Motion coordination control of planar 5R parallel quadruped robot based on SCPL-CPG

Abstract

The parallel leg of the quadruped robot has good structural stiffness, accurate movement, and strong bearing capacity, but it is complicated to control. To solve this problem, a series connection of parallel legs (SCPL) was proposed, as well as a control strategy combined with the central pattern generator (CPG). With the planar 5R parallel leg as the research object, the SCPL analysis method was used to analyze the leg structure. The topology of CPG network was built with the Hopf oscillator as the unit model, and the CPG was the core to model the robot control system. By continuously adjusting the parameters in the CPG control system and changing the connection weight, and the smooth transition between gaits was realized. The simulation results show that the SCPL analysis method can be effectively used in the analysis of parallel legs, and the control system can realize the smooth transition between gaits, which verifies the feasibility and effectiveness of the proposed control strategy.

Keywords

Planar 5R parallel leg parallel quadruped robot central pattern generator Hopf oscillator gait smooth transition

Introduction

Legged robots have a wide range of applications in post-disaster rescue, field exploration, and military fields due to their good flexibility and strong environmental adaptability. In the last few decades, domestic and foreign scholars have done a lot of research on legged robots (such as biped robots, quadruped robots, hexapod robots, etc.), and have made important progress. Among them, quadruped robot has become a research hotspot in the field of robot due to its good adaptability, stability, and strong load capacity. Research mainly focuses on mechanical structure, motion control, and gait planning. The research of mechanical structure is mainly to optimize the leg structure. The leg structure mainly has three forms: series leg structure, parallel leg structure,^1,2 and hybrid leg structure. Among them, the parallel leg structure has the advantages of good rigidity, accurate movement, and strong bearing capacity because of its closed-loop structure. At present, the closed vector polygon projection method, complex vector method, matrix method, and other analysis methods are commonly used in the kinematic analysis of the parallel leg structure. The research of motion control and gait planning algorithm has always been a difficult point, mainly because the parallel leg is composed of multiple joints, which may cause high-dimensional continuous control problems. Therefore, the design of the motion controller of the parallel legged quadruped robot becomes the key. At present, the motion control methods of quadruped robot mainly include: the control method based on static stability, the control method based on dynamic model, and the control method based on the biological neuromodulation mechanism and behavior characteristics. The static and stable control method is mainly based on the zero moment point (ZMP) theory to plan the gait trajectory of the robot.³ But this kind of motion control method is difficult to realize complex motion. The control method of dynamic model mainly includes virtual model control and inverse dynamic control.⁴ Teng et al.⁵ proposed a quadruped robot control framework based on the fusion of the full-scale virtual model and the dynamic model. Yu et al.⁶ proposed an adaptive fuzzy full-state feedback control. Among them, the motion control method based on the CPG has become the research hotspot of bionic control due to its ability to spontaneously generate stable signals and strong coupling ability.

In biology, CPG is a kind of neural network, which can generate rhythmic neural activities without receiving rhythmic input through the combined action of excitatory neurones and inhibitory neurones, thus stimulating the rhythmic movement of animals.⁷ In the motion control of the robot, the biological CPG motion control characteristics are transformed into a mathematical model, which is more better than the traditional control method as being applied to the motion control of the robot.^8,9 The oscillating unit models in the CPG control network can be divided into two categories: neuron-based models and nonlinear oscillator-based models. The former mainly includes Matsuoka¹⁰ model, Kimura et al.¹¹ model, Rulkov¹² model, etc., while the latter mainly includes Kuramoto phase oscillator,¹³ Van Der Pol relaxation oscillator, Hopf harmonic oscillator, etc. Among them, the Hopf oscillator can control the motion of the robot using the simple input parameters and the coupled output stable rhythm signal. The parameters are independent and the waveform is easy to adjust. Therefore, it is widely used in the CPG control model of the robot.¹⁴ In CPG control network consisting of a Hopf oscillator, the generated signals are usually used for robot joint control, and the signal waveform determines the actual implementation trajectory of each joint angle in each cycle. There are two types of joint control of robot: position control and torque control. The former provides the CPG output signal to the desired joint angle of the feedback controller,¹⁵ and the latter directly controls the motor to generate torque with the CPG output signal.¹⁶ The CPG control method is applied to the robot control, and the result is good. The CPG algorithm has been used to realize the smooth transition between different gaits of the hexapod robot^17,18 and walking under different terrains.¹⁹ Liu and Zhang²⁰ and Liu et al.²¹ realize the movement of a quadruped robot in different gaits by using a CPG control network composed of different oscillators. Liu et al.⁸ and Santos et al.²² applied CPG control strategy to the motion control of the biped robot.

In addition to the single CPG control, the combination of CPG and other optimization algorithms is also applied to robot motion control. Wang et al.²³ and Ajalloeian et al.²⁴ combined CPG with a Virtual Model Control (VMC) in a compound control method. Liu et al.⁸ and Kasaei et al.²⁵ proposed a robot control strategy which combines CPG and Zero Moment Position (ZMP). Hasanzadeh and Tootoonchi²⁶ and Xue et al.¹⁶ proposed a hybrid control strategy based on fuzzy control and CPG. Gay et al.²⁷ designed a controller that uses a neural network to receive external feedback within the CPG control system. Thor et al.²⁸ proposed a network based on the combination of CPG and Radial Basis Function (RBF) to realize online adaptive control.

In this paper, an analysis method based on SCPL is proposed for the leg structure of a planar 5R parallel quadruped robot, which converts the parallel leg mechanism into a series structure for analysis. A CPG control system with Hopf oscillator as the core is established to control the gait and gait transition of a quadruped robot. Through the co-simulation experiment, the feasibility of the SCPL analysis method and the stability and effectiveness of the control algorithm are verified.

The kinematic analysis of robot

Build mathematical model of the leg structure

The leg structure of the quadruped robot adopts a planar 2-DOF 5R parallel mechanism, which has the characteristics of high stiffness and strong load carrying capacity. The schematic diagram and prototype of the robot leg mechanism are shown in Figure 1. The leg structure of the quadruped robot is composed of five connecting rods, and the original moving parts are rods $AB$ and $AD$ . The two RRR (where R is the active hinge and R is the passive hinge) ends of the moving branch chain are connected with the body and the hinge points $C$ of the rotating pairs respectively to form a single closed-loop structure and the joints are all rotating pairs. To reduce the singularity of the mechanism in the workspace, two RRR moving branch chains are arranged symmetrically. Combining the configuration characteristics of the mechanism that the number of motion branch chains is the number of degrees of freedom of the parallel mechanism. The planar 5R parallel mechanism has two driving pairs, so two driving motors are required. The connecting rod $AB$ and $AD$ are of equal length and coaxially connected to the body, to ensure that the two driving motors in the body are in a compact position, which can reduce the momentum of inertia of the legs, improve the control accuracy of the foot movement, and have better dynamic performance. The point $P$ is the moving platform of the planar parallel mechanism and can be used as the foot end of the quadruped robot. The foot end is wrapped with a circular rubber pads and contacts with the ground in a point contact mode. Four identical planar 5R parallel legs are symmetrically installed on the body to form a quadruped robot with a total of eight degrees of freedom.

