Unlocking fast robotic locomotor propulsion through dynamic spine-leg synergy

Abstract

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. This work broadens our understanding of the essence of synergistic locomotion in animals and is potentially applicable for the design and control of embodied intelligence–driven robots.

INTRODUCTION

Legged machines, with their agility and versatility, offer notable advantages in assisting humans with exploration (1), search and rescue operations (2), and environmental monitoring (3) in unstructured natural environments. Taking inspiration from the architecture and sensorimotor control of animals is an intuitive and effective approach to develop such advanced systems (4). To adapt to changing habitats, animals have evolved various modes of locomotion, regulated by the intrinsic dynamic synergy embedded in their neural and musculoskeletal system (5–9). This dynamic synergy enables them to exhibit efficient locomotor propulsion through interactions with their surrounding environments, which forms a core part of embodied intelligence (10–12). Quadrupeds transition from low-speed to high-speed locomotion through speed-dependent changes in the neural coordination of spinal circuits that drive gait transitions (13), and the spine plays an important role as a store of elastic strain energy (14–16). Moreover, there is a precise synergy between the spine and leg motions during free running in small mammals (17, 18). The spine reaches its flexion and extension limits just before the hind legs make contact with the ground and immediately after they lift off, respectively. The role of dynamic spine-leg synergy in high-speed locomotion is evident. It allows quadrupeds to increase stride length, speed up transitions between gait cycles, and optimize energy conversion.

Achieving quadruped-like maneuverability has been a long-held human desire and has garnered widespread interest among the robotics community. While legged locomotion of quadruped robots has been extensively explored (19–30) and has demonstrated dynamically proficient gaits, the general absence of functional spine in these systems has impeded the implementation of spine-leg synergistic propulsion (31). This represents a fundamental limitation preventing quadruped robots from achieving locomotor performance comparable to their biological counterparts. The integration of functional spines has been recognized as a promising approach for enhancing locomotion capabilities in quadruped robots. Existing studies have demonstrated the functional advantages of the spinal mechanism, including underactuated or passive spines (32–34), active hinged spines (35–40), cable-driven tensegrity spines (41–44), and fluid-driven spines (45–49). Current spinal morphologies in quadruped robots present two major limitations. First, they frequently use a single revolute joint to mimic spinal motion, failing to replicate the critical mechanical coupling and intervertebral coordination essential for biological locomotion efficiency. Second, existing cable-driven or fluid-actuated spinal designs struggle to execute motion commands with precision under dimensional constraints, consequently failing to leverage dynamic spine-leg synergy for maximizing locomotion efficiency. To date, a robotic spine that effectively mimics the coupling mechanism of biological spinal joints and is readily controllable has not yet been developed.

To synergize spine-leg movements, various control strategies have been developed, including model-based controllers such as impedance control (32, 50, 51) and geometric mechanics (38), as well as model-free methods such as central pattern generators (35, 36), Bayesian optimization (43), and reinforcement learning (52). Using these specialized controllers, robots with diverse spinal morphologies demonstrate the critical role of the spine-leg synergy in facilitating locomotor propulsion. However, to date, no cohesive framework exists to systematically investigate the synergy between spine and leg movements in dynamic quadrupedal systems featuring diverse spinal morphologies.

To address the limitations of existing robotic physical structures and fully explore the dynamic spine-leg synergy, we drew inspiration from nature. Comparative biomechanics across species has revealed that small cursorial mammals exhibit exceptionally pronounced sagittal spinal flexion-extension (up to 120°) compared to larger mammals (53, 54). Spinal motion contributes ∼50% to stride length in small mammals, confirming the evolutionary benefits of spinal-driven propulsion (55). Therefore, we focus on the dynamic spine-leg synergy in small quadrupeds, taking the sengi (Macroscelididae), one of the fastest small (<500 g) quadruped mammals (56, 57), as a model. To our knowledge, current research has not yet replicated fast spine-leg synergistic propulsion at the scale of the sengi (∼150 mm), and three key challenges remain. First, mechanical innovations under scale constraints are required to design robots that effectively incorporate the essential morphological features and synergistic locomotion observed in their biological counterparts while maintaining simple control. Second, it is challenging to synergize spine and leg movements in a small-scale complex multibody dynamic system. Third, to elucidate how dynamic spine-leg synergy influences locomotor propulsion and its generalizability, it is essential to develop an extended dynamics framework that can uncover the shared principle of spine-leg coordination governing synergistic locomotion across robotic quadrupeds with diverse morphologies.

Inspired by small-scale quadruped mammals represented by the sengi and drawing on the lessons learned from developing previous generations of robotic rats (42, 58–61), we developed FLEXOR, which mimics the sengi’s morphology and synergistic locomotion through a dual-joint coupled spine and elastic leg mechanisms. The spinal mechanism uses unidirectional rotation to achieve biologically rhythmic flexion-extension while amplifying the actuator torque, facilitating rapid motion within a limited dimension and simplifying control. Our multiparameter dynamics model predicts the optimal dynamic synergy between the robot’s spine and legs. This synergy is characterized by a slight lag of the hind-leg backswing phase behind the spinal extension phase, within a multidimensional parameter space. We transferred this synergy pattern to a real-world FLEXOR, enabling efficient torque propagation through body segments. This, in turn, aligns the ground reaction force (GRF) favorably for forward propulsion, effectively enhancing stride length. By leveraging this optimal synergy and its structural advantage, FLEXOR achieves fast and stable dynamic locomotion with a maximum speed of 7.13 body lengths (BL)/s and a substantially reduced cost of transport (CoT), which provides insights toward self-propulsion mediated by spine-leg synergy. Moreover, we developed an extended dynamics framework and demonstrated that the optimal spine-leg synergy robustly generalizes across diverse spine-leg morphologies and physical scales. Movie S1 summarizes the main contributions of this work. Although FLEXOR was designed on the basis of the locomotor characteristics of sengis, the abstracted principle of spine-leg coordination is common among legged vertebrates. This work has the potential to bridge the gap between the locomotor capabilities of legged robots and natural animals and also sheds light on the principles underlying natural intelligence for locomotor propulsion in animals.

RESULTS

Bioinspired embodied robotic system

The design of FLEXOR (Fig. 1A) draws inspiration from the functional anatomy and dynamic locomotion characteristics of small-scale quadruped mammals represented by the sengi. The exceptional locomotor performance of sengis principally stems from several key determinants: extensive vertebral flexibility facilitating spinal flexion-extension, specialized hind leg musculoskeletal adaptations, and highly coordinated dynamic coupling between spinal and leg movements (see “The morphological characteristics and locomotor properties of sengis” in the Supplementary Materials). The sengi’s spine plays a crucial role in agile locomotion and is composed of multiple vertebrae, which makes the design of a similar spinal mechanism challenging. Here, we propose a method to determine the number and distribution of key movement joints (KMJs) in the spine from sengi’s dynamic locomotion, thereby enabling the mechanical equivalent of a biological spine to achieve lifelike spinal movements and be easily deployed robotically (see “Model for extracting the spinal KMJs” in Materials and Methods). On the basis of the extracted KMJs, the spine is designed with dual joints that divide the torso into three segments: the thorax, lumbar, and pelvis, respectively, and the arrangement of the leg joints is also aligned with the anatomical features of the animal (Fig. 1, B and C) (55). Figure 1D illustrates the periodic variations of the biological spinal angles and robotic spinal angles, along with sengi’s footfall pattern in the top row. The dynamic locomotion of the sengi predominantly relies on the forward propulsion generated by its hind legs pushing off the ground. The hind legs are characterized by elongated tibiae, deep knee joints, and long tarsal bones, which improve gait flexibility and stability (62). To benefit from these features, we incorporated the bone proportions of the sengi’s legs into the design of FLEXOR. We embodied these morphological and locomotion characteristics into our bioinspired robotic prototype (Fig. 1E). Using the robot as a test platform (63–65), we aim to assess how dynamic spine-leg synergy affects locomotor propulsion.

Fig. 1. Robotic systems for understanding dynamic spine-leg synergy.

