1. Introduction

2. Quadruped Robot Model

3. Quadruped robot flight phase motion parameterization

3.3. Flexible spine quadruped robot flight phase motion parameterization

The overall center of mass of the torso composed of the centers of mass of the anterior and posterior torsos is selected as the object of the analyzed mass point, and the motion process of this mass point in the two-dimensional plane under a specific force is analyzed and described parametrically to represent the flight phase motion process of the flexible spine quadruped robot in a similar manner, which is shown in the schematic diagram in Figure 8(a). In order to describe the parameterized flight phase motion process of the flexible spine quadruped robot, the trajectories of the spine joints in the whole motion process and the ground reaction force in the take-off motion process need to be clarified first.

3.3.1. Spinal joint trajectory planning

Referring to the changing rule of flexible spine in cheetah running set the spine joints only in the take-off motion occurs obvious stretching action, while in the flight phase segment spine joint angle remains unchanged. The flexible spine joints are set to have the locked/unlocked function, and this locked/unlocked function of the flexible spine has been applied to the physical prototype in the research of other domestic counterparts, which is feasible [19]. The motion of the spinal joints is shown schematically in Figure 8(b).

Let the angle of the spinal joints at the start point be ${\phi }_{s0}$ and the stiffness of the joint torsion springs be. the anterior trunk will be subjected to a joint torque $M_s\left(t\right)$ due to the bending of the torsion springs, and $M_s\left(t\right)$ can be expressed as:

$$M_{\mathrm{s}}\left(t\right)=-k_{\mathrm{s}}·{\phi }_s\left(t\right)=J_{\mathrm{bF}}{\ddot{\phi }}_s\left(t\right)=\frac{1}{3}{m}_{\mathrm{bF}}{l}^2{ }_{\mathrm{bF}}·{\ddot{\phi }}_s\left(t\right)$$

(17)

The solution corresponding to the spinal joint angle ${\phi }_s\left(t\right)$ can be found as:

$${\phi }_s\left(t\right)=A\sin \left({\omega }_{\mathrm{n}}t+\phi \right)$$

(18)

where $A$ is the amplitude of the vibration, ${\omega }_n$ is the circular frequency, and it is easy to know that $\phi$ is the initial phase of the vibration. According to the initial condition, ${\phi }_s\left(0\right)={\phi }_{s0}$ ,it can be solved that the amplitude of the vibration $A={\phi }_{s0}$ and the initial phase $\phi =\pi /2$ .

From the circular frequency ${\omega }_n$ of the system, the period $T$ of the vibration can be written:

$$T=\frac{2\pi }{{\omega }_{\mathrm{n}}}$$

(19)

The time of the angular motion of the joint is half of its vibration period and at the same time is equal to the time of the take-off motion $\mathsf{\Delta }{t}_{TO\_flex}$ . Let the initial speeds of the center of mass at its starting point in the horizontal and vertical directions be $v_{STx\_flex}$ , $v_{STy\_flex}$ . Based on Equation (10), the load factor ${\beta }_{flex}$ as well as the gait frequency $f_{flex}$ can be solved for the flexible spine quadruped robot under this motion. Then：

$$\frac{T}{2}=\sqrt{\frac{{\pi }^2{m}_{\mathrm{bF}}{l}_{\mathrm{bF}}{ }^2}{3k_{\mathrm{s}}}}=△t_{\mathrm{TO}\text{\_}\mathrm{flex}}=\frac{{\beta }_{\mathrm{flex}}}{f_{\mathrm{flex}}}$$

(20)

The stiffness $k_s$ of the torsion spring of the spinal joint is:

$$k_{\mathrm{s}}=\frac{{\pi }^2{m}_{\mathrm{bF}}{l}_{\mathrm{bF}}{ }^2{f}_{\mathrm{flex}}{ }^2}{3{\beta }_{\mathrm{flex}}{ }^2}$$

(21)

