In nature, many quadruped animals exhibit exceptional performance in high-speed locomotion, such as the cheetah, which is one of the fastest land runners. From a bionic perspective, designing agile and high-speed quadruped robots has been a hot topic in mobile robotics research. So far, researchers have developed representative high-performance quadruped robots like StarlETH, HyQ, and MIT Cheetah, making significant progress in areas such as compliant actuation, rough terrain adaptation, flexible control, and high-speed motion. However, most bionic quadruped robots adopt a “single trunk with four limbs” structure, ignoring the spinal structure commonly found in quadruped animals, which plays a crucial role in high-speed movements. For instance, Hildebrand compared the running postures of cheetahs and horses, suggesting that the large flexion and extension of the cheetah’s spine during running contribute to its longer stride length relative to body length compared to horses. Additionally, Gray found that when a cheetah’s leg strength is insufficient, back muscles contract to enhance leg power, meeting energy demands. Given the importance of the spine, researchers worldwide have recently explored spine-equipped quadruped robots.
In terms of structural design, spines are categorized into localized and holistic types. Localized spines primarily consist of discontinuous rigid rotational joints. For example, Kim et al. used a single rotational joint as the spine and designed the front-to-back body length ratio at 0.68:1 based on animal models to better mimic the effects of spinal motion. Wang et al. employed two rotational joints to design a 2-DOF spine-equipped quadruped robot, analyzing the coordinated motion of the spine and legs within the dynamics framework, revealing that the spine can increase speed by extending stride length and flight phase duration. Holistic spines mainly comprise continuous flexible units. In the first-generation MIT Cheetah prototype, a differentially driven spine made of elastomeric materials was added between the trunk and hind legs, achieving a running speed of 22 km/h through cable coupling of spine flexion and hind leg motion. Gao et al. proposed a 3-DOF tensegrity structure as a flexible multi-joint spine for quadruped robots, consisting of discontinuous compressive elements and continuous tensile elements, allowing free support and enhancing robot flexibility and adaptability. Qian et al. designed a variable-stiffness continuous bionic spine using cables and springs, leveraging underactuation for large flexible swings to meet the needs of quadruped robots adapting to complex terrains.
In motion control, compared to traditional quadruped robots, the presence of a spine implies more degrees of freedom and more complex motion patterns, posing greater challenges for high-speed control. Li et al. pre-defined the effects of spinal motion on a single rigid body and integrated them into model predictive control (MPC) to solve for ground reaction forces (GRF). Zhang et al. proposed a multi-model hierarchical control method based on task decomposition and neural system mechanisms, incorporating central pattern generators, virtual models, and leg-spine coordination models to achieve coordinated control under different gaits. Cai Changrong built upon the spring-loaded inverted pendulum (SLIP) model, constructing triangular motion models and height feedforward control models, and introduced the spine’s impact on stride length to facilitate the transition from SLIP to spine-equipped quadruped robots.
In summary, current localized spine designs are relatively simple, while holistic spines are difficult to control. Moreover, control methods for spine-equipped quadruped robots are primarily migrated from traditional quadruped robots, but increased DOF and motion complexity present new challenges for model integration. Additionally, most analyses of spinal effects are confined to comparative experiments without in-depth theoretical discussion. Inspired by bounding cheetahs, we design a cable-driven bionic double-joint spine and three-segment flexible legs in Section 1, with the overall structure conforming to cheetah physiological characteristics and motion states approximating cheetah motion mechanisms. In Section 2, we establish a time-based finite state machine, construct a single rigid body dynamics model, and propose a control method for the bounding gait. In Section 3, based on geometric relationships and algorithmic derivations, we analyze the contribution of spinal participation to bounding motion. In Section 4, comparative simulation results verify the effectiveness of the bounding control and the positive contribution of the spine.