(A) Computer-aided design drawing of FLEXOR, which is divided into three main parts: front and hind torso with elastic legs and a dual-joint coupled spine. (B) Motion frame sequences of the sengi. Spinal posture in each frame was extracted to analyze the changing patterns. The top panels illustrate the spinal postures of sengi during three phases within one gait cycle. The number and distribution of KMJs were extracted using the multilink equivalent mathematical model. (C) Functional anatomy of the sengi. KMJs were mapped onto the anatomical diagram of the sengi (adapted from the original skeletal image by Polyoutis, licensed under CC BY-SA 4.0; https://creativecommons.org/licenses/by-sa/4.0/). We defined the acute angle formed by the vertical line of the simplified biological pelvis and the lumbar as ${\mathrm{\psi }}_1^{\prime }$, the obtuse angle formed by the vertical line of thorax and the lumbar as ${\mathrm{\psi }}_2^{\prime }$, and the total angle of the spine as ${\mathrm{\psi }}_3^{\prime }$. Geometrically, ${\mathrm{\psi }}_1^{\prime }$ and ${\mathrm{\psi }}_2^{\prime }$ are supplementary. (D) Angles of equivalent biological and robotic spinal joints (${\mathrm{\psi }}_1$, ${\mathrm{\psi }}_2$, and ${\mathrm{\psi }}_3$) as functions of the gait cycle. (E) The KMJs of the robotic system inspired by the sengi.

On the basis of the abstracted biological features, FLEXOR is designed to be a compact, lightweight, and modular high-dynamic robotic system. It boasts several distinctive attributes, setting it apart from other quadruped robots of similar size (43, 60, 66). First, its spine is capable of transforming simple and efficient unidirectional rotation into swift and rhythmic bidirectional spinal flexion-extension. Second, FLEXOR features elastic legs capable of energy storage and release, which also provides compliance in the dynamic system. Third, FLEXOR replicates not only the morphological characteristics of the sengi within the constraint of scale but also the kinematics of the sengi without increasing control complexity. The main structural components of the robot were fabricated using three-dimensional (3D) printing technology, and metal shafts and bearings were used at rotational and linear joints throughout the body to diminish friction and increase transmission accuracy. This manufacturing methodology strikes a balance between the expediency of component fabrication and the precision of assembly, facilitating rapid prototyping iterations. Movie S2 demonstrates functional components and dynamic locomotion of FLEXOR.

Small-scale cursorial quadrupeds, represented by sengis, exhibit more pronounced rhythmic sagittal spinal bending during high-speed locomotion compared to larger quadrupeds (53). The sengi’s lumbar vertebrae have laterally expanded transverse processes with anteriorly oriented tips and pronounced caudal notches, enhancing attachment areas and mechanical advantage of spinal flexors and extensors to facilitate greater flexion-extension motion range (54). This adaptation facilitates enhancing self-propulsion, which allows the spine to function as the critical source of propulsion during high-speed locomotion (67). To replicate the neural-controlled rhythmic spinal motion of sengis during high-speed running, we propose a dual-joint coupled spinal mechanism that acts as a simple mechanical oscillator, mimicking the function of the biological neural oscillator (Fig. 2A). The design of the spinal mechanism addresses three critical considerations: First, the flexion-extension movements should exhibit rhythmicity consistent with the biological spine (Fig. 1D). Second, it needs to facilitate rapid transitions between spinal flexion and extension. Last, the spinal mechanism should amplify the torque of the actuator while maintaining a compact size. The bioinspired dual-joint coupled spinal mechanism converts rotational input into rhythmic flexion-extension via symmetrically arranged sublinkages mimicking dorsal and ventral musculature. These artificial muscles exhibit controlled concentric contraction and eccentric elongation during respective flexion and extension phases, with mechanical constraints ensuring biologically accurate motion ranges. Notably, this design is a simplified yet effective representation of the complex biological spine system. Our dual-joint coupled spinal design achieves biologically meaningful curvature matching and attains a sagittal plane bending range of 120°, comparable to that of the sengis’ spines, while overcoming the output torque limitations of cable-driven multiple-pitch joint systems (32, 41, 43). We used six sets of sublinkage systems to convert the unidirectional rotation of the actuators into the swing angles ${\mathrm{\psi }}_1$ and ${\mathrm{\psi }}_2$ on both the upper and lower sides of the spine. Specifically, the rotation of the actuators serves as the input for the first linkage system (crank rocker; ① in Fig. 2A), with its output then serving as the input for the second linkage system (rocker slider). Subsequently, the output of the second linkage system serves as the input for the third linkage system (slider rocker; ② in Fig. 2A), where the rocker of the third linkage system represents the swing angle output of the spinal mechanism (③ in Fig. 2A). The same three sets of linkage systems produce swing angle output (①, ④, and ⑤ in Fig. 2A), symmetrically positioned on the upper side of the spine. To achieve biomimetic spinal rhythmicity, we calculated the dimensions of each linkage of the above three systems using a particle swarm optimization algorithm (68) with the objective of ${\mathrm{\psi }}_1+{\mathrm{\psi }}_2=\mathrm{\pi }$ (see “Mathematical modeling of the spinal mechanism” in the Supplementary Materials). The spinal multilink motion chain amplifies the driving torque before transmitting it to the fore and hind legs (fig. S15A). Movie S4 demonstrates the rapid motion of the spinal mechanism.

Fig. 2. Overall design of FLEXOR.

(A) Dual-joint coupled spine. This mechanism replicates the neuromuscular control of spinal flexion and extension through a simple mechanical oscillator, which mimics the function of its biological neural counterpart. It features two motion chains that are coordinately actuated through one unidirectional rotation, facilitating the dynamic transition between spinal flexion and extension. The two motion chains are indicated by ①, ②, and ③, located below the spine frame, and ①, ④, and ⑤, located above the spine frame. ① represents unidirectional rotation. ② and ④ represent translational motion, and ③ and ⑤ represent the rotational motion output. When the spine is at maximum flexion and extension, ② and ④ are at the extreme positions of the motion range, and ${\mathrm{\psi }}_1$ and ${\mathrm{\psi }}_2$ are supplementary at this moment. (B) Elastic hind legs of FLEXOR. The transition between underactuated and fully actuated configurations is achieved through the arrangement of the cables.

Inspired by the leg skeletal-muscle-tendon network structures of the sengi, we designed the robot’s legs using three-segment configuration with elastic compliance (fig. S6). Taking cues from the prominent tibial tuberosity of sengis, which confers a mechanical advantage for knee extension (69, 70), we extended the spring attachment point distally beyond the knee joint to amplify propulsive force during extension. Furthermore, biomechanical analyses reveal a metatarsal-to-femur length ratio of 1.07 (71)—a well-established morphological signature of cursorial adaptation in sengis (fig. S4E), which has been systematically incorporated into our robotic design. In cursorial locomotion of animals, the hind legs act as the key “motor” of locomotion, propelling the body forward, whereas the forelegs function primarily to absorb ground impact as the hind legs generate forward propulsion (72). The knee and ankle joints are mechanically coupled through biarticular muscles in biological systems (73, 74). To replicate this biological linkage, we implemented a parallelogram mechanism (75) at both joints, complemented by a tension spring and compression spring assembly to emulate the function of biological muscle-tendon units. This design provides serial elastic behavior (76–80), thereby enhancing the robot’s locomotion efficiency and stability (74). During quadrupedal running, the knee joint primarily generates the necessary horizontal propulsion (81). Biological studies reveal that quadruped mammals exhibit scapular rotation relative to the thorax during running (55), inspiring our three-segment foreleg design in FLEXOR where an active joint integrates scapulothoracic and shoulder motions to fulfill cushioning and balancing functions while maintaining a passive elbow joint to preserve natural buffering functionality. For the hind legs, which play a pivotal role in running, we devised three knee-joint configurations: passive, underactuated, and fully actuated, denoted as FLEXOR-P, FLEXOR-U, and FLEXOR-F, respectively, to explore how leg configuration affects dynamic spine-leg synergy. The hind legs were designed to 90 mm (the sum of the individual segments), achieving 60-mm standing height that balances ground clearance and hip joint torque while maintaining propulsion efficiency. Hind leg workspace analysis of FLEXOR-F confirms complete coverage of biological foot trajectories (fig. S7), demonstrating functional biomimicry of propulsion mechanics. Figure 2B illustrates distinct hind leg configurations (see “Mathematical modeling of the leg mechanism” in the Supplementary Materials). For a detailed design motivation and results, please refer to “Comparative kinematics and dynamics between robotic and biological systems” in the Supplementary Materials.

