot.
$\mathbf{v}^{cmd}$ , $\boldsymbol{\omega}^{cmd}$	Commanded linear and angular velocities.
$\mathbf{v}$ , $\boldsymbol{\omega}$	Measured body linear and angular velocities.
$\boldsymbol{\tau}_{n}$	Normalized joint torque.
$\boldsymbol{\tau}_{n,last}$	Normalized joint torque at the previous control step.

Table 14

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Quadruped Locomotion Robotics Sim-to-Real Transfer RL for Legged Locomotion

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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.

Abstract

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