Structural Design of the Cable-Driven Quadruped Robot with Double-Joint Spine
The cheetah’s spinal motion during high-speed locomotion involves multiple closely connected vertebrae, divided from head to tail into cervical, thoracic, lumbar, sacral, and caudal parts. Since adjacent spinal joints exhibit similar motion and cervical and caudal movements are limited, the entire spine is simplified into three parts connected by two spinal joints with pitch DOF, forming a three-segment trunk. The legs are characterized by a three-segment design. Based on bionic principles, the overall structure of the bionic quadruped robot is designed as follows. This robot consists of one double-joint spine, four legs, two spinal joint motors, four hip joint motors, four ankle joint motors, and several pulley sets. The spinal joint motors are mounted on the bottom surface of the waist, driving the spinal joints distally via cables; the hip joint motors are mounted on the inner side of the hip joints for direct drive; and the ankle joint motors are mounted on the top surface of the waist, driving the ankle joints distally via cables.
Referring to the elbow joint design principle of the robot LIMS (A Low Inertia Manipulator with high Stiffness and Strength), we design a double-joint bionic spine. The structure mainly comprises the chest, waist, sacrum, and several pulleys. From existing research, double-joint spine models exhibit excellent performance in both stability and speed. To better simulate the elastic characteristics of biological spines, the front and rear spinal joints are driven by two separate motors, employing the same cable arrangement scheme and working principle. The cable wraps around the motor shaft and splits into two branches connected to fixed and moving pulleys, forming a closed loop. Through positive and negative rotation of the motor shaft, the chest and sacrum pitch up and down relative to the waist. This mechanism utilizes pulley principles to achieve torque amplification, motion reduction, and tension amplification effects.
For torque amplification, assuming the input torque from the motor to the cable is $T_{in}$, the output torque acting on the chest or sacrum is:
$$T_{out} = n T_{in},$$
where $n$ is the number of cable wraps around the pulleys, ignoring friction.
For motion reduction, the output linear displacement $\Delta x_{out}$ of the spinal joint is less than the input linear displacement $\Delta x_{in}$ of the motor shaft:
$$\Delta x_{out} = \Delta x_{in} / n.$$
For tension amplification, from the above equations, the equivalent stiffness of the cable at the output end can be expressed as:
$$K_{out} = \frac{T_{out}}{\Delta x_{out}} = \frac{n T_{in}}{\Delta x_{in} / n} = n^2 \frac{T_{in}}{\Delta x_{in}} = n^2 K,$$
where $K$ represents the overall elastic coefficient of the cable, satisfying $K = T_{in} / \Delta x_{in}$.
In terms of kinematics, based on the geometric relationships, the lengths of the inner and outer cables between pulley sets, $l_1$ and $l_2$, relate to the spinal joint rotation angle change $\Delta \theta$ as:
$$l_1 = n \theta_1 r_{sp} = n (\theta_{o1} – \Delta \theta) r_{sp},$$
$$l_2 = n \theta_2 r_{sp} = n (\theta_{o2} + \Delta \theta) r_{sp},$$
where $\theta_1$, $\theta_2$, $r_{sp}$, $\Delta \theta$ are as labeled in the figure, and $\theta_{o1}$ and $\theta_{o2}$ are the initial angles of the inner and outer pulley set angles.
From these, the relationship between the driving motor rotation angle $\Delta \theta_{ac}$ and the spinal joint motion $\Delta \theta$ is:
$$\Delta \theta_{ac} = -\Delta l_1 / r_{ac} = \Delta l_2 / r_{ac} = n \Delta \theta \cdot r_{sp} / r_{ac},$$
where $r_{ac}$ is the radius of the spinal joint motor shaft, satisfying $r_{ac} < r_{sp}$.
From the equations, the inner and outer cables move antisymmetrically, meaning the cable arrangement length remains unchanged during spinal joint rotation, avoiding slack and eliminating the need for additional tensioning devices.