Prediction of the optimal dynamic spine-leg synergy

Dynamic spine-leg synergy is one of the most efficient manners for quadrupeds to achieve fast and stable locomotion (82, 83). We used FLEXOR as a robotic agent to test the hypothesis that an optimal dynamic spine-leg synergy pattern exists, maximizing the robot’s stride length within one gait cycle for enhancing locomotion performance.

The trade-offs made by robots during the transition from the stance phase to the flight phase determine the locomotor propulsion. For instance, an elevated aerial height may increase the leg load and vertical displacement of robots without guaranteeing an increase in forward stride length. On the other hand, a lower aerial height results in a shorter flight phase, which directly leads to an insufficient forward stride length. When selecting appropriate dynamics, robots must balance these trade-offs. During dynamic locomotion, several variables determine the locomotor propulsion of robots, including hardware design, kinematics, and the conditions of ground contact. Kinematic parameters include motion frequency, joint amplitude, and synergy pattern between the spine and legs movements. Here, the synergy of spine and legs is represented by the amount the hind-leg backswing phase lags behind the spinal extension phase. Hardware parameters include size and mass. Considering the dynamic characteristics of FLEXOR’s locomotion, we established a multiparameter dynamics model of FLEXOR, as shown in Fig. 3A.

Fig. 3. Multiparameter dynamics model predicting the optimal dynamic spine-leg synergy.

(A) Optimization loop of parameters in dynamics model. Key state parameters are distilled from FLEXOR’s dynamics model and mapped to joint control parameters. The control parameters are optimized through dynamic programming algorithm to explore the maximum stride length. (B) Mapping between parametric space and stride length. The spatial curves represent the joint angles over time, where color indicates stride length. Each variable has an upper and lower bound when all other variables are held constant. Our model predicts the optimal spine-leg dynamic synergy region, indicated by the dark red area in the graph and enclosed by the solid white line. (C) Effect of the phase lag variation between the spine and hind legs on stride length, with other variables held constant. (D) Effect of frequency variation on stride length. (E) Effect of the amplitude variation of the hip and knee joints on stride length.

These variables all interact in locomotion. Ultimately, locomotion performance is contingent upon the alignment of multibody dynamics, robot hardware, and ground conditions. We predicted the optimal dynamic synergy between the spine and legs of the robot within its high-dimensional parameter space using a dynamic programming optimizer, with the objective of maximizing stride length. Among all state parameters of the dynamic system, we distilled parameters that play key roles in dynamic locomotion from the robot’s motion states, namely, the spinal joint angle ${\mathrm{\theta }}_{\text{ms}}$, hip joint angle ${\mathrm{\theta }}_{\text{mh}}$, and knee joint angle ${\mathrm{\theta }}_{\text{mk}}$. This 3D parameter space is called the “dynamic spine-leg synergy region.” Our multiparameter dynamics model delineates the boundaries of the dynamic spine-leg synergy region by constraining these three joint angles. These angles exhibit temporal variation, and they are regulated by motion frequency, joint motion amplitude, and synergy of the spine and legs.

Figure 3B illustrates how stride length (represented by the color bar) varies under the influence of distilled three control variables: spinal joint angle (${\mathrm{\theta }}_{\text{ms}}$), hip joint angle (${\mathrm{\theta }}_{\text{mh}}$), and knee joint angle (${\mathrm{\theta }}_{\text{mk}}$), which directly govern the robot’s running locomotion. In this 3D coordinate system with ${\mathrm{\theta }}_{\text{ms}}$, ${\mathrm{\theta }}_{\text{mh}}$, and ${\mathrm{\theta }}_{\text{mk}}$ as axes, the spatial curves represent the exploration process of optimal dynamic spine-leg synergy (movie S5). Each curve captures the temporal evolution of joint angles over a 1-s interval. These spatial curves exhibit periodic oscillations along the ${\mathrm{\theta }}_{\text{ms}}$ axis, a characteristic resulting from our specialized spinal mechanism design where the actuator achieves flexion-extension through unidirectional rotation, completing one full revolution per gait cycle. As demonstrated in Fig. 3C, by maintaining constant parameters while adjusting the phase lag between spine and legs, we identified a quasiplanar region. The curves within this region distribute periodically along the ${\mathrm{\theta }}_{\text{ms}}$ axis, with vertical displacements along the ${\mathrm{\theta }}_{\text{ms}}$ axis indicating variations in spine-leg phase lag (positive and negative regions are annotated at the bottom of Fig. 3C). Notably, the deep red segments correspond to optimal synergy conditions at a phase lag of π/10 between spinal extension and hind leg backswing, yielding maximum stride length. Similarly, Fig. 3D displays curve clusters generated by frequency modulation while holding other parameters constant. Frequency variation exclusively affects curve length—higher frequencies elongate the curves and intensify their red coloration, indicating stride length augmentation. Furthermore, increasing the motion amplitudes of knee and hip joints also enhances stride length (Fig. 3E), visualized through progressive color shifts toward deeper red as joint angles amplify. The simulated GRF visualization reveals how FLEXOR leverages optimal dynamic spine-leg synergy to modulate ground contact dynamics. By strategically channeling the GRF vector near the center of mass (c.m.), the robot substantially improves its locomotor propulsion (movie S14 and fig. S30).

Robophysical investigation of the optimal dynamic spine-leg synergy

To further substantiate the proposed hypothesis and evaluate the efficacy of dynamic spine-leg synergy, we conducted a series of controlled robophysical experiments. In these experiments, we incorporated various hind-leg configurations to gain deeper insights into the influence of both active and passive mechanical properties of the legs on dynamic spine-leg synergy. We then executed control group experiments over a single gait cycle by varying two sets of parameters: synergy patterns between the spine and hind legs (phase lags) and hind-leg configurations. To generate distinct synergy patterns, we tested three configurations—FLEXOR-P, FLEXOR-U, and FLEXOR-F—by adjusting the hind-leg backswing phase in relation to the spinal extension phase. Motion trajectories were recorded using a motion capture system, with markers affixed to the anterior torso, posterior torso, and spine to monitor the movement of each segment. The GRFs were measured using a force plate equipped with a pressure sensor array (fig. S21). Figure 4A illustrates the experimental setup and workflow.

Fig. 4. Motion and force characterization of robophysical dynamic spine-leg synergy.

(A) Experimental setup and data processing workflow. The motion and force characterization include changes in robot posture over time, variations in the GRFs over time, and stride length. (B) Experimental results of FLEXOR with passive hind-leg configuration. Time-labeled overlaid snapshots illustrate the changes in robot posture, while normalized GRF curves are enclosed in fan-shaped envelopes to display the component of GRFs along the x and z axes. Bar charts represent stride length with error bars indicating the upper and lower bounds and the median (circles) for three experimental groups. The spine-leg synergy patterns are for phase lags of (a) −3π/10, (b) π/10, and (c) π/2. Experimental results of FLEXOR with underactuated (C) and fully actuated (D) hind-leg configuration.