Figure 1.

The schematic diagram of the robot leg mechanism and prototype, (a) the schematic diagram of the robot leg mechanism, (b) the prototype of the quadruped robot.

The coordinate system $xAy$ is established in the robot leg mechanism diagram, the origin $A$ is located at the axis of the two driving rods, and the length of each rod is set as $l_{i} (i = 1 ~ 4)$ , and the length of the $CP$ section of the connecting rod $BP$ is $l_{22}$ . The angular displacements of each joint are the $θ_{1}$ , $θ_{2}$ , $θ_{3}$ , and $θ_{4}$ as shown in Figure 1(a) (the counterclockwise direction is taken to be positive, and all joint angles noted in the figure are negative).

Location analysis

From Figure 1(a), the vector loop equation of the parallel leg can be obtained as:

\vec{l_{1}} + \vec{l_{21}} = \vec{l_{4}} + \vec{l_{3}}

(1)

The two RRR motion branch chains at point $C$ to form a single closed-loop structure. Using the closed vector polygon projection method, the foot-end position equation $(x_{P}, y_{P})$ can be obtained as:

{\begin{matrix} x_{P} & = l_{1} \cos θ_{1} + l_{2} \cos θ_{2} \\ = l_{4} \cos θ_{4} + l_{3} \cos θ_{3} + l_{22} \cos θ_{2} \\ y_{P} & = l_{1} \sin θ_{1} + l_{2} \sin θ_{2} \\ = l_{4} \sin θ_{4} + l_{3} \sin θ_{3} + l_{22} \sin θ_{2} \end{matrix}

(2)

The joint angular displacement $θ_{2}$ at point $B$ can be obtained from equation (2):

θ_{2} = 2 \arctan (\frac{V + \sqrt{U^{2} + V^{2} - W^{2}}}{U - W})

(3)

where:

\begin{matrix} U = 2 l_{21} (l_{1} \cos θ_{1} - l_{4} \cos θ_{4}); \\ V = 2 l_{21} (l_{1} \sin θ_{1} - l_{4} \sin θ_{4}); \\ W = l_{21}^{2} - l_{3}^{2} + (l_{1} \cos θ_{1} - l_{4} \cos θ_{4})^{2} + \\ (l_{1} \sin θ_{1} - l_{4} \sin θ_{4})^{2} ° \end{matrix}

The joint angular displacement $θ_{3}$ at point $D$ can be obtained from equation (2):

θ_{3} = \arctan \frac{l_{1} \sin θ_{1} + l_{21} \sin θ_{2} - l_{4} \sin θ_{4}}{l_{1} \cos θ_{1} + l_{21} \cos θ_{2} - l_{4} \cos θ_{4}}

(4)

Velocity analysis

By taking the derivative of equation (2) with respect to time $t$ , the velocity equation $(\overset{\cdot}{x_{P}}, \overset{\cdot}{y_{P}})$ of the plantar point $P$ can be obtained as:

{\begin{matrix} \overset{\cdot}{x_{P}} = - \overset{\cdot}{θ_{1}} l_{1} \sin θ_{1} - \overset{\cdot}{θ_{2}} l_{2} \sin θ_{2} = \\ - \overset{\cdot}{θ_{4}} l_{4} \sin θ_{4} - \overset{\cdot}{θ_{3}} l_{3} \sin θ_{3} - \overset{\cdot}{θ_{2}} l_{22} \sin θ_{2} \\ \overset{\cdot}{y_{P}} = \overset{\cdot}{θ_{1}} l_{1} \cos θ_{1} + \overset{\cdot}{θ_{2}} l_{2} \cos θ_{2} = \\ \overset{\cdot}{θ_{4}} l_{4} \cos θ_{4} + \overset{\cdot}{θ_{3}} l_{3} \cos θ_{3} + \overset{\cdot}{θ_{2}} l_{22} \cos θ_{2} \end{matrix}

(5)

The joint angular velocity $ω_{2}$ and $ω_{3}$ at points $B$ and $D$ can be obtained from the equation (5) respectively:

{\begin{matrix} ω_{2} = \overset{\cdot}{θ_{2}} = \frac{\overset{\cdot}{θ_{1}} l_{1} \sin (θ_{1} - θ_{3}) + \overset{\cdot}{θ_{4}} l_{4} \sin (θ_{3} - θ_{4})}{l_{21} \sin (θ_{3} - θ_{2})} \\ ω_{3} = \overset{\cdot}{θ_{3}} = \frac{\overset{\cdot}{θ_{1}} l_{1} \sin (θ_{2} - θ_{1}) + \overset{\cdot}{θ_{4}} l_{4} \sin (θ_{4} - θ_{2})}{l_{3} \sin (θ_{2} - θ_{3})} \end{matrix}

(6)

Acceleration analysis

By taking the derivative of equation (5) with respect to time $t$ , the acceleration equation $(\overset{\cdot \cdot}{x_{P}}, \overset{\cdot \cdot}{y_{P}})$ of the plantar point $P$ can be obtained as:

{\begin{matrix} \overset{\cdot \cdot}{x_{P}} = - \overset{\cdot \cdot}{θ_{1}} l_{1} \sin θ_{1} - \overset{\cdot}{θ_{1}^{2}} l_{1} \cos θ_{1} - \overset{\cdot \cdot}{θ_{2}} l_{2} \sin θ_{2} - \overset{\cdot}{θ_{2}^{2}} l_{2} \cos θ_{2} = \\ - \overset{\cdot \cdot}{θ_{4}} l_{4} \sin θ_{4} - \overset{\cdot}{θ_{4}^{2}} l_{4} \cos θ_{4} - \overset{\cdot \cdot}{θ_{3}} l_{3} \sin θ_{3} - \\ \overset{\cdot}{θ_{3}^{2}} l_{3} \cos θ_{3} - \overset{\cdot \cdot}{θ_{2}} l_{22} \sin θ_{2} - \overset{\cdot}{θ_{2}^{2}} l_{22} \cos θ_{2} \\ \overset{\cdot \cdot}{y_{P}} = \overset{\cdot \cdot}{θ_{1}} l_{1} \cos θ_{1} - \overset{\cdot}{θ_{1}^{2}} l_{1} \sin θ_{1} + \overset{\cdot \cdot}{θ_{2}} l_{2} \cos θ_{2} - \overset{\cdot}{θ_{2}^{2}} l_{2} \sin θ_{2} = \\ \overset{\cdot \cdot}{θ_{4}} l_{4} \cos θ_{4} - \overset{\cdot}{θ_{4}^{2}} l_{4} \sin θ_{4} + \overset{\cdot \cdot}{θ_{3}} l_{3} \cos θ_{3} - \\ \overset{\cdot}{θ_{3}^{2}} l_{3} \sin θ_{3} + \overset{\cdot \cdot}{θ_{2}} l_{22} \cos θ_{2} - \overset{\cdot}{θ_{2}^{2}} l_{22} \sin θ_{2} \end{matrix}