The obtained stiffness $k_s$ , amplitude $A$ , and initial phase $\phi$ of the torsion spring of the spine joint are brought into Equation (18), and the derivation of the time is made, which means that the angular position of the spine joint and the trajectory of the angular velocity with time can be obtained:

$$\left\{\begin{array}{l}{\phi }_s\left(t\right)=\left\{\begin{array}{c}{\phi }_s{ }_0\sin \left(\frac{f_{\mathrm{flex}}}{{\beta }_{\mathrm{flex}}}\pi t+\frac{\pi }{2}\right)t\in \left[0,{\beta }_{\mathrm{flex}}/f_{\mathrm{flex}}\right] \\ -{\phi }_s{ }_{0}t\in \left[{\beta }_{\mathrm{flex}}/f_{\mathrm{flex}},{\beta }_{\mathrm{flex}}/f_{\mathrm{flex}}+△t_{\mathrm{FL}\text{\_}\mathrm{flex}}\right]\end{array}\right.\\ {\dot{\phi }}_s\left(t\right)=\left\{\begin{array}{c}\frac{{\phi }_s{ }_0\pi f_{\mathrm{flex}}}{{\beta }_{\mathrm{flex}}}\cos \left(\frac{f_{\mathrm{flex}}}{{\beta }_{\mathrm{flex}}}\pi t+\frac{\pi }{2}\right)t\in \left[0,{\beta }_{\mathrm{flex}}/f_{\mathrm{flex}}\right]\\ 0t\in \left[{\beta }_{\mathrm{flex}}/f_{\mathrm{flex}},{\beta }_{\mathrm{flex}}/f_{\mathrm{flex}}+△t_{\mathrm{FL}\text{\_}\mathrm{flex}}\right]\end{array}\right.\end{array}\right.$$

(22)

3.3.2. Ground Reaction Force

In order to obtain the ground reaction force which is only affected by the flexible spine compared to the rigid torso quadruped robot, the solution process is based on the ground reaction force of the rigid torso quadruped robot. Firstly, according to the described flying phase process of the rigid torso quadruped robot, the final obtained trajectories of the center of mass position, velocity, torso attitude angle, and angular velocity of the rigid torso over time are converted into the trajectories of the generalized position and generalized velocity variables over time in the complete dynamics model based on the geometric relationships. The geometric relationship between the above several physical quantities is schematically shown in Figure 9(a) set the hip and knee angles of the front leg to maintain fixed values throughout the process. The specific values of both are set here as shown in equation (23). After determining the trajectories of the variables of the generalized position, the generalized velocity is then derived for the obtained generalized position with respect to time.

$$\left\{\begin{array}{l}{\alpha }_{\mathrm{hFL}\text{\_}\mathrm{rigid}\left(\mathrm{hFR}\text{\_}\mathrm{rigid}\right)}=\frac{10}{9}\pi \\ {\alpha }_{\mathrm{kFL}\text{\_}\mathrm{rigid}\left(\mathrm{kFR}\text{\_}\mathrm{rigid}\right)}=\frac{7}{9}\pi \end{array}\right.$$

(23)

The input torque $u_{\mathit{r}i\mathit{g}id}\left(t\right)$ for each joint of the rear leg of the transition segment can be found by bringing the ground reaction force of the rigid torso quadruped robot based on the Fourier series fit with the derived generalized position and velocity trajectories into the planar multi-rigid body model (Equation (18)).

3.3.3. Flight Phase Motion Parameterization

Since the position of the total center of mass is related to the position of the centers of mass of the front and rear trunks, the motions of the respective centers of mass of the front and rear trunks under the corresponding forces are first analyzed in the take-off motion. Subsequently, the motions of both are synthesized by mass distribution to obtain the motion of the total center of mass in the takeoff motion. A schematic diagram of the force and the corresponding velocity analysis for the front and back trunks is shown in Figure 9(b). The velocity $v_{bB}$ of the center of mass of the posterior torso is only affected by its own gravity $G_{bB}$ and the ground reaction force ${\lambda }_{flex}\left(t\right)$ , while the velocity of the center of mass of the anterior torso is not only affected by its own gravity $G_{bF}$ and the ground reaction force ${\lambda }_{flex}\left(t\right)$ , but also by the elastic force $F$ corresponding to the release of flexible spine, and the two kinds of velocities are expressed as and , respectively. The final combined velocity of and is the velocity of the anterior trunk center of mass in takeoff motion.