In biological organisms, the neuromusculoskeletal system requires a pretensioning mechanism when initiating specific actions, especially those characterized by high dynamics (84, 85). To emulate this mechanism, we initialized the robot in a spinal extension state. The robot initiated movement from this state and executed a complete spinal flexion-extension cycle. Simultaneously, the hind legs completed a cycle of forward and backward swings. The spinal extension phase (π) was held constant, while the initiation of the backswing phase was varied from 7π/10 to 3π/2 in intervals of π/10, producing nine distinct spine-leg synergy control groups. The experimental results for the three hind-leg configurations across the nine synergy patterns are summarized in fig. S24A. FLEXOR-P fell backward in all synergy patterns, while FLEXOR-U demonstrated forward strides in three synergy patterns with phase lags of 3π/10, 2π/5, and π/2. FLEXOR-F exhibited forward strides in eight synergy patterns with phase lags ranging from −π/5 to π/2. Given the sufficient phase lag samples for the forward stride (without backward falls) in FLEXOR-F, which suggests more general synergy laws under this configuration, we propose a synergy index to quantify the quality of different phase lags in FLEXOR-F. The synergy index of the FLEXOR-F for all phase lags is shown in fig. S24B and defined as follows

$${\text{Synergy}}_{\text{index}}=\mathrm{\eta }\frac{L_{\text{stride}}\left(\mathrm{\varphi }\right)-\text{min}[L_{\text{stride}}\left(\mathrm{\varphi }\right)]}{\text{max}[L_{\text{stride}}\left(\mathrm{\varphi }\right)]-\text{min}[L_{\text{stride}}\left(\mathrm{\varphi }\right)]}$$ (1)

In this equation, $\mathrm{\varphi }$ represents the phase lag, $L_{\text{stride}}\left(\mathrm{\varphi }\right)$ represents the stride length, and $\mathrm{\eta }$ represents the success rate of completing forward strides. This synergy index is established to form a percentage-based metric for quantifying stride length variations across different spine-leg synergy patterns. It is structured such that an increased forward stride length yields a higher value. The main body of the equation is a normalized fraction representing stride length, with the left side multiplied by a success factor to characterize the stability of forward stride at this phase lag. Three representative categories were selected from all spine-leg synergy patterns for discussion. These categories consist of the hind-leg backswing phase preceding the spinal extension phase (with a phase lag of −3π/10), the hind-leg backswing phase slightly lagging the spinal extension phase (with a phase lag of π/10), and the hind-leg backswing phase substantially lagging behind the spinal extension phase (with a phase lag of π/2). Movies S6 to S8 demonstrate the motion and force characterization of FLEXOR-P, FLEXOR-U, and FLEXOR-F under distinct phase lag patterns.

The experimental results indicate that the phase lag differences between the hind-leg backswing and spinal extension phases substantially influence the movement performance of FLEXORs. When the hind-leg backswing phase preceded the spinal extension phase by a phase lag of −3π/10, all three robot configurations (FLEXOR-U, FLEXOR-P, and FLEXOR-F) fell backward [Fig. 4, B(a), C(a), and D(a)]. This occurred because the combined torques generated by the spine and the hind legs, which are transmitted to the ground, directed the GRFs along the negative x axis, leading to a negative stride length. When the phase lag increased to π/10, FLEXOR-U and FLEXOR-P continued to fall backward, while FLEXOR-F moved forward, due to the positive horizontal GRFs component that generated sufficient forward propulsion [Fig. 4, B(b), C(b), and D(b)]. This configuration also enabled FLEXOR-F to achieve the optimal stride length and a synergy index of 100%. With a larger phase lag of π/2, FLEXOR-P fell backward, while FLEXOR-U and FLEXOR-F moved forward [Fig. 4, B(c), C(c), and D(c)]. In this case, spinal extension occurred substantially before the hind-leg backswing, causing FLEXOR-F’s knee extension to miss the opportunity for effective interaction with the ground. Experimental observations revealed that when FLEXOR-F’s knee extended, the foot had already lifted off the ground, thereby hindering the generation of adequate GRFs and limiting the displacement of the c.m. In contrast, for FLEXOR-U, the energy released by the knee upon contact with the ground in this phase lag pattern actually optimized the forward component of the GRFs, due to the leg mechanism’s characteristics. However, as the knee joint could not produce motion, FLEXOR-P continued to generate displacement along the negative x axis. Compared to the π/10 phase lag, FLEXOR-F’s stride length decreased by 31%, with the synergy index dropping to 73%.

A comparative analysis of FLEXOR’s performance with a specific hind-leg configuration across various phase lag patterns reveals several key trends. First, as the phase lag shifted from negative to positive values, FLEXOR-U and FLEXOR-F transitioned from falling backward to making forward strides. Although FLEXOR-P did not consistently achieve forward strides across all phase lag patterns, its tendency to fall backward diminished as the phase lag neared positive values, and the stride length gradually approached zero. Second, regarding force dynamics, the transition from negative to positive phase lag values led to a gradual decrease in the component of GRFs along the negative x axis, accompanied by an increase along the positive x axis. Notably, after attaining its maximum synergy index, FLEXOR-F did not demonstrate a greater stride length or an increased GRFs component along the positive x axis with additional increases in phase lag.

To validate our multiparameter dynamics model of FLEXOR against these physical findings, we performed a closed-loop quantitative analysis across three representative spine-leg phase lags (−3π/10, π/10, and π/2). At the optimal phase π/10, the simulated c.m. trajectory demonstrated high fidelity to the experiment, with both reaching a peak vertical excursion of ∼0.1 m (fig. S22, A and B), and the model closely followed the experimental trends of the GRF-derived torque about the c.m. and the torque impulse during the stance phase (fig. S22, C and D). The same close agreement held at −3π/10 and π/2 (fig. S23). The root mean square error for the GRF-derived pitch torque is 0.065 Nm at π/10, 0.048 Nm at π/2, and 0.086 Nm at −3π/10, comparable across the three phases, so that the agreement does not degrade as the phase lag is varied. The corresponding Pearson correlation coefficients of the torque profiles are 0.90, 0.81, and 0.97, confirming that the overall trend is well reproduced at all three phases. Although the model predicts a slightly higher theoretical upper bound due to idealized stiffness and damping, it maintains closely matched temporal scaling and in-phase torque trajectories. Consequently, these results confirm that our analytical model accurately captures the phase-dependent spine-leg synergistic dynamics of the physical robot across varying coordination patterns rather than only its macroscopic performance.

Optimal dynamic spine-leg synergy enhances locomotion performance

To further validate the effectiveness of the optimal dynamic spine-leg synergy, we analyzed FLEXOR’s dynamic running performance with varied phase lag patterns. In the design of FLEXOR gait, we prioritized the preservation of synchronized hind-leg movement, which is a hallmark of the sengi’s distinctive half-bounding gait. The gait of the sengi combines alternating foreleg motion with synchronized hind-leg movement, where the hind legs serve as the primary source of propulsion, while the spine coordinates the overall motion. For FLEXOR, we incorporated synchronized foreleg motion to enhance stability, resulting in an adapted bounding gait that includes double-stance phases to simplify dynamic control. This approach effectively minimizes the instability associated with alternating foreleg motion, providing a robust platform for investigating spine-leg synergy in robotic locomotion. For FLEXOR-P, all phase lag synergy patterns led to backward falls; hence, we excluded this configuration from the dynamic running experiments. For FLEXOR-U, three phase lag patterns achieved forward strides (without backward falls), as noted in the previous section. We deployed these phase lag patterns in the running experiments and included a control group without spine actuation. For FLEXOR-F, eight phase lag patterns achieved forward strides. We selected three typical patterns representing the hind legs phase preceding, slightly lagging behind, and substantially lagging behind the spinal extension phase, with phase lags of −π/5, π/10, and π/2, respectively. Similarly, we added a control group without spine actuation.

Figure 5 (A to C) depicts FLEXOR-F’s running locomotion with phase lags of −π/5, π/10, and π/2 at 2.2 Hz, respectively (movie S9). In addition, the corresponding footfall patterns are depicted in fig. S28 (A to C). The maximum speed and optimal stability of the c.m. were achieved at a phase lag of π/10. We used two metrics to characterize the dynamic running stability of the robot: the range of c.m. displacement along the Z axis and the pitch angle of the spine frame. At a phase lag of π/10, Fig. 5D shows a 23.73% reduction in maximum displacement along the Z axis compared with the results for −π/5. Severe slippage observed at π/2 resulted in the maximum Z-axis fluctuation range similar to that at π/10. In addition, at π/10, the maximum pitch angle of the spine frame decreased by 61.40 and 20.34% compared with the results for −π/5 and π/2, respectively (Fig. 5E). Footfall patterns at π/10 exhibited clear cyclic consistency, contrasting with those of −π/5 and π/2, indicating a more stable gait. Compared with the speeds at −π/5 and π/2, the robot’s speed at π/10 increased by 89.75 and 237.24%, respectively (Fig. 5F). At a frequency of 3.5 Hz, the average speed at π/10 reached 0.784 m/s (7.13 BL/s) within 1 s (see movie S3 and “Analysis of maximum speed running experiment” in the Supplementary Materials). Telemetry analysis confirmed high-fidelity kinematic tracking at frequencies up to 3.5 Hz, yielding a minimal average phase lag error of only 1.09% (fig. S27). Peak motor currents (∼1.2 A) remained well below the actuators’ continuous limits without any torque saturation or clipping, ensuring the robust physical execution of the spine-leg synergy. To further understand the contribution of leg actuation to this high-speed performance, we evaluated the underactuated configuration, FLEXOR-U (fig. S29 and movie S10). A direct comparison reveals a substantial performance difference, with the fully actuated FLEXOR-F achieving a running speed 40% faster than FLEXOR-U. This contrast explicitly highlights the pivotal role of active knee joint dynamics in self-propulsion during dynamic locomotion (86).