(7)

The joint angular accelerations $ε_{2}$ and $ε_{3}$ at points $B$ and $D$ can be obtained from the equation (7) respectively:

{\begin{matrix} ε_{2} = \overset{\cdot \cdot}{θ_{2}} = [\overset{\cdot \cdot}{θ_{1}} l_{1} \sin (θ_{1} - θ_{3}) + \overset{\cdot}{θ_{1}^{2}} l_{1} \cos (θ_{1} - θ_{3}) + \\ \overset{\cdot}{θ_{2}^{2}} l_{21} \cos (θ_{3} - θ_{2}) + \overset{\cdot \cdot}{θ_{4}} l_{4} \sin (θ_{3} - θ_{4}) - \\ \overset{\cdot}{θ_{4}^{2}} l_{4} \cos (θ_{3} - θ_{4}) - \overset{\cdot}{θ_{3}^{2}} l_{3}] / (l_{21} \sin (θ_{3} - θ_{2})) \\ ε_{3} = \overset{\cdot \cdot}{θ_{3}} = [\overset{\cdot \cdot}{θ_{1}} l_{1} \sin (θ_{1} - θ_{2}) + \overset{\cdot}{θ_{1}^{2}} l_{1} \cos (θ_{1} - θ_{2}) + \\ \overset{\cdot}{θ_{2}^{2}} l_{21} + \overset{\cdot \cdot}{θ_{4}} l_{4} \sin (θ_{2} - θ_{4}) - \overset{\cdot}{θ_{4}^{2}} l_{4} \cos (θ_{2} - θ_{4}) - \\ \overset{\cdot}{θ_{3}^{2}} l_{3} \cos (θ_{3} - θ_{2})] / (l_{3} \sin (θ_{3} - θ_{2})) \end{matrix}

(8)

Series connection of parallel legs

According to equation (3), the variation of joint angular displacement $θ_{2}$ only depends on the length of the fixed rod and the joint angular displacements $θ_{1}$ and $θ_{4}$ , and there is no coupling between the parameters. Since each moving branch chain of the parallel mechanism can be regarded as a series mechanism, and the moving platform of the planar 5R parallel mechanism is also the common executive end of two RRR branch chains, the freedom of the moving platform is the same as that of the single moving branch chain. To facilitate the definition of the joint of the parallel leg, the leg mechanism of the quadruped robot is simplified as shown in Figure 2.

Figure 2.

Series connection of parallel legs.

After the serialization of the leg structure, the influence of the rod $A^{'} D^{'}$ and $D^{'} C^{'}$ movements is not taken into account. The connecting rod $A^{'} B^{'}$ and “BC” are defined as thigh and calf, respectively. The revolute pair of points $A^{'}$ and $B^{'}$ are defined as the hip joint and the knee joint, respectively, where the rotation angle of the hip joint can be regarded as the angular displacement $θ_{1}$ of the driving rod $A^{'} B^{'}$ . Without considering the hip joint and knee joint rotational coupling in the serial legs, the knee joint rotation angle is only related to the angular displacement $θ_{4}$ of the driving rod $A^{'} D^{'}$ . The plantar is still defined as the end of the moving branch chain consisted of connecting rod $A^{'} B^{'}$ and $B^{'} P^{'}$ .

Built CPG control network model

CPG oscillation unit modeling

The Hopf oscillator has a fast conversion speed and strong robustness to disturbance. Therefore, it has a harmonic limit cycle with stable structure. Its parameters also have clear physical meanings. The frequency and amplitude of the output signal are independently adjustable, which can be used to generate the rhythm signal of a quadruped robot. In this paper, hopf oscillators are chosen as the core of CPG motion control system. Hopf oscillator is used in a single CPG oscillation unit, and its mathematical expression is as follows:

{\begin{matrix} \overset{\cdot}{x} = - ω y + μ x (m - r^{2}) \\ \overset{\cdot}{y} = ω x + μ y (m - r^{2}) \end{matrix}

(9)

Where $x$ and $y$ are the two state variables of the Hopf oscillator. $x$ is the phase of the excited neuron of the oscillator and $y$ is the phase of the inhibited neuron, and $r^{2} = x^{2} + y^{2}$ ; $m$ is the amplitude coefficient of the oscillator, and $A = \sqrt{m}$ , $m > 0$ ; $ω$ is the oscillation frequency of the oscillator, which is a constant; $μ$ is the speed at which the oscillator converges to the limit cycle, and $μ > 0$ .

The equilibrium point and limit cycle characteristics of the Hopf oscillator are analyzed. Set $x = r \cos θ$ , $y = r \sin θ$ , then $θ = \arctan (\frac{y}{x})$ , $r^{2} = x^{2} + y^{2}$ .

where:

{\begin{matrix} \overset{\cdot}{x} = \overset{\cdot}{r} \cos θ - r ω \sin θ \\ \overset{\cdot}{y} = \overset{\cdot}{r} \sin θ + r ω \cos θ \end{matrix}

(10)

Equation (9) can be rewritten as:

{\begin{matrix} \overset{\cdot}{r} = μ (m - r^{2}) r \\ θ^{\cdot} = ω \end{matrix}

(11)

Then the origin point $O (0, 0)$ of the oscillator in equation (9) is the unique singularity, and $ω$ , $μ$ , and $m$ are constants.

To determine the behavior of the orbit in the neighborhood of the singularity point $O (0, 0)$ of the nonlinear system in equation (9), according to the O. Perron theorem,²⁹ and to the cubic nonlinear continuous function $G (x, y)$ and $H (x, y)$ , and let:

{\begin{matrix} G (x, y) = x (x^{2} + y^{2}) \\ H (x, y) = y (x^{2} + y^{2}) \end{matrix}

(12)

Then $G (0, 0) = H (0, 0) = 0$ , $G (x, y)$ and $H (x, y)$ have a first order continuous partial derivative in the neighborhood of $O (0, 0)$ , as $\forall δ > 0$ , such that:

{\begin{matrix} lim_{r \to 0} \frac{G (x, y)}{r^{1 + δ}} = lim_{r \to 0} \frac{r^{2} r \cos θ}{r^{1 + δ}} \\ = lim_{r \to 0} r^{2 - δ} \cos θ = 0 \\ lim_{r \to 0} \frac{H (x, y)}{r^{1 + δ}} = lim_{r \to 0} \frac{r^{2} r \sin θ}{r^{1 + δ}} \\ = lim_{r \to 0} r^{2 - δ} \sin θ = 0 \end{matrix}

(13)

O. Perron theorem is to linearize the nonlinear system, then the matrix is:

A = [\begin{matrix} μ m & - ω \\ ω & μ m \end{matrix}]

(14)

It is still the coefficient matrix corresponding to the linear part of equation (9).

