A Trainable CPG Framework for Quadruped Locomotion

Xiyan Liu; Yide Liu; Guoyong Mao; Shaoxing Qu

cs.ROOct 27, 2026

A Trainable CPG Framework for Quadruped Locomotion

Authors: Xiyan Liu, Yide Liu, Guoyong Mao, Shaoxing Qu

Organizations: The authors are with the State Key Laboratory of Fluid Power & Mechatronic Systems, Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, Center for X-Mechanics, Department of Engineering Mechanics, Zhejiang University, 38 Zheda Road, Hangzhou 310027, China.

Abstract

We present CPG-ENN, a trainable quadruped locomotion framework that combines explicit neural coordination with learned motion realization. A symmetry-based eight-neuron central pattern generator (CPG) supplies gait rhythms, while a pretrained, leg-shared execution neural network (ENN) maps rhythmic signals and trajectory parameters to joint-angle references without online inverse kinematics. A single feedback policy modulates six physically interpretable trajectory variables. The controller tracks planar velocity and yaw-rate commands across walk, trot, pace, bound, and pronk without gait-specific learned controllers or transitionspecific rewards. Simulation yields 99.39% transition success across all 20 directed gait pairs and all source-target speed combinations from {0.5, 1.0, 1.5, 2.0} m/s. Ablations show high cross-gait transition feasibility without transition exposure; exposure mainly refines gait acquisition and velocity transients. Comparisons with AllGaits and Walk These Ways demonstrate accurate gait-speed regulation, and Unitree Go2 experiments validate five gaits and their transitions.

Figures & tables

Fig. 1: Go2 gaits: walk, trot, pace, bound, and pronk. Red: flight phases and foot clearance (bound: 4.1 cm; pronk: 5 cm).

Fig. 2: Overview of the CPG–ENN framework. The CPG coordinates gait rhythms, and a single feedback policy adjusts six motion parameters. The pretrained ENN, shared across all four legs and fixed during policy training, maps rhythmic signals and motion parameters to joint-angle targets tracked by a proportional–derivative (PD) controller.

Fig. 3: Phase relationships of the five gaits generated by the eight-neuron CPG. Numbers denote normalized leg-phase offsets within one gait cycle.

Fig. 4: Foot-trajectory parameters for offline ENN training. (a) Body height

h

, ground clearance

g_{c}

, ground penetration

g_{p}

, and step length

l_{s}

in the sagittal plane. (b) Planar trajectory rotation by

\psi

about the

+z

-axis of the hip-abduction frame.

Fig. 5: Shared ENN architecture. Each leg’s 17-D input combines six global features (5-D gait encoding and body height) with eleven per-leg features. All four legs share one MLP, whose three joint-angle outputs per leg are concatenated into a 12-D reference.

Term	Expression	Weight $w_{k}$
Forward velocity $r_{v_{x}}$	$\displaystyle\exp\!\left(-\frac{(v_{x}^{\mathrm{cmd}}-v_{x})^{2}}{\sigma}\right)$	$4.0$
Lateral velocity $r_{v_{y}}$	$\displaystyle\exp\!\left(-\frac{(v_{y}^{\mathrm{cmd}}-v_{y})^{2}}{\sigma}\right)$	$1.0$
Yaw rate $r_{\omega_{z}}$	$\displaystyle\exp\!\left(-\frac{(\omega_{z}^{\mathrm{cmd}}-\omega_{z})^{2}}{\sigma}\right)$	$1.2$
Velocity achievement $r_{\mathrm{va}}$	$\displaystyle\frac{1}{3}\sum_{\nu\in\{x,y,\omega\}}\exp\!\left(-\frac{\|e_{\nu}\|}{\max(\|c_{\nu}\|,\epsilon_{\nu})}\right)$	$0.6$
Energy $r_{E}$	$\displaystyle\sum_{j}\|\tau_{j}\dot{q}_{j}\|$	$-10^{-4}$
Action rate $r_{\Delta a}$	$\displaystyle\left\\|\tilde{\mathbf{a}}_{t}-\tilde{\mathbf{a}}_{t-1}\right\\|_{2}^{2}$	$-0.01$

TABLE I: Reward terms used for policy training

PPO Parameters	Value
Batch size	$98{,}304\;(4096\times 24)$
Mini-batch size	$24{,}576\;(4096\times 6)$
Number of epochs	$5$ per update
Clip range	$0.2$
Entropy coefficient	$0.005$
Discount factor $\gamma$	$0.99$

TABLE II: Policy optimization and domain-randomization settings

Fig. 6: Velocity tracking under piecewise-random commands. Columns show walk, trot, pace, bound, and pronk; rows show body-frame forward velocity $v_{x}^{b}$ , lateral velocity $v_{y}^{b}$ , and yaw rate $\omega_{z}^{b}$ . Dashed curves show commands; colored curves and shading indicate the temporal mean and $\pm 1$ standard deviation within centered sliding windows, without crossing command changes.