Fig. 5. Experimental results of the dynamically running FLEXOR-F.

(A to C) A sequence of motion frame snapshots of FLEXOR executing spine-leg synergy with phase lags of −π/5, π/10, and π/2 at 2.2 Hz. (To present the frame snapshots, the robot’s wiring was removed.) (D) c.m. displacement along the Z axis over time. The shaded area represents the range of values for the three experimental groups, and the solid line indicates the average. (E) Variation in the spine pitch angle over time. The shaded area represents the range of values for the three experimental groups, and the solid line indicates the average. (F) Bar graph of dynamic running speed. Error bars represent the variance for the three experimental groups. The pentagon symbol represents the maximum average velocity (0.784 m/s) at 3.5 Hz within 1 s. (G) Bar graph of the CoT. (H) Bar graph of the time spent in the flight and stance phases.

Furthermore, to quantitatively evaluate the energetic efficiency of the identified synergy, we conducted a system-level energetic analysis across the tested phase lags. The CoT, defined as CoT = P/(mgv), was used as the primary efficiency metric, with the total electrical power (P) calculated using real-time voltage and current feedback from all actuators. The experimental results reveal a distinct correlation between macroscopic locomotor performance and system-level energy efficiency (Fig. 5G). At the optimal phase lag of π/10, the robot simultaneously maximized forward speed and minimized its CoT to 8.9, maintaining an average power of ∼10 W and a lowest recorded energy expenditure of 30 J/m (fig. S31). Deviating from this synergy caused severe energetic penalties and sharp velocity drops. These energetic alignments explicitly demonstrate that the performance improvements observed in FLEXOR are driven by effective mechanical power transfer facilitated by precise kinematic timing rather than merely relying on increased brute-force actuation to overcome system dynamics. This efficient energy transfer is directly reflected in the robot’s aerial state. As shown in Fig. 5H, the optimal phase lag of π/10 yields the longest flight phase and the shortest stance phase. By minimizing energy dissipation during ground contact and maximizing the aerial phase, this temporal profile serves as the mechanistic foundation for the robot’s high-speed and efficient forward propulsion.

In summary, despite being influenced by inherent real-world factors such as communication latency, sensor noise, system identification inaccuracies, and actuator model mismatches, the physical running tests successfully validated the outcomes and trends predicted by our simulation. Ultimately, the real-world results confirm that the optimal dynamic spine-leg synergy empowers the fully actuated FLEXOR-F to achieve substantially enhanced locomotion performance by simultaneously maximizing forward speed and energetic efficiency.

Robophysical ablation study

To explicitly isolate the contribution of the proposed dual-joint coupled spinal mechanism to overall locomotor performance, we conducted a systematic robophysical ablation study. We compared the proposed FLEXOR platform (fig. S32B) against a physical single-joint baseline configuration (fig. S32A). Both robotic variants were strictly matched in total mass, overall geometry, and maximum actuation power. To ensure a fair comparison, we evaluated each morphology at its own optimal phase. A fine-grained nine-point phase sweep confirmed that the single-joint baseline reaches its maximum stride length at π/10, matching the optimum of the dual-joint configuration. This controlled within-platform comparison ensures that any observed performance variations are directly attributable to the spinal coupling architecture itself, eliminating confounding factors such as leg compliance or motor power density.

The experimental results demonstrate that the dual-joint coupled spine is the fundamental driver of the robot’s high-speed and efficient locomotion (fig. S32, C to E). When evaluated at the operational frequency limit of 3.5 Hz, the single-joint baseline configuration achieved an average running speed of 0.572 m/s with a CoT of 6.946 (fig. S32C). In contrast, the dual-joint coupled configuration substantially outperformed the baseline, achieving a 31.6% increase in average speed. Concurrently, the dual-joint coupled spine substantially reduced the CoT by 32.2%, indicating a profound improvement in biomechanical transport efficiency. Furthermore, an analysis of the system-level energetics confirms that this speed advantage is not the result of merely injecting more raw power to overcome system dynamics. During a continuous 2-s locomotion trial at the 3.5 Hz limit, the single-joint baseline consumed an average power of 13.169 W, resulting in a total energy expenditure of 26.280 J (fig. S32, D and E). Operating within the exact same duration and frequency, the dual-joint configuration demanded lower average power (11.747 W) and less total energy (23.581 J). These comparative findings indicate that a conventional single-joint spine is insufficient to reproduce the highly efficient spine-leg synergistic propulsion observed in FLEXOR. The proposed dual-joint coupled mechanism is thus causally necessary for simultaneously maximizing forward velocity and locomotor economy.

To elucidate the underlying mechanism by which the dual-joint coupled spine outperforms the single-joint baseline, we performed controlled phase-resolved experiments at 2.2 Hz, a stable frequency at which both morphologies run reliably across phase lags. We measured the GRFs of both configurations on the same force platform and analyzed them in two layers (fig. S34). The first layer concerns coordination. Both configurations converge on the same near-zero optimal phase lag (π/10), and at this optimum, they exhibit nearly identical GRF-vector orientation trajectories during stance (fig. S34C), indicating that the two spines apply force with the same timing and direction. Consistently, the single-joint baseline ran fastest at π/10 (0.265 m/s) among the tested phases, with speeds dropping to 0.127 and 0.150 m/s at −π/5 and π/2, respectively (fig. S33 and movie S11). It thus shares the same near-zero optimum as FLEXOR, indicating that the optimal coordination is a generalizable property rather than a feature specific to the dual-joint design. The second layer concerns force magnitude, where the two morphologies diverge. At this shared optimum, the dual-joint coupled spine generates substantially larger GRFs, with the peak resultant force, averaged across the two hind legs, rising from 2.82 to 5.19 N and the resultant GRF impulse per stride from 0.58 to 0.85 N·s (fig. S34, D and E). Because the force orientation is nearly identical, this difference reflects an amplification of magnitude rather than a redirection. The amplification arises from the intrinsic force-transmission property of the closed-chain multilink architecture, in which unidirectional actuation and multilink torque amplification enhance the actuator output torque at high frequencies (fig. S15A). It is not driven by greater energy input, consistent with the lower power and energy already demonstrated in the ablation (fig. S32, D and E).

Extended dynamics framework

Our robophysical experiments provide foundational evidence for the efficacy of spine-leg synergy across both dual-joint and single-joint configurations. Building upon these findings, we developed an extended dynamics framework to further generalize this principle across a broader range of physical and morphological constraints (for detailed mathematical modeling, see “Modeling of the extended dynamics framework” in Materials and Methods).

Through systematic simulations across diverse spine and leg morphologies, we observed that maximum forward velocity typically converges around a near-zero phase lag. To evaluate the robustness of this pattern against specific modeling assumptions, we incorporated asymmetric noncosine trajectories (fig. S38) and multisegment leg morphologies (fig. S39). These variations yielded consistently similar optimal phase lags, suggesting that the synergy relies on global kinematic coordination rather than simplified mathematical rules or specific morphological representations. Furthermore, comprehensive sensitivity analyses regarding spinal bending amplitude (fig. S40), overall body size (fig. S41), and actuation torque limits (fig. S42) revealed that while varying these physical parameters naturally affects the absolute forward speed, the optimal phase lag consistently remains anchored within a near-zero band. When considered alongside our independent robophysical evaluations, these multidimensional analyses suggest that the highly efficient spine-leg synergy may function as a generalizable physical phenomenon. This indicates its potential relevance across diverse scales and architectures, beyond the specific constraints of individual platform designs (movies S12 and S13).