Where the characteristic equation of $A$ is:

\begin{matrix} [\begin{matrix} μ m - λ & - ω \\ ω & μ m - λ \end{matrix}] = λ^{2} - 2 μ m λ + \\ (μ m)^{2} + ω^{2} \\ = λ^{2} + p λ + q = 0 \end{matrix}

(15)

It can be known from equation (15) that $λ = μ m + i ω$ , and the three parameters $p$ , $q$ , and $Δ$ are:

{\begin{matrix} p & = - 2 μ m < 0 \\ q & = (μ m)^{2} + ω^{2} > 0 \\ Δ & = p^{2} - 4 q = - 4 ω^{2} < 0 \end{matrix}

(16)

According to the $p - q$ parameter judgment method, it can be known that the singularity $O (0, 0)$ of equation (9) is the unique unstable focus.

According to the parameter determination method shown in Figure 3, it can be quickly determined that the values of the three parameters of equation (9), $p$ , $q$ , and $Δ$ . They are all located in the unstable focus area of the graph $p - q$ , thus the type of the singularity point $O$ is the unstable focus.

Figure 3.

$p - q$ graph.

According to equation (11), when $0 < r < \sqrt{m}$ , $\overset{\cdot}{r} > 0$ , then $r$ monotonically increases and tends to $\sqrt{m}$ ; When $r > \sqrt{m}$ , $\overset{\cdot}{r} < 0$ , then $r$ monotonically decreases and tends to $\sqrt{m}$ . Therefore, the trajectory lines of equation (11) spirally tend the closed trajectory $r = \sqrt{m}$ in the counterclockwise direction, and $r = \sqrt{m}$ is an isolated closed trajectory and a limit cycle.

Let the annular region surrounded by inner and outer curves $Γ_{1}$ and $Γ_{2}$ be $S$ . If the circumference of the circle is taken as:

{\begin{matrix} Γ_{1} : r = \frac{\sqrt{m}}{2} \\ Γ_{2} : r = 2 \sqrt{m} \end{matrix}

(17)

When $t$ increases, the trajectory lines of equation (11) enter the interior from the outside of the annular domain $S$ in both $Γ_{1}$ and $Γ_{2}$ , and there is no singularity in the annular domain $S$ . According to Poincaré annual region theorem,^30,31 equation (11) has only a unique closed orbit $r = \sqrt{m}$ , which is also a unique and stable limit cycle, as shown in Figure 4.

Figure 4.

The limit cycle of the Hopf oscillator ( $m = 0.4$ ).

The nonlinear differential equations of Hopf oscillator are solved, and the state quantities are plotted in Figure 4. It shows that different initial values converge to the same limit cycle. Except for the singularities point $O$ , with different system state variable initial values, they can converge to the stable limit cycle $r = \sqrt{m}$ . Therefore, Hopf oscillator can switch the motion mode of the quadruped robot. After receiving mode switching instruction, it can automatically converge from the current motion mode to the target motion mode through parameter adjustment. This ensures that the robot has good adaptability. That is, when the external force makes the robot deviate from the target state of motion, it can automatically recover.

The Hopf oscillator outputs a stable periodic oscillation signal, while the movement of a quadruped robot is a periodic rhythmic movement. Therefore, the oscillation signal generated by the oscillator can be used for joint control of a quadruped robot. The rising edge of the output signal corresponds to the swing phase of the leg, and the falling edge corresponds to the stance phase of the leg, as shown in Figure 5.

Figure 5.

The phase of leg joint.

Figure 5 shows that the rising edge and falling edge of the oscillation signal used for joint control have the same period, so that the period of the swing phase and the stance phase of the robot are equal within a gait period. Under different walking gaits, the legs are standing on the ground for different times in a gait cycle, so the load factor $β$ is introduced to improve the oscillator frequency $ω$ in the mathematical expression of Hopf oscillator as follows:

{\begin{matrix} ω = \frac{ω_{stance}}{e^{- ay} + 1} + \frac{ω_{swing}}{e^{ay} + 1} \\ ω_{stance} = \frac{1 - β}{β} ω_{swing} \end{matrix}

(18)

where $ω_{stance}$ and $ω_{swing}$ are the frequency of stance phase and swing phase, respectively. The parameter $a$ is the amplification factor of the phase of the inhibitory neuron, which is generally a large normal number and determines the switching speed from the swing state to the stance state. $β$ is the load factor, and is the proportion of the stance phase in the whole gait cycle. Figure 6 shows the output curve of the state variable $x$ in the walk gait when $β = 0.75$ , $a = 100$ , $ω_{swing} = 5 π$ .

Figure 6.

The output curve of state variable x when $β = 0.75$ .

CPG control network

Biological CPG is composed of several central neurons, which are composed of three basic interneurons: excitatory, lateral inhibitory, and terminal cross inhibitory. The neurons inhibit each other and can generate rhythmic signals and generate rhythmic movements through self-excited oscillations. Multiple oscillating units form a CPG network through phase coupling, and the different topologies of the network can change the output mode of the oscillating signal, to realize a variety of gaits.

CPG can automatically generate stable oscillating behavior in the absence of high-level control signals and external feedback, which can regulate the rhythmic movement of CPG. CPG control network is used to control the movement of a quadruped robot. It can not only adjust the phase difference between the legs, but also realize the phase coupling between the hip joint and the knee joint.

Since the planar 5R parallel quadruped robot has a total of eight active degrees of freedom, eight Hopf oscillators are used to construct a layered CPG control network to control the hip joint and knee joint of the four legs respectively, as shown in Figure 7. The first layer of the control network adopts a fully symmetric network topology structure, which forms the coupling relationship between each leg and hip joint. The second layer is composed of hip and knee oscillators in each leg to form a single leg internal coupling relationship. The output signals $x$ and $y$ of CPG are used as the position control of each joint. That is, using this control network to generate the angle control signals of each joint.

Figure 7.

Topological structure of CPG network.

The mathematical model of CPG network topology composed of Hopf oscillators is as follows:

{\begin{matrix} \overset{\cdot}{x_{i}} = - ω_{i} y_{i} + μ x_{i} (m - r_{i}^{2}) + \\ \sum_{j = 1}^{4} R (θ_{i}^{j}) x_{i} \\ \overset{\cdot}{y_{i}} = ω_{i} x_{i} + μ y_{i} (m - r_{i}^{2}) + \\ \sum_{j = 1}^{4} R (θ_{i}^{j}) y_{i} l, \\ r_{i}^{2} = x_{i}^{2} + y_{i}^{2} \\ ω_{i} = \frac{ω_{stance}}{e^{- a y_{i}} + 1} + \frac{ω_{swing}}{e^{a y_{i}} + 1} \\ ω_{stance} = \frac{1 - β}{β} ω_{swing} \end{matrix} i = 1, \cdot \cdot \cdot, 4

(19)

where $x_{i}$ and $y_{i}$ are the output signals of the oscillator corresponding to four legs, respectively. $R (θ_{i}^{j})$ is the connection weight matrix of the CPG network, which is used to adjust the phase difference between the oscillators to realize different movement gaits of the quadruped robot.

