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Adv."],"published-print":{"date-parts":[[2026,9,25]]},"abstract":"<jats:p>Quadrupeds in nature achieve agile locomotion through a rhythmic flexing of the spine in coordination with leg movement. This dynamic synergy enhances their speed and stability, reflecting a kind of physical intelligence encoded in their bodies. However, identifying this spine-leg synergy and embodying it in robotics to enhance locomotor propulsion remains challenging. To address this, we developed FLEXOR (fast legged robot with a flexible spine for optimal running), which uses dual-joint coupled spine and elastic legs to capture dynamic spine-leg synergy in rapid propulsion of small-scale quadrupeds. Through simulation and physical experiments of FLEXOR, we identified an optimal dynamic spine-leg synergy that maximizes locomotor capabilities by aligning the ground reaction force (GRF) for effective forward propulsion. Moreover, the dual-joint coupled spine amplifies the actuator torque, thereby increasing both the GRF magnitude and the propulsive output without additional energy input. By harnessing this synergy and its structural advantage, FLEXOR achieves a substantial increase in speed with a reduced cost of transport, outperforming state-of-the-art quadruped robots with flexible spines. Through an extended dynamics framework and robophysical validation, we further demonstrate that the optimal spine-leg synergy generalizes robustly across diverse spine-leg morphologies and physical scales. 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