For leg design, based on the Advanced Spring Loaded Pantograph (ASLP) structure, we propose a more lightweight and stable Modified Advanced Spring Loaded Pantograph (MASLP) structure. In the MASLP structure, a knee spring is installed at the knee joint, with one end fixed to the leg and the other end pulled by a cable crossing the knee and ankle joints, replacing the diagonal spring in the ASLP structure. This ensures stability while making the structure more compact. The MASLP structure includes two elastic elements: a parallel spring and a knee spring, for vibration absorption and energy storage. The parallel spring is generally slack and only tensioned when the foot lands with significant impact, providing buffering. The knee spring is pre-loaded, supporting stable standing; when the leg folds, the knee spring stores energy via cable pull, and when the leg extends, it releases energy.
The hip and ankle joints are actuated. The hip joint controls the overall rotation of the leg module, and the ankle joint controls leg folding to generate ground clearance. The ankle joint cable drive scheme uses a cam and cable to separate the drive motor from the joint, reducing leg mass and lowering the drive load on the spinal and hip joints. The ankle joint angle $\phi_a$ indicates the leg folding degree; a smaller $\phi_a$ means higher folding. Based on geometric constraints, $\phi_a$ is expressed as:
$$\phi_a = \arccos \left( \frac{l_1^2 + l_2^2 – l_3^2}{2 l_1 l_2} \right),$$
where $l_1 = HN$, $l_2 = NA$, $l_3 = AH$. Differentiating both sides:
$$d\phi_a = \frac{2 l_3}{-l_1^4 – (l_2^2 – l_3^2)^2 + 2 l_1^2 (l_2^2 + l_3^2)} dl_3.$$
Assuming the cam radius is $r_c$, the relationship between the knee motor output angle $\Delta \alpha$ and leg folding degree $\Delta \phi_a$ is derived as:
$$\Delta \phi_a = \frac{2 l_3 r_c}{-l_1^4 – (l_2^2 – l_3^2)^2 + 2 l_1^2 (l_2^2 + l_3^2)} \Delta \alpha.$$
Hybrid Control Method for the Bounding Gait
The bounding gait is a common high-speed gait for quadruped animals. Based on cheetah bounding motion, the bounding cycle of the quadruped robot is divided into four phases according to leg ground contact: front leg support phase (S1), first flight phase (S2), hind leg support phase (S3), and second flight phase (S4). The left and right legs touch down or lift off simultaneously in each phase, supporting or swinging together. The spine is in a contracted state during S1 and an extended state during S3.
Based on the motion sequence, we propose a time-based finite state machine to coordinate control models under different states, executing specific control methods at specified times. The switching mechanism is as follows:
| Phase | Trigger Condition | Spine | Front Legs | Hind Legs |
|---|---|---|---|---|
| S1 | $t = 0 + \xi T$ | Contracted | Support | Swing |
| S2 | $t = T_{st} + \xi T$ | Hold | Swing | Swing |
| S3 | $t = (T_{st} + T_{sw})/2 + \xi T$ | Extended | Swing | Support |
| S4 | $t = (T_{st} + T_{sw})/2 + T_{st} + \xi T$ | Hold | Swing | Swing |
where $t$ is the current system time, $T_{st}$ is the single leg support time, $T_{sw}$ is the single leg swing time, $T$ is the cycle time, and $\xi$ is a positive integer.
Given the design, the mass of the bionic quadruped robot is concentrated at the waist, with the chest, sacrum, and legs being relatively light, so limb inertial effects are neglected in analysis. Since the bounding gait only considers the pitch DOF around the y-axis, the double-joint spine quadruped robot is simplified to a single rigid body 2D model. The state variables are represented as $\Omega = [p_c, \phi, \dot{p}_c, {}^B \omega] \in \mathbb{R}^{12}$, where $p_c \in \mathbb{R}^3$ is the center of mass position, $\phi$ is the trunk RPY orientation angle, $\dot{p}_c \in \mathbb{R}^3$ is the CoM velocity, and ${}^B \omega \in \mathbb{R}^3$ is the angular velocity in the body frame $\{B\}$. Variables without superscripts are in the world frame $\{W\}$.