Joint mapping function

With the above analysis of the serialization of the legs, the signal $x_{i}$ acts on the connecting rod $AB$ and the rotation angle of the revolute $θ_{1}$ , it’s also used as the hip joint angle control signal $θ_{hi} (t)$ , $θ_{hi} (t) = x_{i}$ . $y_{i}$ acts on the connecting rod $AD$ and the rotation angle of the revolute $θ_{4}$ . Considering the influence of the coupling between the hip joint and the knee joint, with given joint angular displacements $θ_{1}$ and $θ_{4}$ , the knee joint angular displacements $θ_{2}$ can be obtained through equation (2). Then it transforms into the knee joint angle control signal $θ_{ki} (t)$ through the transformation equation of equation (20).

The leg structure of the quadruped robot after SCPL adopts the full knee symmetrical arrangement. In other words, knee joints are used for both front and rear legs. Through analyzing the gait of dogs, horses, and other animals, it shows that there is a fixed phase relationship between the hip joint and knee joint. In the pre-swing phase, as shown in Figure 8 interval $(1, 1.4)$ , the hip joint and the knee joint have the same direction of rotation. In the post-swing phase, the direction of rotation of the hip joint is opposite to the knee joint, and the direction of rotation of the hip joint remains the same. During the stance phase, the hip joint swings in the opposite direction from the swing phase, and the knee joint angle remains the same throughout the stance phase.

Figure 8.

Phase relationship between hip and knee joint.

According to the phase relationship between the hip and knee joint during the swing phase and stance phase of the legs, the transformation equation of the joint angular displacement $θ_{2}$ of the knee joint to the control signal $θ_{ki} (t)$ of the joint angle can be expressed as:

θ_{ki} (t) = {\begin{matrix} \frac{A_{k}}{A_{h}} θ_{2 i}, θ_{2 i} \geq 0 \\ 0, θ_{2 i} < 0 \end{matrix}

(20)

where $A_{h}$ and $A_{k}$ represent the swing amplitude of the hip and knee joint after the serialization of the legs, respectively. According to the characteristics of series legs, the step length and plantar lift height of the robot are determined by $A_{h}$ and $A_{k}$ respectively. $θ_{2 i}$ is the angular displacement of the knee joint obtained from the transformation of the output signal $y_{i}$ through equation (3).

The connection weight of CPG control network

As shown in Figure 7, the four oscillators formed by the CPG network topology at the hip joint are connected with full symmetry, and the oscillators are coupled to each other to ensure the stability and phase-lock relationship of the oscillator output signals. The connection weight matrix $R (θ_{i}^{j})$ can change the phase relationship between the oscillators by adjusting the parameter $θ_{i}^{j}$ to realize different gaits of the quadruped robot. $R (θ_{i}^{j})$ can be expressed as:

R (θ_{i}^{j}) = [\begin{matrix} \cos θ_{i}^{j} & - \sin θ_{i}^{j} \\ \sin θ_{i}^{j} & \cos θ_{i}^{j} \end{matrix}]

(21)

where $θ_{i}^{j} = 2 π (φ_{i} - φ_{j})$ , $θ_{i}^{j}$ represents the phase difference between the $i$ and $j$ oscillators. $φ_{i}$ represents the phase of the $i$ leg relative to the base leg (in this paper, the front-left leg 1 is set as the base leg, $φ_{1} = 0$ , as shown in Figure 9).

Figure 9.

The relative phase of walk and trot gait: (a) walk gait and (b) trot gait.

In Figure 9, in walk gait, the four legs lift alternately in the alternating order of $1 \to 3 \to 2 \to 4 \to 1$ , and the lift time of each leg is delayed by $1 / 4$ gait cycles relative to the previous leg, which is a four-beat gait, and the load factor is $β = 0.75$ . In trot gait, the diagonal legs are in the same phase. In other words, the front-left leg 1 and rear-right leg 3 simultaneously raise and fall, and the leg lifting order is $1 & 3 \to 2 & 4 \to 1 & 3$ . Compared with the front-left leg 1 and rear-right leg 3, the leg lifting time of front-right leg 2 and rear-left leg 4 is delayed by half of the gait cycle, which is a two-beat gait, and the load factor is $β = 0.5$ .

In summary, with the phase $φ_{1} = 0$ of the front-left leg 1 as the initial condition, different gaits can be described by the relationship between the phase $φ_{3}$ of the rear-right leg 3 and the load factor $β$ . The phase determined conditions between each leg are as follows:

Algorithm of gait smooth transition

In this paper, the common four-legged animal, dog, and horse movement gait are researched to study the smooth conversion algorithm of a parallel quadruped robot between walk and trot gaits.

To verify the effectiveness of the proposed CPG control system, the combination simulation platform of MATLAB/Simulink and ADAMS is used as the experiment platform. The CPG control system of the planar 5R parallel quadruped robot is built in MATLAB/Simulink. The virtual prototype of the robot is modeled in ADAMS. The co-simulation platform is completed through the connection of the control system and the virtual prototype model. The co-simulation platform provides an experiment simulation platform for the typical gait motion of the robot and the smooth transition simulation.

Walk gait

When the quadruped robot moves in the walk gait, the CPG model parameters are set as follows: $a = 100$ , $β = 0.75$ , $ω_{swing} = 5 π$ , $φ_{3} = 0.25$ , $μ = 10000$ . As shown in Figure 10, the control curves of the four-legged hip joint and the single-leg hip joint and knee joint are obtained by co-simulation. There is a stable phase difference between the hip joint signals of the four legs, and the knee joint signal of a single leg also satisfies the motion law of the hip and knee joints analyzed above. Figure 11 is the simulation animation screenshot of the four legs alternately lifted by the quadruped robot as moving in the walk gait. It shows that the four legs are out of phase, which satisfies the alternate lifting sequence $1 \to 3 \to 2 \to 4 \to 1$ between the four legs of the planar 5R parallel quadruped robot. It is possible to achieve continuous and stable walk gait.

Figure 10.

The joint control signal during the walk gait: (a) the hip joint control signal of the four legs during the walk gait and (b) the joint control signal of the single leg during the walk gait. Note: To facilitate observation, take the simulation time of $0 ~ 5 s$ .

Figure 11.

The moving simulation during the walk gait: (a) leg 1 lifting, (b) leg 3 lifting, (c) leg 2 lifting, and (d) leg 4 lifting. Note: The legs described in each picture correspond to the red and blue marks.

Figure 12 shows the displacement curve of the body’s centroid in the direction of the coordinate axis $x / y / z$ during the walk gait. The negative direction of $x$ is the forward direction of the robot. In the $10 s$ simulation time, the robot walks forward $1757.9 mm$ . $y$ direction is the lateral deviation direction of the robot, with a maximum deviation of $39.3 mm$ . The negative direction of $z$ is the gravity direction of the robot. The initial height of the robot’s centroid relative to the ground is $196 mm$ . The maximum fluctuation quantity of the height of the centroid relative to the initial state during walking is $7 mm$ . It shows that the fluctuation rate in the vertical direction is $3.57 %$ .Therefore, given the above CPG model parameters, the robot can achieve a walk gait with stable movement and keep walking in a straight line.