DISCUSSION

To address the three key challenges outlined in the fourth paragraph of the introduction, FLEXOR, as a robotic system, revolves around three primary aspects: nimble embodied design, optimized spine-leg synergistic movements, and elucidated general underlying principles of synergistic locomotion. From these results, we distill a shared principle of spine-leg synergy. For quadrupedal running with active spinal flexion-extension, a near-zero phase lag between hind-leg backswing and spinal extension consistently maximizes forward velocity and energetic efficiency by aligning the GRF vector. This principle concerns the timing and direction of force application alone and is shared across spinal morphologies, as confirmed by the convergence of the single-joint and dual-joint configurations on the same optimum (fig. S33) and by the consistency of the optimal phase lag across a wide range of simulated conditions spanning morphology, scale, and actuation (figs. S37 to S42). The closed-chain dual-joint architecture then acts on top of this shared principle, amplifying the magnitude of the force achievable at the optimal timing.

Figure 6 shows the maximum speeds (in body lengths per second) of robotic quadrupeds versus the body mass. For contextualization, tables S2 and S3 provide a comparison of FLEXOR-F’s running speed against other platforms, with specific notes on experimental protocols and morphological differences (e.g., tail exclusion) to facilitate an objective assessment. Thanks to its structural advantages and optimal dynamic spine-leg synergy, FLEXOR demonstrates the highest speed among quadruped robots with a flexible spine. Although the FLEXOR’s speed lags far behind that of the sengi, we attribute this discrepancy to the fact that the mass-specific peak torque (defined as the maximum torque normalized by the mass of the actuator and its transmission) cannot match that of real biological muscles (87). On the one hand, FLEXOR may deepen our understanding of how animals use dynamic spine-leg synergy to optimize locomotor propulsion. On the other hand, FLEXOR sheds light on the path for developing more agile embodied systems, especially those integrating interjoint synergy for dynamic locomotion, from terrestrial legged locomotion to aquatic swimming and aerial flight.

Fig. 6. Statistics of the maximum velocity and body mass of quadruped robots with rigid and flexible spines on logarithmic scales.

“ns” represents the FLEXOR without spine actuation.

Many studies in the field of bioinspired robots focus on mimicking the morphology of biological organisms (88–91), relegating the imitation of motion patterns solely to the control system rather than integrating it into the robot’s physical design. However, this approach struggles to use the physical intelligence embedded within the body to push the boundaries of motor performance. In addition, achieving animal-like motion solely through control systems typically demands extensive manual tuning and resource consumption, hindering the development of efficient embodied intelligence (10). Prior research demonstrates that natural terrestrial locomotion combines intrinsic mechanical movements of the musculoskeletal system with neural regulation (64, 92). It has become the consensus to use the control systems in robots to emulate the functions of the central nervous system in animals (87), which underscores the importance of replicating animal locomotion paradigms by integrating the inherent mechanical movements of the musculoskeletal system into a robot’s physical design. We mapped the biological sengi onto FLEXOR and abstracted a design principle, enabling FLEXOR to not only mimic the key morphological characteristics but also incorporate the inherent dynamic characteristics of spinal flexion-extension. This enables FLEXOR to accurately model the locomotor performance and body kinematics of the sengi and provides a crucial foundation for exploring the essence of dynamic spine-leg synergy in biological systems.

The developed FLEXOR platform features a dual-joint coupled spinal mechanism capable of executing the rhythmic, rapid spinal movements of real sengis. This synergistic interjoint motion does not depend on a central controller but emerges from a morphologically optimized mechanical structure that enables decentralized, distributed motion control, which can be regarded as embodiment (10) of the rhythmic spinal coordinated movement in animal locomotion. Although prior studies have successfully used spinal functionalities to enhance quadrupedal locomotion performance, these spines are either directly driven by the reciprocating rotational motion of actuators, e.g., a spine with one revolute joint that entirely delegates locomotor propulsion to control systems (35, 36) or relies on mechanisms such as cable-driven (41–43) or fluid-driven (21, 47) systems, which pose challenges in deploying precise spine-leg synergy patterns. Our proposed spinal mechanism transforms unidirectional rotation into swift spinal movements. This innovative design offers several advantages. First, it simplifies spine control by consolidating spinal flexion-extension movements under a unidirectional motion, setting it apart from other multijoint spine controls (38, 93). Second, it amplifies torque, allowing motors with low torque to be embedded into the mechanism, thereby enabling a compact size for the robot. Third, the spinal mechanism functions as a precise transmission device capable of simulating various synergistic patterns with the legs.

Compared to the independently actuated serial dual-joint spine, the proposed dual-joint coupled spinal mechanism needs simpler and smoother actuator inputs to achieve the same spinal flexion-extension, avoiding frequent acceleration and deceleration caused by actuator reversals. Furthermore, the torque required by the actuators in our spinal mechanism is only half that of the torque required by the independently actuated serial pitch-joint spine. Because our spine actuation mechanism eliminates the need for frequent motor reversals, backlash is consistently compensated by unidirectional force application. This avoids errors induced by repeated directional switching, enabling more precise and stable position control. This effective and compact spinal mechanism can be regarded as a “quadrupedal motion accelerator” for FLEXOR.

The spine transmits torque to the hind legs, which interact with the ground to generate locomotor propulsion (83). Anatomical evidence indicates that the hind legs are crucial for generating quadrupedal propulsion (71). While the fully active leg configuration introduces a higher total power input compared to the underactuated case (primarily due to the additional active knee joint), our energetic analysis demonstrates that power alone does not guarantee speed. The observation of a minimized CoT exclusively at the optimal phase lag confirms that the performance gain stems from efficient kinematic coupling. In high-speed locomotion, the active-passive characteristics of the leg must be synergistically coordinated with spinal motion to maximize the effective transmission of mechanical work rather than the system merely overcoming dynamics through increased actuation authority.

Using FLEXOR as the robotic agent, our comparative study, integrating numerical and robophysical experiments, reveals that the dynamic spine-leg synergy optimizes quadrupedal locomotor propulsion and energetic efficiency through body-environment interactions. Our extended dynamics framework for quadruped robots serves as a valuable tool for understanding how this synergy principle generalizes across diverse spine-leg morphologies and physical scales. By identifying and implementing the control regimes that faithfully replicate the kinematics of high-speed running observed in sengis, we can obtain insights into the intrinsic dynamic spine-leg synergy patterns required to achieve efficient locomotor propulsion. Our results demonstrate that this mechanically optimized dynamic spine-leg synergy plays a crucial role in fast and stable locomotion. Although FLEXOR’s design is inspired by small-scale quadruped mammals such as the sengi, the synergistic locomotion principles it represents are fundamental biomechanical strategies present to varying degrees among many quadruped animals (94–96). Hence, understanding these synergy patterns is essential for developing more advanced embodied intelligence in legged robots.

The embodied system represented by FLEXOR may enable more efficient execution of robot deployment tasks, especially in resource-constrained scenarios such as search and rescue (97) and environmental monitoring (98). In this study, we exclusively examined the impact of dynamic spine-leg synergy to augment FLEXOR’s locomotion capabilities while neglecting spinal elastic elongation-contraction, which is a common mechanism in animals (99). Research on quadruped mammals has revealed the involvement of the tail and head in maintaining postural stability during fast running (74, 100). Therefore, integrating head, spine, tail, and elastic elongation-contraction mechanisms to expand the whole-body robophysical model for exploring their synergistic patterns holds great potential for pushing the limits of quadrupedal locomotor propulsion.

MATERIALS AND METHODS

Model for extracting the spinal KMJs

We captured frame sequences depicting the cursorial locomotion of the sengi. Considering that quadrupedal mammals exhibit sagittal-plane spinal bending motion with a unified curvature variation direction during running, unlike the undulating lateral bending often seen in reptiles (101–103) and their bioinspired robotic counterparts (104, 105), our extracted KMJ model of the spine uses variable-curvature arcs to preprocess spinal motion. This approach enables broader applicability to biological running patterns characterized by sagittal bending, distinguishing it from previous methods (59, 106). Specifically, each frame of locomotion was analyzed to extract the current spinal posture in the sagittal plane, and circular arcs were fitted to represent the spinal posture. Subsequently, the equations of these arcs were derived. When translating these arcs into linkage systems, the primary challenge lies in determining the number and distribution of joints required. Although a higher number of joints theoretically allows for a more precise fit of the circular arcs, practical deployment requires careful consideration. Increasing the number of joints complicates the mechanism, leading to dimensional growth in the robot and substantially reduced reliability. Hence, selecting the optimal number of spinal joints needs to balance these trade-offs. We identified the number of joints that not only offers an accurate approximation for biological spine but also ensures ease of engineering implementation.