The dynamics equation of this single rigid body model is described as:
$$\dot{\Omega} =
\begin{bmatrix}
\dot{p}_c \\
\dot{\phi} \\
\ddot{p}_c \\
{}^B \dot{\omega}
\end{bmatrix}
=
\begin{bmatrix}
\dot{p}_c \\
R {}^B \omega \\
\frac{1}{m} \sum_{i=1}^{n} F_i – g \\
{}^B I^{-1} (R^T \sum_{i=1}^{n} r_i \times F_i – [{}^B \omega]_{\times} {}^B I {}^B \omega)
\end{bmatrix},$$
where $R \in SO(3)$ is the rotation matrix of $\{B\}$ relative to $\{W\}$, $\ddot{p}_c \in \mathbb{R}^3$ is the CoM acceleration, $F_i \in \mathbb{R}^3$ is the ground reaction force (GRF) acting on the foot contact point $p_{fi} \in \mathbb{R}^3$ with $i \in \{1,2,3,4\}$ denoting left front, right front, left hind, and right hind legs, $m$ is the robot mass, $g$ is gravitational acceleration, ${}^B I \in \mathbb{R}^{3 \times 3}$ is the trunk moment of inertia, $r_i = p_{fi} – p_c$ is the relative position from CoM to foot contact point, and $[{}^B \omega]_{\times} \in \mathbb{R}^{3 \times 3}$ is the skew-symmetric matrix of ${}^B \omega$, satisfying $[{}^B \omega]_{\times} {}^B I = {}^B \omega \times {}^B I$.
For spinal pitch control, according to the motion sequence, the front and rear spinal joints need to move symmetrically and synchronously. Before front foot touchdown, the spine remains extended; after touchdown, it contracts to the lower limit and holds; after hind foot touchdown, it extends to the upper limit and holds until the next cycle triggers. To ensure spinal motion is confined to S1 and S3 without affecting other phase planning, the spinal joint must move to the limit position within the first 50% of the support phase and then wait for the next phase. The spinal joint angle range is set to $(-\pi/6, 0)$, corresponding to contracted and extended limits, with initial and final velocities set to zero. Using a 3rd-order Bezier curve for trajectory interpolation, the spinal motion planning function is:
$$\theta_{sp_j,d} = \theta_{sp_j,\sigma} + \sigma_1 \cdot \frac{2 \pi}{6 (0.5 T_{st})^3} t^3 – \sigma_1 \cdot \frac{3 \pi}{6 (0.5 T_{st})^2} t^2,$$
$$\dot{\theta}_{sp_j,d} = \sigma_1 \cdot \frac{3 \pi}{(0.5 T_{st})^3} t^2 – \sigma_1 \cdot \frac{3 \pi}{(0.5 T_{st})^2} t,$$
where
$$\sigma_1 =
\begin{cases}
1 & t \in S1, \\
-1 & t \in S3, \\
0 & t \in S2 \cup S4,
\end{cases}$$
and $\theta_{sp_j,\sigma}$ is the inherited value from the previous phase, initially zero, with $j \in \{1,2\}$ for front and rear spinal joints. The planned trajectory is shown in the figure.
From the cable drive relationship, the desired trajectory for the spinal joint drive is:
$$\theta_{sp_{ac},d} = n r_{sp} \cdot \theta_{sp_j,d} / r_{ac},$$
$$\dot{\theta}_{sp_{ac},d} = n r_{sp} \cdot \dot{\theta}_{sp_j,d} / r_{ac}.$$
Thus, the spinal joint drive input torque is:
$$\tau_{sp_j} = k_p (\theta_{sp_{ac},d} – \theta_{sp_{ac}}) + k_v (\dot{\theta}_{sp_{ac},d} – \dot{\theta}_{sp_{ac}}).$$
For leg control, based on ground contact, leg motion is divided into support and swing states, each with different control methods. For support state control, it primarily seeks optimal GRF, then mapped to leg drive joints. We propose a fusion control method based on periodic impulse planning and MPC. First, the impulse conservation law is used to parameterize a stable bounding gait to obtain desired GRF. Then, combined with the dynamics model and MPC, optimal input GRF is solved.