Fig. 7: Policy outputs versus target velocities (left to right): step length

l_{s,t}

and height

h_{t}

versus forward velocity; trajectory orientation

\psi_{t}

versus lateral velocity; and left–right step-length difference

\Delta l_{s,t}

versus yaw rate. Colors denote gaits.

Fig. 8: Gait transitions: (a) walk $\to$ trot $\to$ pace $\to$ bound $\to$ pronk; (b) reverse sequence. Rows show forward velocity, policy height $h_{t}$ and measured base height, and roll/pitch. Solid curves show the mean, and shading indicates $\pm 1$ standard deviation across time-aligned rollouts with 10 randomly sampled sets of switch times. Vertical lines mark gait switches.

Fig. 9: Forward-speed tracking across five gaits: CPG–ENN (Ours), AllGaits, and WTW. (a–e) Walk, trot, pace, bound, and pronk. Dashed diagonals indicate ideal tracking. Filled/open markers denote in-distribution (ID)/out-of-distribution (OOD) commands relative to each controller’s gait-specific training range. Crossed markers denote failed trials, displayed at zero; stars denote trials with no forward motion. WTW excludes walk (N/A).

Fig. 10: Gait–speed transition benchmark. (a–c) CPG–ENN; (d–f) AllGaits; (g–i) WTW. Columns show forward-speed RMSE over 0–2 s after switching, speed-pair success rate, and gait-pair success rate. Rows and columns within each matrix denote source and target conditions, respectively. Each speed-pair cell summarizes supported gait transitions across switch times; each gait-pair cell summarizes speed pairs across switch times. RMSE uses only each method’s successful trials; success is defined in Sec. IV-A . Blue indicates lower RMSE or higher success; red indicates the opposite. Speed-matrix diagonals include gait changes at constant speed. Dashes mark excluded same-gait cases; hatching marks unsupported WTW transitions involving walk.

(a) Transition exposure and CPG transition rule
Variant	Success	$T_{\rm acq}$	$v_{x}$ IAE	$\Delta q_{\rm des}^{\rm sw}$	Tilt RMS
	(%) $\uparrow$	(s) $\downarrow$	(m) $\downarrow$	(rad) $\downarrow$	(rad) $\downarrow$
w/o Stage-1 curriculum	97.5	0.810	0.2562	0.5452	0.0391
w/o trans. exposure + initial-state reset	98.5	0.798	0.2971	0.4279	0.0467
w/o trans. exposure + intrinsic dynamics	98.0	1.191	0.5828	0.3357	0.0796
w/o trans. exposure + common-time	99.5	0.772	0.2924	0.5474	0.0401

TABLE IV: Ablation of transition exposure, CPG transition mechanisms, and bounded policy parameterization.

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Sep 22, 2026J/K move · Enter open · S save

SG-CPG: Severity-Gated Central Pattern Generators for Adaptive Quadruped Locomotion under Continuous Actuator Degradation

Adarsh Kumar Kosta, Kaushik Roy

Department of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47906, USA

Dynamical Systems Fault-Tolerant Control Adaptive Control Quadruped Locomotion

Open paper AskPDF arXiv

An animal with a weakened limb does not necessarily switch its gait, instead it unloads the affected limb, re-coordinates the remaining limbs, and scales its response with injury severity. This graded adaptation allows locomotion to persist despite partial loss of limb strength, rather than requiring a discrete transition between healthy and failed. Inspired by this behavior, we propose SG-CPG, a central pattern generator (CPG) for quadruped locomotion under continuous actuator degradation. SG-CPG preserves a frozen healthy CPG policy and introduces two severity-driven gates: a residual gate that re-coordinates all four legs and an amplitude gate that progressively shortens the weakened leg's stride as degradation increases. We emulate progressive degradation through two mechanisms: lowering the joint torque ceiling (ceiling mechanism) and scaling its low-level controller gains (gain mechanism), representing distinct forms of actuator weakening. Our simulations on a Unitree Go2 show that SG-CPG maintains a trot gait with 100% survival across an omnidirectional command schedule under 95% joint strength loss while tracking commands within 8%. Under a lowered torque ceiling, removing either severity path, the residual's severity observation or the amplitude gate, raises clipping at the weakened joint from 4.4% to 13.6% and 26.3% of steps at an 80% loss. On a real Go2, SG-CPG survives 28 of 29 forward and turning trials with up to 93% calf torque degradation. These results show that severity-gated adaptation can extend a healthy locomotion policy to progressive actuator degradation without treating the fault as a discrete failure.

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