Initially, our approach involved fitting the spinal arcs using a two-link mechanism connected by a single revolute joint. First, the extracted circular arcs underwent preprocessing, involving translation and rotation of all equivalent arcs to align them in a common coordinate system. The first and last end points of each arc were aligned with the x axis, and the chord connecting these end points was bisected vertically by the y axis (fig. S8A). Subsequently, we selected a random set of link lengths (L_j1, L_j2), where L_j1, L_j2 ∈ (0, L), with L representing the average length of all spinal arcs. The first and last end points of the two-link mechanism were aligned along the x axis, with their connecting line bisected by the y axis. The process of fitting the spinal arcs using the two-link mechanism yielded a deviation, defined as the sum of the shortest distances from the end points of the two-link mechanism and the revolute joint to the arc. To gauge the accuracy of the linkage system in fitting the arcs, we defined the total fitting deviation ratio of all frame sequences as ${\text{Dev}}_{\text{ratio}}$

$${\text{Dev}}_{\text{ratio}}=\frac{\text{min}{\displaystyle \sum _{i=1}^N}{\displaystyle \sum _{k=1}^{n+1}}{d}_{\mathit{ik}}}{N(n+1)L}$$ (2)

Here, $d_{\mathit{ik}}$ represents the minimum distance of the end points in the linkage system and the intermediate revolute joint to the arc. n represents the number of links, and N represents the number of motion frames. All link length sets of the linkage system were traversed, and every corresponding ${\text{Dev}}_{\text{ratio}}$ was calculated.

To streamline the engineering implementation and minimize computation during fitting arcs, we introduced a constraint requiring equality between the first and end links in equivalent three-link configuration (with two revolute joints). Similarly to the preprocessing for the two-link case, we positioned the end points of the three-link along the x axis. Randomly selecting a set of link lengths (L_j1, L_j2, L_j3 = L_j1), we calculated the deviation resulting from fitting the arc, following the same procedure as in the two-link configuration.

Figure S8B indicates a lower minimum value of ${\text{Dev}}_{\text{ratio}}$ for the three-link configuration than for the two-link configuration. It is inferred that a continuous increase in the number of spinal joints leads to a gradual decrease in ${\text{Dev}}_{\text{ratio}}$. However, increasing the number of spinal joints poses engineering challenges and complicates robot control. To integrate whole-body joint motors within the actual size of the sengi (150 mm) and account for motor size constraints to meet the driving torque, the number of spinal motors should not exceed three. Most existing spine designs are monoarticulated, simplifying control but compromising biomimetic similarity. Given these conclusions and to more closely replicate the characteristics of the sengi’s spinal motion, we modeled the spine as a dual-joint, three-link system, equivalent to the thoracic cavity, lumbar vertebrae, and pelvic cavity. The link-length ratio was defined as 1:3:1. We note that this rod-length optimization is a design-stage step rather than a causal component of the locomotor mechanism. It determines how faithfully the robot reproduces the sengi’s spinal kinematics but does not itself generate or transmit force during running and is therefore not the source of the observed speed increase or CoT reduction.

Modeling of the extended dynamics framework

To extend our findings beyond the FLEXOR platform, we developed an extended dynamics framework for quadruped robots with multijoint spines to investigate generalized spine-leg synergies across diverse vertebrate morphologies. This framework adopts a planar motion assumption, focusing on the sagittal plane, where the dominant kinetic energy transfer and propulsion occur during high-speed bounding and galloping in legged vertebrates (18). We hypothesized that the spinal postures of quadrupeds during dynamic locomotion, despite their varying spinal morphologies, can be effectively represented by arcs of variable curvature (Fig. 7A). By fitting these arcs with a multijoint robotic spine and minimizing the fitting deviation (Eq. 2), we determined the joint angles between the spinal linkages. While the specific 1:3:1 spinal link-length ratio of the physical FLEXOR robot was rigorously derived by optimizing the fit to the backbone deformation trajectories of a running sengi, this extended framework is designed to verify the generality of the proposed methods. We categorized distinct spinal morphologies based on the number of spinal linkages. For even-numbered linkages, these can be perceived as a special case of odd-numbered linkage segments, where the linkage length of the middle segment is $l_{\mathrm{\xi }}=0$. For the three-segment spine (n = 3), we retained the biologically derived 1:3:1 link-length ratio to match the physical FLEXOR platform. However, to prove that the optimal synergy generalizes beyond this specific morphology, we applied Monte Carlo sampling to generate randomized linkage proportions for all other spinal configurations (n ≠ 3). To ensure uniformity across these randomized trials, we stipulated that each randomly selected link’s length is a proportion of the total spinal length, denoted as $p_i=\frac{C_i}{{\displaystyle \sum _{k=0}^{n-1}}{C}_k}(i=0,1,\dots ,n-1)$, where $C_i\in [n-2,n+2]$ and $n=2\mathrm{\xi }+1,\mathrm{\xi }\in {\mathrm{\mathbb{N}}}^{\ast }$ represents the number of spinal linkages (Fig. 7B).

Fig. 7. Extended dynamics for quadrupedal systems with varied spinal and leg morphologies.

(A) Modeling spinal postures of fast-running vertebrates using variable curvature arcs. (B) Spine characterization. Variable curvature arcs map to a multijoint linkage system to determine joint angles. Linkage lengths are randomly assigned within the range $[l_{i,\text{min}},l_{i,\text{max}}]$, where n is the linkage count. (C) Leg characterization. Three typical leg configurations (revolute-prismatic, dual-revolute, and triple-revolute) with swing bias are introduced for biological alignment and synergy validation. (D) Unified joint trajectories and synergy definition. Joint trajectories follow cosine or noncosine asymmetric profiles. $A_{\mathrm{\theta }}$ denotes the joint angle amplitude, and ${\mathrm{\eta }}_{\text{ext}}$ represents the duty cycle of the extension phase. Dynamic synergy is defined by the phase lag between hind-leg backswing and spinal extension. (E) Extended dynamics framework for evaluating locomotion performance and synergy robustness across diverse physical constraints. (F) Velocity versus phase lag across distinct spinal morphologies. Solid lines indicate means; shaded areas represent the range across six experimental groups. (G) Dynamic stability versus phase lag. The top panel shows the stability index, and the bottom panel displays the c.m. Z-axis displacement (blue; left axis) and spinal pitch angle (orange; right axis). Solid lines indicate means; box plots summarize the six experimental groups.

For the legs, we selected three typical configurations commonly used in the field of legged robotics to validate the robustness of the synergy principles: a revolute-prismatic joint model, a dual-revolute joint (two-segment) configuration, and a triple-revolute joint (three-segment) configuration (Fig. 7C). Using the revolute-prismatic joint model as an example, to ensure the foot-end trajectory aligns with biological locomotion characteristics, we introduced a swing bias $B_{{\mathrm{\theta }}_{\text{HL}}}=\frac{\mathrm{\pi }}{2}+{\mathrm{\lambda }}_{\text{bias}}\cdot A_{{\mathrm{\theta }}_{\text{HL}}}$, representing the angle between the bisector of the maximum hip joint angle and the perpendicular line of the spinal linkage connected with hind legs. When the leg extension length $\mathrm{\Delta }{l}_{\text{HL}}$ is zero, the hind leg is fully contracted.