When the quadruped robot performs bounding motion, GRF $F_i = (F_{xi}, F_{yi}, F_{zi})$ is the main control input, with $F_{yi}$ ignored in 2D motion. To obtain a stable periodic orbit for this complex system, the desired GRF $F_{i,d}$ satisfies:
$$\int_0^T \left( \sum_{i=1}^4 F_{zi,d}(t) – m g \right) dt = 0,$$
$$\int_0^T \tau_d(t) dt = 0,$$
where
$$\tau_d(t) = \sum_{i=1}^4 F_{xi,d}(t) \times r_{zi,d}(t) – F_{zi,d}(t) \times r_{xi,d}(t),$$
and $F_{zi,d}(t) \geq 0, \forall t \in [0, T]$. The planned complete GRF trajectory consists of 3rd-order Bezier curves, with coefficients determined based on impulse scaling principles.
After obtaining the desired GRF trajectory, control inputs for each step need to be solved based on the robot state. The MPC method can incorporate system constraints and handle discontinuous control theories, suitable for complex issues like repeated discontinuous ground contact and underactuation during flight in legged robots. To facilitate convex optimization in MPC, the nonlinear dynamics model is first linearized, and a quadratic programming cost function is constructed:
$$\min l(\Omega, F) = \sum_{k=0}^{N-1} \| \Omega_k – \Omega_{k_d} \|_{Q_x} + \| F_k – F_{k_d} \|_{R_u},$$
where $Q_x$ and $R_u$ are positive definite weight matrices, $\Omega_k$, $\Omega_{k_d}$, $F_k$, $F_{k_d}$ are the state vector, desired state vector, input GRF, and desired input GRF at prediction step $k$, respectively. Besides dynamics and initial conditions, constraints include tangential forces within the friction cone and non-negative normal forces:
$$|F_{xi}| \leq \mu |F_{zi}|, \quad |F_{zi}| \leq \mu |F_{zi}|, \quad 0 \leq F_{z,min_i} \leq F_{zi} \leq F_{z,max_i},$$
where $\mu$ is the friction coefficient, the friction cone is approximated, and all contacts are assumed point contacts.
In the designed quadruped robot, since the knee joint is pulled by cables, the main control object for the support state is the hip joint. Specifically, the hip joint torque is computed as:
$$\tau_{hip_i} = J_i^T R^T F_i,$$
where $J_i$ is the Jacobian matrix derived from the partial derivative of the foot position relative to the hip joint with respect to the hip joint angle, and $F_i$ is obtained from the MPC solution.
For swing state control, a classic method combining Raibert’s heuristic and velocity-based capture point formula is used to determine foot placement, and then a 3rd-order Bezier curve fits the foot swing trajectory.
Analysis of Spinal Impact on Bounding Motion
Based on the above, we analyze the advantages of spinal design in bounding motion. For comparison, the designed double-joint spine robot is denoted as QRAS (Quadruped Robot with Active Spine), and a general quadruped robot without spinal joints is introduced as QRIS (Quadruped Robot with Inactive Spine), with physical and control parameters identical except for the absence of spinal joints.