Figure 12.

The displacement curve of the centroid during the trot gait.

Trot gait

When the robot moves in the trot gait, the CPG model parameters are set as follows: $a = 100$ , $β = 0.5$ , $ω_{swing} = 5 π$ , $φ_{3} = 0$ , $μ = 10000$ . As shown in Figure 13, the control curves of the four-legged hip joint and the single-leg hip joint with knee joint are obtained by co-simulation. There is a stable phase difference between the hip joint signals of the four legs, and the knee joint signal of a single leg also satisfies the motion law of the hip and knee joints analyzed above. Figure 14 is the simulation animation screenshot of the four legs alternately lifted by the quadruped robot when moving in the walk gait. The diagonal legs of the robot are in the same phase, satisfying the alternate lifting sequence is $1 & 3 \to 2 & 4 \to 1 & 3$ between four legs, which can achieve a continuous and stable trot gait.

Figure 13.

The joint control signal during the walk gait: (a) the hip joint control signal of the four legs during the trot gait and (b) the joint control signal of the single leg during the trot gait. Note: To facilitate observation, take the simulation time of $0 ~ 5 s$ .

Figure 14.

The moving simulation during the trot gait: (a) leg 1&3 lifting and (b) leg 2&4 lifting. Note: The legs described in each picture correspond to the red and blue marks.

Figure 15 shows the displacement curve of the body’s centroid in the direction of $x / y / z$ during the trot gait. As described in Section 4.1, the negative direction of the coordinate axis $x$ is the forward direction. The robot moves forward $6617.8 mm$ in a trot gait within $10 s$ of simulation time. $y$ direction is the lateral deviation direction of the robot, and the maximum deviation is $227.0 mm$ . The negative direction of $z$ is the gravity direction. The initial height of the robot’s centroid above the ground is $196 mm$ . The maximum fluctuation quantity of the height of the centroid relative to the initial state during walking is $12.9 mm$ , and the fluctuation rate is $6.6 %$ . Therefore, given the above CPG model parameters, the robot can realize stable movement in the trot gait and keep walking in a straight line.

Figure 15.

The displacement curve of the centroid during the trot gait.

Smooth transition of Walk → Trot → Walk gait

To reduce energy consumption when moving on the ground, quadrupeds need to change their forward speed and movement gait according to the actual situation. The energy consumption of the quadruped robot is also the main reason affecting its development. Therefore, the gait of the robot should be converted according to the actual environment. In this paper, the gait conversion of the robot is realized by changing the network characteristics of CPG. In the equation (21), it shows that the connecting weight matrix $R (θ_{i}^{j})$ is a function of the phase $θ_{i}^{j}$ . By adjusting $R (θ_{i}^{j})$ , the phase relationship between each leg can be changed to realize the gait transition. When the robot is in the walks, trot and its gait transition, the initial values of CPG model parameters are set as follows: $a = 100$ , $β = 0.75$ , $ω_{swing} = 5 π$ , $φ_{3} = 0.25$ , $μ = 10000$ . As shown in Figure 16, the control curve of the hip joint of each leg of the robot in different gait and conversion process is obtained by co-simulation. Figure 16 shows that although the signals have a stable phase difference, the conversion between the robot gaits can be realized. However, at the $2 s$ transition point $A$ starting from the “walk-trot” to gait transition, it takes $1 s$ to finish the entire transition from the walk to the trot gait. As for the transition of “trot-walk” gait from the point $B$ , it takes $1.57 s$ to fully transition from the trot to the walk gait. In addition, the hip joint signal output by the CPG model appears sharp points when entering conversion at the points of $2 s$ and $6 s$ , which may cause impact on the motor.

Figure 16.

The hip joint control signal of the four legs during the transition of walk → trot → walk gait.

Figure 17 is the displacement curve of the body’s centroid in the direction of $x / y / z$ during the gait transition of the quadruped robot in “walk → trot → walk” gait. The simulation time is $8 s$ , in which $0 ~ 2 s$ is the walk gait, and the robot advances $305.9 mm$ . $y$ is the lateral deviation of the robot, with a maximum deviation of $9.8 mm$ . The negative direction of $z$ is the gravity direction, and the maximum fluctuation quantity of the height of the robot centroid is $7.2 mm$ during the movement. In different gaits, the robot centroid displacement in all directions and the fluctuation rate in the $z$ direction are shown in Table 1.

Figure 17.

The displacement curve of the centroid during the transition of walk → trot → walk gait.

Table 1.

transition of walk → trot → walk gait displacement in $x / y / z$ axis.

	$x$ $(mm)$	$y$ $(mm)$	$z$ $(mm)$	$z$ -volatility
Walk $(0 ~ 2 s)$	305.9	9.8	7.2	3.7%
Trot $(2 ~ 6 s)$	2300.3	124.6	18.1	9.2%
Walk $(6 ~ 8 s)$	640.1	107.1	7.8	3.9%

Algorithm 1. The phase determined determined conditions
$\begin{matrix} if φ_{4} \neq 0 \\ { \\ φ_{2} = β - φ_{3}; \\ φ_{4} = β; \\ } \\ else \\ { \\ φ_{2} = φ_{3}; \\ φ_{4} = φ_{1}; \\ } \end{matrix}$

Table 1 shows that in the process of robot simulation, there is a large lateral deviation when it moves in the trot gait. Through the analysis of the simulation process and the deviation curve in $y$ direction as shown in Figure 18, the robot has a small range of lateral deviation during the gait transformation of “walk → trot” at the point of $2 s$ . When the gait transition completes at the point of $3 s$ , the axis of the fuselage forms a certain included angle with the $x$ axis due to the influence of the fuselage swing. When entering the gait transition, the fuselage direction at the points of $2 s$ is used as the initial direction. Therefore, the forward direction of the robot after the gait transition is toward the initial direction of the fuselage, so there is a large lateral deviation. Choosing a suitable transition point can avoid the influence of the orientation of the fuselage during the transition on the walking trajectory of the robot. When the angle between the fuselage axis and the $x$ axis is small (such as at the points of $1.81 s$ and $6.2 s$ ), the gait conversion is carried out. Figure 19 shows that the trajectory deviation caused by fuselage swing can be significantly reduced when the gait conversion is carried out at the current conversion point.

Figure 18.

The displacement curve of the centroid in the $y$ direction during the transition of walk → trot → walk gait.

Figure 19.

The displacement curve of the centroid in the $y$ direction after adjusting the conversion point.