Figure 7D illustrates the motion trajectories of the spine and leg joints. By default, these trajectories follow a cosine function over time with adjustable angular amplitudes; however, by modifying the proportion of the extension phase, they can also be configured as noncosine, asymmetric trajectories. The variable $\mathrm{\varphi }={\mathrm{\phi }}_K-{\mathrm{\phi }}_L$ is used to characterize the phase lag in the interaction between spine and leg movements, where ${\mathrm{\phi }}_K$ and ${\mathrm{\phi }}_L$ represent the initial phases of the cosine functions for the spine and hind leg movements, respectively. We propose an extended dynamics model (Fig. 7E) of quadruped robots with a multijoint spine, formulated in the form of Lagrangian dynamics equations

$$\mathit{M}\left(\mathit{q}\right)\ddot{\mathit{q}}+\mathit{C}(\mathit{q},\dot{\mathit{q}})\dot{\mathit{q}}+\mathit{G}\left(\mathit{q}\right)=\mathit{Bu}+{\mathit{J}}^{\mathrm{T}}{\mathit{F}}_{\text{GR}}$$ (3)

In Eq. 3, $\mathit{q}={[x_{{\mathrm{s}}_{\mathrm{\xi }0}},z_{{\mathrm{s}}_{\mathrm{\xi }0}},{\mathrm{\theta }}_{{\mathrm{s}}_{\mathrm{\xi }}},{\mathrm{\theta }}_{j_1},{\mathrm{\theta }}_{j_2},\dots ,{\mathrm{\theta }}_{j_{n-1}},{\mathrm{\theta }}_{\text{HL}},{\mathrm{\theta }}_{\text{FL}},l_{\text{HL}}]}^T$ is the generalized coordinates, and ${[{\mathrm{\theta }}_{j_1},{\mathrm{\theta }}_{j_2},\dots ,{\mathrm{\theta }}_{j_{n-1}}]}^T$ can be determined by the curvature of the spine $K$ and the number of spinal segments $n$, denoted as ${\mathbf{\Theta }}_j(K,n)$. $\mathit{M}\left(\mathit{q}\right)$ is defined as an (n + 5) × (n + 5) mass matrix, while $\mathit{C}(\mathit{q},\dot{\mathit{q}})$ represents the Coriolis force matrix, and $\mathit{G}\left(\mathit{q}\right)$ denotes the gravitational vector. Furthermore, the matrices $\mathit{B}$ and ${\mathit{J}}^{\mathrm{T}}$ serve to map the joint drive forces and torques $\mathit{u}$ and the ground reaction forces ${\mathit{F}}_{\text{GR}}$ into the generalized coordinate space.

On the basis of the rotary-prismatic joint leg configuration, we evaluated the locomotion performance of 4961 distinct spine-leg synergy patterns across various spinal morphologies. By inputting the spinal and leg motion parameters into the extended dynamics model, we calculated the robot’s forward velocity, longitudinal displacement $\mathrm{\Delta }{z}_{{\mathrm{s}}_{\mathrm{\xi }0}}$ of c.m., and pitch angle ${\mathrm{\theta }}_{{\mathrm{s}}_{\mathrm{\xi }}}$ of the centroid coordinate system. Figure 7 (F and G) illustrates speed and dynamic stability across different spinal morphologies as the phase lag varies. Here, dynamic stability is determined by the maximum longitudinal displacement and the maximum pitch angle, represented by a stability index $S=a\cdot e^{\frac{\text{max}\left(\mathrm{\Delta }{z}_{{\mathrm{s}}_{\mathrm{\xi }0}}\right)}{\text{BL}}}+b\cdot e^{\frac{\text{max}\left({\mathrm{\theta }}_{{\mathrm{s}}_{\mathrm{\xi }}}\right)}{\mathrm{\pi }}}$ (Fig. 7G). To ensure an unbiased evaluation metric, the weights a and b were derived rather than empirically assigned. The weights a and b are 1 and 0.78, respectively (the complete theoretical derivation is detailed in the Supplementary Materials). A smaller value of $S$ corresponds to increased stability in the robot’s locomotion. Considering the optimal speed range, we focused exclusively on the stability index where the phase lag is between −π/3 and π/3, indicated by the orange region in Fig. 7G. In proximity to a phase lag of zero, termed optimal region (Fig. 7G and figs. S35 to S37), the dynamic systems of various spinal morphologies achieve optimal speed and dynamic stability.

FLEXOR prototype design and fabrication

To mimic the size and weight of real sengis, FLEXOR is engineered with specific dimensions: a body length of 110 mm, a leg length of 60 mm, and a mass of 338 g. The robot’s overall structure is symmetric, with detailed parameters outlined in table S1. Most FLEXOR’s components, including the front and hind torso, spine, and legs, are fabricated in nylon using 3D printing. FLEXOR is outfitted with a total of six (seven for FLEXOR-F) servomotors, of which the spine is redundantly driven by two servomotors (DYNAMIXEL, XL330–077-T). Furthermore, leg movements are actuated by four (five for FLEXOR-F) servomotors (DYNAMIXEL, XC-M181-T). For the knee joints in different hind-leg configurations, FLEXOR-F uses a single motor to control the movement of both hind-leg knee joints, but this motor does not work in FLEXOR-U and FLEXOR-P. The Supplementary Materials present further details of the design and fabrication for the spine and legs.

Actuators and low-level actuator control setup

Two servomotors (DYNAMIXEL, XL330–077-T) drive the spinal mechanism of FLEXOR and are positioned at the center of the spine frame. Since both servomotors redundantly drive the spine, synchronization is ensured by aligning them with the same positional reference and maintaining mirrored symmetry in their angles. At the shoulder joints of the forelegs, the DYNAMIXEL XC-M181-T servomotors directly actuate the humerus, while at the hip joints of the hind legs, they directly actuate the femurs. On FLEXOR-F, the DYNAMIXEL XC-M181-T servomotor at the hind torso controls the flexion and extension of both knees by rotating the winding wheel. Actuators are operated in position mode at a frequency of 250 Hz through the OpenRB-150 controller. This controller interfaces with a PC via USB communication, with the PC sending specific motor commands to the OpenRB-150 through an Arduino interface. All motor control commands are transmitted as feedforward signals.

The multilink spinal mechanism converts unidirectional rotation into periodic oscillations of the spine in the sagittal plane. Given limitations in computational resources, directly mapping the desired spinal output angle to the servomotor’s input angle via inverse kinematics poses a challenge. Consequently, a Fourier fitting method is used to approximate the input-output relationship of the spinal mechanism, taking into account the periodic nature of robotic locomotion. Ensuring a sufficiently high level of fitting accuracy, the inverse of the fitting function is derived, enabling efficient calculation of the desired spinal output angle. This process streamlines the generation of motor angle signals suitable for actuators.

We denote the servo angle as ${\mathrm{\theta }}_{\text{servo}}$ and the spinal output angle as ${\mathrm{\psi }}_3$. Given the rigidity of each component in the spine structure, the mapping from servo angle to spinal output angle is expressed as ${\mathrm{\psi }}_3=\mathrm{\mu }\left({\mathrm{\theta }}_{\text{servo}}\right)$. Therefore, within the range of 0° to 360°, 720 equidistant points are sampled, resulting in $[{\mathrm{\theta }}_{\text{servo}}^1,{\mathrm{\theta }}_{\text{servo}}^2,{\mathrm{\theta }}_{\text{servo}}^3,\dots ,{\mathrm{\theta }}_{\text{servo}}^{720}]$, corresponding to spinal output angles $[{\mathrm{\psi }}_3^1,{\mathrm{\psi }}_3^2,{\mathrm{\psi }}_3^3,\dots ,{\mathrm{\psi }}_3^{720}]$. Because this function is periodic, a Fourier fit is used, resulting in ${\mathrm{\psi }}_3=a_0+a_1\text{cos}\left({\mathrm{\omega }\mathrm{\theta }}_{\text{servo}}\right)+b_1\text{sin}\left({\mathrm{\omega }\mathrm{\theta }}_{\text{servo}}\right)$, where $a_0=180$, $a_1=-29.48$, $b_1=56.32$, and $\mathrm{\omega }=0.01745$. As the fitting result is in trigonometric form, the inverse can be obtained using auxiliary angle formulas, yielding ${\mathrm{\theta }}_{\text{servo}}={\mathrm{\mu }}^{-1}\left({\mathrm{\psi }}_3\right)$.

Supplementary Materials

The PDF file includes:

Supplementary Text

Figs. S1 to S43

Tables S1 to S3

Legends for movies S1 to S14

Other Supplementary Material for this manuscript includes the following:

Movies S1 to S14