For QRIS, the lever arm between GRF and the single rigid body CoM is:
$$r’_{xi}(t) = p_{f,xi} – p_{c,x},$$
$$r’_{zi}(t) = p_{f,zi} – p_{c,z}.$$
For QRAS, although limb mass is neglected, the geometric effects of spinal motion cannot be ignored. During bounding motion, spinal motion causes the CoM position of the single rigid body model to change. Thus, combined with spinal motion planning, the lever arm for QRAS is:
$$r_{xi}(t) = r’_{xi}(t) – l_{sp} (1 – \cos \theta_{sp_j}(t)),$$
$$r_{zi}(t) = r’_{zi}(t) – l_{sp} \sin \theta_{sp_j}(t).$$
Assuming $t_2 > t_1$ and $t_1, t_2 \in S1$, $t_4 > t_3$ and $t_3, t_4 \in S3$, from the equations:
$$|r_{xi}(t_2) – r_{xi}(t_1)| – |r_{zi}(t_2) – r_{zi}(t_1)| < 0,$$
$$|r_{xi}(t_4) – r_{xi}(t_3)| – |r_{zi}(t_4) – r_{zi}(t_3)| < 0,$$
where
$$r_{xi}(t_2) – r_{xi}(t_1) < 0, \quad r_{xi}(t_4) – r_{xi}(t_3) > 0,$$
$$r_{zi}(t_2) – r_{zi}(t_1) > 0, \quad r_{zi}(t_4) – r_{zi}(t_3) < 0.$$
From impulse conservation and the above, compared to QRIS, the vertical force $F_{zi,d}$ for QRAS remains unchanged, but the horizontal force $F_{xi,d}$ needs adjustment:
$$F_{xi_f,d}(t) = \frac{ – \tau_d(t) + F_{zi_f,d}(t) \times r_{xi_f}(t) }{ r_{zi_f}(t) } \quad t \in S1,$$
$$F_{xi_h,d}(t) = \frac{ – \tau_d(t) + F_{zi_h,d}(t) \times r_{xi_h}(t) }{ r_{zi_h}(t) } \quad t \in S3,$$
where $F_{xi_f,d}(t)$ and $F_{xi_h,d}(t)$ have opposite directions.
Combining the equations, the relationship between horizontal forces for QRAS and QRIS is:
$$|F_{xi_f,d}(t) – F’_{xi_f,d}(t)| – |F_{xi_h,d}(t) – F’_{xi_h,d}(t)| > 0,$$
where
$$F_{xi_f,d}(t) – F’_{xi_f,d}(t) < 0, \quad F_{xi_h,d}(t) – F’_{xi_h,d}(t) < 0.$$
Thus, in GRF planning, spinal participation reduces the required horizontal forces for front and hind legs, with a greater impact on the front legs, which is valuable for speed improvement and energy reduction.
To better describe the contribution of spinal participation to bounding motion, define the contribution degree $\Psi$ for the first cycle as:
$$\Psi = L_{oc} – L’_{oc},$$
where
$$L’_{oc} = \int_0^{T_{st}} F’_{xi_f}(t) dt + \int_{\frac{T_{st} + T_{sw}}{2}}^{\frac{T_{st} + T_{sw}}{2} + T_{st}} F’_{xi_h}(t) dt,$$
$$L_{oc} = \int_0^{T_{st}} F_{xi_f}(t) dt + \int_{\frac{T_{st} + T_{sw}}{2}}^{\frac{T_{st} + T_{sw}}{2} + T_{st}} F_{xi_h}(t) dt.$$
Simulation Results and Analysis
To test the effectiveness of the proposed quadruped robot structure with double-joint spine and control, and verify the positive impact of spinal motion on bounding, the model is converted to XML format in SolidWorks and imported into Simulink, using Simscape toolboxes to complete joint connections. The simulation starts with the robot’s four legs supporting on the ground, specifying desired acceleration and velocity. After controller initialization, based on periodic impulse planning, desired GRF trajectories are obtained. Then, the time-based finite state machine is activated to enter the motion sequence, computing desired states and forces for the next time step. Finally, desired and current values are input into the MPC model for solving optimal control inputs. Simulation parameters are listed in the table below.
| Parameter | Value |
|---|---|
| Mass $m$ (kg) | 6.5 |
| $I_{xx}$ (kg·m²) | 0.020 |
| $I_{yy}$ (kg·m²) | 0.131 |
| $I_{zz}$ (kg·m²) | 0.094 |
| Waist length $l_{bd}$ (m) | 0.4 |
| Waist width $w_{bd}$ (m) | 0.18 |
| Spine length $l_{sp}$ (m) | 0.20 |
| Upper leg length $l_{sc}$ (m) | 0.18 |
| Lower leg length $l_{hu}$ (m) | 0.24 |
| Foot length $l_{ra}$ (m) | 0.15 |
The initial velocity is 0, desired velocity is 2 m/s, desired acceleration is 1 m/s², and simulation time covers 12 bounding cycles.