The break points and phase changes in the output signal during the gait conversion process should be avoided to reduce the impact on the mechanical and electrical systems of the robot. With continuous adjustment of the phase $φ_{3}$ of the rear-right leg and the phase of each leg also changes continuously. Taking the change time of 2 s as an example, the phase $φ_{3}$ of the rear-right leg can be expressed as:

φ_{3} = {\begin{matrix} 0.25, t \leq 2 \\ - 0.125 t + 0.5, 2 < t < 4 \\ 0, 4 \leq t \leq 8 \\ 0.125 t - 1, 8 < t < 10 \\ 0.25, 10 \leq t \leq 12 \end{matrix}

(22)

The effectiveness and reliability of the continuous regulation method mentioned in equation (22) are verified through simulation tests. During the smooth transition of “walk → trot → walk” gait, the initial values of CPG model parameters are consistent with those of the gait transition. The simulation results are shown in Figures 20 and 21. Smooth gait transitions can realize. Figure 21 shows the smooth transition process of the robot gait.

Figure 20.

The hip joint control signal of the four legs during the smooth transition of walk → trot → walk gait.

Figure 21.

The moving simulation during the smooth transition of walk → trot → walk gait: (a) walk gait ( $t = 0.645 s$ ), (b) smooth transition of walk → trot gait ( $t = 2.975 s$ ), (c) trot gait ( $t = 4.530 s$ ), (d) trot gait ( $t = 5.550 s$ ), (e) smooth transition of trot → walk gait ( $t = 9.295 s$ ), and (f) walk gait ( $t = 10.865 s$ ).

The smooth gait transition process of “walk → trot → walk” can be divided into two steps: the first step is the smooth transition of “walk → trot”“walk → trot” gait. The robot first moves in the walk gait, and then the legs are out of phase, and the leg lifting sequence is $1 \to 3 \to 2 \to 4 \to 1$ . Then it run to the point of $2 s$ to enter gait transition, stay in the “walk → trot” gait smooth conversion process within $2 ~ 4 s$ . The rear-right leg phase $φ_{3}$ continuously changes as $0.25 \to 0$ , and the load factor $β$ continuously changes as $0.75 \to 0.5$ . As in the interval period of “walk gait → trot gait” in Figure 20, relative phase of each legs changes continuously, without output signal conversion breakpoint and phase mutation. Smooth transition of gait is achieved and the smooth transition of the gait is completed at the point of $4 s$ . The trot gait begins at the points of $4 s$ . At this time, the diagonal legs are in the same phase, and the leg lifting sequence is $1 & 3 \to 2 & 4 \to 1 & 3$ . The trot gait ends at the point of $8 s$ .

The second step is the smooth transition of the “trot → walk”“trot → walk”“trot → walk” gait. The robot uses the trot gait within the $4 ~ 8 s$ , and ends trot gait at the end of the point of $8 s$ and enters gait smooth conversion process. The rear-right leg phase $φ_{3}$ continuously changes as $0 \to 0.25$ , the load factor $β$ is continuously changes as $0.5 \to 0.75$ , without output signal conversion breakpoint, and phase mutation. Smooth transition of the gait is achieved. It is in the smooth transition process of “trot → walk” gait in $8 ~ 10 s$ , and it ends the transition and enters the walk gait at the point of $10 s$ . According to the above analysis and simulation results, the CPG controller with the Hopf oscillator as the core can achieve a smooth transition between the walk gait and the trot gait by adjusting the continuous change of the rear-right leg phase $φ_{3}$ .

Figure 22 shows the displacement curve of the body’s centroid in the direction of $x / y / z$ during the smooth transition of the quadruped robot from “walk → trot → walk” gait. The simulation time is $12 s$ , in which $0 ~ 2 s$ refers to the walk gait, the robot advances $311.0 mm$ in the walk gait. $y$ is the lateral deviation of the robot, with a maximum deviation of $9.5 mm$ . The negative direction of $z$ is the gravity direction, and the maximum fluctuation quantity of the height of the robot centroid is $6.7 mm$ in the process of motion. In different gaits, the robot centroid displacement in all directions and the fluctuation rate in $z$ direction are shown in Table 2. Therefore, the robot can realize walk and trot gaits and the smooth transition between gaits and the transition process is smooth and stable, without breakpoints and phase changes.

Figure 22.

The displacement curve of the centroid during the smooth transition of walk → trot → walk gait.

Table 2.

smooth transition of walk → trot → walk gait displacement in $x / y / z$ axis.

	$x$ $(mm)$	$y$ $(mm)$	$z$ $(mm)$	$z$ -volatility
Walk $(0 ~ 2 s)$	−311.0	9.5	6.7	3.4%
Walk → trot $(2 ~ 4 s)$	−601.4	28.6	16.4	8.3%
Trot $(4 ~ 8 s)$	−2582.7	224.3	15.3	7.8%
Trot → walk $(8 ~ 10 s)$	−912.4	21.5	14.4	7.3%
Walk $(10 ~ 12 s)$	−387.3	14.8	7.2	3.6%

Table 2 shows that the forward displacement of the robot during the “walk → trot”“walk → trot” gait transition is much lower than that of during the transition of “trot → walk”“trot → walk” gait. By analyzing the velocity and displacement curves of the robot’s centroid in the forward direction in Figure 23, the average velocity is $v_{wa 1} = 0.155 m / s$ during the initial $0 ~ 2 s$ period of walk gait. In $4 ~ 8 s$ of trot gait, the average velocity is $v_{tr} = 0.646 m / s$ . In $8 ~ 10 s$ of walk gait, the average speed is $v_{wa 2} = 0.194 m / s$ . During the transition of gait, the initial velocity is the velocity of walk gait motion at the point of $2 s$ , which is $v_{2 s} = 0.116 m / s$ , and the velocity at the end of the transition is the velocity at the point of $4 s$ , that is $v_{4 s} = 0.469 m / s$ . In the transition of gait, the initial velocity is the trot gait velocity at the point of $8 s$ , $v_{8 s} = 0.629 m / s$ , and the end velocity is the trot gait speed at the point of $10 s$ , that is $v_{10 s} = 0.236 m / s$ . It shows that the two gait transitions have the same gait transition time. The movement displacement of the two gait transitions in the same time is very different because the velocity is different at the beginning of the gait transition.

Figure 23.

The velocity and displacement curve of the centroid in the $x$ direction during the smooth transition of walk → trot → walk gait.

Conclusions

The main contribution of this paper is to use the closed vector polygon projection method to analyze the kinematics of the parallel legs, and propose a series analysis method of the parallel legs to simplify the processing of the parallel legs. The characteristics of the balance point and limit cycle of the Hopf oscillator are analyzed. In view of different walking gait, the leg stance time in a gait cycle is different, the oscillator frequency is improved, and the duration of the swing phase and the stance phase are adjusted. According to the characteristics of the Hopf oscillator, the motion of the robot is controlled by the oscillator coupling output stable rhythm signals. Based on the biological CPG control mechanism, a CPG control system with Hopf oscillator as a unit is successfully constructed to realize the four-legged coordinated motion control of a planar 5R parallel quadruped robot. Combining the leg raising laws of walk and trot gaits, the phase determination conditions between each leg are obtained. And then through the continuous adjustment of the control parameters, the continuous change of the phase relationship between the oscillators is realized. It solves the situation that the joint control signal breaks when the asynchronous state changes, reduces the impact on the mechanical and electrical system when the breakpoint occurs, ensures the continuity of motion, and realizes the smooth transition between the asynchronous states. In the simulation experiment, the stability of the motion and the smoothness of the gait transition were demonstrated in the two gaits, which verified the feasibility of the analysis method and the effectiveness of the control strategy.