Let the horizontal and vertical initial velocities of the robot's total center of mass as well as the front and rear torsos at the starting point be $v_{STx\_flex}$ , $v_{STy\_flex}$ . then, according to the ground reaction force ${\lambda }_{flex\_x}\left(t\right)$ , ${\lambda }_{flex\_y}\left(t\right)$ of the flexible spine object obtained in the previous section, we can obtain the trajectory of the center of mass velocity of the front and rear torsos' centers of mass in the presence of the ground reaction force with time:

$$\left\{\begin{array}{l}{v}_{\mathrm{bBx}}\left(t\right)=v_{\mathrm{STx}\text{\_}\mathrm{flex}}+{\displaystyle \underset{0}{\overset{t}{\int }}\frac{{\lambda }_{\mathrm{flex}}{ }_{\_\mathrm{x}}(t)}{m_0} }dt=v{`}_{\mathrm{bFx}}\left(t\right)\\ v_{\mathrm{bBy}}\left(t\right)=v_{\mathrm{STy}\text{\_}\mathrm{flex}}+{\displaystyle \underset{0}{\overset{t}{\int }}\frac{{\lambda }_{\mathrm{flex}\text{\_}\mathrm{y}}(t)-m_0\mathit{g}}{m_0} }dt=v{`}_{\mathrm{bFy}}\left(t\right)\end{array}\right.$$

(24)

Since the quadrupedal robot with a flexible torso is reduced to a prime model, the angle between its rear side torso and the horizontal direction cannot be obtained, so this quantity is set equal to the torso attitude angle of the quadrupedal robot with a rigid torso. It can be obtained：

$$\left\{\begin{array}{l}v{*}_{\mathrm{bFx}}\left(t\right)=-\frac{1}{2}{\dot{\phi }}_{\mathrm{s}}(t)l_{\mathrm{bF}}·\sin \left[{\phi }_{\mathrm{s}}\left(t\right)+{\phi }_{\mathrm{b}\text{\_}\mathrm{rigid}}\left(t\right)\right]\\ v{*}_{\mathrm{bFy}}\left(t\right)=\frac{1}{2}{\dot{\phi }}_{\mathrm{s}}(t)l_{\mathrm{bF}}·\cos \left[{\phi }_{\mathrm{s}}\left(t\right)+{\phi }_{\mathrm{b}\text{\_}\mathrm{rigid}}\left(t\right)\right]\end{array}\right.$$

(25)

Velocity and positional trajectories of the total center of mass of the torso in the horizontal and vertical directions can be obtained：

$$\left\{\begin{array}{l}{v}_{\mathrm{cx}\text{\_}\mathrm{flex}}\left(t\right)=\frac{m_{\mathrm{bB}}·v_{\mathrm{bBx}}\left(t\right)+m_{\mathrm{bF}}·v_{\mathrm{bFx}}\left(t\right)}{m_0}{v}_{\mathrm{cy}\text{\_}\mathrm{flex}}\left(t\right)=\frac{m_{\mathrm{bB}}·v_{\mathrm{bBy}}\left(t\right)+m_{\mathrm{bF}}·v_{\mathrm{bFy}}\left(t\right)}{m_0}\\ x_{\mathrm{c}\text{\_}\mathrm{flex}\text{\_}\mathrm{tran}}\left(t\right)=x_{\mathrm{ST}\text{\_}\mathrm{flex}}+{\displaystyle \underset{0}{\overset{t}{\int }}{v}_{\mathrm{cx}\text{\_}\mathrm{flex}}\left(t\right) }dt{y}_{\mathrm{c}\text{\_}\mathrm{flex}\text{\_}\mathrm{tran}}\left(t\right)=y_{\mathrm{ST}\text{\_}\mathrm{flex}}+{\displaystyle \underset{0}{\overset{t}{\int }}{v}_{\mathrm{cy}\text{\_}\mathrm{flex}}\left(t\right) }dt\end{array}\right.$$

(26)

Since the flexible spinal joints are in the "locked" state in the flight phase, the motion of the flexible torso center of mass in this process is the same as that of the rigid torso center of mass in the flight phase.

In summary, the parameterization of the flying phase process of the rigid torso quadruped robot is completed. As shown in Table 3.

4. Flight Phase Motion Experiment

5. Discussion and Conclusions

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