In terms of speed performance, the horizontal displacement and velocity comparisons between QRAS and QRIS during bounding motion show that both robots’ actual velocities fluctuate around the desired value due to periodic impulse planning, but QRAS has smaller velocity variations than QRIS, indicating better velocity tracking performance. Calculating the mean squared error (MSE) of the velocity curves, QRAS has an MSE of 0.125, while QRIS has 0.237, meaning spinal introduction reduces velocity MSE by 47.3%.
In attitude performance, alternating GRF on front and hind legs causes trunk pitch. The pitch angle $\phi_p$ during accelerated motion shows that compared to QRIS, QRAS has larger pitch angle variations but smaller fluctuations, with the median always near the balance point zero. Thus, during acceleration, QRAS exhibits more periodic regularity and stability. The phase plot of QRAS during accelerated motion shows trajectories converging to a closed periodic orbit, indicating good motion stability.
In impulse performance, reflected by GRF, which directly affects motor drive and power consumption, is crucial. In planning, when desired acceleration $acc = 1$ m/s², the planned horizontal GRF for both robots during acceleration shows that QRAS requires smaller horizontal desired GRF than QRIS, with a greater difference in front legs, consistent with theoretical analysis. From contribution degree calculation, $\Psi = 12.268 – 7.245 = 5.023$ N·s. To further study QRAS speed improvement, horizontal desired GRF $F_{xi,d}$ is computed for different desired accelerations. As acceleration increases, hind leg horizontal GRF gradually increases, while front leg horizontal GRF decreases, increasing forward impulse. At higher accelerations, the front leg force direction may reverse in the latter half of the support phase to better enhance forward impulse.
In actual control, GRF $F_i$ is solved in real-time by the finite state machine and MPC. Statistics of GRF maxima during acceleration are shown in the table below.
| Part | $F_{xi}$ (N) QRAS | $F_{zi}$ (N) QRAS | $F_i$ (N) QRAS | $F_{xi}$ (N) QRIS | $F_{zi}$ (N) QRIS | $F_i$ (N) QRIS |
|---|---|---|---|---|---|---|
| Front Leg | 84.848 | 96.613 | 128.534 | 111.043 | 126.185 | 168.087 |
| Hind Leg | 115.404 | 131.141 | 174.688 | 115.368 | 131.099 | 174.633 |
Spinal participation reduces extreme forces, mainly affecting front legs. The total horizontal impulse over 12 motion cycles is summarized in the table below.
| Robot Type | $L_{fr}$ (N·s) | $L_{hi}$ (N·s) | $L$ (N·s) |
|---|---|---|---|
| QRAS | 29.581 | -20.663 | 8.918 |
| QRIS | 43.036 | -34.212 | 8.824 |
Although total impulse is similar, QRAS requires smaller horizontal forces for the same speed increase. To verify QRAS superiority in accelerated motion, setting desired acceleration to 1.3 m/s² shows QRIS gradually becomes unstable, while QRAS successfully achieves the speed target.
Conclusion
Based on bionic principles, we design a double-joint spine quadruped robot model. The spinal joints, based on pulley group cable drive, achieve pitch motion of chest and sacrum with output torque amplification, transmission reduction, and tension amplification effects. The three-segment flexible legs, based on a novel MASLP structure, are lightweight, stable, and absorb vibrations and store energy. We propose a hybrid control method for the bounding gait, establishing a time-based finite state machine according to cheetah motion characteristics. Based on the single rigid body model, different control strategies are proposed for the spine and legs. Then, combining structural design and control methods, we theoretically analyze the impact of spinal design on bounding motion, defining an impulse-based contribution formula to measure the gain from spinal participation. Finally, comparative simulations with and without spinal participation prove that adding spinal design achieves similar impulse with smaller horizontal forces, and improves velocity tracking performance, attitude control, and stability.