In future work, intelligent optimization algorithms will be combined to improve the robot’s adaptive walking control under unstructured terrain, and low-cost sensors will be used to detect the body’s attitude and ground contact. At the same time, the effectiveness of optimized algorithms and low-cost sensor detection will be verified on physical prototypes.

Footnotes

Handling Editor: Chenhui Liang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the National Natural Science Foundation of China (No.51965029)

ORCID iDs

Mingfang Chen

Kangkang Hu

References

Kau

Schultz

Ferrante

, et al. Stanford doggo: an open-source, quasi-direct-drive quadruped. In: 2019 international conference on robotics and automation (ICRA), Montreal, QC, Canada, May 2019, pp.6309–6315, .

Blackman

Nicholson

Ordonez

, et al. Gait development on minitaur, a direct drive quadrupedal robot. In: Robert E. Karlsen

Gage

Shoemaker

, et al. (eds.) Unmanned systems technology XVIII. Baltimore, MD: International Society for Optics and Photonics, 2016, pp.98370I. Vol. 9837.

Yanqin

Wei

Chuan

, et al. A turning gait strategy for quadruped robot based on COM adjustment. Control Eng China 2021; 28: 2280–2285.

Buchli

Kalakrishnan

Mistry

, et al. Compliant quadruped locomotion over rough terrain. In: 2009 IEEE/RSJ international conference on intelligent robots and systems, St. Louis, MO, USA, 2019, pp.814–820.

Teng

Yibin

Xuewen

. Design and verification of real-time plantar force optimization for quadruped robots in dynamic gait. Robot 2019; 41: 307–316.

, et al. Adaptive fuzzy full-state and output-feedback control for uncertain robots with output constraint. In: IEEE transactions on systems, man, and cybernetics: systems, 2020, pp.6994–7007.

Ijspeert

. Central pattern generators for locomotion control in animals and robots: a review. Neural Netw 2008; 21: 642–653.

Liu

Yang

, et al. Rhythmic-Reflex hybrid adaptive walking control of biped robot. J Intell Robot Syst 2019; 94: 603–619.

Zhong

Chen

Jiao

, et al. Locomotion control and gait planning of a novel Hexapod robot using biomimetic neurons. IEEE Trans Control Syst Technol 2018; 26: 624–636.

10.

Matsuoka

. Sustained oscillations generated by mutually inhibiting neurons with adaptation. Biol Cybern 1985; 52: 367–376.

11.

Kimura

Fukuoka

Cohen

. Adaptive dynamic walking of a quadruped robot on natural ground based on biological concepts. Int J Rob Res 2007; 26: 475–490.

12.

Rulkov

. Modeling of spiking-bursting neural behavior using two-dimensional map. Phys Rev E 2002; 65: 041922.

13.

Acebrón

Bonilla

Pérez Vicente

, et al. The Kuramoto model: a simple paradigm for synchronization phenomena. Rev Mod Phys 2005; 77: 137–185.

14.

Yang-wei

Zeng

Dong-biao

, et al. Waveform control algorithm for pectoral fin of robotic stingray based on Hopf oscillator. J Zhejiang Univ (Eng Sci) 2019; 53: 1354–1362.

15.

Liu

Chen

Zhang

. Coupled Van Der Pol oscillators utilised as central pattern generators for quadruped locomotion. In: 2009 Chinese control and decision conference, Guilin, China, June 2009, pp.3677–3682.

16.

Xue

Chengju

Qijun

. Adaptive walking control of biped robot based on fuzzy-CPG. Control Theory Appl 2020; 37: 2525–2534.

17.

Chen

Ren

Zhang

, et al. Smooth transition between different gaits of a hexapod robot via a central pattern generators algorithm. J Intell Robot Syst 2012; 67: 255–270.

18.

Bal

. Neural coupled central pattern generator based smooth gait transition of a biomimetic hexapod robot. Neurocomputing 2021; 420: 210–226.

19.

Sartoretti

Shaw

Lam

, et al. Central pattern generator with inertial feedback for stable locomotion and climbing in unstructured terrain. In: 2018 IEEE international conference on robotics and automation (ICRA), Brisbane, QLD, Australia, May 2018, pp.5769–5775.

20.

Liu

Zhang

. Dynamic properties of VDP-CPG model in rhythmic movement with delay. Math Biosci Eng 2020; 17: 3190–3202.

21.

Liu

Jia

. Hopf oscillator based adaptive locomotion control for a bionic quadruped robot. In: 2017 IEEE international conference on mechatronics and automation (ICMA), Takamatsu, Japan, August 2017, pp.949-954.

22.

Santos

Alves

Moreno

. Biped locomotion control through a biomimetic CPG-based controller. J Intell Robot Syst 2017; 85: 47–70.

23.

Wang

Liu

Liang

, et al. Locomotion control for quadruped robot combining central pattern generators with virtual model control. In: 2019 IEEE 15th international conference on control and automation (ICCA), Edinburgh, May 2019, pp.399–404.

24.

Ajallooeian

Pouya

Sproewitz

, et al. Central pattern generators augmented with virtual model control for quadruped rough terrain locomotion. In: 2013 IEEE international conference on robotics and automation, Karlsruhe, Germany, May 2013, pp.3321–3328.

25.

Kasaei

Simões

Lau

, et al. A hybrid ZMP-CPG based walk engine for biped robots. In: Third Iberian robotics conference, Sevilla, Spain, 2017, vol. 694, pp.743–755.

26.

Hasanzadeh

Tootoonchi

. Adaptive optimal locomotion of snake robot based on CPG-network using fuzzy logic tuner. In: 2008 IEEE conference on robotics, automation and mechatronics, Chengdu, China, September 2008, pp.187–192.

27.

Gay

Santos-Victor

Ijspeert

. Learning robot gait stability using neural networks as sensory feedback function for central pattern generators. In: 2013 IEEE/RSJ international conference on intelligent robots and systems, Tokyo, Japan, November 2013, pp.194–201.

28.

Thor

Kulvicius

Manoonpong

. Generic neural locomotion control framework for legged robots. In: IEEE transactions on neural networks and learning systems, August 2020, pp.1–13.

29.

Xiangzheng

Binxiang

Kaiyu

. Theory, method and application of nonlinear ordinary differential equations. Hunan: Hunan University Press, 2006.

30.

Qishao

Linping

Zhuoqin

. Ordinary differential equations and dynamical systems. Beijing: Beihang University Press, 2010.

31.

Jinjun

. Nonlinear differential equation. Beijing: Science Press, 2011.